Logic, Methodology and Philosophy of Science IV

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 74

Editors

A. HEYTING, Amsterdam J. KEISLER, Madison A. MOSTOWSKI, Warszawa A. ROBINSON, New Haven P. SUPPES, Stanford Advisory Editorial Board

Y. BAR-HILLEL, Jerusalem K. L. DE BOuvERE, Santa Clara H. HERMES, Freiburg i, Br. J. HINTIKKA, Helsinki J. C. SHEPHERDS ON, Bristol E. P. SPECKER, Ziirich

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM • LONDON

LOGIC, METHODOLOGY AND PHILOSOPHY OF SCIENCE IV PROCEEDINGS OF THE FOURTH INTERNATIONAL CONGRESS FOR LOGIC, METHODOLOGY AND PHILOSOPHY OF SCIENCE, BUCHAREST, 1971

Edited by

PATRICK SUPPES Stanford University, Stanford, US4 LEON HENKIN University of California, Berkeley, USA ATHANASE JOI A Academie Roumaine, Bucarest, Roumaine GR. C. MOISIL Universite de Bucarest, Bucarest, Roumaine

~c ~

~ 1973

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PREFACE This volume constitutes the Proceedings of the Fourth International Congress for Logic, Methodology and Philosophy of Science held in Bucharest, Rumania, from August 29 to September 4, 1971, under the auspices of the Division of Logic, Methodology and Philosophy of Science of the International Union of History and Philosophy of Science. The papers published in the present volume are texts of addresses given by invitation of the Program Committee. The Program Committee was an international committee appointed by the Executive Committee of the Division of Logic, Methodology and Philosophy of Science. Its members, representing a number of different countries, consisted of the following individuals: Patrick Suppes (Chairman), Yehoshua Bar-Hillel, Leon Henkin, Mary Hesse, Athanase Joja, Stephan Korner, A. A. Markov, Grigore Moisil, Nicholas Rescher, J. F. Staal, and Wolfgang Stegmilller. The editors of this volume served as the Executive Committee of the Program Committee. In accordance with previous congresses sponsored by the Division, the work of the Congress was divided into a number of sections. The various section committees, who worked with the Program Committee in organizing the activities of each section, including both the invited addresses and invited symposia as well as the selection of contributed papers, were made up of the following individuals, with the chairman being the first named individual: Section I (Mathematical Logic), Andrzej Mostowski, Y. L. Ershov, Kurt Schutte; Section II (Foundations of Mathematical Theories), A. A. Markov, Ronald B. Jensen, Laszlo Kalmar, Joseph Shoenfield; Section III (Automata and Programming Languages), Michael Rabin, V. M. Glushkov, Grigore MoisH, John C. Shepherdson; Section IV (Philosophy of Logic and Mathematics), Georg Kreisel, Hans Hermes, Dag Prawitz; Section V (General Problems of Methodology and Philosophy of Science), Wolfgang Stegmiiller, Pavel V. Kopnin, Imre Lakatos; Section VI (Foundations of Probability and Induction), Jaakko Hintikka, Bruno de Finetti, Wesley C. Salmon; Section VII (Methodology and Philosophy of Physical Sciences), Adolf Griinbaum, Brian Ellis, Martin Strauss; Section

x

PREFACE

VIII (Methodology and Philosophy of Biological Sciences), Morton O. Beckner, Aristid Lindenmayer; Section IX (Methodology and Philosophy of Psychological Sciences), Aleksey N. Leontiev, F. Bresson, R. Duncan Luce; Section X (Methodology and Philosophy of Historical and Social Sciences), Herman Wold, Miron Constantinescu, W. H. Dray; Section XI (Methodology and Philosophy of Linguistics), Solomon Marcus, John Lyons; Section XII (History of Logic, Methodology and Philosophy of Science), Mary Hesse, Athanase Joja, Gunther Patzig. The Congress was jointly sponsored by IUHPS and the Academy of the Socialist Republic of Rumania. The arrangements for the holding of the Congress in Bucharest were carried out by the Organizing Committee, which consisted of the following individuals: Athanase Joja (President), Petre Botezatu, Georg Ciucu, Dumitru Ghise, Crizantema Joja, Stefan Milcu, Grigore Moisil, Mihai Neculce, Radu Negru, Octav Onicescu, Dionisie Pippidi, Gabriel Sudan, Serban Titeica, Solomon Marcus, and Constantin Popovici (General Secretary). The editors have confined themselves to arranging the volume and handling various technical matters relating to publication, without attempting detailed editorial treatment. By and large, the choice of notation and symbolism has been left to the individual authors. The full program of invited addresses and symposia is printed at the end of the volume. On behalf of the Program Committee and the Organizing Committee, the editors wish to thank the many persons both in Rumania and in other parts of the world who contributed their generous assistance in determining the program and arranging for the holding of the Congress. The Editors

September 1972

ELEMENTARY LOGIC GR. C. MOISIL Institute of Mathematics, Rumanian Academy, Bucharest, Rumania

1. We shall consider a typified logic of propositions. A well-formed formula of Type I (I-wff) is defined recursively: 1. Propositional variables P, q, r,pl' ... , are I-wff's; 2. If A, B, are I-wff's then (A & B), (A v B), (I A), (A ~ B), (A +-+ B), etc., are I-wff's. Among the wff's of Type I the valid well-formed formulas (vl-wff) are defined syntactically and semantically. The syntactical definition. Suppose given the definition of a demonstration; i.e., a sequence of I-wff's E l , ... , E; (1) such that each E, is an axiom or results from the precedent ones by a transition rule. A vl-wff is a I-wff which can be imbedded in a demonstration. The semantical definition. With every I-wff E we associate a function IE: IE:

L~ ~

L2 ,

where L 2 = {O, I}. A vl-wff is a I-wff whose associated function is the constant 1. 2. Using the metalinguistic variables we can avoid the substitution rule. Among the transition rules we note the modus ponens A,A~B

----B

(2)

and the rule of adjunction, used by A. Heyting A,B A&B

(3)

4

GR. C. MOISIL

but some other rules are used in the nonformalized reasoning. Let us consider the rule A,B,A

~ (B~

C)

C

(4)

Such a rule seems to be valid; for instance the tree

has the structure

and the three I-wff's on the top of the tree (*) are instances of the axioms; the transition rule used in (*) is the modus ponens. When we are saying that a rule of transition (5)

is valid, we think of the following definitions. Metatheoretical definition. For each substitution (1, if (1 transforms AI, ... , An into vI-wff's then (1 transforms B into a vI-wff. First syntactical definition. One can construct a tree like (**) having at the top Au ... , An; each I-wff results from the I-wff's which are above it by one of the transition rules. The metatheoretical definition gives the following semantical definitions. First semantical definition. For each substitution of the variables of the functions fAt' ... ,fAn by functions of new variables, for which the functions fAt' ... .I», become the constant 1, fB becomes the constant 1. Second semantical definition. This definition differs from the first by the fact that the substitution (1 is not arbitrary, the functions defining (1 being not arbitrary but associated with I-wff's. Third semantical definition. For each assignment of values in L 2 to the variables, for which the values of J,rA1 , ... ,fA n are 1, the value of fB is 1.

5

ELEMENTARY LOGIC

Fourth seman tical definition. The associated inclusion fAt n ... nfA n CfB

(6)

(AI, ... , An) It (B)

(7)

is true in L 2 • Following G. Gentzen we shall write (5) in the form

and we shall call (7) a II-wff. With each II-wff SIl, for instance (7), we will associate a function (8)

or (9)

being the implicational connective in L 2 • Fifth seman tical definition. fSIl is the constant 1. Given a Boolean algebra B, we can associate with a l-wff a function IE.B: fE.B:B n -+ B. ,-+'

Sixth-tenth seman tical definitions. These definitions are to be obtained from the first-fifth semantical definitions using fAt. B' ... ,fAn. B, fA. B instead offAt , ••• ,IAn,fn· It is possible to give different syntactical definitions to the validity of

II-wff; we will call vll-wff''s the valid II-wff's. Second syntactical definition. We will call demonstration any sequence of II-wff's: (10)

in which each Sp is an axiom or results from the precedent ones by a transition rule. A vII-wff is a II-wff which may be imbedded in a demonstration. The third,fourth and fifth syntactical definitions for the validity of II-wff's require for the validity of (7) the validity of the I-wff's: (AI & ... & An)

-+

(AI -+ (A 2 -+ ..• -+ (A"

(11)

B

-+

B) ...)

iA 1 v ... viAnvB.

(12)

(13)

G. Gentzen introduced another kind of II-wff's, which he called 'Sequenzen' (14)

6

GR. C. MOISIL

the inclusion associated with such a form being 1',(

J~ 1

( ] ••• (]

fA n

C

fB l

U ... U

(15)

fBm .

The function associated with (14) and the different semantical definition for the II-wff's having the structure (14) are obtained from those related to the Form (7) by writing BI v ... v B m and fB: U ... U J»; instead of B andfB, respectively. 3. The transition rules of a demonstration of Type (10) were given by G. Gentzen. We shall write these rules in the form (SP, ... , S:I) itx (TIl)

(16)

Sp, ... , S:I, T" being II-wff's. Let us give some examples: «SL

, SD It (T I ») iiI «S~(l)' ... , S;(S» It (T I »)

(17)

«SL

, SD It (T ») iiI «SA, Sf,

, SD It (T »)

(18)

«SA, SA, Sf,

, SD It (T I ») iiI «SA, SL

, sD It (P»)

(19)

«Sf,

I

, S~)

It (T

I

) ,

I

(T!, Uf, ... , U~)

itx «Sf, ... , S:, U], ... ,

It (WI»)

U~)

It (WI»).

(20)

These III-wff's will be called Perm m , Renf n H Replll and Enchlll· For a better understanding of the idea of vIII-wff's, one must think of the Ill-wff's which will be called Ded m : «AI' ... , An, B)

It (C») itx «AI' ... , An) It (B

--+

C»).

This is the famous Deduction Theorem. The validity of this III-wff in classical sentence calculus must be proved. 4. It is easy to extend these ideas, defining the (N + l)-wff's by (S~, ... , S~) N+I (T~)

(21)

si., ..., S;, TN being N-wff's. With each N-wff may be associated a function and an inclusion; it is easy to define the valid N-wff's, the ideas of demonstration, demonstration from hypothesis, etc. 5. The usefulness of the calculus of the second type was clearly shown by the work of G. Gentzen for the classical and the intuitionistic propositionallogic.

ELEMENTARY LOGIC

7

The use of propositional logic of higher order is very fruitful for developing nonclassical propositional calculi too. There exist logics in which no I-wff is valid. These logics may be developed as calculus of vll-wff's. The calculus whose models are all general lattices, which will be called the strictly positive logic, has the axioms (A & B) It (A)

(A) It (A v B)

(A & B) It (B)

(B) It (A VB)

(A, Bht (A &B)

(I)

(A v B) It (A, B)

using II-wff's of the Form (14). The strictly symmetric logic introduces a negation, that is a connective N with (see Section 6 below): (A) t{(NNA) (A) It (B))

itr (NB) It (NA)).

(II)

The logic of quantum theory having the models introduced by Garrett Birkhoff and J. von Neumann has the Axioms (I), (II) and one of the following

It (A, NA)

(A, NA) It

(III)

that express the principle of excluded middle and the principle of contradiction. The pure positive logic which introduces the implication in two forms: as a relation of consequence of any type 'rr' ... , 'N" ... and as a connective '--+', related by the modus ponens and the deduction theorem:

It is the implicationallogic of Henkin; it is the Heyting logic, restricted to the connective '--+' alone and has as models the Hilbert algebras of the school of Bahia Blanca. The pure modal logic is obtained from the pure positive logic by dualization, introducing a new connective '-' which is the exception; A - B will be read "A excepted B." The axioms of this logic are (IV) and (V):

8

GR. C. MOISIL

In this calculus modalities may be introduced; impossibility (TJ), nonnecessity (y), possibility (p,) and necessity (v) are defined by

11 (A -+ (P-p) (yA) 11 ((P -+ p)-A) (p,A) 11 (1)1)A) (vA) 11 (yyA) . (1)A)

(VI)

The positive logic of Hilbert-Bernays, which is the Heyting calculus restricted to the connectives '&', 'v' and '-+', has the Axioms (I), (IV). The models of this calculus are all residuated lattices. The calculus whose models are the bi-residuated lattices, the second residuation being the exception '-' with the Axiom (V), named the general modal logic, was considered by us in 1942. We introduced also the two axioms

(A & B)

-+

C) It (A

-+

C) v (B -+ C)

(A-B) & (A-C» It (A-(BvC».

(VII)

The models of this calculus are the linearly ordered sets; this calculus, the linear (or special) modal logic, was developed by A. Monteiro and the author. The three-valued Lukasiewicz logic (with an implication connective which is not the Lukasiewiczian one) may be characterized by the Axioms (I), (IV), (V), (VI), (VII), (VIII): (vA -+ vB, pA -+ pB) It (A -+ B).

(VIII)

A negation connective N, which satisfies (II), may be introduced. The classical logic is characterized by (I), (IV), (V), (VI), (VII) and one of the axioms (IX): (vA -+ vB) 'it (A -+ B) (p,A -+ pH) 'it (A -+ B) (A) 'it (vA) (p,A) 'it (A) (p,A) 'it (vA)

It (A, 1)A) (A, yA) 'it.

(IX)

9

I!LIlMIlNTARY LOGIC

For the Lukasiewicz n-valued and f)-valued logics similar systems may be given. 6. An important relation between N-wff's is the equivalence, defined by +-+ (TV» -+ (SN) -+ (TN» ( SN) N+1 N+2 N+1 (SV) N71 (TN» N+2 (TN) N+1 (SN»

(21) (22)

-+ (TN) (T'V) -+ (S"i» -+ (SN) +-+ (TN». ( SN) N+l ' N+l N+2 N+1

(23)

The use of equivalences of lind order allows us to give to the axioms of each propositional calculus a form which translates the algebraic structure of its models. We shall consider the propositional calculus which has as models the lattices. The axiom system of a lattice: a rvb se b rva aub=bua an (b n c) = (a n b) n c au (b u c) = (a u b) u c an (a u b) = a a u (a n b) = a is translated by the following system of the strictly positive logic:

tt (B & A) (A & (B & C» tt (A & B) & C) (A & (A v B» tt (A) (A & B)

(A v B)

tt (B v A)

(A v(BvC» tt(A v B)vC) (A v (A & B»

tt (A).

Another system of axioms for the propositional calculus in the strictly positive logic corresponds to the axioms of lattice theory in terms of the inclusion relation. These axioms express the reflexivity and the transitivity of the inclusion relation 'c', and the properties

a rvb cz a a rvb cz b

acaub bcaub

if a c b and ace then a c b n c if ace and b c c then au bee.

Their translation in the propositional calculus are the first four II-wff's (1) and the vlII-wff's:

It (B), (A) It (C» iD: (A) It (B & C» (A) It (C)' (B) It (C» iD: (A v B) It (C».

(A)

The last two axioms (I) introduce a 'sequence relation' in the lattice: (a1' ... , a,,) <s; (b1, ... , bm) ,

10

GR. C. MOISIL

which signifies The properties of the 'sequence relations' are extensions of the properties of order relations. 7. The most general laws which are valid in the Chrysippian logic as well as in all non-Chrysippian logics are the extensions to an arbitrary type of the principles (17)-(20) given in Section 3. Let us write these principles; bold letters stand for sequences; for instance S = (Sl' ... , S.). Id N +1 is

where ffi, "p are permutations and ffiS = (S'I'(l)' ... , S'I'('»' Renf aN+2 and Renf CN+2 are «SN) N"11 (TN» N-;2 «SC, SN) N-;l (TN» «SN) N"11 (TN» N-:2 (CSN)

a. (TC, TN».

RepaN+2 and RepcN+2 are «(S{f, S{f, SN) N""t1 (TN» N""t2 «S~, SN) N"11 (TN» «S·) N""t1 (T{f, TC, TN» N"12 «SN) N"11 (Ttf, TN». Ench 1N +2 , Ench 2N +2 are

These sequences are valid in each of the logics cited above, the definition of the validity being one of the definitions mentioned in Section 1, conveniently adapted. These eight rules define the elementary logic. 8. It is important to observe that wff's like (A) It (A), itr «A) It (A», ~ itr «A) It (A», ... are different.

11

ELEMENTARY LOGIC

We must remark that the sequence (Sr, ... , Sf) N"11

translates the idea of incompatibility of Sf, ... , S~; therefore (SN) N"11

is the negation of SN. We gave in MOISIL (1971a) a few demonstrations in this elementary logic. Many elementary properties of the arithmetic of natural numbers were proved by means of elementary logic in MOISIL (1971b). We began the algebraic study of the models of the elementary logic in MOISIL (1969). In another paper I will compare these ideas to the work of A. A. Markov's school. References MOISIL, Gr. C., 1969, Studio algebrico della logica elementare, Atti della Accademia delle Scienze dell'Istituto di Bologna, Classe di Scienze fisiche. Anno 257 Rendiconti, tomo 6, pp. 33-56 MOISIL, Gr. C., 1971a, La Iogique elementaire, in: Logique, automatique, informatique, -: ed. Gr. C. Moisil, (Editions de l'Academie de Ia Republique Socialiste de Roumanie) pp, 31-39 MOISIL, Gr. C., 1971b, Essais sur les logiques non-Chrysippiennes, Editions de I'Academie de la Republique Socialiste de Roumanie, pp. 707-738 0

PARTIAL ORDERINGS OF THE FAMILY OF w-MODELS

A. MOSTOWSKI University of Warsaw, Warsaw. Poland

ROSSER (1937)was the first to prove that there exist many nonisomorphic w-models for higher-order arithmetic, i.e., models whose arithmetical parts are standard. The discovery of Rosser showed that it is impossible to characterize axiomatically the concept of a set of integers even if we accept the absolute concept of an integer. co-models for higher-order arithmetic were discussed in several papers during the past few years. In the present paper I discuss the family of co-models of second-order arithmetic (called hereafter A2 for short) viewing it as a set partially ordered by some natural relations such as the inclusion relation. or relation of elementary embeddability. The high degree of noncategoricity of A2 is witnessed by the degree of complication of the ordered structures thus obtained. After having finished my paper I was left with two impressions. The first es that unless we assume the continuum hypothesis we know next to nothing about models of A 2 of power c; the second is that methods known at presint allow us to discuss co-models of A z but not of a system obtained from A z by dropping the axiom scheme of choice. Both these remarks suggest the direction in which the present work could be continued. 1. Preliminaries

Arithmetic of second order A 2 is a two-sorted theory. Objects of the first sort are called integers and objects of the second sort are called sets of integers. We assume the usual axioms of Peano with the axiom of induction formulated as a single sentence containing one set variable. Besides these axioms we assume the axiom of extensionality and two axiom schemes: the scheme of the axiom of comprehension and the scheme of the axiom of choice. To simplify the exposition we shall use in A2 the same symbols of arithmetic and of set theory as are used in informal expositions. The pairing func-

14

A. MOSTOWSKl

tion y+ (x+y)(x+ y+ 1)/2 will be denoted by lex, y). For a set X c OJ we shall denote by x(n) the set of those x for which len, x) eX. Owing to the existence of a one-one correspondence between integers and their pairs we can speak of sets of integers as if they were binary relations. Thus we can use (informally or in A 2 ) expressions like 'X is a transitive relation', 'X is a well ordering', 'X is a mapping of OJ into OJ' instead of a more pedantic 'the set of pairs (x, y) such that lex, y) e X is a transitive (or well ordering or functional) relation'. Let S be a set of sentences containing all the axioms of A 2 • We denote by OJ(S) the family of models of S whose integers coincide with the usual integers. These models are called co-models of S. The family of denumerable co-models of S will be denoted by OJ*(S). Since an «i-model is completely determined by the family of its sets we shall identify it with this family. p-models are co-models such that for every set X in the model, X satisfies in the model the formula 'X is a well ordering' if and only if X is really a well ordering. The family of arbitrary or denumerable p-models of S will be denoted by peS) or by P*(S). A further classification of co-models has been proposed by Enderton (see ENDERTON and FRIEDMAN, 1971) but we shall not discuss it in the present paper. We denote by M o the principal model, i.e., one whose integers are the usual integers and whose sets are arbitrary subsets of OJ. 2. Prime models Many investigations concerning models of A 2 were stimulated by the following result due to Gandy (unpublished) and Putnam (BoYD," HENSEL and PUTNAM, 1969): There exists a prime co-model, i.e., a model M in P(A 2 ) which is a submodel of any other model M' in P(A 2 ) . The prime model is of course denumerable and the axiom of constructibility V = L in its arithmetical form is valid in it. Its sets coincide with sets definable in the so-called ramified analysis. It follows that the intersection of all p-models (the 'hard core' of p-models in the terminology of KREISEL, 1965) consists of the elements of the prime p-model. A contrasting situation for «i-models has been discovered by Friedman (unpublished): each co-model for A 2 contains a proper submodel which is also an OJ-model. It is also easy to show (cf. Lemma D in the Appendix) that the 'hard core' for OJ-models consists of all hyperarithmetical sets and is thus not a model for A 2 •

15

PARTIAL ORDERINGS

3. Partial orderings of w*(8) We shall consider the following relations which partially order the set w(8): the usual relation c; the relation - w 1 , does it necessarily have an elementary extension of power c?

20

A. MOSTOWSKI

(2) If the answer to (1) is no, can y(M) be an aleph with a limit index? (3) Can one prove without any additional hypothesis, such as CH, that there is more than one eo-model of A z of the same power> WI? Problem (3) suggests the following problem which may be of some interest to set theorists: (4) Let R be the set of all real numbers, no, nl' ... a sequence of integers ~ 0 and let jj: Rnl --+ R for i = 0, 1, ... Let us say that the sequence (fO'/I' ...) is full on a set X c R if every real number a can be represented in the formjj(Xl' ... , x nI.) where i is an integer and Xj E X for j = 1, ... , n.. If CH is false, does there always exist a set X of power c on which the sequence (fO'/I' ...) is not full? ERDOS, HAJNAL and RADO (cf. 1965, Theorem 17A, p. 165) have proved that if CH is true, then there exists a function F: RZ --+ R which is full on every nondenumerable set X. It is easy to explain the connection of (4) to the problem of co-models of power c: If the answer to (4) would be positive, then so would be the answer to Problem (3). For either CH is true and we apply the Keisler theorem alluded to above obtaining 2C different models. Or CH is false and there exists a set X of power c on which the sequence (fO'/I' ...) consisting of all Skolem functions for the principal model M o and their iterations are not full. The values of the functions jj for the arguments of X form then a proper elementary submodel of M o of power c.

Appendix

I. Proof that if M, M 1 E w(A z ) and P E M 1 , MeM1 , then M 1 contains (in the sense of relation e) a family F such that all sets which are hyperarithmetical in P belong to F. Remark. The result is known from the literature (see, e.g.,

GRZEGOR-

and RYLL-NARDZEWSKI, 1958, p. 203), even without the assumption that M 1 contains (in the sense of e) another co-model. We give the proof in order to make the paper self-contained. Let L be the language of the second-order arithmetic and let L' arise from L by adding to L one set constant C. Assume moreover that M, M 1 E w(A z ), MeM I and P E MI' Let us denote by Dc(P) the description of P in L', i.e., a set consisting of all the sentences p E C where p is an element of P and all the sentences 1(.0 E C) where p is not an element of P. The set of all axioms of

CZYK, MOSTOWSKI,

PARTIAL ORDERINGS

21

A2 being recursive and the set De(P) being recursive in P, we infer that there exists a formula K of L without bound set variables and with exactly two free variables which both are set variables such that for any set X in M 1 the condition M 1 f:: K[X, P] is equivalent to the following: X contains the Godel numbers of all axioms of logic, of all axioms of A 2 and of all sentences in Dc(P) and is closed with respect to the usual rules of proof and the rule co. According to one of the possible definitions of hyperarithmetic sets, a set T is hyperarithmetic in P if and only if there is a formula H of L' with exactly one free (number) variable such that for each integer n ifn E T, then Hen) E Cnw(Dc(P)),

if n non E

T, then (iH(ii)) E Cn.;(Dc(P)) ,

Consider the formulae G(x): x is the Giidel number of a formula of L' with just one free number variable and with no free set variables; W(X, x, y): G(x) & (y)[K(Y, X) -t Sb(x, y) E Y]; W(X, x, y): G(x) & (Y)[K(Y, X) -t NegSb(x, y) E Y].

Here Sb and Neg are self-evident abbreviations for arithmetical terms which denote operations on Godel numbers corresponding to the substitution of numerals for the free number variables and to the operation of forming the negation of a formula. Since the axiom of comprehension is valid in M 1 we infer that there exists a set U in M 1 such that for each integer n, k nE

U(k)

== M 1 f:: W[P, k, n].

We claim that U has the desired property, i.e., that for any set T hyperarithmetic in P there is an integer k such that T = Let T be defined by (*), let H# be the Godel number of H and put

o».

T' = {n:M 1

f::

W[P, H#, n]}.

Obviously T' is an element of M 1 and T' = UC H # ) , Thus in order to prove I it will be sufficient to show that T' = T. First let nET. The first of formulae (*) is equivalent to M o f:: W[P, H#, n]. However we can replace here the principal model M o by any co-model because W is a general formula and no bound set variable occurs in the scope of the initial general quantifier «Y)'. This proves that n e T'.

22

A. MOSTOWSKI

Next let n non the same way

E

T. Using the second of the formulae (*) we infer in M1

~

W[P,H#,n].

If n were an element of T' we would also have M 1 F W[P, H#, n]. Now let V be the set of all sentences of L' which are true in M when C is interpreted as P. This set, closed under the usual rules of proof and under the rule 00, contains axioms of A2 and the set De(P). Denoting by V# the set of Godel numbers of sentences in V we thus obtain M 1 F K[V#, Pl. We can now omit the quantifier '(Y)' in formulae Wand Wand replace 'Y' by 'V#'. In view of M 1 F G[H#] we obtain from the formulae given above M1

F (Sb(H#, n) E V#)

and

which is not true because M 1 f= I is thus proved.

M1

eft E V#)

Il, Proof that if ME 00* (A2 ) , M 1

F (NegSb(H#, n) E V#),

- i(Neg(p)

E

V#).

(J(A2 ) , P E M 1 , M8M1 , then M 1 contains (in the sense of relation 8) the family of all sets which are hyperarithmetical in P. We keep the notation introduced in Part I and notice first that the conditions (a) M 1 F W[P, k, n] and Mol= W[P, k, n] E

are equivalent. The implication from right to left follows from the absoluteness of ilt formulae in arbitrary eo-models and is therefore true even without the assumption that M 1 is a {J-model. The converse implication follows likewisefrom the absoluteness of 1:1 formulae in arbitrary (J-models. In the same way we show that M1

F

W[P, k, n] is equivalent to M o F W[P, k, n],

M1

F R[P, k] is equivalent to M o F R[P, k],

(b) (c)

where R(X, x) is the formula (Y)[W(X, x, y) v W(X, x, y)]. Let U1 be a set in M 1 such that

n E Ufk) == M 1

1= R[P, k]

& W[P, k, n].

We claim that the family {UfO>, UP>, ...} coded by U1 consists exactly of sets which are hyperarithmetical in P. We distinguish two cases.

23

PARTIAL ORDERINGS

Case 1: M 1 F= "lR[P, k]. In this case U~k) is empty and the result is obvious. Case 2: M 1 F= R[P, k]. In this case n E U~k)

==

M

1

F=

W[P, k, n].

(d)

F=

(e)

We first show the equivalence n non

E

Ui k )

==

M1

W[P, k, n].

Since we are dealing with Case 2, we obtain M 1 1= W [P, k, n] v W[P, k, n], hence n non E Ul k ) implies M11= W[P, k, n]. Conversely, if the last formula holds, then M1

F=

K[X, P] --+ NegSb(k, n)

EX

for each set X in MI' In particular we can take X = V# where V is defined as in Part I. Since, as we have shown in Part I, M 1 1= K[V#, P), we obtain M1

F

Neg Sb(k, n)

E

V#.

Formalizing in M 1 the usual proof of consistency of V we infer M1

F Neg(a) E V#

--+

"I (a E V#)

for each a which is the Godel number of a sentence whence, by taking a = Sb(k, 11), M 1 non

F

W[P, k , n],

i.e., n non E Uik ). The equivalence (e) is thus proved. From (a), (b), (d), and (e) we derive now 11

n non

E

U~k)

E

Ul k)

== M o F == M o F

W[P, k, 11], W[P, k, 11].

The first formula shows that Ul ) is a lIt-set relatively to P; the second formula shows the same for the complement of Ui k ) , hence Ul k ) is hyperarithmetical in P. Next, we show that for each set T which is hyperarithmetical in P there is an integer k such that T = Ul k ) . Let T be defined by the formulae (*). We first show that M 1 F R [P, H#]. In view of (c) it is sufficient to show this for the principal model and this is evident since M o F= W[P, H#, m] or M o F W[P, H#, m] according as m e Tor m non E T. Now we see that the set T' = {n: M 1 F= W[P, H#, belongs to M 1 and is equal to Ui H # ). We show exactly as in Part I that T = T. The proof of Part II is thus finished. k

nn

24

A. MOSTOWSKJ

Remark. The assumption that M I is a p-model is essential for the above proof. To see this we notice that by a theorem of KLEENE (1959) there is a set Z which is not hyperarithmetical but is definable in the form n

E

Z

== (there is a hyperarithmeticai set X such that M o F H[X, nD,

where H is a formula without bound set variables. Let now M be an wmodel such that Z non E M. We claim that no code of the family of hyperarithmetical sets belongs to M. For if C were such a code, then the formula above would yield the equivalence n E Z == M F (Ex)H[C<xl, n] and Z would belong to M.

Ill. Proof of the theorem p. 17. We let again L' be the language obtained from L by adding a new setconstant C and let be a sequence consisting of all sentences of L'. We shall need an operation on formulae of L called the operation of relativization of set quantifiers to C. Whenever F is a formula, we denote by np the number of its free number variables and by mp the number of its free set variables. For any formula F of L we denote by Fc the formula resulting from F by replacing each subformula of the form (X)[ X ... ] or of the form (EX)[ ... X ...] by an expression (x)[ ... C(x) ] or (Ex)[ ... C(x)... ]; in both cases x is a' new variable. We admit now the assumptions (1) and (2) formulated on p. 17. and denote by k o a code of mo which belongs to mi' Let be a set consisting of the following sentences: (i) All axioms of A 2 ; (ii) Sentences J (0, ji) E C for p such that p E k o ; (iii) Sentences I (J(O, p) E C) for p such that p non E k o ; (iv) Sentences

r

(XI' ... ,

x np ) (XI , ... , X mp)[ / \

l.;j .. n p

(Xj

= Pj) & ==

/\

l.;i.;mp

X E C(O)@;)]

(x)fx E Xi

& FCw) - Fe],

where F is an arbitrary formula of L. The intuitive content of (iv) is that the family coded by mentary submodel of the family coded by C. We can easily prove

C(O)

is an ele-

25

PARTIAL ORDERINGS

(A) If r has an co-model Q, then the reduct of Q to the language L belongs to w(A 2) and the denotation CQ of C in Q is a code of an co-model m2 E w*(S), C~O) = k o , k o E m2 and m., e'm2' To see this we merely notice that, by (i), the reduct of Q to the language L is an w-model of A2 • By (ii) and (iii) k o = ChO ) and hence k o E m2 and m.,ems: In view of (iv) the set k o codes an elementary submodel of m2 whence m o 8' m2 which also proves that m 2 E w*(S). (B) There is an w-model Q of r such that Q8M*. Let k be a code of m, which belongs to M. We can assume (since this is just a matter of a suitable 'rearrangement' of sets k(O), k(l), ...) that k(O) = k o . Now we can easily verify that the required model of is M with C interpreted as k.

r

(C) If F is a finite set of sentences of L', then the set CnfJ)(ruF)#, i.e., the set of Godel numbers of formulae derivable from r u F by means of the usual rules of proof and the rule w, is lIt in k o .

This follows from the remark that the set J'# is recursive in k o • We shall now prove the main lemma: (D) If E is a set such that for no j = 0, 1,2, ... the set E(j) is hyperarithmetical in k o , then there exist two sequences hn , Fn , n = 0, 1, 2, ... , satisfying conditions (a)-(g) given below. If moreover E E M*, then the sentence stating the existence of these sequences is true in M*. (a) If n > 0, then Fn_1 C Fn; F; is a finite set of sentences; (b) If n > 0, then hn_1 C hn; hn E w nxn; (c) Fu F; has an oi-model; (d) If n > 0, then either A n_1 E F; or ClA n- 1 ) E Fn; (e) If n > and an existential sentence (Ex)H(x) belongs to T's» Fn_1 , then there is p such that the sentence Hep) belongs to r u Fn; ---=7-.'---;----;-:-~ (f) If i,j < nand hnCi,j) non E then the sentence lei, hn(i,j)) E C belongs to Fn; (g) If i,j < nand hn(i,j) E E(}) then the sentence i(JU, h~-;(i,j)) E C) belongs to Fn.

°

ts»,

We shall use induction. Take F o = 0 = h o ' Then (a), (b) are obvious, (c) results from (B) and (d)-(g) are satisfied vacuously. Let us now assume that n ~ 0 and that Fn , h; have been defined. We start by constructing 2n+ 1 integers a(i, n), i < nand b(n,j), j ~ n which will be taken as values of hn +1 (i, n) and hn+1 (n, j).

26

A. MOSTOWSKI

e».

Consider the n + 1 sets E(O), ••• , None of them is hyperarithmetical in k o and hence none of them is strongly representable in Cn"lr u Fn). In particular the formula x E C(O) does not strongly represent E(~) which implies that there exists a (least) integer a(O, n) such that either a(O, n) E E(n) but the formula a(O, n) E C(O) does not belong to Cnw(ru Fn) or a(O, n) non E E(n) but the formula I (a (0, n) E C(Ol) does not belong to Cnw(r u Fn). In the former case, we adjoin to F; the sentence 1(1 (0, a(O,n)) E C) and in the latter we adjoin the sentence 1(0, a(O,-~)) E C. Let the resulting set be Fn , o- By completeness theorem the set u F n , 0 has an co-model and the set Fn,o satisfies conditions (f), (g) for i = 0, j = n. Now we replace in this construction E(n) by E(n-1) and F; by Fn, 0, C(O) by We obtain a set Fn,l such that ru Fn,l has an co-model and the set Fn , l satisfies (f), (g) for i = 1, j = n. Continuing in this way we construct n integers a(O, n), ... , a(n-l, n) and sets Fn. o C ... C Fn,n-l such that rUFn,n_l has an co-model and Fn, n-l satisfies (f), (g) for i = n, j = n, Proceeding similarly we construct sets Fo,n C F1,n C ... C Fn,n and integers ben, 0), ben, 1), ... , ben, n) such that if k e; n, then Fn,n_l C Fl,n, ru Fl,n has an co-model and Fl,n satisfies (f), (g) for i = n, j::::; k. Thus if F is a finite set of sentences which contains Fn,n, then F ;2 Fn, T u F has an co-model and satisfies (f), (g) for i < n, j :::;; n. Thus if we put

r

c».

hn+t(i,j) = hn(i,j)

for

(i,j)Enxn,

hn+t(i, n) = a(i, n)

for

i

< n,

hn+ 1(n,j) = b(n,j)

for

i

« n,

we shall have (b), (f) and (g) satisfied for the integer n+ 1 whatever superset of Fn , n we take as F n+! . We select now any co-model M of ruFn,n and adjoin to Fn,n the sentence An if An is true in this model and IAn otherwise. The resulting set F~ and any of its supersets clearly satisfies (d). Next we take care of condition (e). Let (ExDHi(Xi), i = 1,2, ... , s be existential sentences in F~ which do not belong to F n . Since these sentences are true in an co-model M, there exist integers PI' ...,Ps such that M F Hi(pD. We add these sentences to F~ obtaining a finite set Fn+ 1 which clearly satisfies (a), (c), (d)-(g). Since (b) is satisfied by hn+1 the inductive proof of the first part of (D) is finished. If E E M* then the construction can be carried out in M* and since

PARTIAL ORDERINGS

27

the concepts occurring in the statement of the theorem are absolute, we obtain the second part. Using (D) and (C) we are able to prove the theorem. Let E be a code of the family of all sets which belong to m l but are not hyperarithmetical in k o . By Part II of the Appendix such a code can be found among elements of M*. Hence we can assume that E E M*. By (D) we find two sequences h«, F; in M* satisfying conditions (a)-(g). Put F = Ps: U Fn • By OREY'S n

theorem (1956) (which is valid in M*) we find an co-model Q with a code in M* such that all sentences of F are valid in Q. By (A) the denotation CQ of C in Q is a code of an co-model m2 which belongs to w*(S). We claim that m, and m2 are ko-disjoint. First of all k., E m, by assumption and k o E ms by (A). Let X be a set in m l n m 2 and assume that it is not hyperarithmetical in k o. By definition of E there is an integer q such that X = Since X E m2 and since CQ is a code of m2' we see that X = C~) for some r. Now let n = max(q,r)+1 and s = hn(r,q). We have to consider two possibilities: either s E E(q) or s non E E(q). In the former case we use (g) and obtain Jtr , s) ¢ CQ whence s ¢ C~) = X which is a contradiction because X = e». In the latter case we use (f) and obtain a similar contradiction. The theorem is thus proved.

e».

References BoYD, R., G. HENsEL and H. PUTNAM, 1969, A recursion-theoretic characterization of the ramified analytical hierarchy, Transactions of the American Mathematical Society, vol. 141, pp. 37-62 ELLENTUCK, E., 1972, A minimal w-model for strong second order arithmetic, Fundamenta Mathematicae, vol. 73, pp, 125-131 ENDERTON, H. B. and H. FRIEDMAN, 1971, Approximating the standard model of analysis, Fundamenta Mathematicae, vol. 72, pp. 175-188 ERDOS, P., A. HAJNAL and R. RADO, 1965, Partition relations for cardinal numbers, Acta Mathematica Hungarica, vol. 16, pp. 93-196 GRZEGORCZYK, A., A. MOSTOWSKI and Cz. RYLL-NARDZEWSKI, 1958, The classical and the co-complete arithmetic, Journal of Symbolic Logic, vol. 23, pp. 188-206 KEISLER, H. J., 1971, Model Theory for Infinitary Logic, (North-Holland, Amsterdam) KLEENE, S. C., 1959, Quantification of number-theoretic functions, Compositio Mathematica, vol. 14, pp. 23-40 KREIsEL, G., 1965, Model-theoretic invariants; Applications to recursive and hyperarithmetic operations, in: The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley.ed, J. W. ADDSON, L. HENKIN and A. TARSKI(North-Holiand, Amsterdam), pp. 190-205

28

A.

MOSTOWSKI

MOSTOWSKI, Ao, 1961, Formal system of analysis based on an infinitistie rule of proo/, in: Infinitistic Methods, Proceedings of a Symposium of Foundations of Mathematics, Warsaw, 1959 (pergamon Press, New York), PP. 141-166 MOSTOWSKI, A., 1972, A transfinite sequence of co-models, Journal of Symbolic Logic vol. 37, PPo 96-102 OREY, So, 1956, co-consistency and related properties, Journal of Symbolic Logic, vol. 21, pp. 246-252 Rosssa, J. B., 1937, GiJdel theorems for nonconstructive logics, Journal of Symbolic Logic, vol. 2, pp. 129-137

Added in proof: Professor Sacks has informed me that he knows the solution of problem (3).

A SURVEY OF DECIDABlLITY RESULTS FOR MODAL, TENSE AND INTERMEDIATE LOGICS 1

D. M. GABBAY Stanford University, Stanford, California. USA

1. Systems considered The systems we are concerned with are mainly propositional modal and tense logics and intermediate logics (i.e., systems between the intuitionistic propositional calculus and the classical propositional calculus). We shall also consider first-order theories based on these logics, in particular theories based on the intuitionistic predicate logic. 2. The language Syntactically, modal and tense logics are concerned with the addition of one or two additional unary connectives to the language of classical logic. The language of intuitionistic logic I and its extensions is, of course, the same as that of classical logic. Let us denote the additional modal unary operator by L and the additional tense operators by G and H. So, for example, the set of wff of the modal language is defined by the usual induction clauses with the additional clause that if A is a wff so is LA. 3. The basic systems We are not concerned here with the interpretation of these systems or the motivations for investigating them; our aim is to present the main methods that can be employed in solving the decision problem of a given system. Emphasis is laid on methods with examples of results. 1 This research has been supported in part by National Science Foundation Grant NSFGJ-443X.

30

D. M. GABBAY

Let us just mention that LA reads: 'A is necessarily true', GA reads 'A will always be true' and HA reads 'A was always true', The basic modal system, call it L, has the following axioms: (1) All substitutions (in the enlarged language) of the truth functional tautologies. (2) The inference rules of detachment and substitution. (3) The rule: I-A I-LA

+-+ B +-+ LB'

This boils down to adding one additional unary operator L (to the classical language) that is invariant under provable equivalence and has no other special properties. The basic tense system is Kt of Lemmon; Kt has Axioms (1) and (2) (for the tense language) and also

(4)

G(A

1\

B)

+-+

(GA

1\

GB)

H(A

1\

B)

+-+

(HA

1\

HB).

(5) The rules: I-A I-HA,I-GA

-------

(6)

'" A

-+

H '" GA

'" A

-+

G '" HA.

These axioms give G, H the minimal properties of tense operators, namely, that time is given by a binary earlier-later relation R (that is not even required to be transitive). The basic intermediate logic is the intuitionistic calculus, We assume familiarity with the system. 4. Modal systems Let us describe in this section some of the main modal systems considered in the literature. The smallest among the philosophically interesting systems is the system C-2 of Lemmon, with Axioms (1) and (2) above and Axioms (7) and (8) below: (7) The rule: I-A I-LA

-+ -+

B LB'

MODAL, TENSE AND INTERMEDIATE LOGICS

(8)

(LA /\ LB)

+-+

31

L(A /\ B).

C-2 is considered as the system with the minimal properties of the 'necessity' operator. K is the extension of C-2 with (9). (9) The rule:

I-A

--,=LA' 52 (respectively KT) is the extension of C-2 (respectively K) with Axiom

(10) below and the axiom L(A ~ A). Where (10) LA ~ A. 84 (respectively K4) is the extension of KT (respectively K) with (11). (11) LA ~ LLA. The system KB resp 85 is the extension of K resp 84 with (12). (12) A ~ L ~ L ~ A. The system 84.1 is the extension of 84 with (13). (13) ~ L ~ LA v ~ L ~ L ~ A. The system 84.2 is the extension of 84 with Axiom (14). (14)

L

L

~

~

A v L "'" LA.

The system S4.3 (resp K4.3) is the extension of 84 (resp K4) with (15)

L(LA

~

B) v L(B /\ LB

~

A).

5. Tense systems The main tense systems considered in the literature are the following: (a) The system Kt4 for transitive time, obtained by adding to K, the analog of Axiom (11) for G and H. (b) The system Kt4.3 for linear time, obtained by adding to Kt4 the analog of Axiom (15) for G and H. (c) The system KtR for rational time obtained by adding to K, 4.3 the axioms: GGA ~ GA ~ G(A /\ ~ A) HHA

~

HA

~

H(A /\

~

A).

(d) The system for real numbers 'time'. Other systems may be considered such as binary-tree (i.e., binary forking) time, integer time, etc. We shall mention more systems later.

32

D. M. GABBAY

6. Intermediate logics The intermediate logics that are well known are, e.g., KC (with the axiom ,.., A v ,.., ,.., A), LC (with the axiom (A -4 B) v (B -4 A)), KP (with the axiom (,.., A -4 (B v C)) -4 ((,.., A -4 B) v (,.., A -4 C))), Dm (with the axiom):

and others. 7. Decidability by algebraic methods Consider the Lindenbaum algebra of the propositional system L. This is nothing but a Boolean algebra (with operation symbols s , v, -4, ,..,) with an additional unary operator L. One can now define an algebraic model for the system L, by assigning elements of the algebra to the atomic propositions and extending the assignment (or valuation) to all formulas by performing the respective algebraic operations that correspond to the connectives. Stronger systems, e.g., 82 or 84, have algebraic models that fulfill more special properties (in fact, the properties expressed by the axioms). The algebraic models corresponding to intuitionistic logic are the wellknown Heyting (or Brouwerian) algebras. The first decidability results were obtained by investigating these algebras: DEFINITION 17. A system X is said to have the finite model property (fmp) if whenever X f-r- B, there exists a finite algebra (of the appropriate kind) where all the theorems of X are valid and B is not valid. THEOREM 18 (HARROP, 1958). If X is finitely axiomatizable and has the Imp then X is decidable.

The proof is simple since we can enumerate both the theorems and the nontheorems. McKINSEY (1941) showed that 82 and S4 have the fmp, Jaskowski showed that I has the fmp. McKinsey-Tarski investigated the algebraic models for modal logics and for a long time these methods were the main tool used to obtain decidability in modal logics. More examples of the applications of this method are:

MODAL, TENSE AND INTERMEDIATE LOGICS

33

THEOREM 19 (BULL, 1966). Every extension of 84.3 for which Axiom (9) is valid has the fmp and therefore is decidable if finitely axiomatizable. THEOREM 20 (McKAY, 1968). Every extension of the intuitionistic proposi-

tional calculus with axioms without disjunction has the fmp and is therefore decidable if finitely axiomatizable. We do not intend to give details on how these results are proved because later we shall describe the filtration method which is used to prove the fmp and which is essentially the same as the algebraic one. Let us just mention the general idea: if X fr B, take the Lindenbaum algebra of X and take some finite subalgebra, containing all subformulas of B, such that (if possible) it is closed under the operations and fulfills the axioms of X (of course B is not valid (under the natural assignment) in this algebra». 8. Discussion All the modal and intermediate propositional systems listed above and a good deal of the tense systems above have the fmp and therefore are decidable, though many of the proofs are not algebraic but use the selective filtration method (to be described later). This method (of proving decidability through the fmp) has its limitations. First, it does not apply at all, i.e., the definitions are useless, to predicate logic, and second, even for propositional calculi we cannot go too far, as the following theorems show: THEOREM 21 (MAKINSON, 1969). There exists an extension of modal KT which is finitely axiomatizable and lacks the fmp. THEOREM 22 (FINE, 1970; KUZNECOV-GERCIU, 1970). There exists an extension of intuitionistic logic which is finitely axiomatizable and lacks thefmp. THEOREM 23 (GABBAY, 1971). (a) There exists a decidablefinitely axiomatizable extension of modal KT which lacks the fmp. (b) There exists a decidable extension of the intuitionistic logic which lacks the fmp. A stronger method of proving decidability, called the reduction method (GABBAY, 1969) via a theorem of RABIN (1969) shall be described later.

34

D. M. GABBAY

Meanwhile let us study the filtration method (LEMMON-SCOTT, 1966) and the selective filtration method (GABBAY, 1972a) which can be used to obtain the fmp, for many systems and which is also an important tool in the applications of the reduction method. 9. The Kripke semantics We assume familiarity with the Kanger-Hintikka-Kripke semantics for modal logics and the Kripke semantics for intuitionistic logic. To fix our notation let the structures be of the form (S, R, 0, D t ) where S is the set of possible worlds, R the accessibility relation, 0 the actual world and D t , t E S is the classical model associated with the world t. Let us denote the truth value of a formula A at a world t by IIAII,. A is valid at a structure (S, R, 0, D t) if IIAllo = I. X is complete for a class of structures M if all theorems of X are valid in any structure of M and if X fr- B then B is not valid in some structure of M. X is strongly complete for M if for every theory 0 and a formula B if f,

DEFINITION

1.1.

(i) cP E S if F= 3 xq; (ii) for i(f}) < w, cP E Srl"' if q; E S'I (iii) for I(f}) < co, q; E S'I'"' if there exists a disjoint sequence ' and rj be in ro>w. We have (i) if "P implies "P' and "P E S" then "P' E S" (ii) if 1jJ E S" then at least one of"P /\ "P', "P /\ I"P' E sq (iii) there exists a w-formula 1p*(Y) such that for any b E M, Ip(X; b) E S" if and only if F= Ip* [b] (iv) Rank e and Deg e exist (v) if"P implies "P' then either Rank e < Ranke', or Rank1jJ= Rank e' and Deg "P ~ Deg "P' (vi) either Rank e /\ "P' = Rank e or Rank e /\ I"P' = Rank "P

(vii) if Ip'(x; y) ELJ n then either Rankn"P /\ "P' < Rankn"P, or Rankn"P /\ I "P' < Rank, "P, or Degn"P /\ "P' < Deg, "P, or Degn"P r; I "P' < Deg, "P (viii) if Ip'(x; y) E LJ n there exists 1p#(X; y) such that for any hE M, Rankn"P /\ Ip' (x; b) = Rankn"P if and only if F= Ip# [a; bj. PROOF: (i) and (ii) are immediate by induction of 'fj. It is clear that (iii) is true for 'fj = (). Suppose that (iii) is true for 'YJ = ." <m) . From Section 2, of SHELAH (1971) we can form O'(x; y') such that for every

a' E M there exist a E M and n < 2 such that O'(x; a') is equivalent to O(x; a)n for some 0 ELJ/(T)' and conversely for any O(x; y) E LJ/(T)' n < 2, and a E M, there exists a' EM such that O(x; a)n is equivalent to O'(x; a'). Let yO, yt, ... be sequences of variables, pairwise disjoint and disjoint from y,such that I(Y')

= 1(jJ') for n < w. Let OO(x; yO) be O'(x; yO) and for each n < w let On+l(X; yO, be 0'( x;y -n+l) /\ I 0'(x;y. -0) F or eac h n < w, I et "Pn{" -n) " · , ,,n+l) f v:y-0 , ... ,y

be a w-formula such that F= "Pnfa; aO, ••• , an] if and only if"P r. on(x; aO, ••• , an) EST"<m). Let 1p*(Y) be a w-formula equivalent to 3y03 yl ... ("P 0 /\"P1 /\ .•.); elementary manipulations show that such Ip* must exist. If F= Ip* fa] then clearly there exist aO, at, ... such that for all n < wF=~[a; aO, ... , an] whence "p /\on(x; aD, ..., il") E ST"<m). Since (on(x; aO, ... , il")ln < w) is obviously disjoint, "P E sr"<m+ I). If "P EST" <m+l) then there exists a disjoint sequence (lplli < w) in LJ~T) such that "P r; Ipl E ST"<m) for each t < w. Choose bO,

hi E M such that for each i < 2n, Ipl is a Boolean combination of 0' (x; /;0), ... , 0' (x; 7}I). From (ii) without loss of generality we may ••• ,

suppose that for each i < 2n and j ~ I, either

Ipl

implies O'(x;bJ) or

Ipl

48

A. H. LACHLAN

implies iO'(x;Z,J). Now we can choose ijo, ... , (1'-1 in turn such that for each k < n there exists i < 2n such that CPi implies O'(x; ijk) A irJ'(x; jjk-1) A

and there exist at least

2n - k - 1

"lrJ'(x; aO), values of i < 2n such that CPi implies ••• A

"lrJ'(x; jjk) A "lrJ'(x; jjk-1) A It is clear from this that l= 3 yo 3ji1 ... 3yn-1( tp°(a; yO)

A

"lrJ'(x; aO).

•.. A

tp1(a; yO, ji1) A

••. A

lpn-1 (a; yO, ... , yn-1)).

Since n is arbitrary and M is ~ 1-saturated, 1= cP* [a]. We have shown that (iii) holds for 'fJ = r" <m + 1) if it holds for 'fJ = r" <m). Since (iii) is given for 'fJ = and holds for 'fJ = r" (0) if it holds for 'fJ = r, (iii) holds for all 'fJ E w>w. For (iv) suppose for proof by contradiction that tp = cp(x; a) E STr.(m) for all m < w. For each m there exists lim such that tp A rJ' (x; bm), tpAirJ'(x;b m) EST"<m>. From (iii) and the fact that Mis ~1-saturateditfol­ lows that there exists bE M such that 11' A 0' (x, iJ), tp A "l 0' (x, b) E ST"(m) for all m < w. Thus tp can be partitioned, by an instance of rJ'(x; y), into two formulas having the same property as 11'. It follows that rJ'(x; ji) is unstable by Section 2 of SHELAH (1970). Thus if tp E ST there is a greatest m < w such that 11' E ST"(m>. This is enough to show that Rank tp is well defined. Let tp E ST"(m> - ST"<m+l> and suppose for proof by contradiction that for arbitrarily large n there exists a disjoint sequence t{o 1-types over A, then T has > ~o (IAI + 1)-types, whence T has > ~o countable models. Thus without loss of generality we may assume that there are ~ ~o 1-types over any finite A. From 3.5 of VAUGHT (1961)

if A c

IMI

and

IAI
n) /\ rpl (x; c)i = Rank.e,, whence j = 0 by (i). This completes the proof. Let liE M realize Pro' Let I c M satisfy the following conditions for each n < OJ

52

A. H. LACHLAN

(i) a E I implies F On[a; bn] and F 'Pn[a] (ii) for any formula f{I(x; y), a E I, and C E (1- {a}) v Rngb- such that F w such that cfIn is defined and RanknV'n 1\ On(X; (tin) 1\ On(X; [jn) = RanknV'n. From Choice 2.2 (iii) and (vi), it follows that

rl

is unique. Further, if

n > 0, and a' is the initial segment of a~n such that lea') Rank.i , V'n-I

1\

On_1 (x; a')

1\

=

I(En-I), then

On_I (x; lin-I)

~ Rank n_1V'n I\On(X; a~n) I\On(X; lin) = (RanknV'n I\On(X; £l~n) I\On(X; b"))/n

= (RanknV'n)ln = Rank n_1V'n = Rank n_t V'n-I where the last equality holds by Choice 2.1. Thus 'YJn :=> 'YJn-t. Let 'YJ = U {'YJn In < w}, we call aTJ a quasi-index of 1.

A set J c M is called independent if each singleton subset is indiscernible with index in 8, and for any formula tp(x; y), a e J, and c e (J - {a}) u U {Rngc Ia E 8} such that F= tp [a; c] it is the case that Rank, V'n 1\ tp(x; c) = Rank, V'n for all n < w. Suppose J is independent, it is easy to see that there is a unique mapping F:J -+ 8 such that for any 'a e S, F-I(ti) is an indiscernible set, possibly empty, with index a. Further, if ao, ... , an_ 1 are distinct members of J then the type realized by n such that (G(m)jn, G(m-1)ln, ... , G(m-n)ln) = ('YJol n, 'YJll n, ... , 'YJnln).

To see that suitable ('YJd i < w) can be found consider a fixed n. It is enough to observe that, since G(j)(k) ~ C(k) < w for all j, k < to, as m varies through w there are only a finite number of values possible for (G(m)ln, G(m-1)ln, ... , G(m-nln). Let O~.o(x, y) and Oto(x, y') denote 3y'O*(x; y, y') and 3yO*(x; y, y'), respectively. For each n > 0 let O~.n(x, y) and Otix, y') denote

55

NUMBER OF COUNTABLE MODELS

3y'(O~,n_l (X,y') "O~,o(y',y» and 3y(O~.n_t(x.j') " O!.o(y,y'», respectively. From the way ao, a~, at , a~, '" were chosen it follows that if m > n then FO~.n[am+l' am_n] and F0t,n[am+l' a~_n]' Also, it is easy to prove by induction on n that if {a} is indiscernible and either 1= O~.n [a, b] or F 01,n[a, b] then {b} is indiscernible. It is necessary to know that if {a}, {b} are both indiscernible then a, b realize the same type, but this should be clear from the discussion of indiscernible sets above.

2.3. There exist b t, b~, b':«, b':«, ... e M such that, {btl is indiscernible, {b~, b':«, b'-.2' ... } = B is independent with F(b'-.l) = a'll for each i < w, and 1= OL [b lo b'-.l] for each i < co, LEMMA

PROOF: We regard b l ' b~, b'-.t, b':«, ... as unknowns in M. Let To be a set of closed formulas expressing the indiscernibility of {bd. Consider a proposition P of the form

(Fq;[b'-.m; C])

~

Rank.e, "On(x; a'lm l(n+1» "q;(x; c) = Rankn"Pn

c

where q;(x;)I) eL1 n, m and n are < co, and E (B- {b'-.m}) uU {Rngala EEl. By Lemma 1.1 (viii) P is expressible by a first-order formula. Further, because Deg e, "On(x; a'lm!(n+l) = 1, P and Fq;[b'-.m; C] together imply that Rankk"Pk "Ok(X; a'lm!(k+l) "q;(x; c) = Rankk"Pk for all k < co, Let I', consist of all first-order formulas expressing propositions such as P. Note that the members of I', will contain parameters from B uU [Rngji la e E}. Finally, let T 2 = {OL(bt , b'-.I)1 i < w}. From the definition of independent set and the ~ t-saturation of M it is sufficient to show that the set T = To uTt uT2 is consistent. Let .1 be a finite subset of T in which at most b t, b~, ... , b'-.k occur from Bu{b t}. For n < w let m be chosen such that

(G(m)ln, G(m-1)ln, ... , G(m-n)ln) = (17ol n,

, 17nl n).

For all sufficiently large n, letting b t = am+t, b~ = a~, , b'-.k = a~_k will satisfy .1, whence .1 is consistent. Thus T is consistent which proves the lemma. Returning to the proof of the theorem let us now suppose that M is Alia-saturated where A ~ ~t. Then we can find an independent set J c IMI such that IJI = A and IF- t(ii"t)1 = A for each i < w. Recall from the discussion of independent sets above that the type realized by a sequence (co, ... , cn_t) of distinct members of Jdepends only on (F(co) , ... , F(cn_ t ) . By Lemma 2.3, given any A c U {F-t(a'lt)! i < w} such that IAII Fr! (a'll)I

56

A. H. LACHLAN

= 1 for each i < w, there exists b such that {b} is indiscernible and for each i < w F0T,t[b, b;] where b, is the unique member of A II F-l((j'1t). Such b is said to select A. Consider particular i < wand let 0T,tCY, x) be in Ll j • From Lemma 1.2 Rank j 1J'j 1\ O!.t(b, x) < Rank j 1J'j whence F-l((j'1t) can be extended to an indiscernible set I c M such that 11- Fr ! ((j'1I) I = ~o and Fi<J?T,I[b, a] for 'each a E F-l(a"l). From 5.9 of SHELAH (1970), F <J?T,I[b, a] for only finitely many a E F-l((j'1I). Thus a particular b selects at most 2110 subsets A. Hence for all sufficiently large A there are at least AliO L-types over J whence T is not superstable. This contradiction of our original hypothesis completes the proof of the theorem.

References BALDWIN, J. T. and A. H. LACHLAN, 1971, On strongly minimal sets, Journal of Symbolic Logic, vol. 36, pp. 79-96 LACHLAN, A. H., A property of stable theories, Fundamenta Mathematicae (in press) MORLEY, M. D., 1965, Categoricity in power, Transactions American Mathematical Society, vol. 114, pp. 514-538 MORLEY, M. D., 1970, The number of models of a countable theory, Journal of Symbolic Logic, vol. 35, pp. 14-18 SHELAH, S., 1969, Stable theories, Israel Journal of Mathematics, vol. 7, pp. 187-202 SHELAH, S., 1971, Stability, the finite cover property, and superstability; model-theoretic properties of formulas in first-order theory, Annals of Mathematical Logic, vol. 3, pp. 271-362 VAUGHT, R. L., 1961, Denumerable models of complete theories, in: Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics (pergamon Press, New York), pp, 303-321

COUNTABLE MODELS WITH STANDARD PART M. MORLEyI Cornell University, Ithaca, N. Y., USA

One may consider co-logic as the study of those models where one specified part is isomorphic to the natural numbers, (w,

8) A - p-~P;P => A - n;HJIHH,rJ;p; 9) A ~pmBApm ~lBpm; 10) A - p-n;H.1lHHJ.\p Apm ~ 1 A Apm

o1 (B).

~

A

~ 1 B);

A;

B.

Oraonrenae pm-CBO,rJ;HMOCTH MO)J{HO ObillO ObI CtIHTaTb aHaJIOrOM peKypCHBHOH nepetIHCJlHMOCTH P;IDI m-CBOp;HMOCTH, Op;HaRO cBoHCTBa

9, 10

nOKa3bIBaIOT, 'ITO 3TO ObIJIO ObI He yp;OBJIeTBOpHTeJIbHO. Ho npeacne

norrsrraexca

nOHHTb, oYP;eT JIH OTHOllleHHe

JIOrOM oTHorueHHIO 'A-peKypcHBHo B

B' (A ~ T B)?

rrocnezmero

ecmr

H

(N\A)

BblllOJIHeHO

peKypCHBHO B

m-CBOp;HMOCTH.

CBOHCTBO :

B.

A

~mB

IIo3TOMY cztenan

ED (N\B)

~ m B ana-

3aMeTHM, 'ITO P;JIH

peKypCHBHO

B

B,

TO

3TO Ba)J{HOe CBPIICTBO He cnpaseanaao p;JIH MHHHMaJIbHOe H3MeHeHHe rrpaxozraa

K onpeneneaaro: oyp;eM roaoparr., 'ITO eCJIH

A-

A

(anecs

C EB D

~

A m-peKYPcu8110 8 B (A ~ m B), {2xlxeC}u{2x+ l!xeD}).

3aMeTIIM cpasy, tITO

12) A ~mB(N\A) «r», 13) A ~mBA ~m(N\B).

Teneps yxce JIerKO onpezternrrs nOHHTHe m-nepetIHCJlHMOCTH: MHOA - m-nepe1JUCIlUMO 8 B (A ~ pm B), eCJIH A ~ pm B ED (N\B). 14) A ~ pm B A ~ pm (N\B). 0003HatIHM tIepe3 mjA MHOmeCTBO (AEB (N\A»)pm. 15) A ~lmjA; 16) mjA ~ mj(N\A); 17) A =mB=>mjA ==mmjB. ECJIH tIepe3 dm(A) 0003HatIaTb m-CTeneHb A, TO CBOHCTBO 17 nOKa3bIaaer, tITO MO)J{HO onpezremrrs onepanaro mj aa MHOmeCTBe scex m-ete-

meCTBO

72

to. JI. !!PllIOB

neaea TaK: mjd",(A) ~ d",(mjA). ECJIH d - m-CTeneHI>, TO mjd Ha30BeM m-CKa'lKOAt d. 18) ECIDIdmo6aHm-CTenem,TOd<mjd,TOeCT1>d ~ mjd,Ho i(mjd ~ d). 19) B

A ¢> d",eB) ~ mjd",(A). 18) II 19) onpaazrsraaror

~ P'"

CBOHCTBa

eCTeCTBeHHOCTb orrpenenenaa

m-

cxanxa.

2. Tlepexonaa

K

onpenerreaaro

BII~HO,

srro

B nO('TpOeHlIlI O. Hs onpenerreaaa

CaMOK IIepapxIIII.

IICllOJTh30BaHa CIICTeMa 0003HaqeHIIH MlIHII

npyrne

MO>IIY>o a &xEMy}P"'.

Ilonaraex nanee m

E(a) ~

xanecme M 1

m

{XIX ~/IIMa};

1)

3aAtettaHUR:

B

aE 0

)J;IDI mo6oro

In

lIla) ~

{XI(N\X) E k(a)}'

ECJIII IICnOJTh30BaTb OTHomeHIIe

MO>I mo6oe

percypcasaoe

~ '"

BMeCTO

~ III' TO

MHOllieCTBO. 3TO He

In

In

In

2) B npeztsrnymax patiorax BMeCTO E(a), lIla) IICnOJTh30BaJIOCI>, nO-BIIll;lIMOMy, MeHee ynaxaoe 0603HaqeHIIe E(a~, [J(a~' 3) TOJThKO B IIyHKTe (3) orrpeneneaaa lIMeeTCH nexoropoe OTJIIIqIIe OT CT~apTHOro reneps nocrpoeaaa ranepaparpxeraaecxoa aepapxaa C IIC-

II3MeHIIT KJIaCCOB E(a)'

nOJTh30BaHIIeM T-CKatn
1)

E(I)

reneps

In

= lI(l) -

In

In

OCHOBHbIe CBOHCTBa IIepapXlIII

xnacc scex

2)

E(2) -

3)

ECIDI

aeO, a ¥= 1,

4)

ECJIlI

a, bE 0, a < 0 b,

KJIaCC

peKypClIBHbIX MHQ)KeCTB;

scex peKypcIIBHO nepe~CJIlIMbIXMHOllieCTB; In

In

TO E(a):P lIla); m

m

TO E(a) u

m

.ILJm

m

mo6oro /II

6)

U E(a)

aeD

a EO, E(a)

= Ll~ ;

!;;

m

m

JJ(a) ~ E(b) n lIth) (=Ll(b»' BKJIIO-

qeHlIe CTpOrOe;

5)

{E(a)la E O}:

Ll~ ( = E~

n IIg);

73

HEPAPXWI MHO)i(J!CTB m

7) U E(a) = Ef aeO

HlUiMCHblIIlUI

6yJICBa

anrefipa MHO}l{CCTB,

conep-

lalo

)l{aI.QlUI BCC pCKypCHBHO nCpC'lliCJIHMbIC MHO}l{CCTBa; 8) ECJIH a e 0, b = 2a , TO

m

m

m

XELl Cb) ~3Y3Z(YEE(a) &ZE[]Ca) &X:::::: Y EEl Z). m

E(a), a E 0 3aBHCHT HC TOJIbKO OT opna0603HaQCHHH a. 30) (the 'reals') and we call subsets of Cartesian products Xl x X X k with each Xl = OJ or Xi = EJ1t pointsets; E,} ,il,}, L1~ and ~~, n~, a~ are respectively the analytical and projective classes of pointsets (pointclasses) of level n ~ 0.3 We work in ZermeloFraenkel set theory with the axiom of Dependent Choices, ZF + DC, and we explicitly state all additional hypotheses, including the full Axiom of Choice, AC. An inner model of ZF + DC is a transitive class M which contains all ordinals and satisfies all the axioms of ZF + DC. 1 The author is a Sloan Foundation fellow. Work on this paper was sponsored in part by NSF Grant GP-27964. I want to thank my student and collaborator A. S. Kechris for the many stimulating conversations we have held on these problems. 2 Theorem 4 is announced here for the first time and Theorems 11-13, 15, 16 and parts of 14 were obtained after the Bucharest meeting. (Detailed proofs of these will be given in MOSCHOVAKIS (in preparation)). Failure to assign credit to a theorem or conjecture does not mean that it is due to me but that I do not know the correct credit. 3 For a brief list of definitions and a discussion of the basic notions that we need, see Sections 1,2 of MOSCHOVAKIS (1970). In particular L'J = all recursively enumerable sets, I:J = all open sets.

78

Y. N. MOSCHOVAKIS

1. Determinacy With each set A of reals we associate a game as follows: Players 1 and II successively choose natural numbers no, n 1 , n 2, ... (I picks no, II picks nl' 1 picks n2' etc.) and if ex = (no, nl' n2' ... ) E A, 1 wins, if ex ¢ A, II wins. A strategy for player 1 is a function on sequences of numbers of even length (including the empty sequence) into ro, a strategy for player II a function on sequences of numbers of odd length. If I's strategy a is pitted against II's strategy T, the resulting play is where no

= tj(0), nl = .(no), n2 = tj(no, nl), n 3 = .(no, nl' n2), ... , etc.

We call a winning for 1 if VT[a*T E A] and T winning for II if Va[a*. ¢ A] The set A is determined if either 1 or II has a winning strategy. If A is a class of sets, put Determinacy (A) Every set in A is determined. We also abbreviate,

AD



PD



Every set is determined, Every projective set is determined.

Games of this type were first introduced in the literature by GALE and STEWART (1953) where it was shown that closed sets are determined and that, granting the axiom of choice, there exist nondetermined games. The strongest result in ZF + DC to date is THEOREM 1. Determinacy (~~). (J. Paris.) Here ~~ = open sets, ~~ = Fa = countable unions of closed sets, etc. Paris' result is the latest in a sequence of difficult theorems by M. DAVIS (1964) (Determinacy (~g)) and D. A. MARTIN (Determinacy ~~), granting a weakly compact cardinal). On the other hand FRIEDMAN (1970) has shown that Determinacy (AD cannot be shown in Zermelo set theory, i.e., any proof of this must involve some essential use of the replacement schema. If we are willing to assume strong infinity axioms, then the situation is much better. THEOREM 2. If there exists a measurable cardinal, then Determinacy (~D. (MARTIN, 1970). Martin has extended his theorem to prove the determinacy of a reasonably large class of A! sets on the basis of somewhat stronger hypotheses,

79

ANALYTICAL DBFINABILITY

but it was clear from the very first results in this area that axioms of a different type will be needed to establish Determinacy (aD. The strongest known result which points in this direction is THEOREM 3. If Determinacy (aD, then there is an inner model of ZF +AC+3u [uis measurable and the limit ofmeasurabies]. (R. M. Solovay.) As one would expect, PD does not follow from Determinacy (AD, for any n. It is perhaps surprising that the only known proofs of this depend on Theorem 7 below. We state here a rather general result which shows that we can construct inner models and keep fairly good control on the class of determined projective sets.

.E:

THEOREM 4. Assume PD, call an inner model M r:-correct if formulas are absolute for M. For each n ~ 2 there is a smallest .Ei-correct model of ZF+DC, call it M"; M" further satisfies AC, GCH, Determinacy (A:_1) , iDeterminacy (a:+l) and in M" the continuum admits a .1:+1 wellordering (Y. N. Moschovakis).? By the definition, M2 = L and in this case we actually know that L satisfies iDeterminacy (An and that in L the continuum admits a .1~ wellordering. These stronger properties generalize to all even n (see Theorem 15) but apparently not to the odd n. At present there is scant direct evidence for PD; those who accept it as a plausible, working hypothesis do so because of its many and beautiful consequences and the apparent 'naturalness' of the proofs of these consequences. A better understanding of the hypothesis PD would seem to require proofs of at least the following two conjectures. CONJECTURE 1. Determinacy

(aD

can be established in ZF + AC.

CONJECTURE 2. (R. M. Solovay). The existence of strongly compact cardinals implies PD. 2. Prewellorderings and the periodicity phenomenon One of the early concerns of classical Descriptive Set Theory was the establishment of closure and structure properties for the classes 1:1, ai, e.g.,

m,

To simplify matters, we shall take PD as our standard hypothesis in most of the results below. Often finer theorems can be proved from weaker hypotheses. Almost all the results in this paper assume PD and generalize to arbitrary n theorems of ZF +DC which are well known for n = 1 or n = 1,2. I shall not always identify, credit and separate the classical results. 4

80

Y. N. MOSCHOVAKIS

the closure of A} under continuous one-to-one images, the Reduction and Uniformization properties of n} , Suslin's identification of A} with the Borel sets, etc. Some results of this type were proved for the classes n1,1:1, Ai, but nothing nontrivial was shown about the higher levels. The hypothesis that every set is constructible in the sense of Godel, V = L, yields an almost complete structure theory for the projective sets, see ADDISON (1959a, 1959b). It was an exciting discovery that the hypothesis PD also settles most of these questions, and that the answers it gives are different from those implied by V = L. The key to many of these results from either of the hypotheses V = L or PD is the Prewellordering Property for a class of pointsets say = 1:~ or = n~ for some n. A norm on a pointset P is a function tp: P -+ ordinals; we call rp a Tsnorm if there are relations ~r, ~r in r and the dual class (of all complements of relations in 1), respectively, such that

r,

r

r

r

yEP => (Vx)[x ~rY ¢> x ~rY ¢> (x

Put Prewellordering (1) ¢> Each P in

E

P & rp(x)~· rp(y))].

r admits a Tsnorm,

THEOREM 5. Assume PD.. For each odd n ~ 1, Prewellordering (II;) and for each even k ~ 0, Prewellordering (Ei). (The First Periodicity Theorem; MARTIN, 1968; ADDISON and MOSCHOVAKIS, 1968.)5

Of course Prewellordering (I1D, Prewellordering (El) are provable in ZF + DC, and they help explain the similarities in structure between/Z] and E~. The differences in structure between III and 1:1 stem mostly from the fact that III is closed under universal and quantification while E~ is closed under existential real quantification. Theorem 5 allows us to extend many results about III to all II; , odd n and many results about E1 to all 1:f, even k, see ADDISON and MOSCHOVAKIS (1968). For example, put Reduction (1) ¢> For every two pointsets P, Q in P 1 , Q1 in such that

r

P1

S;

P,

Q1 £ Q,

P1

U

Q1

=P

u Q,

r, there are pointsets P 1 n Q1

= 0.

The application of determined games to Descriptive Set Theory was initiated by (1967) which gave a new proof of Reduction (lID using the known Determinacy C1=y). Addison and Martin independently saw that this technique proved Determinacy (aD => Reduction (lID; then Martin and Moschovakis independently proved Theorem 5, with the inspiration for Moschovakis coming from a lecture by Addison on his result. S

BLACKWELL

81

ANALYTICAL DI!FINABILITY

THEOREM 6. Assume PD. For each odd n, Reduction (Il~) and for each even k, Reduction (.ED. (MARTIN, 1968; ADDISON and MOSCHOVAKIS, 1968.)5 If V = L, then for every I ~ 2 Prewellordering (.El) and hence Reduction (.El). Since neither the prewellordering nor the reduction property can hold for both .El and Ill, PD gives a structure theory for the analytical classes which is radically different from that in the inner model L. Roughly speaking, if V = L, then all .E/ with I ~ 2 are very similar to .Ei, while .Et stands apart; while if PD, then all .Ei, with even k are like .E~, and all .E~ with odd n are like .E1. It should be pointed out-that in the inner models M" of Theorem 4 the prewellordering property oscillates finitely often and then settles on the 1: side, e.g., in M3 and M 4 it holds for 1:6, lIt , 1:1, Ill,

.El, 1:~, 1:J, ... 3. Scales and more periodicity Some of the deeper properties of III do not seem to follow from the prewellordering property alone. The more subtle Scale Property that implies them was isolated by analyzing carefully the classical proof of the NovikoffKondo-Addison Theorem (SHOENFIELD, 1967) and that of the MartinSolovay Basis Theorem for .Ej (MARTIN and SOLOVAY, 1969). A scale on a pointset P is a sequence {qJn}nea> of norms on P such that if {X/}ie", is a sequence of points in P, if limitl-+oox/ = x, if for each fixed n and all large i, qJn(x/) = An' then x E P and for each n, qJn(x) ~ An; we call {qJn}ne", a Tvscale, if there are relations Sr, Sy, in rand t; respectively, such that for each n, y, yEP => (Vx)[Sr(n, x, y)

Put

¢>

Sy,(n, x, y)

Scale (F) ¢> Each P in

¢>

(x

E

P & qJn(x) ~ qJnCY))].

r admits a Tsscale,

THEOREM 7. Assume PD. For each odd n ~ 1, Scale (Il~) and for each even k ~ 0, Scale (.ED. (The Second Periodicity Theorem, MOSCHOVAKIS, 1971.) The first corollary of this is the extension of the Uniformization Theorem to the higher levels. Put Uniformization (F) ¢> For each relation Pi«, x) in in such that P*(a., x) => Pt«, x)

r

r, there is some P*(a., x)

(3a.)P(a., x) ¢> (3 !a.)P*(a., x).

82

Y. N, MOSCHOVAKIS

THEOREM 8. Assume PD. For each odd n;?; 1, Uniformization (II;) and for each even k ;?; 2, Uniformization (EI). (MOSCHOVAKIS, 1971.) TmOREM 9. Assume PD. Each nonempty analytical set A ~ &l contains an analytical real. More precisely, if k is even;?; 2, then every nonempty El set contains a 11i real; if n is odd, then there is a fixed 11;+1 real 0(0 such that each nonempty E; set contains a real recursive in 0(0' (MOSCHOVAKIS, 1971, for the general case. For n = 3 this is due to MARTIN and SOLOVAY, 1969, and in this case the result can also be proved under the alternative hypothesis that there exists a measurable cardinal; MANSFIELD, 1971, gave an improvement of the Martin-Solovay Theorem which also generalizes to all odd n granting PD.) Another neat definability result which follows easily from the two periodicity theorems is the classification of {O( E f!ll : 0( is 11;} in the analytical hierarchy. TmOREM 10. Assume PD. If n is odd, then the set {O( Eat: 0( is 11;} is II; but is not E;; if k is even ;?; 2, then the set {ex Eat: ex is 11:} is Ei but is not Ill. In 1968 Martin conjectured that for each 111 game, the winning player has a 11l+ 1 strategy. Martin knew that this implied Theorem 12 below. It turned out that a stronger periodicity theorem actually holds which almost finishes the job of generalizing the structure theory of lIt to II; for all odd n. THEOREM 11. Assume PD. If n is odd and player I wins a II; game, then I has a 11;+1 winning strategy; if k is even and player I wins a El game, "then I has a 111+ 1 winning strategy. (The Third Periodicity Theorem, Y. N. Moschovakis.) Only the second of these assertions is new, the first being a trivial consequence of Theorem 9. We list two corollaries of it here, but Theorem 11 is also essential for the proofs of Theorems 14-16. THEOREM 12. Assume PD. For each odd n, a set of integers A is II; if and only if there is a IILI relation Pt«, i) such that i

E

A¢> (3 ex

E

11;)P(ex, i).

(Y. N. Moschovakis for the general case; for n = 1 GANDY, 1960, SPECTOR, 1960, with no determinacy hypotheses.)

ANALYTIeAL DEFINABILITY

83

THEOREM 13. Assume PD. For each odd n :;;?; 1, a set A is Lf; if and only if A is the recursive one-to-one image ofaII;_l set; hence A is ~~ if and only if A is the continuous one-to-one image of a n~_l set. (Y. N. Moschovakis.) One basic property of 1:} has stubbornly resisted generalization to all odd levels til now. CONJECTURE 3. If PO and n are odd, then each nonempty E; set of reals contains an element which is recursive in some 1:; set of integers. 4. Countable analytical sets It is an easy consequence of Theorem 9 that every countable analytica set of reals contains only analytical reals. The next result gives some more detailed information about these sets. THEOREM 14. Assume PD. If n is odd, then there is a largest countable II: set of reals and if k is even :;;?; 2, then there is a largest countable 1:l set of reals. (For k = 2, SOLOVAY, 1966, and for even k » 2, KECHRIS and MosCHOVAKIS, in press. For n = 1 G. Sacks and A. S. Kechris independently and for all n > 1, A. S. Kechris.) For even k, let Ck be the largest countable 1:l set. SOLOVAY (1966) proved that C z exists from the very weak hypothesis that there are only countably many constructible reals and in fact he established that C2 = L n §to THEOREM 15. Assume PD. For each even k :;;?; 2 let

LA:

=

L(Ck)

be the smallest model of ZF which contains Ct. Then Ck = L k n tJ,l, L k is 1:l-correct and it further satisfies AC, GCR, '§t admits a Lfl well-ordering', Determinacy (~~-1),1 Determinacy (~D. (A. S. Kechris and Y. N. Moschovakis.) The models L k seem to be the correct higher level generalizations of L = L 2 , but more research is needed before we can understand them completely. It is a theorem of ZF + DC that every countable 1:} set of reals contains only L1 ~ reals, see MANSFIELD (1970). We only know a partial generalization of this fact at this time. THEOREM 16. Assume PD. If n is odd and A is a countable L1; set of reaIs, then A contains only L1~ reals. (Y. N. Moschovakis.)

84

Y. N. MOSCHOVAKIS

CONJECTURE 4. If PD and n are odd, then each countable .E~ set of reals contains only LI~ reals.

Proofs of Conjectures 3 and 4 would complete the program of generalizing the definability theory of .Et and u; to all odd levels. References ADDISON, J. W., 1959a, Separation principles in the hierarchies of classical and effective descriptive set theory, Fundamenta Mathematicae, vol. 46, pp. 123-135 ADDISON, J. W., 1959b, Some consequences of the axiom of constructibility, Fundamenta Mathematicae, vol. 46, pp. 337-357 ADDISON, J. W. and Y. N. MosCHovAKIS, 1968, Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences, U.S.A., vol. 59, pp. 708-712 BLACKWELL, D., 1967, Infinite games and analytic sets, Proceedings of the National Academy of Sciences, U.S.A., vol. 58, pp, 1836-1837 DAVIS, M., 1964, Infinite games of perfect information, Advances in game theory, Annals of Mathematics, vol. 52, pp. 95-101 F'JuEoMAN, H. M., 1970, Higher set theory and mathematical practice, Annals of Mathematical Logic, vol. 2, pp. 325-357 GALE, D. and F. M. STEWART,1953, Infinite games with perfect information, Ann. Math. Studies, vol. 28, pp. 245-266 GANDY, R. 0., 1960, Proof of Mostowsk": ccrjccture, Bulletin de l'Academie Polonaise des Sciences, vol. 8, pp. 571-575 KEcHRrs, A. S. and Y. N. MOSCHOVAKIS, Two theorems about projective sets, Israel Journal of Mathematics (in press) MANSFIELD, R., 1970, Perfect subsets of definatuc 0':: .. of real numbers, Pacific 'Journal of Mathematics, vol. 35, pp. 451-457 MANSFIELD, R., 1971, A Souslin operation on IlL Israel Journal of Mathematics, vol. 9, pp. 367-379 MARTIN, D. A., 1968, The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74, pp. 687-689 MARTIN, D. A., 1970, Measurable cardinals and analytic games, Fundamenta Mathematicae, vol. 66, pp. 287-291 MARTIN, D. A., Projective sets and cardinal numbers (to appear) MARTIN, D. A. and R. M. SOLOVAY, 1969, A basis theorem for EJ sets, Annals of Mathematics, vol. 89, pp. 138-159 MOSCHOVAKIS, Y. N., 1970, Determinacy and prewellorderings of the continuum, in: Mathematical Logic and Foundations of Set Theory, ed, Y. Bar-Hillel (NorthHolland, Amsterdam-London), pp. 24-62 MOSCHOVAKIS, Y. N., 1971, Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77, pp, 731-736 MOSCHOVAKIS, Y. N., Descriptive set theory, a foundational approach (North-Holland, Amsterdam) (in preparation)

ANALYI1CAL DEFINABILITY

85

SHOENFIELD, J. R., 1967, Mathematical Logic (Addison-Wesley, Reading, Mass.) SOLOVAY, R. M., 1966, On the cardinality of Ei sets of reaIs, in: Foundations of Mathematics, eds. J. J. Bulloff, T. C. Holyoke and S. N. Hahn (Springer-Verlag, Berlin), pp, 58-73 SPECTOR, c., 1960, Hyperarithmetical quantifiers, Fundamenta Mathematicae, vol. 48, pp, 313-320

ON RECURSIVE UNSOLVABILITY OF HILBERT'S TENTH PROBLEM

YD. V. MATIJASEVIC Leningrad Branch, Steklov Institute of Mathematics, Academy of Sciences of the USSR, Leningrad, USSR

In 1900 David Hilbert delivered before the Second International Congress of Mathematicians his famous lecture entitled, "Mathematical Problems." In his lecture he posed 23 problems, the solutions of which he thought would stimulate the further development of mathematics. The tenth problem was formulated as follows (see HILBERT, 1900): Eine diophantische Gleichung mit irgendwelchen Unbekannten und mit ganzen rationalen Zahlkoe./fizienten sei vorgelegt; man soli ein Verfahren angeben, nach welchem sich mittels einer endlichen Anzahl von Operationen entscheiden ldsst, ob die Gleichung in ganzen rationalen Zahlen liisbar ist. The tenth problem is the only one of the 23 problems which clearly has an algorithmical nature. The problem proved to be rather difficult and only last year it was shown to be unsolvable: There is no algorithm for determining whether an arbitrary diophantine equation has a solution. Here it is immaterial whether we are interested in solutions in integers or in nonnegative integers, because it is well known that these two decision problems are equivalent. In what follows the variables range over natural numbers 0,1,2, '" Here we give a sketch of the history of how the unsolvability of Hilbert's tenth problem was proved. The first work aimed at proving the unsolvability appeared at the end of the 1940's and the beginning of the 1950's. At that time Martin Davis posed the following conjecture:

90

YU. V. MATIJASEVIC

Every recursively enumerable (r.e.) predicate is diophantine, i.e., can be represented in the form

3z1

...

zn[D(al , ... , ak>

Zl' •.. ,

Zn)

= 0],

(1)

where D is a polynomial with integer coefficients. It is evident that Davis' conjecture implies the recursive unsolvability of Hilbert's tenth problem. The following weaker theorem was proved by Davis (1950):

Every r.e. predicate can be represented in the form 3xVYo;x3z 1

...

zn[D(a l, ... , ak> x, y,

Zl' ... ,

z.) = 0],

(2)

where D is a polynomial with integer coefficients.

In fact, this theorem is an improvement of GODEL'S (1931) famous result about arithmetical representation of partial recursive functions. According to this result every r.e. predicate can be represented in the form QlZl ...

Q.zn[G(a l, ... , ak> z.. .. . , zn) = 0],

(3)

where the Q's are existential or bounded universal quantifiers and G is a polynomial with integer coefficients. A very important contribution to the proof of Davis' conjecture was made in DAVIS, PuTNAM and ROBINSON (1961). Taking advantage of Davis' representation (2), they proved the following theorem: Every r.e. predicate can be represented in the form

3zl

...

zn[P(al' ... , ak>

Zl' ... ,

z.)

= R(al' ... , ak> z.. .. . , zn)],

(4)

where P and R are some functions built from variables and particular positive integers by addition, multiplication and also exponentiation.

Now, if we want to transform representation (4) into some diophantine representation (1) of the same predicate we need only to show that the three-place relation given by formula a=~

~

is diophantine. The question of whether this relation is diophantine was studied by JULIA ROBINSON (1952). Among other theorems she proved the following one: Relation (6) is diophantine provided that there exists a diophantine relation £i)(u, e) such that (6) Vuv[£i)(u, v) => v ~ u"]

91

HILBERT'S TENTH PROBLEM

and Vk3uv[~(u,v) &v

> t1'].

(7)

Relations which meet conditions (6) and (7) are said to be relations of exponential growth. The question of whether there exists any diophantine relation of exponential growth remained open for 18 years until in January 1970 I succeeded in finding such a relation (MATIJASEVIC, 1970). This result completes the proof of Davis' conjecture and thus gives the 'negative' solution of Hilbert's tenth problem. The predicate to be a prime number is recursively enumerable, hence we can find a particular polynomial representing this predicate. Moreover, using a technique due to PUTNAM (1960) we can construct a polynomial P with integer coefficients which has the following property: the set of all positive values of P is precisely the set of all prime numbers.: This is an example of an interesting, pure mathematical theorem proved by metamathematical methods. For another example we can take some recursively enumerable but not recursive set 9.R and find a particular polynomial M such that aE9.R~3z1'" z"[M(a,zl' ... ,z")

= 0].

It is clear that there is no algorithm for determining whether given a the equation M(a,zl""'Z") = 0 (8) has a solution in unknowns Zl, ... , z". Hilbert asked about an algorithm for diophantine equations with an arbitrary number of unknowns. But we see that even if we consider equations with n unknowns for some n then already there is no algorithm. It would be nice to find the minimal n with this property. Last year I reported to the ICM in Nice that this minimal n is less than 200. This is a very rough estimate; it follows immediately from original proofs given in the mentioned papers by Davis, Putnam, Robinson and myself. Afterwards some papers appeared whose aim was to simplify the proofs and to give a better estimate for the number of unknowns. First of all we need to find some simple diophantine relation of exponential growth. My original example of such a relation is based on some properties of so-called Fibonacci numbers defined by relations ({Jo = 1

0,

({Jl =

1,

({J"+1 = ({J,,+({J"-l'

A particular specimen of such a polynomial P is exhibited in MATIJASEVIC (1971b).

92

YD. V. MA1UASEVI~

CUONOVSKII (1970), DAVIS (1971), Kochen (see DAVIS, 1971, or J. ROBINSON, 1971), KOSOVSKII (1971), and Schutte (see J. ROBINSON, 1971, or FENSTAD, 1971) showed that some other recurrent sequences have analogous properties; they exhibited diophantine representations for these sequences. DAVIS (1971) and KOSOVSKII (1971) exhibited rather simple diophantine representations for relation (5) (for another diophantine representation for relation (5) see MATIJASEVIC, 1971a, 1971b). Julia Robinson and I (independently of one another) found examples of relations of exponential growth which can be defined with five bound variables. Combining our ideas we were able to define a relation of exponential growth with only three bound variables. Continuing our joint investigation we obtained some simplifications in the proof of the theorem of Davis-Putnam-Robinson about exponentialdiophantine representation (4) of r.e. predicates. Namely, it was shown that the following equivalence holds (see Appendix I): VY~b3z1 ~Cl ... Zn~ [G(al' ... , ak> b, Cl' ... , Cn, y, Zl' ... , zn) ""';c,.

~3r

[r >

=

0]

b+(b+l)b+l(b+l)b+lF)(Cl+ 1 ) ... (c n +l )

& 3Wl

••.

wnl(b: 1) I(C:~ 1) & I

& (b:l) G(al' ... , ai, b, Cl, Here F = F(a l

, ... ,

ak> b , Cl,

... ,

&

(b: ICn~ 1)

1)

, Cn, r, WI, ... , Wn)]/ .

(9)

cn) is such a polynomial that

Val'" aibc, ... CuY~bZl.:_cl ... Zn~

~Cn

[F> IG(Ol' ... , ak, b , Cl, ... , Cn, y, Zl, ... , zn)l].

(10)

For every polynomial G a polynomial F with this property can be easily constructed. JULIA ROBINSON (1952) exhibited an exponential-diophantine representation for the function

(~).

That representation is rather complicated.

We found new simple representations:

(~) =

rem (entier(

if u > k" then

(U~I)k ),u) = entier(rem(u+)k,if+l)).

(11)

Here rem(o, b) is the remainder after dividing a by b, and entierf«) is the largest integer which is less than or equal to 0(.

HILBERT'S TENTH pROBLEM

93

It is clear that the following equivalence holds:

\1'Y,a3Zl'" zn21¢>3cl ... cn\1'Y"'b3zl_~Cl ... zn=EtC n 21.

So using the above formulas and a diophantine representation for relation (5) we can transform an arithmetical representation (3) of an arbitrary r.e. predicate into some diophantine representation (1) of the same predicate, provided that all universal quantifiers in the arithmetical representation are bounded. It is no longer necessary to start from Davis' representation (2). If we want to prove the unsolvability of diophantine equations with a few number of unknowns we need to start from some simple undecidable arithmetical problem. What kind of unsolvable arithmetical problems are known? Here we take an opportunity to give a classification of arithmetical formulas with bounded universal quantifiers (such formulas define recursively enumerable predicates). Let 33 \1'3\1'2 denote the class of all formulas of the form 3ul uz U3\1'V", ... 3w\1'Zl",B/Z"'B,lR = S],

where A, B l , B z , Rand S are polynomials with natural coefficients in the following variables: A = A(a l, , ako U l, U2, U3), B. = Bl(a., ,akoul,uZ,U3,W), B 2 = BZ(al, , ak> Ul, U2, U3' w), R = R(a., , ako U l, U2, U3'V, W, Z., zz), S

= Sta«,

, ak,

U l,

U2, U3, V, w, z., Z2)'

Notations like (12) have an analogous meaning. We say that the class (12) is undecidable if there is no algorithm for determining whether an arbitrary closed formula from this class is true. It follows from GODEL's (1931) results that for some finite sequence Ql ... Qn of quantifiers this class is unsolvable. DAVIS (1950) proved that for some sufficiently large m class 3\1'3'" is unsolvable. R. M. ROBINSON (1956) showed that one can put m = 4; he also proved the unsolvability of the class 3z \1'3 3 • It follows from Myhill's theorem proved by PUTNAM (1960) that the class 3\1'/3 is unsolvable for some sufficiently large I.

94

YU.

v.

MATIJASEVI6

The theorem about diophantine representation of recursively enumerable predicates implies that for some sufficiently large k the class 3k is undecidable. As a 'positive' result we can mention the solvability of the class 3'1 proved by R. M. ROBINSON (1956). This result can be easily generalized to all classes of the form V"3Vn (k, n = 0,1, ... ). I was able to show that the classes 33 '13, 32 '132 , 3'133 , 32 '12 3, 3V3'v'3, 3'1232 , 3'1 33 are undecidable (see Appendix II). Starting from the class 33 '13 JULIA ROBINSON and I were able to show the unsolvability of the class 31 4 • Let. us list now all still undecided classes. These are the following: V"3' 'In,

V"3vm3vn,

V"32 vs«,

V"3'132 'In

(1 = 2, ... , 13; m = 1,2; k , n = 0, 1, ... ). (It can be easily shown that if a class Ql ... Qn is decidable then both VQl ... Qn and Ql ... Qn V are decidable too, so in fact there are only

16

undecided classes, namely, 32 , ... ,3 1 3 , 3'13, 3'123, 32 '13, 3'132 ) . The above-mentioned results are listed in Table 1. The first column of the table is entitled "Unsolvable" but it could be also entitled "Universal" because every r.e. predicate can be defined by a formula from each of the classes listed in the first column. For the classes from the upper part of the column this was proved in corresponding papers by the authors Table 1 UNSOLVABLE

I

UNDECIDED

Ql ... QII (ODdel). 3V3 m (Davis), 3V3 4 , 3 2 V3 3 (R. M. Robinson), 3V' 3 (Myhili-Putnam), 3" (Davis-Putnarn-Robinson-Matija sevic) 3 3V3. 3 2V3 2 • 3V3 3 • 3 2V 23. 3V3V3, 3V 23 2 • 3V 33. 31 4

I

SOLVABLE

3V (R. M. Robinson)

V"3' v«, V"3v m3vn. V"3 2V3vn. V"3V3 2vn. 1= 2•...• 13; m = 1,2; k, n = 0.1, ...

V"3vn.

k, n = 0.1....

95

HILBERT'S TENTH PROBLEM

mentioned. The proofs for all classes from the lower part of the column with exception of class 31 4 are outlined in Appendix II. The proof for class 314 is much more complicated and will be published elsewhere. The result for class 31 4 gives the current least-known number of unknowns in a recursively undecidable one-parameter system (8) of diophantine equations. This result also enables us to construct a polynomial U(a, Zl' ... , Z14' e) with integer coefficients such that every recursively enumerable set Wl can be represented in the form a

E

Wl 3z 1

...

z14[U(a, Zl' ... , Z14' e)

= 0],

(13)

where e is a Godel number of Wl in some enumeration. The universal diophantine representation (13) of r.e. sets can be regarded as a rather simple one since there are only 14 bound variables in it. But the polynomial U is of quite large degree and has 'irregular' coefficients. So one could ask: is it possible to choose a special Giidel numeration in such a way that a universal representation could be constructed with a 'visualizable' polynomial even in many variables? This problem was solve? by KOSOVSKII (1971). The main idea is to take advantage of the existence of polynomials T with the following property: T(x 1, ... , x n) = T(Y1' ···,Yn) ~ Xl = Y1 & ... Sc x; = Yn'

(14)

For any number of variables one can easily construct 'visualizable' polynomials with this property. Let U be some universal polynomial, i.e., such a polynomial that for every r.e. set Wl Wl 3z 1 ... zn[U(a, Zl' ... , Zn, e) = 0], where e is a Godel number of Wl in some enumeration. Let m be the degree of U. We denote by H':,. the following polynomial in variables a, z, (i = 1, ... , n), Pio ...ln, q1o...ln (io+ ... +in ~ m): a

E

H::'

=

L

lo+ ..·+ln 3z1

•.• Zn

3

io+ ... +ln then Zl +zz

= sqrt(Y),

rem(y, sqrt(y»), Zz = sqrt(y) ..:. rem(y, sqrt(y»).

Zl

=

Hence equation (16) is an identity iff equation

F(rem(y, sqrt(y»), sqrt(y) ..:. rem(y, sqrt(y») , Z3,

..., Zn}

=

0

98

YU. V. MATIJASEVIC

is. Iterating such a transformation we shall finally obtain an equation with only one variable which is an identity iff equation (16) is. There are some applications of the theorem about diophantine representation of enumerable predicates to model theory (see, for example, A. ROBINSON, 1971, 1973). For another application of the theorem see KOSOVSKII (1967). Let us in conclusion discuss the formulation of the tenth problem given by Hilbert. He asked about solutions of diophantine equations in integers. This is equivalent to asking about solutions in natural numbers. But also we can be interested in solutions in Gauss numbers (complex integers of the form a+bi) or in rational numbers. (The latter problem is equivalent to considering only homogeneous diophantine equations and asking about nontrivial solutions in integers. This is a very important case). And why did Hilbert ask only about solutions in integers? It seems very likely that Hilbert was sure that there exists a method for deciding whether a diophantine equation is solvable in integers. And this is the most difficult problem. The other two problems (about solutions in rational numbers and in Gauss numbers) can be easily reduced to the problem of solutions in integers. But the former problem has proved to be unsolvable and we can say nothing about the two others. One can formally say that Hilbert's tenth problem is closed, but I believe we cannot be satisfied completely until it is known if there exist algorithms for determining whether an arbitrary diophantine equation has a solution in rational and Gauss numbers. This question has been worked on quite a little, but it still awaits solution. Appendix I Here the proof of formula (9) is given. It is based on the following lemma which can be regarded as a multiplicative version of Dirichlet's well-known principle. LEMMA 1. If slt o ... tk then there exist numbers p and I such that I p Itil pis and p ;;:: S(k+1)-I.

We also need the following generalization of Lemma 1. LEMMA

2. If q :F 0 and

slqtiO ... t ik,

U=l .... ,n)

~

k,

99

HILBERT'S TENTH PROBLEM

then there exist numbers p and 11, ... , In such that l; :::;; k;, pi t;Ii (i = 1, ... , n),

pis and s ( -q

p~

PROOF:

)«k + 1)...

(k n

l

+1»-1

Let s' = sjg.c.d. (s, q). We have:

s'l tl o ...

(i = 1, ... , n).

tiki

Now it suffices to apply n times Lemma 1. Let numbers ai' ... , ak- b, Cl, ... , c; satisfy the right part of equivalence (9). Let Y :::;; b. We have: (r- y)j(b+ I)! . G(a 1, ... , ak- b, C1> ... , Cn, r, WI' ... , wn), (r-y)l(b+1)!' wl(w;-I) ... (Wl-Cl) (i = 1, '''' n).

(21)

By Lemma 2 we can find p and ZI, ... , Zn such that

= 1,

, n),

(22)

(i = 1,

,n),

(23)

(i

Zi:::;; ci plwl-rl

(24)

plr-y,

p~

r-y (b+l)!

(

« C1 +

1) ... (c n+ 1»-1

)

In accordance with (10) and (22) we have: p

~

(

r-y (b+l)!

« C1 +1)...(cn+l»-1 )

>- ( (b + 1)(b+1)((b + l)b+1Fyc1 +1) ... (c n+l)

1)

... (Cn+

1»-1

(b+ I)!

~ ~

)«C1+

(b+ l)b+l . F(a 1,

> (b+l)!·IG(al,

, ako b , C1 ,

••• ,

cn)

,akob,c1, ... ,cn,y,ZI"",zn)l.

(25)

By (21), (23) and (24) and by virtue of (25) G(al' ... ,ak-h'Cl' ... ,Cn'Y'ZI' ""zn) = O.

So if a1> ... , ak, b, C1 , the left part too.

... ,

Cn satisfy the right part of (9) then they satisfy

100

YU. V. MATIJASEVIC

Let now numbers ai' ... , ak, b, Cl, ... , CII meet the left part of (9). We can find numbers Zty (i = 1, ... , h; y = 0, ... , b) such that Zty G(a 1 ,

...

~

(i = 1, ... ,n; y = 0, ... ,b),

Ct

,ak,b,c1"",Ck,y,Zly,,,,,ZIIY) = 0

(27)

(y=O, ... ,b).

(28)

Let d > max{b, C1 , ... , CII}' We put r = h(d!)2-1 where h is so large that the inequality in the right part of (9) is satisfied. We have: r ) ( b+l

=

r(r-l) 1·2·

= (~-1) (~-1)

(r-b) (b+l)

1

2

...

(~-1) b+l

.

For i = 1, ... , n numbers

-1), ..., (~:~ -1)

(Ct+ 1) ! , (r:l

(29)

are pairwise relatively prime integers and by the Chinese Remainder Theorem there are numbers WI' ... , Wn such that Wt == Zty(mod

~:~

-1)

(i

= 1, ... , n; y = 0, ... ,b).

By (27) and (30) r+l -1 Wt(Wt- 1) ... (Wt-Ct) y+l I'

and since numbers (29) are pairwise relatively prime (Ct+ 1)'

hence

(b~ 1) Wt(Wt- 1) ... (Wt-Ct), I

Wt 1) (b +r 1) I(Ct+

(i =

1, ...,

n).

It follows from (28) and (30) that

(~:~

-1) IG(a

1 , ... ,

Ok, b, C1 ,

••• ,

CII' r, WI' ... , WII)

and hence

(b~l) IG(a

1 , ••• ,

Ok, b,

This completes the proof of (9).

Cl> ... ,

CII' r , WI, ... , WII)'

(30)

HILBERT'S TENTH PROBLEM

Appendix

101

n

Here we prove that every r.e. set can be defined by a formula from each of the following classes: 33Y3, 32Y32 , 3Y33 , 32y 23, 3y 232 , 3Y3Y3, 3y 33. By the usual technique these results can be easily extended to representations of r.e. predicates in any number of arguments. All proofs are based on the following MAIN TECHNICAL RESULT.

For every r.e. set 9.n there is a formula Ql with

the following properties: 1) Ql is of the form w

Yy 1711]2 is one of the instructions of the machine. For every instruction and for every couple of letters of the alphabet {*, ixO' ••• , iXm} there is in (47) a disjunctive term of the above form. Q3~ means that the last letter of word (45) is %: y At last,

Q3~

= k(2a+2k+3)+3

& +d(r+1)(y-1)/s-c(%).

is the following condition: y

~k(2a+2k+3)+4.

Formulas Q30, Q3~, Q3~, Q3~ and Q3~ have the form (31). Formulas Q3~, and Q3}" can be transformed into equivalent formulas of this form on the basis of the following lemmas.

Q3~, Q3"

LEMMA

2. If P

~

R then

P

~

Q

~R

Q2+PR

Q(P+R).

~

PROOF:

Q P« ~

P+R 2

~ RP---~ ~

~

P+R P+R Q-R-2 - «: ~ 2-

R-P P+R R-P 222

----~ Q---~--­

I P+R R-P ( P+R)2 ( R-P )2 !Q--2- ~--2-·· Q-·-2- ~ --2'I



LEMMA

3.

Q2+PR

~

Q(P+R).

If numbers P l, ... , P; are pairwise relatively prime then

P 1/Ql & ... & PnlQn Pl· .. PnIQ1 P2'" Pn+P 1Q2 P 3 The proof is clear.

'"

Pn+ ... +Pl .. · Pn- 1Q n '

YU. V. MATIJASEVIC

106

It can be easily checked that due to our special choice of d the numbers

1 +d(r+ 1)(y-2a-2k-3), 1 +d(r+ 1)y,

1 +d(r+ 1) (y-2a-2k-2),

l+d(r+I)(y+I)

are pairwise relatively prime and Lemma 3 can be applied. This completes the proof of the Main Technical Result. We need also the following lemmas. LEMMA 4. If Q =/: 0 then P ~ 0 & QIR & 8 = 0

Y

¢>3x(Q2((R2+282+I)P-82-x-I)-R2 = Q4S

(a = 0 means that a is a full square).

PROOF: P~

O&QIR&S= D ¢>3Xl(R2+2S2+I)(P+1) =

~:

+8 2 ±VS +X+I]

¢>3X[(R2+2S2+1)(P+1)-S2-X-I-

~:

ys]

= ±

¢>

3X[Q2((R 2+ 28 2 + 1)(P+ 1)- 8 2 -x-I) - R2 =

± Q2}iS1

¢>

3x [(Q2((R2+28 2+ 1)(P+ 1)-82-x-1)- R2

=

r

Q28].

LEMMA 5. LEMMA 6.

n

V 3x[P/= 0] ¢>3x[P1 ... P" = 0].

1=1

THEOREM 1. Every r.e. set 9Jl can be represented in the form

ae9Jl¢>3krsVy.;;,3x[Ql(a,k,r,s,y,x) = 0] where Ql is a polynomial with integer coefficients. PROOF: The theorem follows immediately from the Main Technical Result and Lemmas 4, 5 and 6. THEOREM 2. Every r.e. set 9Jl can be represented in the form

ae9Jl¢>3rtVY"r3kx[Q2(a,r,t,y,k k,x) = 0] where Q2 is a polynomial with integer coefficients.

HILBERT'S TENTH PROBLEM

PROOF: It

107

is not difficult to understand that the following equivalence

holds: w

a E':m3rtVy",3k[t-k = 0 &P < t &

V [S8tl:-kl] 1+1

denotes the result of substitution fJ for ct in ~). In fact, if a E':m then in accordance with the Main Technical Result we can choose numbers k, r and s such that s = 0 and s > k 2 and put t = s+k; on the other hand, the condition t-k = 0 &k2 < t implies that ([~Ip]

k = t-(entier(ytW

and thus k does not depend on y. Then, the condition t-k

= 0 &P
3y B(x, y))

:::>

3u 'v'x(i A (x)

:::>

3z(Tl (u,

X,

z) & B(x, Uz)))

or Markov's scheme: 'Ix(A(x) v iA(x))&ii3xA(x)

:::>

3xA(x).

However, not all traditional constructivists adopt these principles. 3) Ramified constructive analysis is a sort of stepwise build up of constructive mathematics. We start from arithmetical propositions. At the next step we study formulas of the previous language as an object of investigation together with the notion of validity for these formulas (PHAN DIN ZIEU, 1967). 4) Still another case is constructive mathematics with general inductive definitions (KREISEL and TROELSTRA, 1970; TROELSTRA, 1968). As an example we give the following. Let us enumerate finite sequences of natural numbers with the help of natural numbers in a constructive one-one correspondence. The operation of concatenation of two finite sequences is interpreted in this numeration as a two-place primitive recursive function x * y. A oneplace finite sequence with the single member x is interpreted by the natural number (x). Letfbe a primitive recursive function and let x be a natural number. Let us define the relation B (f, x) (read 'f bars x') by the following inductive clauses: (1) f(x) = 0 ::::> BC!, x) f(x)

=1=

0", 'v'nB(f, x

* (n»)

:::>

BC!, x).

(2)

The inductive character of this definition suggests the possibility of using the following inductive principle: 'v'x(J(x)

= 0::::> A(x))&'v'x(J(x)

=1=

0", 'v'nA(x*(n»)::::> A(x)) :::> 'Ix(BCf,x)::::> A(x)).

(3)

114

A. G. DRAGALIN

B(!, x) is an example of a relation which is meaningful in a constructive sense but not an arithmetical one. The necessary condition for application of a general inductive definition in constructive mathematics is its accordance with 'recursiveness' in constructivism. For our definition, if rp is a recursive function which gives an object rp(x) with some property A for every x such that I(x) = 0; and "P(e, x) gives us an object with property A for x, iffunction {e} (a recursive function with Godel number e) gives objects {e}(n) for every x * 'fIJ), where q;, 1jJ ELI and is closed under conjunctions (q; & 'fIJ), q;, 1jJ E L 2 • Then we define the extending hierarchy of languages L 1 , L 2 , ... where L n+1 contains all formulas of L n , formulas (q; ::::> 'P), sp ; 'fIJ E Ln and is closed under &. L w is the union of all L; (n < w). Beginning from L W +1 the quantor \Ix arises. The hierarchy L W +1 ' L w+2 , ••• , L w+n ••• , L w+n+1 contains all formulas of L w+n , formulas (q;::::> 'fIJ), q;, 'fIJ E L w+n and is closed under & and v. At last, L 2 w is the union of all L w+n • Thus, L 2 w consists of all arithmetical formulas without connectives v and 3. The relation of truth for formulas L 1 and L 2 is defined in some special manner (MARKOV, 1968) and further validity of an implication is treated in a deductive way: (q; ::::> 'fIJ) is valid in the next language if there is a deduction 'fIJ from q; in the previous language with the help of some precise list of deduction rules. Validity of generality is treated from the point of view of intuition of generality: Vxq;(x) is valid if we can make clear the validity of formulas q;(O), q;(1), .•• and so on. We shall write Tr, (q;), if q; E L" q; contains no parameters and q; is valid in L i • Let L 2 w+1 consist of all arithmetical formulas. The relation Tr 2 CD+1 (q;) for closed formulas q; E L 2 w+1 is defined by SHANIN'S (1958) translation realizability using Tr 2W(q;). In analogy with realizability, for every q;e L; such that Trl(q;) we can confront some number r coding information for establishing Trl(q;). For example, if q; is 'fIJi ::::> 'P2 then r is the Godel number of a deduction 'P2 from 'fIJI, if q; is Vx 'fIJ (x), then r is the number of the function {r} which gives us information about validity 1jJ(n) for every n. We shall call such a number r 'a support of validity' and write (rTrlq;). It is very natural to specialize the intuition of generality in the following strong form: for q; e L l Trl(q;) if and only if when there is r (rTrlq;). Let us consider a semiformal system LM of sequents in L 2 w • Sequents

116

A. G. DRAGALIN

LM are of the form r -+ ,1, where ,1 is a single formula of L 2w or several formulas of L 1 • LM contains no cuts and besides the usual logical rules has the rule of infinite induction (in a recursive form) and a special implication rule: (cp => tp) => IX, cpr ~ tp, ,1; IXr -+ ,1 (cp => tp) => IX, r -+ ,1 where cp, tp E L 1 and IX E £1 or IX is (cp' => tp/), tp", tp' E L 1• There is the following criterion for validity in L 2w. THEOREM 1. Tr2w(cp) iffwhen cp is deducible in LM. Moreover, for every deduction

3V and a logical constant ..L ('false') in the usual way. The negation is defined as A :::> ..L. The denotation Q(x 1 ... x; I SI ... Sn), where Xl' ... , X n are distinct variables and SI' ... , S are terms or functors, respectively, means the result of a simultaneous substitution of S l ' ... , S; in places of free occurrences Xl> ... , X n in Q with obvious renaming of closed variables if necessary. 5.13. Logic LS is intuitionistic predicate calculus with three sorts of variables.

CONSTRUCTIVE MATHEMATICS AND MODELS

119

Admissible objects for substitution of variables u, v, 11', '" are the terms, for substitution of variables a, b, c, ... are the functors, containing no lawless variables; and for substitution of variables Ct, fJ, y, ... only the same sort of variables 0(, fJ, y, ... (J.

=

5.1.4. LS Equality Axioms. 1) u = u; 0( = 0(; 2) u = v & u = 11' ::J V = 11'; y::J P = y; 3) u = V::J (I, u) = (j, v); 0( = p::J (e , u)

= P& ':1. = (P, u).

5.1.5. LS Arithmetical Axioms.

1) Su = Sv ::J U = v; 2) I Su = 0; 3) A(o) & 'v'u(A (u) ::J A(Su)) ::J'v'uA(u); 4) (Aut, v) = t(ulv); 5) (Ruvr, t)(o) = t; 6) 3a(a E K & VU(1.1 ... eLn(au "I: 0 & ¥- ((1.1' ... , eLn) & &i(eL, Ek7u):::> A(eL 1, ... , eLn, aU-d)));

5)

V(1.1 ... (1.n ("I: ((1.1' ... , (1.n) :::> :::> 3bV(1.1 ..•

OCn("I:

3aA((1.1' ... , eLn, a) ((1.1' ... , (1.n):::>

3uA(oc 1, ... ,

(1.n,

(b)u)).

Here in Schemes (3), (4), (5) all lawless parameters of A are exhibited. Formulation of LS is finished. Scheme (4) is given in its strong form, indicated by TROELSTRA (1970). 5.2. Now we come to the description of JCTa> theory. JCTOJ theory deals with the interpretation of LS. It is the theory of finite-type functionals (GoDEL, 1958; GRZEGORCZYK, 1964b), strengthened by the functionals for continuous operations (KREISEL and TROELSTRA, 1970). An algorithmical model for lCTa> is given below. 5.2.1. Type structure is defined inductively. 1) 0 is a type, K is a type; 2) if a and T are types, then raT1is a type; 3) if a and T are types, then (o-r) is a type. For every type a we fix variables x" , y" ... of this type, sometimes omitting the upper index. Variables of type 0 run over natural numbers, variables of type (o r) run over some functions, each is defined for every object of type (J and has objects of type T as its values. Thus, for example, objects of type (00) are the sequences of natural numbers. We denote variables of type (00) as a, b, c, ... Variables of type K we denote also as a, b , C, ••• and sometimes as e.], '" Objects of K type are the sequences of natural numbers, representing continuous operations

CONSTRUCTIVE MATIIEMATICS AND MODELS

121

of Brouwer and Kreisel (KREISEL and TROELSTRA, 1970). Then, objects of type (0'1") are the ordered pairs from objects of 0' and 1" type, respectively. 5.2.2. The notion of polynomial of type 0' is defined inductively. 1) There is some class of polynomial-constants of fixed type. Polynomial-constants have special definitions, namely: 0 is of type 0; S is of type (00); for any type e, 0', 1" the constants Eei; of type ((O'(1"e)) ((O'1')(O'e))) and IIa T of type (0'(1"0')); for any 0',1" the constant D a T of type (0'(1"[0'1"])); for any 0' and 1" constants DIan D2 a T of types ([0'1"]0') and ([0'1"]1"), respectively; for any type 0' the constant R a of type (0'((0'(00')) (00'))); constant KO of type (oK), constant K' of type ((oK)K), constant K2 of type (K(oK») and for every type 0' constant

K~

of ((00') (((oK) ((00')0')) (KO'))). In some

cases we shall omit the type indication in constant index. 2) Variable x" of type 0' is by definition a polynomial of type 0'. 3) If G is a polynomial of type (0'1") and g is a polynomial of type 0', then (G, g) is a polynomial of type 1". 4) If G is a polynomial of K type and g is a polynomial of type 0, than (G,g) is a polynomial of type o. Polynomial of (... ((tl> (2 ) , ( 3), ... , tm ) type we denote as t2 ... tn) or as tt(t2 ... tn)'

«.

5.2.3. Atomic formulas of JCTOJ may be written as (F = G), where F and G are the polynomial of the same type. Formulas of JCTOJ are built from atomic ones using logical connectives V & ~ V3 and logical constant .L ('raIse') in the usual way. Quantors V and 3 are used for variables of all types. 5.2.4. Logic JCT~ is a many-sorted intuitionistic predicate calculus with variables for every type. Further in stating axioms we omit denotation types of variables. 5.2.5. Equality Axioms of JCT(J):

1) x = x; 2) x = y & x = Z :::l Y = Z; 3) XI = X2 & Yt = Y2 :::l (x t, Yt) = (X2' Y2)' 5.2.6. JCT OJ Arithmetical Axioms: 1) Su = Sv :::l U = 2) I Su = 0; 3) A(o) & Vu(A(u)

4)IIxy

=

x;

V;

:::l

A(Su)) ~ VuA(u);

122

A. G. DRAGALIN

5) Exyz = xz(yz); 6) D1(Dxy) = x; 7) D 2(Dxy) = y; 8) D(D 1x)(D 2x) = x; 9) Rxyo = x; 10) Rxy(Sz) = y(Rxyz)z.

The given axioms allow us to develop the JCT OJ A-notations theory (TAIT, 1967) and the theory of primitive recursion (GRZEGORCZYK, 1964b).

5.2.7 Axioms for K: 1) (KOu, v) = Su; 2) (K1x, 0) = 0; 3) (K1x, U* v) = «x, u), v); 4) VuA(KOu) & Vx(VuA(xu)::::> A(K1x»)::::> VeA(e); 5) K2(K1x) = x; 6) K3zy(KOv) = zv; 7) K3zy (K 1x) = yx(AuK 3zy(xu»). Axiom (1) corresponds to 5.1.7, (2); Axioms (2) and (3) correspond to 5.1.7, (3); Axiom (4) corresponds to 5.1.7, (4); Axioms (6) and (7) realize in JCTOJ the principle of 'recursion by definition of continuous operation'. The formulation of JCT OJ is finished. Note, JCYO> does not contain a choice axiom and does not contain any kind of axioms like those of extensionality.

5.3. Interpretation of LS in JCYOJ. Define in JCT OJ the primitive-recursive function, 'distinguishing' elements of finite sequence. Precisely [(xo, ... , Xn>]i = Xi if i ~ n. 5.3.1 Let us introduce a notion of 'information of degree n': Inf(n, x) = h & (Vi < n) (lh[x]i = n & (Vj < lhr) ([[X]ilJ = [i]j»). The significance of this definition is that in the model we define in JCT OJ for LS all the lawless sequences are numbered by natural numbers. Information x = (xo, ... , Xn-2> reports about values of sequences with numbers 0,1, ... , n-1. Namely, if 0( has a number i (i < n), then O(U) = [[x]dj for j = 0, 1, ... , n -1. Thus information x of degree n reports about the values of the first n sequences at the first n points. (Vj < lhi) ([[x]tlj = [i]j) condition means that some initial values of ex are already defined by number i of sequence ex and information x must be in accordance with these data. Introduce the following relation: information U2 is an extension of information U 1 : U1 ~ U2 == 3nl n2(Inf(nl' Ul) & Inf(n2' U2) & n 1 ~ n2 & (Vi < nl)3v([ul]i * v = [U2]i»)'

== lhx

CONSTRUCI1VE MATHEMATICS AND MODELS

123

Two primitive-recursive functions hl(x, k) and h2(x,y) enumerating for the fixed information x degree n every information y, x £;; y degree (n+ 1), using k, may easily be built. Precisely, in JCT w: l)h l(x,k) =I: 0::::> 3nInf(n,x); 2) Inf(n, x) ::::> Vk(Inf(n+l, hl(x, k» & x£;; hl(x, k)); 3) Inf(n, x) & Inf(n+ 1, y) & x £;; Y::::> y = hi (x, h2(x, y».

Further consider the primitive-recursive function n(u), assigning information degree n for every finite sequence u, lhu = n. Namely, n(o) = 0, n(u * k) = hI (n(u), k). 5.3.2. Let us assume that variables of every sort are ordered as natural numbers. We make correspond to the nth lawless variable the (3n + l)th variable 0(* of type 0 in JCT w; to the nth variable u for numbers of LS we make correspond the 3nth variable of type 0 in JCT w; we make correspond to the nth variable a for sequences given by law in LS the 3nth variable of type (00) in JCT w. Abusing notation we shall denote corresponding variables by the same symbols in LS and in JCTw. But note that lawless variables of LS are represented in JCT w by variables of type o (so to say a sequence 0( has a number 0(* in JCT W ) . 5.3.3. We correspond for every term t of LS a polynomial t:" of type o in JCT w and for every functor of LS a polynomial j.; of type (00); t:" andf~ have only parameters from t and fand may be one additional variable w of type 0, which is not contained in t nor f. This additional variable is exhibited in notation t:",f~. Namely, t:" andf~ are built in JCT w just the same way as t andfin LS with the exception of lawless parameters. We define primitive-recursive q;(w, u, z), such that 1) q;(w, u, z) = [u)z, if Z < lhu; 2) q;(w, u, z) = [[n(w)]u]z, if z ~ lhu, u < lhn(w), z < lhn(w); 3) q;(w, u, z) = 0 in other cases. For getting t:" (f~) we replace every occurrence of lawless parameter 0( in t(f) by AZq;(W, 0(* , z). The sense of this procedure is that lawless parameters are calculated in accordance with information n(w) about O('s in point w. If in calculating t:" we did not melt a Case (3) of definition we say that t:" is meaningful in w. More precisely, we define a formula !t:" in the language JCT{J) which says this. 5.3.4. For every formula A of LS we define a formula [x, w, A), of JCT W (read 'x realizes A in point w') by induction on building A. The formula

124

A. G. DRAGALIN

[x, IV, A] contains only parameters of A and may be two additional variables x and w. These variables do not occur in A and are noted in [x, w, A]

Explicitly, w is a variable of type 0 and the type of x is defined by A. If we denote the type of x as AO then AO may be calculated with help of recursive rules: 1) A is atomic of LS, then (t = r)O is K; (a = (3)0 is 0; (K(f»)O is K. 2) (B & C)0 is [BO CO]; 3) (Bv C)O is [K(oBO)(oCO)]; 4) (B :::::> C)0is (BO(oCO»);

5) .L is 0; 6) (VuB)O is (oBO); 7) (VaB)O is ((oo)BO); 8) (Va B)O is (oBO); 9) (3uB)0 is [K(oBO)(oo)]; 10) (3aB)O is [K(oBO) (0(00»)]; 11) (3a B)O is [K(oBO)(00)].

»:

Here [ear] is an abbreviation of[e [aT]], and' D 3 D~, D~ are polynomials of types (e(a(r[eaTJ))), ([ear] a), ([ear]e), ([ear]T), respectively, and such that in JCT(J) D3xyz = Dx(Dyz),

D~x = DIx,

Dix = D I(D2x),

D~x = D 2(D2x).

w is an abbreviation for 3z (v = w * z). The precise definition of [x, w, A] contains the following inductive items:

V ~

1) A is atomic, then [x, w, t = r] is ("Iv ~ w)(xv i: 0 :::::> !t~ & !r; & (t~ = r;»); [x, W, a = (3] is x = 0 & a* = (3*; [x, IV, K(f)] is Vu(xu = fu); 2) [x, IV, B & C] is [DIx, W, B] & [D 2x, w, C); 3) [x, w, Bv C] is Vuv((D~x)u i: 0 :::::> (Dix)u = (Dix)(u * v») & Vuv ((D~x)u i: 0 :::::> (D~x)u = (D~x)(u * v») & ("Iv ~ w)((D~x)v = 1:::::> [(Dix)v, v, BJ) & (Vv~ w)((Dix)v > 1 :::::> [(D~x)v, v, CJ);

4) [x, 5) [x, 6) [x, 7) [x, 8) [x, 9) [x,

w, B :::::> C) is Vy(Vv ~ w)([y, o , B] :::::> [xyv, o , C); w, 1.] is 1.; w, VuB] is Vu[xu, w, B]; w, "laB] is Va[xa, w, B]; W, "laB] is Va* [xa*, w, B]; w, 3u B] is Vuv((Dgx)u i: 0 :::::> (D~x)u = (Dix)(u * v»)

CONSTRUCTIVE MATHEMATICS AND MODELS

125

& Vuv(D~x)u =I- 0 ::> (D~x)u = (D~x)(u * V») & ("Iv ~ w) (D5x)v =I- 0 ::> [(D~x)v, v, BJ(ul(D~x)v);

10) [x, w, 3a B] is Vuv(D8x)u =I- 0 ::> (D~x)u = (D~x)(u * v») & Vuv(D5x)u =I- 0 ::> (D~x)u = (D~x)(u * v») & (V'll ~ w) (D8x)v =I-

0 ::>

[(D~x)v, v, BJ(al(D~x)v);

11) [x, w, 31XB] is Vuv(D~x)u =I- 0 ::> (D~x)u = (D~x)(u * v») & Vuv(D~x)u =I- 0 ::> (D~x)u = (Dh)(u * v)) & ("Iv ~ w)«D5x)v =I- 0 ::> [(D~x)v, v, Bl(IX*I(D~x)v»). Let us say that formula A of LS is realizable in the model, if there is a polynomial F of type AO, which contains only parameters from A and such that in JCT CD the formula [F, A] is deducible. (Here is the denotation of the number for the empty finite sequence.)

g E A o • =1=

5.4.3. For the building the model of JCT W we must still prescribe some natural numbers of respective sets for constant polynomials of JCT ... , Xl) such that VnVx I ... VXtP(n,f(n, Xl' ... , Xl), f(n+ 1, Xl' ... , Xt), Xl' ... , Xl), is called a solution to this lfp. Remark 1. If, in Definition 1, we suppose P to be an arbitrary general recursive (instead of primitive recursive) predicate, then the associated

130

YU. T. MEDVEDEV

problem ~ can be shown equivalent, informally, to a ~* which is a Ifp in the initial sense. For in that case we can find a primitive recursive predicate R such that P(n,p,q,xI' ""xk)3yR(n,p,q,x l, ""XbY) for each set of values of n, p, q, Xl' ... , Xb and if, furthermore, we consider another primitive recursive predicate p* satisfying the condition P*(n, p, q, Xl' ... , Xk)

==

r

3r3s3y3z(p = 2

•

3'

& q = 2"' 3%& R(n, r, s, Xl' ... ,Xk, z)),

then each solution to the lfp ~*(XI' ... , Xk) will enable us to construct a solution to ~(XI' ... , Xk)' and conversely. Examples of lfp's. The decision problem for any given recursively enumerable set E can be represented in the form of lfp. To see this, first note that nEE 3xH(n, x) where H is primitive recursive. Next consider a 3-ary primitive recursive predicate P such that P(n,p, q) holds if and only if p and q are G6del numbers of (n + 2)-tuple (k o , ... , k n, Yn) and (n+3)-tuple (k o , ... ,kn,kn+1 , Yn+I ) ' respectively, where Yn+l > Yn and k, = 1 3x(x ~ Yn+l & H(i, x)) for i = 0, ... , n+ 1. Then it is easily seen that from each solution to ~ we can obtain a decision procedure for E, and that the converse is also true. Similarly, the problems of separability of recursively enumerable sets and the problems of extension of partially recursive functions can be shown equivalent to some suitable lfp's. In order to define the operations on Ifp's we shall need Some notational conventions. We take poo+1 ... p~m+l, where p/s are the primes numbered in an increasing order, to be the Godel number of an ordered (m + i)-tuple (no, ... , nm ) . If X is the Godel number of (no, ... , nm ) , then we write L(x) = m and x(i) = ni for i ~ m, x(i) = 0 for i » m; if X is a G6del number of no ordered tuple, then we take L(x) = 0 and x(i) = 0 for any i. We shall write (x(i»)<j) simply as x(i,j). Predicate F(u) ('ll is afinite function') is defined by the condition: F(u)

==

V iV j(i, j

~

L(u(O») & U(O,i) = u(O,})

U(l,i)

= U(l,}»).

Predicate !u(x) ('x is in the domain of u') is defined by the condition: !u(x) == F(u) & 3i(i ~ L(u(O») & X = U(O.i»); for any u, x and i such that i ~ L(u(O») & x = U(O,i) is true we; use u(x) to designate U(l,i). Three further predicates L1 (u, e), V(x, y) and l(x, b, d) are defined as follows:

== F(u) & F(v) & Vx[!u(x) = (!v(x) & u(x) = vex, y) == L(x) ~ L(y) & Vi(i ~ L(x) X(i) = y(i)) lex, b, d) == L(x) = d & Vi(i ~ d X(i) ~ b). L1(u, v)

=

=

vex))]

131

INTUITIONISTIC NUMBER THEORY

We shall also use a Godel numbering of all ordinals which are less than Cantor's ordinal 8 0, by means of all natural numbers. We suppose that the Godel number of the least ordinal is O. The Godel number of an ordinal 0( is written as oc. The order relations 0( -< fJ and 0( -< fJ will occasionally be regarded as relations among oc and 7f (in which cases we shall employ the same symbols -< and 5). They can be represented by some primitive recursive predicates. We are going to define 'P v Q, 'P & C, 3x'P(x), Vx'P(x) and'P =>« Q where, 'P, 0, 'P(x) are any given lfp's, possibly with some free variables (besides x in case of 'P(x» not indicated for the sake of simplicity, and where 0( is any given ordinal -< 80' It means defining five operations on the predicates P, Q, P(x). We shall simply describe, in each case, the properties which a predicate R is to possess in order to be a result of the corresponding operation, but it would be easy to specify R as a predicate primitive recursive in P and Q (or in P(x), as the case may be). DEFINITION 2.

m = 'P v Q is the lfp associated with R defined thus:

R(n,p,q) == (p(O)

=

O&q(O) =

o & P(n,p(l),q(1»)v(p(O) & lO)

=

1

= 1 & Q(n,p(1), q(1»).

m = 'P & Q is the lfp associated with R defined thus: R(n, p, q) == P(n, pro), g(O» & Q(n, p(l), q(1».

DEFINITION 3.

m = 3x'P(x) is the lfp associated with R defined thus: R(n,p, q) == (P(O) = q(O» &P(n,p(1), q(1),p(O».

DEFINITION 4.

DEFINITION 5. (Auxiliary predicates's is a nth partial solution to and's is a partial solution to 'P').

Pen, s) == n = Ov [n > 0 & L(s) = n & Vi(i:::;; n-l => P(i,

':P'

S(i),S(l+l»)]

pes) == 3nP(n, s). DEFINITION 6. m

=

Vx'P (x) is the lfp associated with R defined thus:

R(n, p, q) == Vi [i :::;; n => (L(p(i» = n & V (p(l!, q(i» & Vi(i :::;; n

+ 1 => p(n+ 1, q(i), i»)]

The following notions are a fundamental part of Definition 8 of implication to be given below. We define inductively a predicate N(~) ('~ is a t-net'i, connected with $ three numbers (the breadth B($), the depth D(~),

132

YU. T. MEDVEDEV

the order Om, of ~), a predicate C(s, ~) ('S is a component of ~') and also a predicate E(~, 'YJ) (''YJ is an extension of f). Informally, these will enable us to deal with finite 'branching' sets of the ordered tuples whose Godel numbers s satisfy C(s, ~) with a ~ such that N(~) is true. DEFINITION

7. (1) N(~) is true if

B(~) = ~(o),

Dm =

Om

;(1),

1. In this case

L(~) =

=

C(s, ;) == I(s,

0,

~(o),

;(1».

(2) N(~) is true if the following assertions are true

Lm =

VX[I(X,

=

;(0),

3,

F(~(3»,

~(1»

(!;(3)(x) & N(;(3)(X») & 0(~(3)(X»)

-< ~(2) & B (~(3)(x»)

=

;(0)

& D(~(3)(x») ~ ;(1»)].

In this case Bm =

~(o),

Dm =

Om =

~(l),

~(2),

C(s,~) == 3x(I(x, ~(o), ~(1» & C(s, ;(3)(X») & vex, s»). (3) If N(~) and N('YJ) are true and if Om E(;, 'YJ) ==

< B(1') &

B(~)

= 0, then

D(~) =

D('YJ) & 0(1') = 0.

(4) If N(~) and N(1') are true and if Om #= 0, then E(~,

'YJ) == B(~) < B('YJ) & D(~) = D(1') & Om = 0(1') & L1(~(3), 1')(3».

It is to be noted that in this definition all the notions are primitive recursive. DEFINITION 8. (Implication ::::>" of order IX). SR = cp ::::>".Q is the lfp associated with R such that R(n, p, q) is true if and only if the following five conditions are satisfied:

Vi

[i ~ n = (N(p(O,i»

& O(p(O,I».:S.

ex & B(p(O,i» = n

=

& F(p(1,i» & Vs(C(S,p(O,I»

Vi [i ~ n+1

=(N(q(O.i»

& O(q(O,i»

-< ex &B(q(O,i»

& F(q(1,i» & Vs(C(s, q(O,i»

Vi[i

~

n

= (E(P(O,i), q(O,i»

=

!p(l,i)(S»))J,

(1)

= n+l !q(l,i)(S»))] '

&L1 (p(1,i), q(l,i))],

(2) (3)

133

INTUITIONISTIC NUMBER THEORY

ViVs(i ~ n+1 & C(s, q(O,I» &P(s) => Q(i, q(I,I)(S»), VNjVsVt[i< j

~

(4)

h+1 & C(s, q(O,I» & C(t, q(O,j»

& P(s) & P(t) & (V (s, t) v V(t, s)

=>

V(q(I,I)(S), q(l,j)(t»].

(5)

It follows immediately from this definition that (a) There exists a solution ofCP =>0 cP; (b) If oc < fl -< 80' and iffis a solution to cP =>ee 0, then f is also a solution to cP => p Q. THEOREM 1. Let cP and Q be some lfp's. If cP has a solution f and if cP => ex Q has a solution g for an oc -< eo, then there exists a solution h to 0, which can be obtained from f and g using some primitive recursive operations and also an operation of unnested recursion up to oc. More specifically, the last recursion has the form of the following definition, in terms of given functions 'ljJ, () and X, of a function ip:

q;(O, s) = 'ljJ(s),

where VaVs[(a+ 1

q;(a+ 1, s) = q;(()(a, s), x(a, s)

-< ex => ()(a, s) -< a+ 1) & ()(a, s) -< aJ.

To prove Theorem 1, besides recursion up to oc one has to use induction up to oc. However, owing to the special form of the latter in our proof, an informal justification for it reduces to that for the former. A similar observation applies to Theorems 3 and 4 below. DEFINITION 9. (Negation lee of order oc). leeCP is cP =>ex cpo where cpo is a fixed lfp without free variables such that pO(n, p, q) is identically false. THEOREM 2. There exist primitive recursive functions U1 (n, a, b, c), Ul(n, a) (2 ~ t ~ 4), Uj(n, x, a) (5 ~ j ~ 6) such that the following is true. Let .t/x) for 5 ~ j ~ 6. Remark 2. The assertions of this theorem correspond to the logical rules of the introduction of ::>, Y and 3 of GENTZEN'S (1935) calculus LJ. Thus from the solvability of ~2 we can infer, using Theorem 1, the following analogy to the rule -+ ::> : /fg{ & ", Q is solvable, so is g{ ::>", (", Q). It is possible to obtain analogies to all the other rules of U by adding to ~C~6 further cases of solvable lfp's. These, however, would be less instructive. From now on we shall be interested in number-theoretic formulas. Such a formula if> is built up of prime formulas of the form t l = t 2 , where t 1 and t 2 are terms, by means of the logical operations. To each prime subformula if>1 = (/)1 (Xl, ... , x m) of if> we make correspond the lfp is replaced by ::> p, and similarly for all negative and all positive occurrences of I. DEFINITION 10. Let e be an ordinal -< eo. A number-theoretic formula (/)(x l , ... , Xk) is said to be a o-formula if there exist a primitive recursive function A(a) and a;general recursive function V(n, Xl' ... , Xb a) such that Y IX (IX -< e = A (OC) -< e) and that for every IX -< e the function f",(n, Xl> ..• , Xk) ='V(n, Xl' ... , xi, ~) is a solution to {if>}~("') where (3(IX) is an ordinal whose Godel number is A (a).

Remark 3. In case if> (Xl, ... , Xk) contains no logical symbols except v, &, 3 and Y, it can be shown (for any choice of e -< eo) to be a e-formula if and only if it is general recursively true (for this notion see HILBERT and BERNAYS, 1939, and also KLEENE, 1952. Kleene's definition refers to prenex formulas, but it admits of an obvious extension to our case). To say that a formula if> is provable in the intuitionistic formal system of number theory is equivalent to saying that the sequent -+ if> is provable in GENTZEN'S (1935) calculus U upon adjoining to it equality =, number-

135

INTUITIONISTIC NUMBER THEORY

theoretic terms, a few axioms of the form T', -+ r 2 with T', and r 2 containing only prime formulas, and the following induction rule: If eJ)(x) -+ eJ)(x'), then I', ct>(O) -+ ct>(t) (x should not occur free in We designate such a calculus by Z.

n.

r,

THEOREM 3. Every formula ct>(x 1 , ... , Xk) provable in the intuitionistic formal system of number theory is a Eo-formula. Furthermore, if y -< Eo is the ordinal of a proof (see GENTZEN, 1939) in Z of the sequent -+ eJ), then Yen, x t , ... , xs; a) required according to Definition 10 can be chosen to be a function primitive recursive in a function !p(c, s) defined by recursion up to y oftheform: !p(O,s) = "p(s), !p(c+1, s) = n(c, s, !p(O(c, s), x(c, s»), where "p, n, 0 and X are primitive recursive and 0 satisfies this condition:

VcVs[(c+ 1

- or , such that eJ) v ,eJ) is not a Eo-formula. On the other hand, there exists a formula eJ)t such that both eJ)t and ,eJ)t are o-formulas for any limit ordinal e -< EO' This fact implies, in particular, that the assertion 'If ct>1 and eJ)t ::> ct>2 are o-formulas then so is eJ)2' is not always true. However, it can be shown to be true if ct>t does not contain any negative occurrences of::> or "]. Lastly we state a limited result, which can be easily obtained from Theorems 1 and 2. Let Z· be the formal system Z under the restriction on the cut rule and the induction rule that cut formulas (respectively, induction formulas) should not contain any negative occurrences of ::> or , . Let A4 = w·\ A3 = w.l2 , A2 = WOO. 4. If a sequent -+ eJ) is provable in Z·, then eJ) is a o-formula for a e -< A4' Furthermore, the function V referred to in Definition 10 can be constructed using a succession of primitive recursive operations and of operations of unnested recursion up to e. THEOREM

Remark 4. Both Theorem 3 and Theorem 4 will remain true upon adjoining any number of general recursive functions to the formalisms of Z and Z·. (The possibility of defining {eJ)}g as some lfp's in this case follows from Remark 1.) References GBNTZEN, G., 1935, Untersuchungen fiber das logische Schliessen, Mathematische Zeitschrift, vol. 39, pp, 176-210

136

YU. T. MEDVEDEV

GENTZEN, G., 1939, Neue Fassung des Widerspruchsfreiheitsbeweises fur die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, neue Folge, Heft 4 (Hirzel, Leipzig), pp, 19-44 Hn.BERT, D. und P. BERNAYs, 1939, Grundlagen der Mathematik, vol. 2 (Springer, Berlin) K.1.EENE, S. C., 1952, Introduction to Metamathematics (North-Holland, Amsterdam)

NONSTANDARD ARITHMETIC AND GENERIC ARITHMETIC· ABRAHAM ROBINSON Yale University, New Haven, USA

1. Introduction My point of departure is the fact that the theory of algebraically closed fields is the model companion of the theory of fields. That is to say (i) every field can be embedded in an algebraically closed field (model consistency) and (ii) if M 1 and M 2 are algebraically closed and M. c M 2 then M 2 is an elementary extension of M., M 1 -< M 2 (model completeness). Similarly, the theory of real closed fields is the model companion of the theory of formally real fields, that is to say, every formally real field can be embedded in a real closed field and the theory of real closed fields is model complete. The two cases are not strictly analogous because, in the first case, the theory of algebraically closed fields is, more particularly, model complete relative to the theory offields. Tha vis to say .if M is a field, and M. , M 2 are two algebraically closed extensions of M then M. is elementarily equivalent to M 2 in the vocabulary of M, (i.e., in a language which includes individual constants for all elements of M. The analogous assertion for the real closed fields relative to the formally real fields is not true, although the theory of real closed ordered fields is model complete relative to the theory of ordered fields. Within the last two years it has become clear that the state of affairs described above is quite general and applies to the class of models, L, of an arbitrary first-order theory K provided K is inductive, i.e., L is closed under unions of chains or, equivalently, K can be axiomatized by '13 sentences, except that, in the general case, the second class of structures, exemplified above by the algebraically closed fields or by the real closed 1

Research supported in part by the National Science Foundation, Grant No. GP-

292181.

A. ROBINSON

~38

fields, or by the real closed ordered fields, need no longer be axiomatizable. Thus, for given K and E as described there exists a subclass G of E which satisfies the following three conditions. (i) Every M E E can be embedded in an M' consistent relative to E. (ii) If M, M' E E and M c: .M' then M complete. (iii) If M'

E

E

G, in words, G is model

< M',

in words, G is model

G and ME E is an elementary submodel of M' then MEG.

Condition (iii) was not mentioned in the cases discussed earlier since it is satisfied implicitly if G is axiomatizable. If, at the outset, we are given a theory H which is not inductive then we define K as the inductive part of H, H V 3 ' H V 3 is by definition the set of '13 sentences deducible from H and formulated in the vocabulary of H. The alternative possibility of defining K as the set of universal sentences, in the vocabulary of H, which are deducible from H does not yield anything essentially different. For given K and E as above, G is defined uniquely as a subclass of E which satisfies conditions (i), (ii) and (iii). The elements of G are called the generic structures of E (or of K or, if K = H V 3 ' of the original H). The theory of G is called the forcing companion of K (or of H) and is denoted by K F (or H''). If G is not axiomatizable then KF possesses models which are not in G. The existence and uniqueness of G was first proved in A. RoBINSON (1971a, 1971b) with the help of forcing arguments. Other proofs have since been given by CHERLIN (1971) and by Ed Fisher (unpublished) who have developed the theory also in other directions. A structure M E 1: is said to be existentially complete in l: if every universal sentence which holds in M holds also in all extensions of M which are contained in l: (cf. EKLOF and SABBAGH, 1970; A. ROBINSON, 1971a, 1971b; and SIMMONS, 1972). The class of these structures (for given K and l:) will be denoted by E. Then G c: E. If G is axiomatizable or if E is axiomatizable then G and E coincide. This happens in the three cases mentioned at the beginning of this section. There are various other subclasses of l: which can be attached to l: in a natural way. They do not form the object of the present paper. Here we shall focus attention on the classes E and G for arithmetic, i.e., when H is the set of all sentences true for the natural numbers, or for the ring of rational integers or for the field of rational numbers. By generic

139

NONSTANDARD AND GENERIC ARITHMETIC

arithmetic we mean the theory of those structures (where the term theory is meant informally). For the first of the cases just mentioned the corresponding class of generic structures was considered briefly in A. ROBINSON (1971a, 1971b). In Section 2 we shall investigate this case in more detail and, in particular, shall discuss the question to what extent the corresponding forcing companion coincides with ordinary arithmetic, and in which way it differs from it. In Section 3 we consider the mutual relationship between the generic structures and the forcing companions of the several arithmetical theories mentioned above. The investigation of these structures and theories for arithmetic is warranted by the fact that they are obtained by a canonical derivation which, for other cases, yields such important classes as the algebraically closed fields and the real closed fields. Reinforcing this point of view, I shall show in Section 4 that, quite generally, the generic structures of a class 1: possess an analogue of the theory of algebraic varieties for algebraically closed fields, which does not exist for 1: itself. For this purpose, we shall require the following reduction lemma (A. ROBINSON, 1971a, 1971b). 1.1 Let K be any inductive theory, 1: its class of models, and G the class of generic structures in 1:. Let A (Xl' ... , Xn), be any predicate (wff) in the vocabulary of K. Then there exist predicates A.",(xt> ... , x n) in the vocabulary of K, whose number may be finite or infinite, such that for any M E 1: and elements al'"'' an E M, M 1= V /\ A.ixl' ... , Xn) if and only •

p

if A(al> ... , an) holds in all generic structures which are extensions of M. For universal sentences, A(x l, ... , x n), the predicate V /\ A,p(xl> ... , x n) •

p

may be replaced by a single (infinite) disjunction. We have in fact, more generally (ROBINSON, 1971c). 1.2. Let E be the class of existentially complete structures in 1:, with K and 1: as in 1.1 and let A (x, , ... , x n ) be any universal predicate in the vocabulary of K. Then there exist existential predicates A.(x 1 , ... , x n) in the same vocabulary such that for any Me 1: and elements al'"'' an EM, M 1= VA,,(al, ... , an) if and only if A(a l, ... , an) holds in all existentially

•

complete structures M' which are extensions of M, M c M' E E. Notice that M' 1= A(a l , ... , an) for all M' such that M c M' E E if and only if this is the case for all M' such that M c M' E G. The formulae V /\ A.",(x l, ... , x n) in 1.1 and V A,(Xl' ... , x n) in 1.2 •

I"

are called resultants for the respective A (Xl ,

•

... , x n) ,

and the A. p and A.

A. ROBINSON

140

are called the components of these resultants. It follows from 1.1 that for any generic M, 1.3.

M

F

A(xj , ... , x n )

==

j, V, I\A,I'(x I'

... , x n )

and it follows from 1.2 that for any universal A(xj, ... , x n) and any existentially complete M, 1.4.

M

F

A(x j, ... , x n)

== V , A,(x t ,

.•. ,

x n) .

We shall also make use of a result of the general theory which states that G is closed under unions of chains. (This follows immediately from the fact that the elements of G are just the structures on which the forcing relation and the satisfaction relation coincide, A. ROBINSON, 1971a). A slight variation of this argument shows that if K is countable and M is a countable structure in :E then there exists an M' E G, M' c M, such that M' is countable and homogeneous. Given such an M, embed M in a countable generic M j , embed M j in a countable and homogeneous M 2 by elementary extension, embed M 2 in a countable and generic M 3 , and so on. This leads to a chain Me M 1 -< M 2 c M 3 -< M 4 c ... where M 2 k+ t E G and M 2 k is homogeneous, k = 1,2,3, ... and all the M; are countable. Then M' = U M 2 k = U M 2 k+ l is countable and generic and homogeneous. 2. The inductive part of the theory of natural numbers Let 91 be the system of natural numbers and let N be the corresponding theory, i.e., the set of all first-order sentences true in N in terms of the relation of equality, =, the operations of addition and multiplication, and the constants 0 and (for convenience) 1. Let :E be the class of models of K = N V 3 and let G c :E and E c :E be the corresponding classes of generic structures and of existentially complete structures, respectively. Since N is a complete theory, :E possesses the joint embedding property, so (A. ROBINSON, 1971) all generic structures are elementarily equivalent. It follows that the theory of these structures, N F , which is the forcing companion of N, is complete. :E consists of the standard model of arithmetic, 91, together with all other models of N V 3 ' RABIN (1962) has shown that, in our terminology, no nonstandard model of arithmetic can be existentially complete. Suppose now that N F = N. Then any model of N F other than N is a nonstandard model of arithmetic. Since every model of N can be embedded in a generic

NONSTANDARD AND GENERIC ARITHMETIC

141

structure it follows that there exists an MEG which is a nonstandard model of arithmetic. But then M is existentially complete. This contradicts RABIN'S (1962) result and shows that N F =1= N. Later, we shall sharpen this conclusion by actually pointing out a sentence which belongs to N F but not to N. G is not axiomatizable. For if it were then G = E. However, 91 E G since 91 F N, but 91 E E. This proves our assertion. Since Eel:, K = N V 3 is satisfied by all existentially complete structures. In particular, NV 3 c N F • We shall now show, as an easy consequence of the theorem of MATIJASEVIC (1970), that a large portion of arithmetic is deducible from NV 3 ' Matijasevic's theorem states that every recursively enumerable predicate A(xo, Xl> ... , x n) is diophantine. This implies in particular that A (given in any form) is equivalent to an existential predicate in 91. Thus 2.1.

A(xo, Xl' ... , X n) == (3Y1) ... (3Yk)AI(xO , Xl' ... , Xn, YI' ... , Yk),

where Al is free of quantifiers, is a theorem of N. And if A(xo, Xl' ... , Xn) is even recursive then, for some A 2 (x o, Xl' ... , X n, Zl' ... , Zj) which is free of quantifiers, A(xo, Xl' ... , Xn) == (VZI )

2.2.

•••

(Vz j)A 2(xO, Xl' ... , Xn, Zl' ... ,

Zj)

is a theorem of N also, and so is 2.3.

(Vxo)(Vxd ... (Vx n)[(3YI)'" (3Yk)A I(xo, Xl'

==

(VZ I) ...

(Vz j ) A 2(xo,

Xl'

, Xn, YI,,,· Yk) , X n , Zl' ... , Zj)].

But 2.3 is logically equivalent to an '13 sentence and so 2.3 is a consequence of NV 3 ' A similar argument shows that if two existential predicates are equivalent to A, and hence to each other, in 91, then their equivalence is deducible from NV 3 ' This shows that the choice of the particular existential predicate which represents A is irrelevant. Consider the axiom of induction for a predicate A(xo, Xl' ... , x n) where Xl' ... , x; are parameters, n ;;::: 0, and the induction is carried out with respect to Xo. This is, in closed form, ('IXI)'" (Vxn)[A(O, Xl' ... , Xn) /\ (Vxo)[A(xo,

Xl' ... ,

Xn)

=' A(xo+1, Xl' ... , Xn)] =' (Vxo)A(xo, Xl' ... , Xn)]

or, equivalently 2.4.

(VXI) ... (VXn)[iA(O, Xl' ... , Xn) v (3y)[A(y, Xl' ... , Xn)

/\ iA(y+ 1, Xl' ... , Xn)] v ('Iz)A(z, Xl> ... , Xn)].

1.42

A. ROBINSON

Now suppose that A denotes a recursive predicate and is, in fact, given in the form of the right-hand side of 2.1. Within N V 3 ' we may then replace the third and fourth occurrences of A in 2.4 by the right-hand side of 2.2. Interchanging the order of negation and quantification and shifting quantifiers we obtain 2.5.

(V'Xl) '" (V'XlI)(V'Yl) ... (V'Yk)(V'Zl) ... (V'z¥3yi) ... (3yD(3z;)

... (3zj)[i Al(O, Xl' 1\

iA 2 (y + l , Xl'

, Xli' Yl' , Xli' zi,

, Yk) V [A 1 (y , Xl' , zj)] V Az(Z, Xl'

, Xli' Y;,

, Xli' Zl,

, yk) , Zj)].

Since 2.4 belongs to N, 2.5 belongs to N..,3' And since 2.3 is a consequence of N V 3 we conclude that the same is true of 2.4. Thus, all instances of the induction scheme for which A(xo, Xl> ... , Xli) is existential and denotes a recursive predicate are consequences of N V 3 ' Let M E E and let M' be a model of N which includes M. Let Y = f(x l> ... , Xli) be any recursive function, and let it be represented by the existential predicate A (Xl , ... , x,; y)

=

(3Z1) ... (3zk)A 1(Xl' ... , Xli' Y, Zl' ... , Zk)

where Al is free of quantifiers. The possibility of this representation follows again from Matijasevic's theorem. Since

belongs to N V 3 it is true in both M and M' and so is

From this we may conclude that A(x 1 , ... , Xli' y), taken in M, defines a function M" ~ M which coincides with the restriction of f(x l, ... , Xli) from M'" to M". Since this function is determined by A within M it follows that it is independent of the particular choice of M'. Observe that every M E E can be embedded in a model of N. While it is impossible to delimit the sentences of N V 3 ' the principles that we have enumerated suffice to show that much of elementary number theory is a consequence of that set. We now turn to the consideration of certain differences between the structures of E and G and the models of N. For this purpose, we first choose some recursivelyenumerable predicate which is not recursive and we represent it by an existential formula, Dtx«, ... , Xli) = (3Yl) ... (3Yk) D l (Xl' ... , x,; Yl' ... , Yk)' More particu-

NONSTANDARD AND GENERIC ARITHMETIC

lady, we may assume that D(x l nomial equation

, •.. ,

DI(XI, ""Xn,YI, "',Yk)

=

143

x,; YI, ... , Yk) has the form of a poly[P(x l, ""Xn'YI, ''',Yk)

= OJ,

where p is a polynomial with integer (possibly, negative) coefficients, in other words, D is diophantine. For the given n, we now choose a Godel numbering of all polynomials q(XI' ... , Xn, Yl, ... , YJ) with integer coefficients so that qz(xl, ... , x,; Yl, ... , YJ.) is the polynomial with Godel number z, where the numbering has been carried out in such a way that Z ranges over all natural numbers. Then the predicate F(z, Y, Xl' ... , x n ) which is supposed to hold if

is primitive recursive (RABIN, 1962). Accordingly, we may assume that F, as it stands, is an existential formula. It is equivalent, in N and hence also in N V3 to a universal predicate F I (z, Y, X I, ••• , x n) . Once again the existence of such representations is guaranteed by Matijasevic's theorem. In the standard model sn we then have, for every natural number m, 2.6.

(Vx l )

...

(Vxn)(3y) F(m, Y, Xl' ... , Xn)

== (3Y1) ... (3YJ)[qm(XI, ... , Xn, YI, ... , YJ)

= 0]

where m = 1 + 1 + ... + 1 (m times) or m is zero-and where the equation qm = 0 has been expressed within the theory of natural numbers. And

since 2.6 holds in sn, it is a theorem of N and a consequence of N V3 ' Consider the predicate

2.7.

J(w) = (VXI) ... (VXn) [ iD(XI, 1\

(3y)F(v, Y, Xl' ... , Xn)" ('v'Zl)

, x n)

== (3'll)['ll ~ w

(Vz n)[(3y) F(v , Y, Zl' ... , Zn)

::> 1 D(ZI, ... , Zn)J]]

where v ~ w is expressed by (3t)['ll+t = wJ. We are going to show that, in any ME 1:, J(m) is false, for any (standard) natural number m. Indeed, we have M

!=

(Vv)[v ~ m ::> V

Hence, M

!=

J(m) would entail

2.8.

!=

('v'x1)

M

...

(VXn)

=0 v o

= 1v

... v o = mJ.

[I Dtx«, ... , x n) == p=o V[(3y)F(p, Y, Xl' ... , Xn)

v ('v'ZI) ... (Vzn)[3y) F (p , y , Zl'''' Zn) ::>1 D(Zl, ... , Zn)]]J.

144

A. ROBINSON

Let t-tl, ... , t-ts be the set of natural numbers

~ m

for which

2.9. (VZI)'" (Vzn)[(3y)F(lli,y,Zt> ""Zn)=>1 D(zt> ""zn)],

i = 1, ... .s

holds in 91. Then 2.9 is a consequence of N V 3 and holds also in M. Put 2.10.

F'(x l, ... , xn ) == (3YI)F(llt, YI, Xl' ... , Xn) V

..•

v

(3Ys)F(ps, Ys,

Xl' ... ,

x n)

where F'(x.; ... , x n) is obtained from the right-hand side of the equivalence by renaming variables and shifting quantifiers to the left so that the resulting formula is existential. If the disjunction on the right-hand side is empty, put F'tx«, ... , x n) = [IX I = Xl V ... vlXn = Xn]. Then 2.8 shows that 2.11.

M

1=

(VXt)

(Vxn)[1 Dix«, ... , x n) == F'(x l, ... , xn)J.

For, if t-t #: t-t/, i = 1,

, s, t-t

~

m then (compare 2.9),

holds in 91. Inspection shows that this is logically equivalent to an existential sentence and so 2.12 holds also in M. Now suppose that, for aI, ... , an in M, M 1= I Dta«, ... , an)' Then, by 2.8, 2.13.

M

m

1= V

1'=0

(3y)F(p,y, at, ... , an)/\ (VZt) ... (Vzn)[(3y) F(Il, Y, Zl, ... , zn)=>1 D(ZI, ... , zn)].

Hence, M 1= F'(a., ... , a,,), by 2.10 and 2.12. Conversely, suppose that M 1= F'(a.; ... , an)' Then this implies that the set {t-tt, ... , t-ts} is not empty. Hence, 2.13 holds and further, M 1= I Dta., ... , an), by 2.8. This proves 2.11. Suppose now that the sentence of 2.11 does not hold in 91. Then, for certain numbers aI' ... , an E 91, 91 1= I [I Dia«, ... , an) == F'(a l, ... , an)]' But this is logically equivalent to an V3 sentence which must therefore be true also in M, contradicting 2.11. Hence 911= (Vx t )

...

(VXn)[1 D(x l, ... , x n) == F'(x l

, ... ,

x n)].

But this would imply that I D is recursively enumerable, so that D would be recursive. This proves that M 1= IJ(m). Now suppose more particularly that M is existentially complete, so that M E E, and let a be any nonstandard element of M. Thus, a #: t-t for all standard natural numbers t-t and, more specifically, a > t-t in M. We propose to show that M 1= J(a).

145

NONSTANDARD AND GENERIC ARITHMETIC

We observe that the implication [(3v)[v:( a r. (3y)F(y, Xl , ... , Xn)A (V'Zl)'" (Az n) [(3y ) F(v , y , Zl'

, Zn)

::lID (Zl"'" Znm] ::lD(x l,

, Xn)

is a theorem of the predicate calculus. Accordingly, it only remains for us to verify that 2.14.

M F (V'xl) ... ('v'Xn)[ ,D(x l, ... , Xn)::l (3v)[v :( a v (3y)F(v, y, Xl' ... , Xn)A (V'Zl) ... (Azn) [(3y ) F(v , y, Zl' ... , Zn)

::ll D(ZI'

, Znm].

For this purpose, we apply 1.2 to A(x l, ... , x n) == ,D(x l, , x n) where we have obtained A from the right-hand side of the equivalence by shifting quantifiers across ,. Thus, A is universal and we have, by 1.4. 2.15.

== VAv(x l ,

M F A(Xl' ... , x n)

..

... ,

x n) .

Here, the components A are existential and in ~ or more generally in any element of 1:, an existential predicate is equivalent to a diophantine predicate. Thus, we may assume from the outset that, for all v, A.(x l, ... , x n) = (3Yl) ... (3Yj)[T,,(XI' ... , Xn, YI , ... Yj)

=

0]

where Tv is a polynomial with integer coefficients. It follows that Tv coincides with one of the qz, Tv = q,(v), say. Hence (compare 2.6)

is a theorem of N. Remembering that F and A are existential we can replace 2.16 by an V'3 sentence. It follows that 2.16 holds in M and so M

F

I D(x l, ... , x n) ==

V (3y)F({(v) , Y, Xl' v

... , Xn).

Thus, for each v,

Also for all a l, ... , an in M (and not only for standard al' ... , an) there is a particular v such that 2.18.

M F I D(a l, ... , an) ::l (3y)F(C(v) , y, aI, ... , an).

But C(v) is a standard natural number and so M F C(v) < a. Hence, from 2.17 and 2.18,

A. ROBINSON

146

M

F' D(a l ,

... ,

/\ (VZ l )

an) ~(3v)[v =:;; a r. (3y)F(v, y, a l ...

, ... ,

all)

(VZ n) [(3y)F(v ,y, Zl' ... , zn) ~, Diz«, ... , zn)]].

This proves 2.14. We have established 2.19. THEOREM. Let M E E. Then M if and only if a is nonstandard.

F

J(a), where J(w) is given by 2.7,

Inspecting 2.7 we see, after the usual modifications, that J(w) is logically equivalent to an V3V formula. Thus, the set of standard (finite) natural numbers is definable in M by an 3V3 formula (= 'J(w)). Observe that in proving 2.19, we had not excluded the case M = 91. Since 91 contains only standard numbers we therefore have 91 F (Vw) [I J(w)] but M F (3w)J(w) for all other existentially complete M. Hence 2.20. THEOREM. There exists an V3V3 sentence which holds in 91 but not in any other existentially complete structure.

Also 2.21. THEOREM. The existentially complete structures, other than 91, are not models of Peano arithmetic.

- For the set defined by J(w) does not possess a smallest element. Finally, consider any Godelization of the formulae, which are based on the vocabulary of N. For any such formula, Y, we may consider the transform eYof Y which is obtained by relativizing the quantifiers of Y with respect to the predicate iJ(w). For any existentially complete structure Min E we then have M F eY if and only if 91 F Y and this is true even if M = 91. Moreover, the set of Godel numbers of the formulae eY is recursive (assuming, as is indeed implicit in the notion of Godelization, that the Godel numbers of the Y form a recursive set). It follows that the set of sentences true in M cannot be arithmetical. For if it were, then the set of sentences true in 91 would be arithmetical and this contradicts Tarski's theorem. The same argument shows that the set of sentences true in all M E S where S is an arbitrary nonempty set of existentially complete structures cannot be arithmetical. In particular 2.22. THEOREM. The set N F is not arithmetical.

This answers a question asked by Martin Davis during the Nice Congress and sharpens an earlier observation (A. ROBINSON, 1971) that N F is not recursively enumerable.

NONSTANDARD AND GIlNllR.IC ARITHMETIC

147

3. Generic structures for the integers and rationals With the same vocabulary as before, let Z be the first-order theory of the ring of rational integers, .3 and Q the first-order theory of the field of rational numbers, Q. We write KN = NV3 ' Kz = ZV3' K Q = QV3 and we denote the corresponding classes of models by EN, LZ' LQ. Thus, EN is the class denoted by L in the preceding section. All elements of LZ are rings and all elements of E Q are fields. The corresponding classes of existentially complete structures will be denoted by EN, Ez , E Q , and the respective classes of generic structures by GN , Gz , GQ • Since N, Z, and Q are complete we have 91 E E.~ (as mentioned earlier) and .3 E Ez , Q E E Q • For the same reason, the structures of GN , Gz , GQ are elementarily equivalent, respectively, and the corresponding theories N F, ZF, QF are complete. Let ME E,,;. We define a structure CM in terms of pairs oc = (a, a'), fJ = (b, b'), ... where a, a', b, b', ... EM as in the definition of the integers from the natural numbers. Thus, IX = fJ if a+b' = b+a', oc+fJ = (a+b, a' +b'), and ocfJ = (aa' +bb', ab' +a'b). Then CM E E z . For let X be any '13 sentence which holds in 3. Then .3 is isomorphic to C91 and X may be expressed as an '13 sentence Y for the elements of the pairs which constitute C91. But then Y holds in 91, hence in M, hence CM satisfies X. Evidently, Cpreserves inclusion. It also preserves elementary inclusion because a sentence about CM can be expressed as a sentence about M. We now wish to show that the set CGN which consists of CGN together with the isomorphs of the elements of that set, exhausts Gz • We propose to show that CGN satisfies the three conditions which characterize the class of generic structures in E z . (i) Let ME LZ. Then M is a model of Z'f'3 and, hence is ordered by the predicate P(z) = (3Xl)(3X2)(3X3)(3X4)[X~+X~+x~+x~ = z] as a definition for the set of nonnegative numbers. Let M N = {z E MIM 1= P(z)}. A sentence X about M N is expressed in M by relativizing the quantifiers of X with respect to the predicate P(z). Since P(z) is existential, eX is an '13 sentence if X is. Since M is a model of ZV3' we may conclude that M N is a model of NV3 ' M N E EN. Let NN C M~ where M~ E GN • Then i;MN C CM~ C i;GN • Hence M, which is isomorphic to i;MN , is contained in CGN • (ii) Let M, M' E i;GN ; where M c. M', We have to show that M -< M'. Byassumption, M is isomorphic to i;MN for some M N EGN and M' is isomorphic to i;M~ for M~ E GN • Since M c M' and both M and M' are elements of ZV3' M N and M~ may be identified with the sets {z E MIM 1= P(z)}

148

A. ROBINSON

and {z E M'IM'F P(z)} as described under (i). Then MNcMj., so M N -< Mj., and CMN -< CM;'. We conclude that M -< M'. (iii) Let M' E CG-;; and let ME Lz such that M -< M'. Determining M N and M;' as in (i) and (ii), by means of P(z) we then have Mj., E GN and MN c M;'. Moreover, M -< M' entails M N -< Mj., as we see when we express the latter condition in terms of M and M'. Hence M N E GN, M E CG N, as required. We conclude that CGN coincides with Gz. It is even easier to show that Ez = CEN which consists of fEN and its isomorphs. Also if we define a mapping y from LZ into EN by v'M = {z E MIM F P(z)}, then, by the same reasoning, yEz = EN and yGz = GN • The connection between the classes of structures associated with Z and Q, respectively, is less satisfactory. For ME LZ' let XM = M' be the field of quotients of M, which is obtained by introducing pairs of integers or 'fractions' a = A(x1>"o,xn)

then A(x 1 , ••• , x n ) E J. An ideal is proper if J :1= J o. A proper ideal J is prime (or irreducible) if it is not the intersection of two ideals including it and distinct from it. Equivalently, J :1= J o is prime if for any finite disjunction of elements of J« which is deducible from J with respect to K, one of the disjuncts belongs to J. If J is an ideal and Y E J o - J then there exists an ideal J' c J o which includes J and excludes Yand is maximal with respect to these properties. Then J' is prime. It follows that every ideal is the intersection of prime ideals. An ideal J is maximal and hence prime if it is not included in any proper ideal different from it. Choosing Y contradictory we see that every J c J o, J a proper ideal, can be embedded in a maximal ideal. Let M" = M x M x ... x M be the n-dimensional space over a structure M E E. Let B(x 1, ... , x n ) be any predicate of n variables in the vocabulary of K. The question whether B(x 1 , ... , x n) holds at a point (el, ... , en> E M" depends, generally speaking, not only on l ' ..• , en> but also on M, i.e., the answer may change if we extend M. Accordingly, we define a point (or compound point) P as an (n+ 1)-tuple (M';l' ... , Cn> where ME E and ;1' ..., ;n are elements of M, and we say that B(P) holds or is true or that P satisfies B if M F B(;l' ... , en). If P = (M, ;1' ..., en> where ME E then we call P an E-point and if MEG we call P a G-point. A general theory of varieties, corresponding to the theory of ideals quoted above is developed in A. ROBINSON (1951, 1963) and, in more detail, in A. ROBINSON (1955). In the present context it is appropriate to develop an alternative theory in which the varieties consist of E-points or, more particularly, of G-points. Let J be a proper ideal in J o. A point P is said to be generic for J if A(P) holds for all A E J and does not hold for any A E Jo-J.

«

4.1. THEOREM. In order that the proper ideal J c Jo possess a generic point which is an E-point it is necessary and sufficient that J be maximal. PROOF: Let P = (M, ;1' ..., ~n> be an E-point and let J = {A E JolA satisfies Pl. We have to show that Jis maximal. Let A' (Xl' ... , X n ) E J« -J. Then I A'(x 1 , ... , x.) may be written as a universal predicate. By 1.2 there exist A,(X1' ... , x n ) E J o such that for any M' E E, M'

F

M

F

iA'(x 1 ,

...

,xn )

In particular, iA'(~1> ... , En)

== VA.(x 1 , • ==

VAvC~I'

•

...

,xn ) •

..., En) .

150

A. ROBINSON

Hence, for some

')I

= p., M

F

Ai~l' ... , ~n),

and so AI' e J. But, for any

M'eE

4.2. and so, if a point P' = <M', ~i, ... , ~~> satisfies all A e J then it does not satisfy A'. Suppose now that J is not maximal and is included in a proper ideal J' =1= J. Choose A'(Xl, ... , x n ) e J' -J. Then K u J U {A' (Xl> ... , x n )} is consistent, there exists a point Po = (Mo , 'YIl' ... , 'YIn>, M o e J: such that AI' and A' are both satisfied by Po, with AI' as in 4.2. Moreover, since AI' and A' are both existential, we may embed M o in an existentially complete M' so that AI" and A' both hold at the point P' = (M', 'YIl, .••, 'YIn>. But this contradicts 4.2 and shows that the condition of the theorem is necessary. Suppose next that J is maximal. Then any point P which satisfies all A e J does not satisfy any A e J o - J. Choose a point Po which satisfies alIA eJ. This can be done since K u J is consistent. Let Po = <Mo , ~1' ..., ~n> and embed M o in an existentially complete M. Then P = (M, ~l' ... , ~n> still satisfies all A E J and none of the A e J o -J. But P is an E-point and so the proof is complete. Let S be a nonempty set of E-points. We define the ideal of S, j(S) by 4.3.

j(S) = {A e JolA is satisfied by all PeS}.

It is not difficult to verify that j(S) is a proper ideal. Any ideal which is obtained in this way will be called an E-ideal. Again, let J be a subset of J o which is consistent with K. Thus, J is included in a proper ideal. The class of E-points which satisfy all A e J is called the E-variety of J, v(J). An E-variety is, by definition, the E-variety of some

proper ideal. Let V be an E-variety. We divide the elements of V into equivalence classes according to the "relation P '" P' if for all A e J o, A(P) and A(P') hold or do not hold, simultaneously. In other words, p", P' if j({P}) = j ({ P'}). Thus V = U V,; where the V. are the equivalence classes just defined. We claim that if P E V, then V, is just the E-variety of the maximal ideal J;, = j({P}). For this purpose, it only remains to be shown that if P' E v(Jv) then P' E V. But this is obvious for if V = v(J) then J, ::::l J and so v(J.) c v(J) and v(Jv) = V•. We have therefore shown that every E-variety V is the union of E-varieties V, of maximal ideals which are contained in it. If V is the E-variety of a maximal ideal J from the outset, then any two of its elements are equivalent, p »: P'. Suppose that there is a prop-

NONSTANDARD AND GENERIC ARITHMETIC

151

er subclass V' of V which is an E-variety. Then we may assume (if necessary, by decomposing V' further as a preliminary step) that V' = vel') where 1'is again maximal. But then, if P' E J' and P '" P' we have P E V' and so V' = V. It follows that the E-varieties of maximal ideals are just the minimal E-varieties. 4.4. THEOREM. Every Esvariety is the disjoint union of the minimal varieties contained in it. Observe that if V = vel) and J' is any maximal ideal that contains J then V' = V(JI) appears in the decomposition of V into minimal E-varieties. We now turn to the consideration of G-points. Since every existentially complete structure M can be embedded in a generic structure and since, in that case, every universal sentence in the vocabulary of M that holds in M holds also in M ', it follows that if P = (M, ~!, ... , ~n> is a generic point for an ideal 1 then P' = (M', ~1' .•. , ~n> is a generic point for the same ideal. Hence, from 4.1, 4.5. THEOREM. In order that a proper ideal J c 1 0 possess a generic point which is a G-point, it is necessary and sufficient that J be maximal. It follows that if we define a G-variety w(J) of a set 1, which is consistent with K as the class of G-points that satisfy the elements of J, then w(l) is just the set of G-points of vel). Also, w(l) is the disjoint union of the maximal G-varieties contained in it, and each one of these is the nonempty intersection of a minimal variety in vel) with the class of G-points. Thus, we do not get any new feature, as long as we consider only existential predicates. However, we shall now show that, within the class of generic structures, the maximal ideal which belongs to any point P,j({P}) is sufficient to determine completely the type of P in the sense of model theory. In detail

4.6. THEOREM. Let P = (M, ~1' ... , ~n> and P' = (M', ~;, ... ~~> be two Gspoints such thatj({P}) = j({P'}). Let A(X1' ... , x n) be any predicate in the vocabulary of K. Then

F

M PROOF:

that M

if and only if

A(;1' ... , ;n)

M'

F

A(~;, ... , ~~).

Let P, pi and A satisfy the assumptions of the theorem. Suppose A(;1' ... , ;n)' Let V 1\ A.,.. (x! , ... , x n) be a resultant for A

F

(see 1.1). Then M

.

F V 1\ ,..

.

,..

A.,..(;!, ... , ~n)

and so, for some v, M

F 1\ ,.. A.,.. (;1 , ... , ;n)' But then for this particular v and for all fl, A ,..(X1' V

152

A. ROBINSON

... ,Xn)Ej({P})=j({P'}). Hence M/~/\Av,l~~, ... ,~~) and, further p

M' F= A(~l' ... , ~n)' Since our assumptions are symmetrical with respect to P and P', the theorem follows.

Reverting for a moment to the decomposition of E-varieties, we saw that if V = v(J), then V = U Vv where Vv = v(Jv), with J; ranging over all maximal ideals that contain J (Theorem 4.4). Let J = {A;.(x 1 , ... , x n)} and let J, = {AVP(x 1 , ... , x n) }. Then we have, as an immediate consequence of 4.4 that, for every existentially complete M, 4.7.

M

F=

/\A;.(x 1 ,

...

;.

,xn)

S

V /\A.p(x 1 , ... ,xn). •

p

Now let {A;'(Xl'"'' x n ) } be a set of arbitrary predicatesin the vocabulary of K and assume, in order to discard trivial exceptions, that there exists a G-point which satisfies all A;.. If P is such a point and P' is another G-point such that p", P', i.e., such that j({P}) = j({P'}) then P' also satisfies all A;., by 4.6. Thus, if {Jv} is the set of all maximal ideals that occur for such P, J. = j({P}), and J. = {AVP(Xl'"'' x n)}, then we still have 4.7, provided M is generic. In particular ifthere is only a single A;. = A, then 4.7 becomes 4.8.

M

F=

A(Xl, ... ,xn)

S

V /\Avp(x v

P

1 , ...

,xn).

This brings us back to 1.3 except that the representation in 4.8 is now canonical, and the equivalence is destroyed if we omit 1\ AVj.l(x1 , .•. , x n ) for f.1 any single 'V. Suppose now that {A;'(Xl'"'' x n)} is the type of a G-point P, i.e., it is the set of all predicates in the vocabulary of K which are satisfied by P. Then any other G-point P' that satisfies all A;. is equivalent to P, P '" P', and so there is only a single corresponding maximal ideal Jv • Moreover, this J; then consists simply of the existential predicates among the A;., to be denoted by A~(Xl' ... , x n). Then the equivalence 4.9.

M

F= /\ A;.(x 1 , ;.

••• ,

x n ) == /\ A~(Xl' ... , x n) j.l

expresses in a compact manner the fact that within the class of generic structures the type of a point is determined completely by the set of existential predicates satisfied by it. In this context, it is appropriate to call the set of all existential predicates satisfied by a point P (which, as we know, is just a maximal ideal if P is an E-point) also its existential type. Thus, 4.10. THEOREM. The type of a G-point is determined entirely by its existential type.

NONSTANDARD AND GENERIC ARITHMETIC

153

Now let M be an infinite generic structure which is also homogeneous (briefly, a homogeneric structure). For the case that K is countable, and possesses infinite models, the existence of such a structure is guaranteed by the arguments at the end of Section 1. Suppose that two points in M, P = (M, t., ... , ~n) and pi = (M, ~~, ... , ~~) have the same existential type. It then follows from 4.10 that they have the same type. Hence, by the basic property of homogeneous structures there exists an automorphism of M which carries (~l' ... , ~n) into (~~, ... , ~~) (and which does not move the elements which are denoted by constants of K). 4.11.

Let M be a homogeneric structure, and let P = (M, ~l' ... , ~n) and P' = (M, ~~, , ~~) be two points in M which have the same existential type. Let r;l' , r;n be any set ofm elements of M. Then there exist r;~, ... ,r;:" in M such that (M'~l''''~n,r;l, ... ,r;n> and (M, ~;, ... , ~~, r;~, ... , r;:"> have the same existential type. THEOREM.

This theorem follows immediately from the preceding discussion, for we may take r;~, ... , r;:" as the images of r;l' ... , r;m in an automorphism of M which carries ~ l> ... , ~n into ~~, ... , ~~. We have stated it here explicitly because it illustrates the fact that we are dealing with a genuine generalization of the theory of ideals and varieties (algebraic sets) in algebraic geometry. Thus, let K be the theory of algebraically closed fields which extend a given countable field F. For this theory, a homogeneric structure M is just a universal domain over F, i.e., an algebraically closed field of infinite degree of transcendence over F. Because of elimination theory, the specification of the prime ideal Q c F[x1, ... , x n] of a point (~l' ... , ~n> in M in the ordinary algebraic sense is equivalent to the specification of the existential type of (M, ~l , ••• , ~n> for if we assert that Q is the prime of (~l' ... , ~n> we mean not only that P(~l' , ~n) = 0 for all p E Q but also that P(~l' ... , ~n) =I- 0 for p E F[Xl' , xn]-Q. In this sense, 4.11 becomes just the statement of the extension lemma of algebraic geometry for generic specializations. Returning to the arithmetical theories considered elsewhere in this paper, we fix our ideas by considering the case K = Z,/3' For this case, an existential predicate A (Xl , ... , x n ) is equivalent to a predicate of the form 4.12. where p is a polynomial with integer coefficients. Now, 4.12 amounts to the assertion that (Xl' ... , x.> can be obtained from a certain hypersurface by projection parallel to some of the axes. The existential type of a point

154

A. ROBINSON

P then specifies the hypersurfaces (in an arbitrarily high number of dimensions) from which P can be obtained in this way. And Theorem 4.10 tells us that this is sufficientfor the complete determination of the type of a point in a generic structure of Z.

References CHERLIN, G., 1971, The model companion of a class of structures, Yale University, (mimeographed) EKLOF, P. and G. SABBAGH, 1970, Model completions and modules, Annals of Mathematical Logic, vol. 2, pp. 251-296 FIsHER, E. R. and A. ROBlNSON, 1971, Inductive theories and their forcing companions, Yale University, (mimeographed) MAIDASEVI(\ Y. V., 1970, Diofantovost perechislimykh mnozhestv, Doklady Akad. Nauk USSR, vol, 191, pp. 279-282 RABIN, M.O., 1962, Diophantine equations and non-standard models of arithmetic, in: Logic, Methodology and Philosophy of Science, Proceedings of the 1960 International Congress, eds, E. Nagel, P. Suppes and A. Tarski (Stanford University Press, Stanford), pp. 151-158 ROBlNSON, A., 1951, On the metamathematics of algebra (North-Holland, Amsterdam) ROBlNSON, A., 1955, Theone metamathematiquedes ideaux, Collection de Logique Mathe. matique, ser. A, (Gauthier-Villars, Paris) ROBINSON, A., 1963, Introduction to model theory and to the metamathematics of algebra (North-Holland, Amsterdam) ROBINSON, A., 1971a, Infinite forcing in model theory, in: Proceedings of the Second Scandinavian Logic Symposium 1970, ed. J. E. Fenstad (North-Holland, Amsterdam), pp, 317-340 ROBINSON, A., 1971b, Forcing in model theory, Proceedings of the International Congress of Mathematicians, Nice, 1970, vol. 1, pp. 245-250 ROBINSON, A., 1971c, A decision method for elementary algebra and geometry-revisited, Symposium in honor of Alfred Tarski, Berkeley (mimeographed, Yale University). ROBINSON, J. B., 1949, De/inability and decision problems in arithmetic, J oumal of Symbolic Logic, vol. 14, pp, 98-114 SIMMONS, H., 1972, Existentially closed structures, Journal of Symbolie Logic, vol. 32, pp. 293-310

MODELS FOR VARIOUS TYPE-FREE CALCULI

D. SCOTI Oxford University, Oxford, Eng/and

Introduction

The question is this: what is the proper logical notion of function? For the discussion it seems to me that we must distinguish logical notions from those axiomatized in mathematical theories. Making such a distinction in no way means that, for our own benefit, we should refrain from investigating the coherence of certain of our ideas with the aid of mathematical models. Thus, for example, notions of proposition and propositional connective can be very greatly illuminated by consideration of the variety of Boolean algebras or more general lattices. Some of us are content to confine attention completely to these models, but this is surely wrong. To make the ascent from mathematics to logic is to pass from the object language to the metalanguage or, as it might be said without jargon, to stop toying around and to start believing in something. The difference between play and faith is well illustrated by geometry (and, also by category theory). Euclidean geometry can provide a field for a lifetime of study-especially with the use of the advanced techniques of algebra and topology-but we can also believe that the geometry of physical space is Euclidean. Euclid did, Newton did too, but now we know better (some of us, that is). Physics has shown that the flow of time together with the presence of matter dictate a far more complex geometry for 'true' space. Alas, we are not yet sure exactly which this geometry is, and the uncertainty no doubt drives many of us back to pure mathematics. The complexity of the situation makes mathematical study and experimentation essential for progress, but the difficulties cannot keep me from entertaining the belief that space has some geometry, a preferred, correct geometry. I, myself, cannot give a very clear or exciting description of this geometry, but I know people who can. As regards logic I have rather more self-confidence.

158

D. SCOTI

Turning to category theory we come closer to the topic ot this paper. One view of a category is that it represents a 'space' of allowed functionsfunctions under composition rather than under function value. The generality attained by freeing functions from the tie to the specifics of argumentvalue pairs and by treating them in relation to one another has excellent mathematical benefits: there are many-what should we call them? nonFregean?-models of the axioms of category theory that cannot be explained through arguments and values. That is fine. I approve. But note that when category theorists speak of functors they go straight back to the use of arguments and values. They have this idea of functional correspondence as part of their logic, I claim. I would like to make this logic clearer, and so would they. They often study functor categories and thereby apply their own theory to itself. It is an excellent idea, but it is no replacement for logic. If and when the so-called category of all categories is fully revealed to us, the story may be different. But I doubt it. Until the gospel is written, I will continue to assert that, in the 'universal' or 'logical' category, function-value can be defined in terms of composition, and so the dispute is a 'semantical' one. Euclid was proven wrong, and so it may be with us Fregeans. In the meantime I think I have an interesting contribution to make to the logical study of functions, in particular as it is cast in the framework of the A-calculus of Church and Curry. In the introduction to his final publication on the A-calculus, Church writes: Underlying the formal calculus which we shall develop is the concept of a function, as it appears in various branches of mathematics, either under that name or under one of the synonymous names, "operation" or "transformation." The study of the general properties of functions, independently of their appearance in any particular mathematical (or other) domain, belongs to formal logic or lies on the boundary line between logic and mathematics. This study is the original motivation for the calculi-but they are so formulated that it is possible to abstract from the intended meaning and regard them merely as formal systems. A function is a rule of correspondence by which when anything is given (an argument) another thing (the value of the function for that argument) may be obtained. That is, a function is an operation which may be applied on one thing (the argument) to yield another thing (the value of the function). It is not, however, required that the operation shall necessarily be applicable to everything whatsoever; but for each function there is a class, or range, of possible arguments-the class of things to which the operation is significantly applicable-and this we shall call the range of arguments, or range of the independent variable, for that function ... It is, of course, not excluded that the range of arguments or range of values of a function should consist wholly or partly of functions. The derivative, as this notion appears

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in the elementary differential calculus, is a familiar mathematical example of a function for which both ranges consist of functions... Formal logic provides other examples; thus the existential quantifier, according to the present account, is a function for which the range of arguments consists of propositional functions, and the range of values consists of truth values. In particular it is not excluded that one of the elements of the range of arguments of a function f should be the function f itself. This possibility has frequently been denied, and indeed, if a function is defined as a correspondence between two previously given ranges, the reason for the denial is clear. Here, however, we regard the operation or rule of correspondence, which constitutes the function, as being first given, and the range of arguments then determined as consisting of the things to which the operation is applicable. This is a departure from the point of view usual in mathematics, but it is a departure which is natural in passing from consideration of functions in a special domain to the consideration of functions in general, and it finds support in consistency theorems which will be proved below.1

Personally, I see no need "to abstract from the intended meaning" and find mere formal systems merely boring. Was Church worried that mathematicians might not consider logical studies worthwhile? or was he still hurting from the inconsistency that Kleene and Rosser had found" in his 'logistic'? The specter of formal inconsistency certainly must lie behind the strange inconsistency of exposition whereby, on the one hand, Church takes such care in explaining that functions need not be everywhere defined, while on the other, he allows so easily for meaningful self-application. The claim for the naturalness in generality is awfully weak, and the formal consistency proof (The Church-Rosser Theorem) is not convincing. I should say that truth is that self-application is allowed because it is so convenient for what Church wanted to do. Indeed, he wanted to do much more and could not. The A-calculus is only a fragment of his inconsistent system, an interesting fragment to be sure, and the consistency proof is comforting. Still, a fragmentary result does not prove that the concept as intended is coherent. In fact, I feel that Church went much too far when he further claimed that only the terms with normal forms are meaningful. Such an extreme stand seems to me to rather spoil the original insight. Curry provides a somewhat different view: Much of the work in illative combinatory logic has been concerned with proving consistency of very weak systems. There is a reason for this. [Combinatory logic] is concerned with notions of such generality that we have no intuitions concerning them. 1 A. CHuRCH, The calculi of lambda-conversion, Annals of Mathematics Studies, vol. 6, 2nd ed., Princeton, 1951, pp. 2-3. 2 References and historical comments are given H. B. CuRRY and R. FEYs, Combinatory logic, vol, 1, Amsterdam, 1958, pp. 273-274.

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D. SCOTI

The paradoxes have shown this; indeed these contradictions have arisen because they involve combinatory situations-including a delimitation of the notion of propositionwhich had not been made explicit. Furthermore these paradoxes have arisen in the work of such minds as Frege, Church, and Quine. This is an argument for the thesis that logical intuitions are not inborn, but arise from experience. The only way of developing new intuitions is by trial and error. The investigations of weak systems, and possibly the discovery of new inconsistencies, is a means to that end. Thus the weakness of these systems does not show that the methods are unsound. We know from the work of Godel that we cannot expect to prove by finitary methods the consistency of really useful systems. Indeed it is possible that we shall... continue to prove stronger and stronger systems consistent and weaker and weaker systems inconsistent, but that we shall always be interested in systems for which neither of these alternatives holds. Whether this is so or not, the study of consistency questions advances our knowledge. In the early stages we are interested in constructive consistency proofs for very weak systems. But as our intuitions develop we may expect infinitistic methods to be applied and to bear fruit."

I find it very difficult to agree with this position. It is not that I disparage experience or trial and error, rather I wish to dispute the claim that we have no intuitions. In the next section of this paper an intuitive construction will be attempted which will lead to some quite specific investigations. The means I shall use are 'infinitistic', but it does not proceed by the kind of formalistic experimentation that Curry seems to have in mind. The discovery that I made was the result of considerable trial and error on my part, but it was on a conceptual level and did not unfold by attempting consistency proofs. It will be in terms of rather definite intuitions that I shall try to convince people to be interested in various systems, and, after my intuition about the A.-calculus is explained, I shall attempt to say something new about the nature of the paradoxes in this context. Before beginning the detailed discussion, there is another general point that needs attention. I refer to the conflict in conception between functions in extension versus functions in intension. Which is the proper approach for logic? The extensional view of function treats them as abstract objects which are completely determined by their argument-value pairs. As it is often said: a function is uniquely determined by its graph; though there is no reason to identify a function with its graph as some kind of class of pairs. In intension, on the other hand, a function is a rule; that is, a function is somehow connected up with its definition. If this was indeed the 'original intention of the founders of our subject', then I submit that they intended to lead us directly into a confusion of use and mention. Though I cannot 3

H. B. CURRY, Combinatory logic, in Contemporary philosophy. A survey, vol. 1

CR. Klebansky, ed.), Firenze, 1968, pp. 295-307.

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agree completely with Quine that intensional 'logic' is entirely a usemention confusion, it is very easy to fall into such error. I hope some day to see a convincing formulation of intensional logic, but I claim that nothing exists today that approaches the level of development of extensional logic. Of course, that is no argument against the use of intensions. My complaint is that so far everything is very fragmentary." Schonfinkel seemed to have something of the idea that his general functions were rules; because he, like Curry, who made the discovery for himself, wanted to eliminate variables from logic in order to get away from the complications of substitution. But substitution is a syntactical problem, while logic is a conceptual one. I do not see that the self-applicable combinators can be justified as providing an 'analysis' of substitution. Their use may make the act of substitution unnecessary, which is nice, but there is much more to them than that. But I do not find an explanation of just what this is in the work of Curry and his students. S Consider Church's A-b-calculus. 6 Values are determined according to whether a term has a normal form or not. (Similar systems have been proposed by Goodman? and Kearns.") That strikes me as being an out-and-out use-mention mistake. Now, I do not mean to say that the formal system does not work somehow, I only wonder why I should consider this system interesting. After all, the deductive rules are (in view of Godel) quite weak. Why make decisions on the basis of such a formalism? What notion has been (partially) axiomatized here? Should the general notion of function be so sensitive to details of choice of description and deduction? Some say yes, I say no. There may be another point of view from which intensional comparisons are reasonable. We do have something of an idea of a proof, a computation, a process. A function could be a 'scheme' for a type of process which would become definite when presented with an argument. The value would be extracted as an end result of the process. Two functions that are extensionally 4 Studies in modal logic have not yet gotten to mathematical problems. The closest approach is investigations in proof theory of intuitionistic logic, but connections with the type-free A.-calculus are not yet very clear. ~ Up-to-date references can be found in H. B. CURRY, T. R. HINDLEY and T. P. SELDIN, Combinatory Logic, vol. 2, Amsterdam, 1972. 6 A. CHURCH, op. cit, pp. 55-60, and CURRY and FEYS, op, cit., p. 93. 7 N. GOODMAN, A simplification 0/ combinatory logic, Journal of Symbolic Logic, vol. 34, pp, 1-12. 8 T. KEARNS, A system ofreduction, Journal of Symbolic Logic, vol. 32.

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D. SCOlT

the same might 'compute', however, by quite different processes. (One is fast, the other slow. We all know examples.) This kind of talk is very loose, though. The stickler is that these 'computations' have to be abstract objects; they have to be more than just types as distinguished from tokens, because equivalences of 'meaning' are usually allowed. (Say, the orders of execution of certain steps may be interchangeable.) The mixture of abstract objects needed for a sweeping theory of this kind would obviously have to be very rich, and I worry that it is a quicksand for foundational studies. In this paper the extensional approach is adopted, and if necessary we shall model intensional notions in terms of suitably chosen extensional objects. Maybe after sufficient trial and error, we can come to agree that intensions have to be believed in, not just reconstructed; but I have not yet been able to reach that higher state of mind. 1. Analysis

In the beginning there were the two values, the true and the false. Why? One answer certainly is that we need at least two discrete, separated, distinguishable, distinct different objects. That much is clear, but why identify these values with the truth values? The answer is: why not? Though explicit reference to the truth values can be eliminated from many discussions, it need not be. If one insists, one can say that certain abstract values 'represent' the truth values without saying that they are the truth values. But since I have no desire to restrict the use of abstract objects, I see no harm in making the identification and feel that it makes life simpler to do so. 1.1.

AXIOM.

true

=1=

false.

Having taken the first step, one must then find ways of using the truth values. An obvious use is in the making (better: recording) of decisions. Decision entails choice, I suppose, and the choice situation will be best expressed by the conditional: Z:::J X,

y.

Once we decide whether the value of z is true or false, we shall then be able to make the corresponding choice of X or y. This is not the material conditional of propositional calculus, because x and y may be any two values, not just truth values." Another way of explaining what we are doing is to say that we allow for definition by cases, which is often written as: 9

The use of the conditional was introduced into algorithmic languages by T. Mc Carthy.

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(z::>x,Y)=

X

{ y,'

if z; if not.

But this is all notation, the real assumptions are given by: (true ,o x, y) (false :o x, y) = y.

1.2. AXIOM. (i) (ii)

=

x;

Such conditional expressions, when iterated, permit the definition of many different truth functions. These functions, along with others, we wish to contemplate as abstract objects. That is effected by functional abstraction, as in this example: 1.3. DEFINITION. (x, y) = AZ' (z

::>

x, y).

Again it could be asked why just this combination is called the ordered pair. The answer is pragmatic: by Axiom 1.2 we see that it works; and it is a simple device requiring no novel primitive notions. The pair is, however, treated as a function, which is going beyond mere combinations of truth values. In general we must take abstraction (AZ') as a variable binding operator which, in the theory, is complemented by the binary operation f(x) of functional application. The basic intuition of argument-value functional correspondence is embodied in: 1.4. AXIOM.

(AZ.~Z~)(x) =

--x--.

Strictly speaking 1.4 is an axiom schema, because any 'well-formed' expression can go in for the ~-part we have indicated schematically. Just which the well-formed expressions actually are, we leave vague at the moment. At least, at first glance, we can guess that we mean to allow all those composed out of A, ::>, true, false, and variables with the aid of functional application. But care must be exercised, since such a type-free naive function theory might just be inconsistent. It is not, and it is one of the main aims of this essay to give intuitive reasons why not. Axioms for equality (=) have not been explicitly stated: equality is assumed to be a common notion with the usual properties. Axiom 1.4 is often taken as the second axiom of conversion (,B-conversion). The first axiom allows for rewriting of bound variables:

AZ· (~ Z

~) =

AW'

(~

w ~),

with the usual side conditions on avoiding the clash of free and bound variables (e-conversion). This is a 'grammatical' matter not specific to the

164

D. SCOTT

ideas under discussion, and we again assume all this as part of our common understanding.l? On the borderline between the general and the specific is the principle of extensionality: 1.5. AXIOM. If for all z,

~

z

~~

= oo.

Z ••• ,

then

AZ' (~~ z e--) = AZ' (oo. zoo.).

We give it a number for emphasis. What is without question a special assumption is the principle offunctionality: 1.6. AXIOM. f = AX ·f(x)

This axiom is often called the axiom of 'I7-conversionY Both 1.5 and 1.6 could be combined into a single statement: f = g whenever f(x) = g(x)for all x.

This is the usual version of the axiom of extensionality, but it seems better to separate out 1.5, which has a more general logical character, from 1.6, which is quite specific to our concept of function. Supposing that we are able to carry through with a uniform type-freeness, then the import of Axiom 1.6 is that everything is a function (f is a free variable). Such austerity! and purity! No other objects but functions are to be found in our universe. But then how neat things will be. It is very tempting to try to make such an arrangement. The only trouble is that intuition may find itself at a loss to answer: what function is the truth value true? But there is no need to be at a loss: 1.7. AXIOM. (i)

true

=

Ax.true;

(ii) false = Ax· false. Said otherwise, the truth values are each identical with their own constant functions. (Given a, the function with constant value a is Ax·a.) The justification? Again the answer is why not? Any number of 'atomic' objects could be accommodated in this way. It is a very simple method of preserving the purity of 1.6. I know of other plans for making identifications, but this seems the simplest and most elegant.P See the discussion in CURRY and FEYS, op. cit., especially Chapter 3, and on p. 90. CURRY and FEYS, op, cit., p. 92. 12 The identification of atoms with their own constant functions is similar to Quine's idea in New Foundations that an individual can be identified with its own unit set. 10 11

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We must make a digression at this point. The classical versions of the A-calculus opt for even greater purity: No atomic elements at all. The truth values and pairs are introduced by: true = AX· Ay·X; false = Ax'AY'Y;

(x, y) = AZ'Z(X)(Y),

These are quite pleasant definitions and I have often enjoyed working with them.P I have recently found that they possess one fault, however: they cause these very basic notions to be inextricably tied up with the idea of function (the type of inextricability can be made precise). The truth values and the conditional (or pair) are more primitive, I feel. In any case, we are presently engaged in an analysis of the (iterative and self-applicative) notion of function; we have long ago come to terms with the truth values. Thus, on conceptual grounds, I prefer not to define them in this functional manner. The construal of pairs as functions and the identifications of truth values with constant functions are quite harmless evocations of functionality. The danger comes in 1.4 as we shall now see. Recalling Church's generosity with self-application quoted in the introduction, let us consider an example often used for illustration:

\7 = AX,X(x) (x). In a straightforward and formal manner, we allow 'x(x)(x)' as a wellformed expression. The problem is to evaluate \7 (V). By formal application of 1.4 we find: \7(V) = V(V)(\7) = V(\7)(\7)(V) =

....

If 1.4 is the only method of evaluation, then \7(V) cannot be reduced to a 'meaningful' normal form. Church seems to have felt that the value should be 'undefined'. I have come to agree, but for a totally different reason. The undefinedness of V(V) will be provable as a theorem. It will not be the case that any two expressions without normal forms are to be put equal; of course, Church did not think so either. If the reader wants, he may say that my method gives a way of making precise discriminations between degrees of undefinedness. The value of V(V), as it turns out in my suggested interpretation, is undefined in the worst possible way. 13

They are used also to good effect in

GOODMAN,

op. cit.

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D. SCOTT

We objectified truth values, because we wanted to 'see' them and to think about them-and because we felt there was something definite about them. The undefined is not often objectified, because there is so little that is definite about it. Indeed it might at first seem that it is almost self-contradictory to objectify it. Let us investigate the matter. A symbol is needed. I have chosen this symbol: 1-. When we write: f(x) = 1-,

we mean that the function f is (fully) undefined at the argument x. Eventually we shall want to prove that V(V) = 1.., so that 1- becomes definable in the calculus. For the present we prefer to regard 1- as a new primitive notion. If 1- is to mean total undefinedness, then degrees of undefinedness can be sorted out with the aid of this definition: 1.8.

DEFINITION. X ~

y iff whenever feY) = 1-, then f(x) = 1-.

The idea is that y is better as regards definedness in case the total undefinedness for any function at y always implies the same for x. This definition provides a comparison in more than degree: if x ~ y, then y is a better version of x, or, as we might say, x approximates y. This can be made more precise: 1.9.

PROPOSITION.

(ii) x (iii) x (iv)f

(i) x

~

x;

y and y ~ z imply x ~ z; ~ y implies f(x) ~ fey); ~ g implies f(x) ~ g(x). ~

PROOF: The first two properties are immediate from the definition. To prove (iii), assume that x ~ y and g(f(y») = 1-. By 1.4 we can find a function h = AZ.g([(z», so that hey) = 1-. But then hex) = 1.., and thus g(f(x» = 1-. For (iv), assume f ~ g. This time define h = AZ' z(x). By (iii), h(f) ~ h(g); then by 1.4 we findf(x) ~ g(x). 0 It will be noted that we have made heavy use of Axiom 1.4. The reason is that ~ has been defined in 1.8 by quantification over the (unknown) domain of functions. The alternative would have been to assume all of 1.9 axiomatically; however, 1.8 is so simple, it was not really worth the effort. Even so, some additional properties of ~ will have to be axiomatic.

1.10. (ii) x

~

AXIOM.

(i) 1-

y and y

~

~

x;

x imply x = y.

TYPE-FREE CALCULI

1.11. AXIOM. f

~

g if for all x, f(x)

~

167

g(x).

The effect of these assumptions is that our universe of objects is partially ordered by ~ in such a way that the operation of functional application f(x) is monotonic in bothf and x. Furthermore, the partial ordering makes the ordering among the functions come out argumentwise.i" Clearly 1.11 renders 1.5 and 1.6 superfluous. We stated 1.5 and 1.6 because they were much more intuitive; 1.11 is a very strong assumption. In this type-free theory, functions are also functionals (that is, operators on other functions). As a consequence of 1.11 we would have h(f) ~ h(g) provided only that f(x) ~ g(x) holds for all x. In a way this means that, in the 'calculation' of h{f), the function f enters only through its function values. This is an intuitive assessment, but I think the remark gives the right impression. Some examples of how the partial ordering works out should make the situation clearer. The element .1 is an object without 'content'. It might be better to say it carries no information-other than its being an element of our universe. Now .1 ~ Ax. .l holds by 1.10 (i). Thus ..1 (x) ~ ..1 by 1.9 (iv). It follows at once that .l = Ax'..1; in other words, as a function, ..1 is the constantly totally undefined function. A function that is not totally undefined is the function (true, false) which takes on at least two distinct values. Using functions like this we can prove: 1.12. PROPOSITION. (i) (ii)

true $ false;

false $ true.

PROOF: Suppose true

~

false, then

true = (true, .l) (true)

~

(true, .i) (false) = ..1;

false = (false, .l) (true)

~

(false, ..1) (false) = ..1.

Thus true = ..1 = false, which contradicts 1.1. Part (ii) is proved similarly. 0 Note that by monotonicity we have the relationships a ~ c and b ~ c, where a = (true, .l); b = (.1, false); c = (true, false).

Clearly any arbitrary finite partial ordering could be isomorphically represented using iterated ordered pairs. In an expression like that for a above, 14 This idea of treating partial functions via monotonicity is taken from R. New foundations for recursion theory, Stanford Thesis, 1964.

PLATEK,

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D. SCOTr

the 1- -part can be regarded as an incompleteness which is open to 'improvement'. And indeed a was improved to c. It is not the case that b is an improvement of a, however, even though they both contain about the same 'amount' of information. The elements a and b are in fact incomparable, just as true and false are. The point of this example was to show in a simple context why the relationship a ~ c is qualitative rather than quantitative. It has more the nature of a part whole relationship, because in a sense the information content of a is 'included' in that of c. The possibility of being able to 'improve' information, should suggest the questions of whether improvement can be iterated and whether there are limits to improvement. Again examples with ordered pairs suffice to make the point. Let ao = 1- and proceed inductively:

an+ 1 = (true, an). Inasmuch as ao ~ a 1 , we can argue that an ~ an+! for all n my monotonicity. Could there be a 'limit' to this sequence. Since ~ is a partial ordering, and since the sequence is monotonically increasing, it would be natural to suppose that a limit would be a least upper bound in the sense of the partial ordering:

If this element existed, then in some sense it would satisfy an 'infinite' equation: a oo = (true, (true, (true, ... Indeed it would be tempting to say that:

»)).

Oa,

= (true, oa,).

But before we can justify such an equation, a further analysis of the notions is required. First, we must ask about least upper bounds. What is the meaning of this construct intuitively? Think of the 'partial' ordered pairs 0 and b we had before. Each had a different 'bit' of information the other lacked. Joining them together gives the pair c as a 'whole'. We would like to write: c = 0 U b. In this case we already knew the desired join c, whereas in the other example was something 'new' that had to be found. Thinking of parts and wholes, there would seem to be no reason why we could not join any number of objects into a whole. Why not try? Trying means assuming a new axiom

Oa,

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we could call the principle of completeness. (As usual an axiom is justified by explicit reference to a few outstanding particular cases.) 1.13. AxIOM. Every collection of elements has a least upper bound in the sense of the partial ordering.

The empty collection

°

is included and we could just as well define:

.1 = U0, where U is the least-upper-bound symbol. (The small symbol U is used for a finite number of terms.) This approach makes Axiom 1.10 (i) superfluous. Axiom 1.11 has an interesting consequence with regard to functions: 1.14.

PROPOSITION.

/U g = ;"x.(J(x) U g(x»).

PROOF: The function on the right (note that we are assuming that it exists) is clearly an upperbound to f and g in view of 1.11. Suppose h is another. Then f(x) £ hex) and g(x) £ hex). Thus f(x) U g(x) £ hex) holds for all x. By 1.1 again, we have Ax·f(x) U g(x) ~ h. 0 This simple result shows once more how functions are 'determined' by their argument-value pairs. Here the 'join' of the information contained in two functions is accomplished in a completely argumentwise fashion. The proposition can be generalized to any number of functions even an infinite collection. A curious question is that of calculating the join of true and false. We might even worry whether these two elements should even possess a join. The a, b, c-example with ordered pairs shows that certain joins exist. We have uncritically assumed in AXiom 1.13 that all collections of elements have a join. The trouble is that we are not familiar with the idea of the join of {true, false}. An even worse example would be the join of all elements. This 'total' join (assuming we can come to accept it) will be denoted by T. It satisfies the characteristic inequality: x£

T.

Returning now to the truth values, if they are so very separate (even: contradictory), their join must be very large. In fact, why not assume that they are as separate as possible? In order to be able to formulate the full force of the principle of separateness, we require the notion of the meet or greatest lower bound of elements. The symbols to be used are nand n. The definition can be given in terms of U:

nX = U{y:y £

x for all x e

Ks.

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D. SCOTT

A partial ordering with U is a complete lattice, and this reduction of n to U, as is well known, is valid in all complete lattices.P 1.15. AxIOM. (i) true U false = T; (ii) true n false =..L. A good answer to one question usually provokes another query. Now that we are thinking of the relation of T and ..L to the truth values, we must return to a consideration of the conditional. There are different ways to select the assumptions, but I find a certain simplicity and regularity in the following: 1.16.

AxIOM.

Let w = (z

If true £ If true $ (iii) If true $ (iv) If true £ (i)

(ii)

=:l X,

y).

z and false $ z, then w = x; z and false £ z, then w = y;

z and false $ z, then w = 1.; z and false £ z, then w = xU y. The reason I like this choice of convention for =:l is by using the values indicated, the function Az· (z =:l x, y) is the minimal extension to the domain of all objects of the natural function on {true, false} defined by Axiom 1.2. (Note that 1.2 is included now as a special case of 1.16 if we grant 1.12.) In making the extension we have chosen the smallest values possible (in the sense of the partial ordering £) which render the function monotonic. Other conventions are possible over those of 1.16. For example, clause 1.16 (iv) might seem more reasonable (or at least: more strict) if we were to put w = T. Call the strict function : =:l. It can be defined in terms of ~, however: (z:=:l x, y) = (z =:l (z =:l x, T), y). Thus it seems quite sufficient to allow => as the more fundamental primitive. And =:l has several useful properties. 1.17.

PROPOSITION.

Let

f = (true, false), then

j(z) = j(f(z» £ z holds for all z. 0 The proof need not detain us since it is an easy argument by cases supplied by the clauses of 1.16. Note that the functionjof 1.17 isjour valued, because in view of 1.15 and 1.16 the values will fall in {..L, true, false, T}. Indeed

l'

The finite inf is of course definable in terms of the general one.

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these four values form a lattice among themselves. The property of the function j displayed in 1.17 is usefully called being a projection. In words, we are assuming in our axioms that the four-valued lattice generated by the truth values is a projection of the whole universe. Another example of a projection, which this time projects the universe onto a quite large subuniverse, is definable in terms of variable ordered pairs. 1.18.

PROPOSITION.

Let F = Af·(f(true), j (falsej],

then this junction is a projection. PROOF: It is obvious from the properties of pairs that F(f) = F(F(f)). What remains is to prove that: (z ::::l j(true),fCfalse)) r;; j(z). This follows by monotonicity ofjby checking out the four cases of 1.16. 0 These two results give only the barest indication of the variety of subspaces available and of their relationships. Note that the range of the projection of 1.18 is the whole collection of ordered pairs of elements of the universe. Axioms 1.15 and 1.16 express in different ways the separation one from the other of the two truth values, true and false. It is possible to insist on an even greater separation of these two elements considering them in relation to all the other elements. Thus in view of 1.18 we could define:

to t"+1

= ..i, = (true,

ttl),

and we would have: to r;;

t 1 r;; ... r;; tPl r;; ... r;;

true.

Would it be conceivable that 'in the limit', so to speak, the equation: 00

true = U ttl "=0

could hold? In this case the answer is no, because each t,,(..i) = ..i; while this is not the case for the (constant) function true. But could there be other partial functions, infinite in number, which when joined together gave true? A more difficult example of a strictly increasing sequence of elements below true is defined by: Uo

U"+l

= .L, = Ax· (true, u,,),

172

D. SC01T

Here we must go to Un(jJ(JJ = 1.. before we see that the limit formula is impossible. But these are special cases. We shall assume for full isolation that no infinite limit can reach the truth values. This is somehow reasonable, because they are assumed as primitive and are not constructed from other parts. The resulting axiom may be called a principle of continuity: 1.19. AxIOM. Let t = true or t = false. Then if X is any set of elements and t !;; U X, then t !;; U X O , for some finite subset X o ~ X. It may seem strange to use the word continuity in connection with 1.19, but the usage is quite justified.!" For a given truth value t, the predicate t !;; x can be considered as a two-valued predicate of x. But the two values are separated, and what we have assumed in 1.19 is that this discrete-valued predicate is continuous-in a suitable sense. Let us try to make this sense more precise. The U-operation works as a kind of limit operation in this space. This is especially intuitive in forming the U of an increasing sequence of elements. More generally we can consider as limits the U of directed sets of elements. (A set X is directed iff for all finite X o ~ X, there exists an y e X with UX o !;; y, Thus a directed set is nonempty .) To be continuous means to preserve limits; formally:

1.20. DEFINITION. A function f is continuous iff for all directed sets X f(~ X) =

U {I(x): x eX}.

We note that since functions are monotonic, the set on the right in 1.20 is also directed. Now what we have assumed about the truth values is that the two-valued monotonic functions:

(T, 1..)

and

(1.., T)

are both continuous. And so are such functions as (x, y) for any given

x and y. Indeed these particular finite-valued functions f have a stronger property: for any set X there is a finite subset X, !;; X such that

ft.J X)

=

feU X o).

Such functions will be called discrete. There are a large number of discrete and finite-valued functions. 1.21. DEFINITION. (i) 10 = (true, false); (ll)

In+1 = Af· Ax.ln(f('n (x»);

(iii)

In = Af·f

16 A complete discussion is given in my paper Continuous lattices, in Toposes, Algebraic Geometry and Logic, Springer Lecture Notes in Mathematics, vol. 274, 1972, pp. 97-136.

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1.22. PROPOSITION. The functions I" form a strictly increasing sequence of projections below the identity function I. They are discrete and finite valued, and the elements of their ranges are hereditarily so. 0 The proof is straightforward from the definitions; the hereditary property means that each element of the range of the function has the same property. In the first instance the facts about 10 are immediate; the formula 1.21 (ii) then makes an inductive argument possible. Thus the range of 10 is 4-valued and that of 11 is 36-valued, the latter being in a one-one correspondence with all t.i. true, false, T }-valued monotonic functions defined on that same 4-valued lattice. These 36 functions again form a lattice, the diagram of which is shown in the figure. (Note that the bilateral symmetry corresponds to the interchange of the roles of true and false.) The diagram for the range of 12 would be quite impossible to draw, but all its elements are explicitly definable by very elementary expressions in our language. The same is true of all the I" for n > 2. The objects in the ranges of the I" are all finite, discrete-better: isolated, in the sense applied to true and false in 1.19. Moreover, they are all expressible. Should there be any other such elements? Maybe. If one wants them, they are conceivable. But-for logic-they are not at all necessary. Certainly for logic we need the true and the false, and, through our analysis, also ..L and T. If logic needs functions (and it does) the other elements follow. Note however in our presentation that 'function' means partial function, and, in the finite domain, this also means monotonic function. Note too that for the sake of simplicity and elegance, functions of one kind can also be of another kind: the value true is at the same time the constant function true. This systematic accumulation (the range of 1"+1 contains the range of III) allows for a move that is not merely one of elegance: namely, the passage to the limit. For it is the case that though there are only a countable number of finite functions in the (combined) ranges of the I", there are a continuum number of possible limits of these functions. We have already seen a few examples. Now the final question is: how much further shall we go? Are there other functions beyond these? Answer: Maybe. But do we need them? Answer: No. The space of limit points is quite big enough-for a great deal of logic. (This remark will be made more precise later.) So then we must seek a way of limiting ourselves to this quite definite totality. As in classical set theory we need a principle offoundation (axiom of foundation):

174

1.23.

D. SCOTT 00

AXIOM.

I = U In. n=O

This is the most obvious and direct form of the axiom; but by utilizing some of its consequences it can be put in a more 'primitive' form. Note first that

follows for all x; hence, as was desired, each element is the limit of finite objects. Furthermore, if x is finite (isolated), then x = In(x) for a suitable choice of n, Thus we see that the finite elements are just those in the ranges of the In. Next for an arbitrary x, since each In(x) is a continuous function, the limit is also. Therefore, all functions of our universe are continuous. This is the key fact to understanding the whole approach. We could have T

T T

T/T~T

TfT

TtT

T/~~T/l~T Tff tt T Ttt

TVXL\:'ZNT ffT

T

T

/\l " " "~/\ \/\ 7\/ fff

fff

fT

Tf

tT

Tt

v,

fT f

fff

ttt

fT L

LTf

tTL

LTt

ttt

ttt

L~NT!XX!L t,

Lf

LL

ttL

J.tt

L~r/L~r/L LfL

LtL

L~L/L L

L

L

The range of In.

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taken this as an axiom, but it seems better to 'discover' it through contemplation of what would happen by assuming that all elements are limits of finite elements. If we assumed continuity, however, the form 1.23 could take would be more elementary:

In l;; F whenever 10 U A.I 'Ax' F(F (f(x))) l;; F. In other words: 1 is the least fixed point of a certain functional operator. From the assumption that all functions are continuous (an assumption which would also imply 1.19), it would follow that the equation of 1.23 holds. This completes our list of axioms. As presented the axioms are categorical. That all models of this (second-order) theory are isomorphic should be clear by now. What may not be so obvious is the existence of one model. The question construed in this mathematical way will be discussed in Section 3. It will be seen that there are many similar models of closely related theories. The purpose of the informal discussion of this section was to show that it is reasonable to conceive of something of this nature to believe in. The space we have been investigating has turned out to be far more definite than would have been imagined at first-and I hope it can also be regarded as natural. But these are value judgments, it is the 'canonical' character of this space that cannot be denied. In this respect it is much like the real numbers (Euclidean space) or Baire space or Cantor space. Indeed I propose to call it logical space, and after a discussion of some necessary model-theoretic niceties, I shall attempt to argue that this is not a misnomer. 2. Summary

As the last section was so long, I propose here to give a quick review of the axioms in a somewhat different order. (The order of Section 1 was necessary, though, for a motivated analysis of the notions.) What turned out was that Logical Space had all along a certain partial ordering l;; • One basic circle of assumptions can be summarized in: (I) Logical space is a nontrivial complete lattice under the partial ordering.

We have put (I) first here because it is easier to say. What started us off, however, was the concept of function. Looked at very abstractly, Logical Space is a function algebra structured by the binary operation of functional application and the reciprocal operator of functional abstraction. We may say that:

176

D. SCOTT

(II) In logical space application and abstraction satisfy the usual laws of conversion and are moreover related to the partial ordering in that the ordering between functions is argumentwise.

The various monotonic laws are consequences. A certain vagueness resides in the question of what can occur within the scope of an abstraction. For the time being we can restrict attention to expressions manufactured from variables, ..L T, u, n, and certain constants by application and abstraction. This restriction will be broadened later. Next we turn to the truth values: (III) Within logical space the true and the false form, together with ..1 and T, a four-element sublattice of isolated elements which are at the same time identical with their own constant functions. It should be remarked that one of the consequences of foundation (Axiom 1.23) is that these four values are the only elements satisfying the equation a = Ax·a. Having exposed the truth values, the conditional function ;:) puts them to work: (IV) On logical space the conditional permits a continuous projection of arguments onto the four element sublattice and then an arbitrary assignment of values corresponding to the two truth values.

Finally the initial projection to the truth values is but the first step of a hierarchy: (V) The whole of logical space is generated by limits of the finite elements which are obtained from the truth values by considering truth-valuedfunctions and then functions with these values,etc.• arranged as the values of an increasing sequence of projections. As a consequence we found that all functions in logical space are continuous. And this answers the query about what is permitted in the scope of the abstraction operator: any expression defining a function continuous in the bound variable determines an element of logical space. The principles summarized in (I)-(V) describe logical space as a space of partial objects. The extremes of undefinedness and overdefinedness (..1. and T) have been objectified so that they may enter in an extensional way into the construction of other objects. These combinations are rendered coherent through the partial ordering [::::;;. The variety of combination is provided by the truth-values, the conditional, and the functional combina-

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tions, which may also be formed into limits. These are operations of a logical nature, and these are all that are permitted. Hence the space can contain only logical elements: it is logical space. 3. Construction

The explicit details of a mathematical construction of models for the theory of logical space have been published elsewhere."? That presentation, however, is rather severe in style, so it may be well to review here the major points. The ideas of continuity and limits figure heavily in the investigations; thus, it is not surprising that a discussion of topology emerges. After all, it is the same story in Euclidean space whether one begins with elementary geometry or elementary algebra-and combinatory 'logic' for the most part is the elementary part of the logical algebra of functions. So I look on this work as an inevitable development and only wonder why it took so long to become clear. A topology can be introduced into any complete lattice. One defines open sets as being those sets U satisfying these two conditions: (i) whenever x E U and x !;; y, then y E U; (ii) whenever X s;; U is directed, and X E U, then X n U =I: 0. The continuous functions in this topology are exactly the continuous functions satisfying the condition of Definition 1.20. The complete lattice becomes a To-space, because each point is uniquely determined by its system of neighborhoods. Let us consider for a moment Ta-spaces and their continuous functions in general. Let 0 be a fixed space (like logical space) in which functions can take values. Suppose X and Yare any two Ta-spaces, where X s;; Y as a subspace (in the technical sense of a topological subspace). Consider this diagram: X £; Y ,/ " f"'./

'" o 1

Here f: X - D is a given continuous D-valued function. From the point of view of the superspace Y, the function f is only partially defined. We could leave it at that, or we could ask whether the value space D has some 'partial' elements sufficient in number to allow for an extension of f to 17

See the paper referenced in note 16.

D. SCO'IT

178

be made to a continuous f: Y -+ D. Indeed, we may ask whether D can be obtained so that this kind of extension is possible no matter whatf: X -+ D and Y may be. What I discovered as a consequence of my work on logical space is that there exists an enormous variety of such D. Aside from the trivial one-element space, the two-element complete lattice LL T} (with its appropriate topology) is the most immediate example. Any finite lattice is also of this kind, but not every infinite lattice. (A complete and atomless Boolean algebra is unsuitable for this purpose, for example.) I was surprised and pleased to find that any such D with the extension property must be a complete lattice with its lattice topology. The extra condition that the lattice must satisfy is one that makes an intimate connection between the lattice structure and the topology: (iii) y = U: YEU}. This equation must hold for all Y E D, where U ranges over all open sets in the sense of (i) and (ii) above. Such lattices I call continuous lattices, and their mathematical theory is highly satisfactory. The study of these lattices is, for me, totally motivated by the desire to found a coherent theory of partial functions (in extension). I feel that I am only doing what is necessary to make things work out properly. As further evidence that the program is progressing well, I found that I could prove that if {Xj : j Ecf} is any given system of To-spaces (including ordinary sets as discrete spaces, note), then there exists a suitable continuous lattice D such that all the X j can be embedded in D as subspaces. (Supposing the X j to be disjoint, then we could make all Xj s;; D.) Thus by virtue of the extension property of D, if f: x, -+ Xj in continuous (which is no restriction in the case of discrete spaces), we may regardfas being a continuous map into D itself, and then a continuous extension

u {n

1: D-+ D

becomes possible. The space D truly is a universe: on that one single space, its homogeneously defined continuous functions can be restricted to produce arbitrary continuous functions on subspaces. We are thus well on our way to a type-jree, suitably general theory of partial functions. (Certain other steps must be taken though, before the type-freeness allows complete freedom of functional combination.) Topological spaces are all very well, but we should not become involved in purely topological considerations for their own sake. What is desired

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is an application of ideas from a familiar and well-developed mathematical theory to a problem of logical analysis. The set problem is to understand partial functions (especially of higher or even type-free types). In particular, if topology is going to help us, we should stop to think whether spaces of functions possess interesting topologies. Such questions have been widely discussed in mathematics and can involve deep results. It is very fortunate that in the present context we need invoke only the easiest notions contained in all the standard text books. This is not to say that future developments will never require more sophisticated techniques, but it is a relief to be able to produce the basic definitions in short order. As we noted, there is a class of richly endowed spaces D, the continuous lattices, which are at the same time To·spaces and complete lattices which enjoy a close connection between the lattice structure and the topology. In suitably chosen such space, part of the endowment consists in the variety of continuous functions f: D - D that are possible. An obvious question, then, is to look at the space [0 - D] of all such continuous functions. But which among many is the appropriate topology? There are so many functionspace topologies that have been proposed. What we find is that it is the simplest one that works, namely: the product topology (often called the topology of pointwise convergence). The usefulness of this topology here is due, no doubt, to the assumption of the completeness properties of D which distinguish it as a continuous lattice. That assumption, relatively speaking, requires of 0 an extensive filling in of gaps. (Naturally, in the finite case, the discrete features of the partial ordering should not be regarded as gaps. In fact, the consecutive intervals are significant jumps serving to isolate the various elements one from another. These isolated elements may similarly exist in infinite lattices.) By so structuring the space of elements, all necessary discriminations between functions can be provided by pointwise properties. (We indicated this in the discussion around Axiom 1.11).Technically speaking, we justify the approach by showing that in general if 0 and D' are two continuous lattices, then so is [D - D'] when given the product topology (= the topology associated with the pointwise partial ordering). Furthermore the construct [D - Of] can be shown to have all the right abstract 'Hom'-properties within the category of continuous lattices and continuous functions. Less abstractly, one can conveniently analyze the neighborhood structure of [D - D'] in terms of the neighborhoods in D and D' and the idea of approximations to partial functions.!" 18

See Continuous lattices, § 3, pp. 111-113.

180

D. SCOTT

Having gained some insight into function spaces, we can restate the problem of finding 'type-free' spaces: Is it possible to find nontrivial spaces D which can be identified (latticetheoretically and topologically) with their function spaces [D --+ D]? If we can make the D sufficiently large continuous lattices, the extension property together with the possibility of self-application shows that a fully type-free system is possible. (Self-application comes about since under the identification an element may at the same time be considered an argument and a function.) Among ordinary spaces, as usually studied in topology books, it does not seem possible to make D = [D --+ D]. However, as I found out in a somewhat indirect way, there are many among To-spaces, an underdeveloped part of topology. Indeed, any given system of (ordinary) spaces can be embedded in such a space. The method is one of passage to the limit as suggested in Section 1. There, please note, we simply assumed that the desired limits existed using the argument that there was no need to exclude objects that seemed to fit in well. That level of analysis is shallow and in risk of producing an inconsistency. Here we now outline the construction which assures consistency and possibly even naturalness. As we already remarked, mere embedding in a continuous lattice, say Do, is cheap. Let D1 = [Do --+ Do],

with the pointwise topology (lattice structure). Is there any chance that Do and D1 can be identified (that is made homeomorphic as spaces and isomorphic as lattices)? It is, in general, very unlikely. But note that Do can be very naturally embedded in D1 : let to each x E Do correspond the constant function with value x, an element of D 1 • This provides a mapping that is a continuous subspace embedding. The map io possesses a partial inverse jo: 0 1

--+

Do,

which is uniquely determined by the two equations: jo(io(X» = x,

io(j(x'»)

~

x',

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181

for all x E 0 0 and x' E 0 1 , In fact, it turns out that jo(x') = X'C.U, the minimum value of the (monotonic) function x', This can be put in words: each constant can be regarded as a function; but which constant most nearly approximates an arbitrary function? Answer: the minimum value which for monotonic functions can be thought of as the total variation of the function. (This only makes sense if one remembers to regard the ~ -relation in 0 0 as an approximation relation.) And the next step? Easy! Inductively define Dn+1 = [On ~ Onl· And embed On into Dn+!? Yes, but with care. (I consider this the most original step in my construction: once I had this straight all else was forced.) We were free at first to construe the elements of 0 0 as being constant. That gave us an embedding into 0 1 , But now when we come to O 2 we are not allowed to be so free with 0 1, which after all is the function space over 0 0 , We should think of the problem as the embedding of [0 0

~

Ool into [0 1 ~ Oil

while keeping in mind the embedding (already fixed) of 0 0 into 0 1, A diagram should make matters clearer: to

00~01

f

-I-

~: -1-1'

00~01 JO

There are obviously two possible functional relations between an f an f':

I' = io of -I«. f = jo 01' io · 0

The first relationship defines the mapping: and the second:

i, :0 2

~

01,

Shortly said: if one knows the approximations between elements (of different spaces), then one knows the corresponding approximations between functions. This procedure can at once be iterated to define in+ 1 andjn+1 in terms of in andj,; Each in embeds On into On+! andj, projects On+1 onto On'

182

D. SCO'IT

We can if we like, identify x E 0" with i,,(x) E 0,,+1' For x' E 0,,+1, we can call j,,(x') E 0" as the best approximation of the higher-type object by a lower-type object. The compositions i, 0 ill and ill 0 j" behave as in the case n = 0 already discussed making the notions of approximation precise. This possibility of shifting types was the main clue to the solution of the problem of self-application. The (higher) types of functions can get very mixed. We have chosen but a selection in the 0". What we note, however, is that this selection is in some good sense cofinal in all the finite types. Indeed 0" is already embedded in On+1 and hence in all Om for m > n. And the later ones project down onto the earlier ones. The types have thus been made cumulative and along the way all mixed types can easily be picked up. (The passage from Russell type theory to Zermelo's cumulative types is very analogous-but it is easier and less messy to do with sets.) The accumulation would not have been possible without the aid of the approximation relation ~. In this respect partial functions are shown to be more convenient than total functions of the ordinary kind. It remains now to pass to the limit and make precise the concept of infinite type. Suppose a selection x; E On has been made for n = 0, 1,2, ... and formed into an infinite sequence:

If the x; have been chosen so as to 'fit together', then we can regard this sequence itself as the limit. But what does it mean, 'fitting together'? For one thing we could ask that the sequence be increasing-except the x; belong to different domains. But this problem is solved by shifting types; to be increasing means just this:

for all n. This is useful to note, but obviously many different increasing sequences can have the same limit. The question is whether we can easily distinguish one special sequence out of all the equivalent ones to be the limit. The answer is to take the maximal one where each X n is a best approximation. That sounds fine if it can be precisely formulated. Recalling that the projections i, give best approximations, we can try restricting to those sequences x where at each stage

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This is stronger than simply being increasing. The totality of all sequences satisfying condition (*) forms, by construction the infinite type space D",. It can be proved to be a continuous lattice comprehending all the Dn. Its lattice structure for x, y ED", is given by the relation: for all n. Further the mappingj",n:D

-+

Dn, where

j",n(x) = Xn,

for xED"" proves to be the correct projection inverse to the obvious embedding of Dn into D",. Indeed there are embed dings and projections imn and jnm for all pairs n, m with 0 ~ m ~ n ~ 00. (Technically: in the category of continuous lattices and continuous maps the space D", is both the direct and inverse limit of the sequence of spaces (Dn>':=o with respect to the two systems (in>':=o and (jn>':=o of connecting maps.) That D", somehow exists as a lattice is not so strange; what is pleasant is its function algebra. Remember that, by design, Dn+ 1 is the function space [D, -+ Dn] over D,; Thus, for x, y ED"" it always makes sense to write X n+ 1 (Yn)

since this belongs to Dn • These elements form a monotonic sequence but may fail to satisfy condition (*). To define z = x(y) in D", we must write: Zn

=

'" }mn(xm+ 1(ym)). LJ

m-en

This formula makes z automatically best. In this way we extend ordinary functional application on pairs in Dn + 1 x Dn to an operation or D",x D",. Here is the place where functional application may at last become selfapplication-if you so desire it. There are clearly many details to check out, but I hope the idea is clear. The final step is to worry about [D", -+ D",]. Must we call this D"'+l and go to higher ordinals? No, for the simple reason that all functions we employ are continuous. Thus suppose f: D", -+ D", is continuous. Then because all our embeddings and projections are continuous, we can define a sequence u ED", where:

184

D. SCOTT

We can calculate out that for all x

f(x)

E

DOll:

= u(x),

where on the left we have ordinary (if you like: set-theoretical) functional application defined on [Doo --. Doo] x Doo , while on the right we have the newly defined operation on Doo x Doo • That is to say, all the functions (continuous!) in DOO + 1 are already perfectly represented in D,,: the two spaces are isomorphic! Functional application has been made type-free, continuous, and comprehensive. In this discussion the initial space Do was any continuous lattice. In the discussion of Sections 1 and 2 we wanted to be more specific. To obtain the desired specialization we have only to take Do as the four-element lattice LL true, false, T}, and the corresponding Doo is a model of the axioms set out above. This gives a very explicit construction of logical space. It seems very special since Do can be arbitrary, but it is not. Suppose that D is an arbitrary continuous lattice with a countable basis for its topology. I conjectured that D could be found as a retract of logical space, and this conjecture was quickly proved.!? Thus logical space is again in another way universal. Even better there is a whole calculus of retracts that allows one to find them (and the corresponding subspaces) in an effective manner. This requires a bit of explanation.t? Let a be a retract of logical space. Its range is a subspace, but we may as well regard it as a (continuous) map of logical space into itself. Therefore we may as well regard it as an element of logical space. The characteristic of a retract is that: a

0

a = a.

Now composition (better: self-composition) is a continuous map, so the retracts of logical space can be thought of as the set of fixed points of a continuous map. Now in general for any complete lattice, the fixed points of a (monotonic) function form a complete lattice. So we may say that the retracts of logical space form a complete lattice. This highly abstract discussion is of interest because we can show that the retracts are closed under so many continuous functions. Here are some examples: Given retracts a and b, define: 19

20

This has been established independently by J. Reynolds and Tang. Retracts are also discussed in Continuous lattices.

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18~

a x b = AU'(a(u, (true», b(u(false»)

a-s-b a~b

=

AU, (u(true):::::l (true, a(u, (false»), false, b(u(false»)

= Au·A,x·b (u(a (x»).

Then each of ax b, a + b, a ~ b are again retracts and these operations are continuous (and definable). What is the significance? Let the range space of a retraction be denoted by: D(a)

=

{x:a(x)

=

x},

which in the case of retractions is always a continuous lattice. Then we can prove that D(a x b) is always isomorphic with the usual Cartesian product of D(a) with D(b). Next, D(a+b) is isomorphic with the least continuous lattice containing the disjoint sum of the two lattices D(a) and D(b). Finally, D(a

~

b)

is isomorphic with the function space [D(a) ~ D(b)]. Thus within logical space we have found 'internal' operations corresponding to many important constructions. And the constructs are continuous. Every continuous function has a least fixed point. Recall the retract /0 (onto Do) and find the least fixed point such that: a

=

/o+(axa)+(a -+ a).

On general principles this can be shown again to be a retract. Now D(a) in this case must be an interesting space. Well, up to isomorphism, we can say that every element of D(a) is either a truth value or a pair or a functionand these cases can be distinguished. Note that the components (coordinates or argument-values) are again from the same space D(a). This means that D(a) is a model for a whole new type-free theory. By choosing combinations of retracts in this way we can construct all sorts of models-explicitly by taking advantage of the universal character of logical space. Once logical space has been constructed, these other constructions become almost automatic.P! 21 The use of the calculus of retracts also allows for proofs of properties of the subspaces to be given formally from the axioms for logical space.

186

D. SCOTI'

Conclusion In Sections 1 and 2 we spoke informally about the axioms for logical space. In Section 3 a construction was supplied proving the consistency of the axioms. This model-theoretic consistency proof applies to the original A-calculus of Church and Curry in particular, but the axioms we have given are much stronger than any usually considered in combinatory logic. This added strength suggests a change of terminology. Though the words 'calculus'and 'Iogic'do have general significance,I would propose calling the systems of Church and Curry A-algebra (or if you like: combinatory algebra). This is in analogy to classical algebra. Their theories are equational theories of (type-free) functions in combination: the algebra offunctions-wthether formulated with A-abstraction or with the so-called combinators. (True, it is a branch oflogic like Boolean algebra; but we can usefully apply mathematical techniques.) What I have done is to introduce something new: limits and topology. Therefore, in analogy with classical mathematics, I would like to call the extended theory A-calculus. Any system of rules can be called a calculus, if you like; but analysis (differential and integral calculus) only took wing from the starting point of algebra after the notion of limit was introduced. This sounds egotistical and is in a way, because I have not discovered anything quite as useful as the integral. But I am not quite mad, as I can show by example. We recall Church's troubles with normal forms and nonnormal forms. These problems can all be completely analyzed in logical space with the aid of limits. 2 2 Not only is V(V) = 1., but one can give a criterion for a A-expression to evaluate out to 1. that shows exactly how normal forms enter. Even more useful is the analysis of the so-called paradoxical combinator.P This is defined as: Y

= Af·(Axf·(x(x»))(Ax·f(x(x))).

This combinator has no normal form but is not meaningless. In fact, for any element f of logical space, Y(f) is the least fixed point of J, the least solution of the equation: x =f(x). In terms of limits we can write: 22 This follows from the continuity of the operations (including A-abstraction) and the construction of logical space. 23 See CURRY and FEYS. op, cit., pp. 177-178.

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187

00

Y

= Af· U !"(T), n=O

where fn denote iterated composition.P" In other words: a certain infinite series can be written in closed form. Such a result seems to me to be very much in the spirit of classical analysis. There must be many other such results.We must develop the methods of proving them not only for logical space but for the many other analogous spaces that can be similarly constructed.r" This especially interesting theorem is due to David Park. The applications of the calculus that seem most promising are to higher-level computer languages which require complicated recursions on quite involved domains. 24

25

THE DANGERS OF COMPUTER-SCIENCE THEORy1 DONALD E. KNUTH Stanford University, Stanford, Calif., USA

The text of my sermon today is taken from Plato (the Republic, vii) who said, "I have hardly ever known a mathematician who was able to reason". If we make an unbiased examination of the accomplishments made by mathematicians to the real world of computer programming, we are forced to conclude that, so far, the theory has actually done more harm than good. There are numerous instances in which theoretical 'advances' have actually stifled the development of computer programming, and in some cases they have even made it take several steps backward! I shall discuss a few such cases in this talk. Last week at the IFIP 71 Congress in Ljubljana, I presented a lecture which had quite a different flavor: I spoke pro-Computer-Science, and I extolled the virtues of the associated mathematical theory. Today, however, I must consider the Methodology and Philosophy of Computer Science, and so I feel it necessary to right the balance and to give an anti-ComputerScience talk. (I hope that by showing the other side of the coin I will not prejudice the sales of the books I have written.) Perhaps the most famous example of misdirected theory has occurred in connection with random-number generation. Many of you know that sequences of pseudorandom numbers are often generated by the rule x n + 1 = (ax n) mod m

for some multiplier a and some modulus m. For many years such sequences were used successfully, with multipliers a chosen nearly at random. The numbers passed empirical tests for randomness, but no theoretical reason for this was known except that number theory was able to predict that the 1 The preparation of this paper was supported in part by the National Science Foundation and the National Research Council. My wife and I wish to thank: our Rumanian hosts for their extraordinary hospitality.

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190

sequence has a very long period before it begins to repeat. Finally, the first theoretical advance was made: It was proved (cf. GREENBERGER, 1961) that the serial correlation between adjacent members of the sequence, averaged over the entire period, is bounded by Ir(xi'

Xi+ 1)1

4

< -a +

16a+28

m

.

As you know, a correlation coefficient of + 1 or -1 indicates a strong dependency, while a random sequence should have 0 correlation. According to this new theorem, if we choose the multiplier to be a

sz

vm

we can guarantee a small serial correlation. As I said, this was the first real theorem about the multiplicative congruential sequences; because of it, people changed the random-number generators they were using, and the result was catastrophic (cf. GREENBERGER, 1965). A multiplier near J! m is always bad when other tests for randomness are considered. For example, the correlation is indeed nearly zero when averaged over the entire period, but the theory did not take into account the fact that the correlation is nearly + lover half the period and -1 over the other half! It turns out that almost all multipliers are better than those near Vm. Yet the horrible sequence x n+1 = (2 1 6+3)xn mod 23 1

is still being supplied by IBM as the standard random-number generator for use on its Systemj360 computers. Such misapplications of theory have been with us since the beginning. From a historical point of view I believe that the very first work on what is now called the theory of computational complexity was the Ph. D. dissertation of DEMUTH (1956), who made a theoretical study of the problem of sorting numbers into order. From a mathematical standpoint, Demuth's thesis was a beautiful piece of work; he defined three classes of automata, and he found reasonably tight bounds on how fast each of these classes of machines is able to sort. But from a practical standpoint, the thesis was of no help. In fact, one of Demuth's main results was that in a certain sense 'bubble sorting' is the optimum way to sort. During the last three years I have been studying the sorting problem in great detail, and I have therefore analyzed about 30 different sorting methods including the bubble sort.

DANGERS OF COMPUTER-SCIENCE THEORY

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It turns out that the other 29 methods are always better in practice, in spite of the fact that Demuth had proved the optimality of bubble sorting on a certain peculiar type of machine. Unfortunately, people still play the same game today with computational complexity theory: Instead of looking for the best way to solve a problem, we first think of an algorithm, and then we look for a sense in which it is optimum! Traditionally the problem in finding an optimum sorting method is to minimize the number of comparisons between data elements while the sorting takes place. The best method, in the sense that it takes fewer comparisons than any other known scheme, was invented by L. R. FORD and S. M. JOHNSON (1959). But their approach has fortunately never been used by programmers, because the complex method used to decide what comparison to make costs much more time than the comparisons themselves. The three most commonly known methods of sorting with magnetic tapes are the balanced merge (MAUCHLY, 1946), the cascade merge (BETZ and CARTER, 1959), and the polyphase merge (GILSTAD, 1960). Cascade merging with n tapes is based on an (n-l)-way merge, (n-2)-way merge, ... , 2-way merge, while polyphase is apparently an improvement since it uses (n -1)-way merging throughout. But DAVID FERGUSON (1964, p. 297) noticed that cascade merging is surprisingly better than polyphase on a large number of tapes: Given N records to sort on n tapes, the following asymptotic formulas for the tape read/write time are valid when Nand n are large:

balanced merge, NlogN/log(n/2); cascade merge, NlogN/log(nn/4); polyphase merge, NlogN/log4. This is all very fine in theory but almost worthless in practice. In the first place, the number of tapes n must be large; but tapes are very unreliable, I have never seen more than about six tape units simultaneously in working condition! More seriously, these formulas are valid only as N goes to infinity, yet all N records must fit on one finite reel of tape. Commercial tape reels are never more than 2400 feet long, and this means that other factors suppressed in the above formulas actually are more important than the leading terms. For practical sizes of N it turns out therefore that polyphase is superior to cascade, contrary to the formulas. Incidentally, Professor Karp of Berkeley has proved a beautiful theorem which shows that cascade merging is optimum in the sense that it minimizes the number of phases, over all possible tape merging patterns. But unfortu-

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D. E. KNUTH

nately this theoretical notion of a 'phase' has no physical significance, it is not the sort of thing anyone would ever want to minimize. This leads me to quote from Webster's dictionary of the English language (pre-1960), where we find that the verb 'to optimize' means "to view with optimism." A few months ago I computed the effect of tape rewind time, which the above formulas exclude, and I discovered to my great surprise that the old-fashioned balanced merging scheme was actually better than both polyphase and cascade on six tapes! Thus the theoretical calculations which have been so widely quoted have caused an inferior method to be used. An overemphasis on asymptotic behavior has often led to similar abuses. For example, some years ago I was preparing part of an operating system where it was necessary to determine whether or not a given record called a 'page' was in the high-speed computer memory. I had just learned the very beautiful method of balanced binary trees devised by the Russian mathematicians ADEL'SON-VfL'SKII and LANDIS (1962), which guarantees that only O(logn) steps are needed to find a particular page if n pages are present. After I had devised a complicated program using this method, I remembered that the computer I was using had a special search instruction which would do the same job by brute force in O(n) steps. Since this instruction operated at hardware speed, and since the memory size guaranteed that n would never exceed 1000, the brute force method was much faster than the sophisticated logn method. Now I should say a few words about automata theory. For many years the theory of automata was developing rapidly and solving problems which were ostensibly related to computers; but real programmers could not care less about the automaton theorems because Turing machines were so different from real machines. However, one result was highly touted as the first contribution of automata theory to real programming, an efficient algorithm that was discovered first by the theoreticians, namely, the HFNNIESTEARNS (1966) construction which showed that a k-tape Turing machine can be simulated by a 2-tape machine with only a logarithmic increase in the execution time. This meant for example that sorting could be achieved on two tapes in O(N(logN)2) steps, which was much better than the O(N2) methods known for two tapes. Well, once again the theory did not work in practice; the Hennie-Stearns construction involves writing in the middle of a magnetic tape, which is rather difficult, and it includes a lot of unused blank space on the tape. As I mentioned before, a single tape is only 2400' long; so the asymptotic formulas do not tell the story. When the Hennie-

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193

Stearns method is actually applied to a tape full of data, almost 40 hours are required, compared with only about 8 hours for the asymptotically slow method. The theory of automata is slowly changing to a study of random-access computations, and this work promises to be more useful. Last week in Ljubljana, S. A. Cook presented his interesting theorem which states in essence that any algorithm programmable on a certain kind of pushdown automaton can be performed efficiently on a random-access machine, no matter how slowly the pushdown program runs. When I first heard about his theorem last year, I was able to use it to find an efficient pattern-matching procedure; this was the first time in my experience that I had learned something new about real programming by applying automata theory. (Unfortunately I found out that MORRIS, 1970, had independently discovered the same algorithm, a few weeks earlier, without using automata theory at all.) Is there any area (outside of numerical analysis) where mathematical theory has actually helped computer programmers? The theory of languages springs to mind; surely the development of programming languages has been helped by the highly sophisticated theory of mathematical linguistics. But even here the theory has not been an unmixed blessing, and for several years the idea of top-down parsing was unjustly maligned because of misapplied theory. Furthermore, too many problems in mathematical linguistics have been shown to be unsolvable in certain levels of generality, and this has tended to make people afraid to look for solvable subproblems. We tend to forget that every problem we solve is a special case of some recursively unsolvable problem! Another difficulty with the theory of languages is that it has led to an overemphasis on syntax as opposed to semantics. You all know the old joke about the man who was searching for his lost watch under the lamppost. His friend came up to him and said, "What are you doing?" "I'm looking for my watch." "Where did you lose it?" "Oh, over there, down the street." "But why are you looking for it here?" "Because the light is much better here." For many years there was much light on syntax and very little on semantics; so simple semantic constructions were unnaturally grafted onto syntactic definitions, making rather unwieldy grammars, instead of searching for theories more appropriate to semantics.

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Of course you know that the theory of languages has by now become ultrageneralized so that it bears little relation to its practical origins. This is not bad in itself, although sometimes it reminds me of a satirical article published a few years ago by AUSTIN (1967); paraphrasing this article, we should not be surprised to find someday a paper entitled "On triplydegenerate prewaffies having no proper normal subwaffie with the pseudo-a; property, dedicated to A. B. Smith on his 19th birthday." Sometimes theories tend to become very baroque! The tendency of modern mathematics to be 'modern' in the sense of 'modern art' has been aptly described in an extraordinary article by HAMMERSLEY (1968) which should be required reading for everyone. At this point I would like to quote from some lectures on Pragmatism (Chapter 2) given by the philosopher William James at the beginning of this century: When the first mathematical, logical, and natural uniformities, the first laws were discovered, men were so carried away by the clearness, beauty and simplification that resulted, that they believed themselves to have deciphered authentically the eternal thoughts of the Almighty. You see that computer science has been subject to a recurring problem: Some theory is developed which is very beautiful, and too often it is therefore thought to be relevant. An article has recently been published by CHRISTOPHER STRACHEY of Oxford University, entitled "Is Computing Science?" (1970). He presents two tables, one which ranks topics now considered part of computer science in order of their relevance to real programming, and another which ranks those same topics in order of their present state of theoretical development. As you might suspect, the two rankings are in opposite order. Perhaps this is the way things should be. Maybe theories are more beautiful and more worthy of development if they are further from reality. Some of the examples I have mentioned suggest in fact that it is dangerous even to try to develop any theory which is relevant to actual computer programming practice, since the record shows that such theories have usually been misapplied. Well, I must confess that I have had my tongue in my cheek, in many of the above remarks. When I first prepared this talk, sitting in beautiful Cismigiu Park, I was not intending to write it down for the published proceedings, and I could not resist the temptation to have some fun giving an unexpected 'methodology' lecture. I have stated the case against computer-science theory as well as I could; but as many of you probably suspect,

DANGERS OF COMPUTER-SCIENCH THEORY

195

I do not really believe everything I said. It is true that theory has often been irrelevant and misapplied; but so what? We get enjoyment and stimulation from abstract theories, and the mental concepts we learn to manipulate as we study them often give us practical insights and skills. On the other hand, practical considerations do not necessarily lead to awkward mathematical problems that are inherently impure or distasteful. In fact I have been spending many years preparing a series of books which attempt to show that there is a great deal of beautiful mathematics which is directly helpful to computer programmers. My experience has been that theories are often more structured and more interesting when they are based on real problems; somehow they are more exciting than completely abstract theories will ever be. References T. M., JIAH,I:lllC, E. M., 1962, DauH a/lZopu¢M opZaHU3al/UU DAH CCCP, T. 146, CTp. 263-266 AUSTIN, A. K., 1967, Modem research in mathematics, Mathematics Gazette, vol. 51, pp. 149-150 BETZ, B. K. and W. C. CARTER, 1959, New merge sorting techniques, in: Preprints of Summaries of Papers Presented at the 14th National Meeting, Association for Computing Machinery (Association for Computing Machinery, Cambridge, Mass.) DEMUTH, H. B., 1956, Electronic data sorting, Ph. D. dissertation, Stanford University FERGUSON, D. E., 1964, More on merging, Communications of the Association for Computing Machinery, vol. 7, p. 297 FORD, L. R., Jr. and S. M. JOHNSON, 1959, A tournament problem, American Mathematical Monthly, vol. 66, pp. 387-389 GILSTAD, R. L., 1960, Polyphase merge sorting, an advanced technique, Proceedings Joint Computer Conference, vol. 18, pp. 143-148 GREENBERGER, M., 1961, An a priori determination of serial correlation in computer generated random numbers, Mathematics of Computation, vol. 15, pp. 383-389 GREENBERGER, M., 1965, Method in randomness, Communications of the Association for Computing Machinery, vol. 8, pp. 177-179 HAMMERSLEY, J. M., 1968, On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities, Bulletin of the Institute for Mathematics and Its Applications, vol. 4, pp. 66-85 HENNIE, F. C. and R. E. STEARNS, 1966, Two-tape simulation of multitape Turing machines, Journal of the Association for Computing Machinery, vol. 13, pp. 533-546 MAUCHLY, J. W., 1946, Sorting and collating, in: Theory and Techniques for the Design of Electronic Digital Computers, ed. G. W. Patterson, vol. 3, lecture 22 MORRIS, J. B., Jr. and V. PRATT, 1970, A linear pattern-matching algorithm, Technical Report No. 40, University of California, Berkeley, Computation Center STRACHEY, C., 1970, Is computing sciences, Bulletin of the Institute for Mathematics and Its Applications, vol. 6, pp. 80-82 ~EJIbCOH-BI!Jl"CKllH-

UH¢Op,ItaI/UU,

SUR UN LANGAGE EQUIVALENT AU LANGAGE DE DYCK M.P. SCHOTZENBERGER Faculte des Sciences, Paris, France

1. Introduction

Le Iangage de Dyck D 2 sur deux paires de Iettres {Y, y, Y' , y'} est comme on sait, defini de facon equivalente, comme la solution de l'equation ~ = 1+ji~y~+Y'~y'~

ou comme la cIasse de 1 pour Ia congruence 1 == jiy == ji'y'

sur Ie monoide libre {y, y, Y' , y'}*. Appelant suivant Eilenberg cone !b'(L) d'un Iangage L, Ia famille des Iangages qui peuvent etre deduits de L par une relation rationnelle, on sait que Ie cone !b'(D2 ) est l'ensemble des Iangages algebriques, Cette propriete n'est pas partagee par Ie Iangage de Dyck D 1 = D 2 (') {ji, y}* sur une seule paire de lettres. On se propose de montrer qu'au contraire la meme propriete est possedee par Ie Iangage L c {Y, y}* defini de facon equivalente comme: -Ie quotient de {Y, y}* = y* par Ia congruence definie par: yy == yyyyyy;

- la solution de I'equation ~ = yy+y~~y;

- Ia partie L de D 1 \1 formee des mots dEDI \1 tels que pour chaque factorisation d = ayb (a, b E Y*) Ie mot jib ait exactement un facteur gauche dans D 1\1 ssi a E 1 u Y*y. On observera enfin que l'equivalence rationnelle de D 2 et L (!b'(D2 ) = ~(L)) entraine celIe de D 2 et de tout Iangage defini par une equation de la forme ~ = a + b~c~d ou cette fois a, b , c et d sont quatre mots verifiant des conditions assez peu restrictives (par exemple que {a, b , c} engendre

M.P. SCHOTZENBRRGRR

198

librement Ie monoide {a, b, c} *). En raison de sa simplicite cette derniere equation peut presenter des avantages techniques dans certains problemes d'equivalence rationnelle avec D 2 • 2. Les differentes definitions de L Nous considerons L c y* comme etant defini par I'equation ~

=

yy+y~~y.

Posant X = {~, y, y} ceci equivaut a L = M () Y*, OU M est la plus petite partie de X* contenant ~ qui soit telle que

f, g E x* ,

Rg EM=> fyyg,

fyUyg EM.

De facon equivalente, M peut etre considere comme le langage obtenu en remplacant x par ~ dans Ia solution de l'equation

=

~

x+yy+y~~y.

Natant comme d'usage Ihl: le nombre d'occurrences de Ia lettre z dans le mot h, nous deduisons immediatement de la definition de M les deux proprietes suivantes :

2.1. e EM=> e E {~}

U

X*(jiy+y~~y)X*

2.2. e EM=> lel y = lel y et, si e

= fg ou f,

g

E X*\~*,

alors

Ifly > Ifly· PREUVE: En effet ces assertions sont trivialement vraies pour e = ~ et si eIIes sont vraies pour un mot e E X* eIles Ie demeurent pour tout mot e' obtenu en remplacant dans e une occurrence de ~ par yy au par YUy. Ecrivons h' E hCJ i- 1 (i = 0, 1) ssi reciproquement il existe des mots f, g E X* tels que h = Jv.s, h' = tt« ou Uo = yy, U 1 = YUy. Rappelons que deux mots a, b e X* ne chevauchent pas ssi il n'y a aucun mot c E Xx* tel que:

a E X*Xc,

et

b e cXX*

a

et

bE

E

cXX*,

au

X*Xc.

Cette condition est satisfaite ssi pour toute relation fag = f'bg' «(, (', g, g' E X*) I'on a I'une des quatre eventualites suivantes: - f'b un facteur gauche de f;

199

LANOAOE EQUIVALENT AU LANOAGE DE DYCK

-f'b un facteur gauche defa etfun facteur gauchef'; - fa un facteur gauche de [' b et f' un facteur gauche de f; - fa un facteur gauche f",

2.3. h eX* => MOIOll

= hallO ol.

PREUVE: Ceci resulte immediatement de ce que deux mots (eventuellement egaux) de {uo, ud ne chevauchent pas et que, de plus, aucun d'eux n'est facteur de l'autre quand iIs sont differents. Soit maintenant == la plus petite congruence sur x* telle que

2.4. M est la classe de ~ pour == et Lest la classe de yy pour la congruence (==) sur y* definie par yy (==) YYYhY. PREUVE: Supposant e E M tel que eOol u eOll c M, et e = Rg (j, g E Y*), iI resulte de 2.3 que ron a encore e'ool u e' 01 1 c M pour e' = fu, g (i = 0, 1). Par induction ceci implique M = {h EX·: h(OOl u ( 1 1 )* = ~}, done M={hEX·:h==~}.

Comme y~~y EyyyyyyoolOol, la deuxieme partie de l'enonce resulte de la premiere. COROLLAIRE 2.5: Le langage Lest rationnellement equivalent au langage N solution de l' equation ~

=

x+y~~y.

PREUVE: Soit ep Ie morphisme de {x, y, y}* sur y* tel que xep = yy, yep = Y, yep = y. 11 est clair que la restriction lplN est une bijection sur L. Reciproquement, si e = fyzg E L (z = Y ou y) iI resulte de 2.5 que ron aRg E Mssi z = y. De meme sifzyg E Lz = youy on aRg E Mssi z = y. 11 en resulte que N = Lij; (avec epcp = 1) ou cp: Y· --. {y, y, x}* est la relation rationnelle remplacant par x tout y suivi d'un y, et par 1 tout y precede d'un y et laissant inchangees les autres lettres. 2.6. Soit e = fyg E M. On a yg = mg ' ou g' selon que f E Y*(y+~) ou E 1 u Y*y. PREUVE: L'enonce est vrai pour e il est encore vrai pour f'u.g',

=~

E

YY· et ou m

E

M ou

et s'il est vrai pour e

E

M2

= f' ~g',

200

M. P. SCHtJTZENBERGER

Rappelons maintenant que Ie Iangage de Dyck D 1 est Ie sous-monoide D* de y* librement engendre par l'ensemble D des mots de yy* teIs que:

- Idly = Idly; -

d

= fg, t;

g

E

yy* => Ifly >

Ifly.

Cette condition implique que deux mots de D ne se chevauchent pas et que tout mot de y* ait au plus un facteur gauche (et droit) dans D. On sait aussi que: Pour toute factorisation fjig d'un mot de D 1 (f, g e Y*), on a jig = d'g' au g' e 1 u yy* et d' e D 1\1, cette factorisation etant unique. On a alors fg'eD l • 2.7. Les deux conditions suivantes (D) et (D)' sur un mot dEDI \1 sont equivalentes : Pour toute factorisation d = fjig (f, g e Y*), (D)

jig a exactement un facteur gauche dans D 1 \1 ssi

f

e 1 u Y*y;

(D)' jig = d'g' ou g' e 1 u yy* et ou d' e D ou e D2 selon que f e Y*ji ou e 1 u Y*y.

PREUVE: En vertu de D 1 = D* on peut ecrire jig = d 1d2 ••• dpg' ou d 1,d2 , ... ,dpeD, g'rf=DY*, done g'e1uyY*. Comme DeDI \1 p est Ie nombre de facteurs gauche dans D 1 \1 de yg.

p~1,

Maintenant, comme D c Y*ji, la condition (D) equivaut que p :::;; 2 avec egalite ssi f e Y*ji, c'est-a-dire a (D)'.

a la condition

PROPRIETE 2.8.: Lest la partie de D 1 \1 definie par (D). PREUVE: La remarque 2.2 montre que LcD L satisfait (D) resulte de 2.6 et 2.7.

C

D 1 \1 et Ie fait que

Reciproquement, on a jiy e D (") L. Supposons que I'implication (D)' => Lsoit etablie pour taus les mots plus courts que Ie motdeD 1 \(1 +yy) satisfaisant (D)'. On peut ecrire d = ljig avec 1= 1. Done, de D, d'apres (D)' et plus precisement, d = jid 1d2y au d 1 , d 2 ED. Comme la condition (D) sur un mot, implique Ia meme condition sur taus Ies facteurs dans D 1 \1 de ce mot, l'hypothese d'induction donne d.; d2 e L, d'ou evidernment,

LANGAGE EQUIVALENT AU LANGAGE DE DYCK

201

3. Equivalence rationneUe de D 2 et de L Comme Lest algebrique et que par consequent L E ft?(D 2 ) , il suffit de montrer que reciproquement D 2 E ft?(L). Pour simplifier Ies calculs on etablira plutot D~ E ft?(N) ou D~ c Y'* (y' = {Y, y, Y' , y' , y" , x}) est defini par l'equation et M par l'equation ~ = x+Y~~y·

Comme ft?(L) = ft?(N) ainsi qu'on I'a vu en 2.6 et comme D~ E ft?(D 2 ) puisque D; est algebrique, l'equivalence des inclusions D 2 E ft?(L) et D; E ft?(N) resulte immediatement de Ia relation D 2 = D;f(i ou f(i est Ie morphisme de Y'* dans lui-meme envoyant y" et x sur 1 et laissant inchangees les autres lettres. De facon inverse, l'inclusion D; E ft?(N) sera verifiee en prouvant l'existence d'un morphisme 1p: (~ u Y')* ~ (~ u X')* (X' = {x, y, y}) tel qu'en posant a = Y1p, a' = y'1p, b = y1p, b' = y'ip, c = y"1p, d = X1p Ies deux conditions suivantes soient satisfaites: (1) ~1p = et 1p est injectif. (2) II existe un langage rationnel R c Y'* tel que D; c Ret que R1p ('\ N soit Ia solution P de l'equation

e

~

=

d+a~b~c+a'~b'~c.

En effet sous ces conditions on aura: D; = (N ('\ R1p)-l1p, puisque D;1p c P, par construction.

Nous commencons par la construction de R. 3.1. D~\x est contenu dans Ie langage R' accepte par l'automate avec 1 et 6 comme hat initial et final respectivement.

g,g'

a6 etats

202

M. P. SCHOTZENBERGER

PREUVE: Soit R o = (y + y') x(y + y') xy". On a R o c R' et on voit sur Ie graphe que fxg E R' => fRog c R'. Done, par induction => fR'g c R' ce qui etablit D~\x c R'. Soit maintenant X" = {~, x, y, y}, et soit == la congruence sur X"* definie par: ~ == x == yy == y~~y.

Sa restriction a X est la congruence etudiee plus haut et notant M' la classe de ~ on voit d'apres 2.4 que N = M'\X"*(~+yy)X"*. Nous choisissons maintenant Ie morphisme 'ljJ et nous posons:

=

a = y'ljJ a'

=

yyxyy;

y''ljJ = yyyxy;

b = y'ljJ = wyxyy b' = y''ljJ = wyyxy

c = y, d=x OU W = yxxy. Ce morphisme est injectif puisqu'aucun mot de A = Y''ljJ n'est facteur gauche d'un autre mot de A. 3.2. Pest contenu dans N et axb' et a'xb appartiennent : X"*hX"* (') M' = 0}.

PREUVE: La premiere assertion resulte de decoule elle-meme du parenthesage :

a~b~c

= Y

a~b~c,

a0 =

{h EM'

a' ~b' ~c EM' qui

(yx(ji(y~(w)y)(XY)Y)~Y) E M'

a'~b'~c = Y (y(jix(ji~(w)y)y)XY) ~y EM'. Pour verifier la seconde assertion, nous notons que d'apres 2.6, on a y~y, O. Par consequent,

~~~ E

axb' == a~b' = yjixY(YHw)y)yxy == yyxy~yxy EO

et a'xb == a'~b

= yjiyx(y~wy)xyy ==

yjijiU~yy E

O.

Fin de fa preuve de D 2 E ~(L). II ne no us reste plus qu'a verifier:

sx

E

(x

U

S) (') N

=>

S

E

P

ou S = R''ljJ. Comme Ie resultat est vrai pour s = x, axbc et a'xbx'c OU les deux derniers sont les mots les plus courts de S, nous pouvons, par induction sur la longueur, nous borner a verifier que tout mot S E S (') N \ {axbxc + a'xb'xc} admet une factorisation S = SlVS' telle que v = axbxc ou a'xb'xc et que SlXS' E S (') N 1 +x.

LANGAGE EQUIVALENT AU LANGAGE DE DYCK

203

Soit done un tel mot s. Considerant l'automate qui definit R', l'hypothese S implique l'existence d'une factorisation s = Sl asz ou S = Sl a's, avec St E A* et Sz E A*\A*(a+a')A*. De plus, la meme hypothese entraine Sz E x(b+b')x c s' ou SlXS' E xuS. On a done S = St rs' OU r E (a+a')x(b+b')xc. Comme rEP si r = axbxc ou r = a'xb'xc il n'y a done qu'a verifier r =Faxb'xc, a'xbxc, ce qui resulte immediatement de 3.2 puisque SEN. s

E

Note bibliographique. Le langage defini par l'equation ~ = d+a~b~c a ete introduit par M. Nivat (These, Paris, 1967) qui l'appele .Jangage de expressions arithmetiques", Un eleve de M. Nivat, Y. Cochet a traite de facon approfondie dans sa These (Rennes, 1971)les langages qui sont c1asse d'une congurence. Enfin une theorie complete des langages rationellement equivalents a Dz est en cours de publication sous les signatures conjoints de L. Boasson et M. Nivat. Nous avons fait dans le present travail de nombreux emprunts aux recherches de ces auteurs.

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~o

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211

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naurea TepMHHOJIOrHH

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aa

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(1 osaasaer G

(npenaxara

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0-

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n, TaKOe ero BbIqnCJIeHHe aa

sm C F-oPaKYJIOM, npa KOTOpOM ,D;JIH BCHKoro n, Masm BhIqnCJIHeT F(n) He 06paLQaHCh K opaxyrry C BonpOCOM "n E G?"

nexoropoa MalIIHHe IIIHHa

(XOTH II MOllieT aanaaars JII06bIe

sonpocsr

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xaer nztea 06'bHBJIHTb ,,6Hympel-me-3a6UCUMoU" BCHKyIO TaKYIO np06JIeMy, ,n;JIH KOTOPOH B03MO)KB0 aBTOCBe,n;eHHe, a ,,6HympeWle-lle3a6UCUMOU" JIIO6yIO, ,n;JIH KOTOpOH 3TO HeB03MOlliHO. TaKOH nonxon K neny HaMH ,n;eHCTBHTeJIbHO HCllOJIb30BaJICH (TpAXTEHEPOT,

1970».

3aMeTHM, O,D;HaKO, srro ,n;JIH

3cPcPeKTlfBHbIX np06JIeM 3Ta lI,n;eH B TaKOM BH,D;e He rO,D;HTCH, H60 BCHKaH 3cPcPeKTHBHaH np06JIeMa aBTOMaTlIqeCKlI II aBTOCBO,D;HMa. B3aHMHYIO He3aBlICHMOCTh e,D;HHHqHhIX sanas HeKOTOpOH: 3cPcPeKTHBHOH MaccoBoH np06JIeMbI 6y,n;eM nOHHMaTh n03ToMY B TOM CMbICJIe, nro ,n;JIH nee He CYLQeCTBYeT aBTOCBe,n;eHHH saasarerrsao 60JIee npocroro, qeM npaMaH

paspemaromaa

nponenypa. OKa3bIBaeTCH, nro eCJIH CJIOlliHOCTh H3-

MepHTh eMKOCTHhIMH ClIrHaJIH3l1PYIOLQHMH, a cpanaenae CHrHaJIH3l1PYIOlQHX npOH3Bo,n;HTh no rropanxy, TO ,,6Hympelllle-He3a«UCUMble" MaCCOBbIe np06JIeMbI CYLQeCTBYIOT. TEOPEMA

D. Ilycms

mozoa O/lR /l1060U

Toxnee,

HMeeT MeCTO

namypansnue uucna KOOUPY1OmcR ynapuoi; 3anUCblO; enxocmuoa rfiYHKl/UU h cytqecmeyem npeouxam F, Y006/le-

m60pR1OUjUU YC/106URM:

212

E. A. TPAXTEHBPOT

(1) cytqecmeyem uauiuua

sn

3UPYlOU/eU S~(n) = h(n); (2) OJlR Jl106ol0 a8mOC8eOe1lUR

(6e3 opasyJla), 8bl'IUCJlRlOU/aR

c:m npeousama r

r

C CU21laJlU-

cytqecmeyem tcoucmauma C

maxan, umo V n[S9Jlr(n) ~ c·h(n)]. IIpRBe,I:\eHHbln panee

noxassrsaer,

npaxep (8bl800UMOcmb H == 8bl800UMOcmb I I H)

'ITO CYI1..\eCTBYIOT MaCCOBhle np06JIeMhI C o-reas rrpOCThIM

aBTOCBe,I:\eHReM; Op;HaKO 3TO RMeJIO MeCTO B CBHSR C TpRBRa.JThHOn BsaRMHon saBRCRMOCTblO RH,[IHBlI,lJ;Ya.JThHhIX aaztax. CJIe,I:lylOI1..\aH

reopeaa

06Ha-

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IOII~lIe "TpIOKH" Tlrn:oro pozta ; B Hen, KaK II asnne, rrpe,I:\IIOJIaraeTCH, 'ITO

narypansasre TEOPEMA

qllCJIa

saKO~pOBaHbI

YHapHO.

E. J(JlR KaJICOOU cmpoeo .M01l0m01l1l0U esacocmnoii ¢jY1lKljUU h

cytqecmeyem npeouuam

r, Y008Jlem60pRlOU/UU YCJl08URM:

(1) Ilpeoutcam T'uueem mouuyto CU21laJlU3upylOU/YlO h, mo ecms oonycxaem euuucnenue (a6COJllOm1l0e) C maKOU cueuanusupyxnuea U ue oonycxaem a6COJllOm1l0l0 8bl'lUCJle1lUR C cueuasusupytoiqei; ,Ue1lbuteU no nopnaxy .

r, xomopoe nyuuie h 8 mOM

(2) Cytuecmeyem a8mOC8eOe1lUe Q npeouxama

cuucne, umo OJlR netcomopoii nOOXOORU/eu xoucmaumu d cnpaeeonueo:

V:,[Snr(n) ~ d- n].

Ilpuuenauu», B

paoore (TpAXTEHBPOT,

1970)

OhIH napaaar yraepacneaaa

2 reopexsr E,

A.

Menepy II M.

YCHJIeHHe IIpHHawre>KlIT

CO,I:\epiKHTCH 60JIee CJIa-

3:'[Snr(n)::::;; dn}. Tlarepcony. HMH me aHOHCRa lIMeHHO:

pOBaH peaynsrar:

Katcoea 6bl 1lU 6blJla 3¢j¢jeKmU81laR ¢jY1lKljUR t, cytqecmeytom npeousamu

r 1 U r 2, OJlR xomopux eunonumomcn YCJl08UR:

1. Kasoea 6bl 1lU 6bwa uaucuua V, 6bl'lUC/lRlOU/aR I', (6bl'lUCJlRlOU/aR

r 2)

V:'[ Sy(n) ~ ten)].

2.

KaKd6a 6bl 1lU 6bl/la YCJl061laR uauiuua

W,

6bl'lUCJlRlOU/aR npu opaxyne

r 1 npeouxam r 2, 1laUOemCR otiuuuan uaucuna V, 6bl'lUC/lRlOU/aR r 2 U rnaxan, 'lmo

V:'[SyCn) ::::;; Swr.(n)].

3. A1laJlOlU'l1l0, eCJlU noueunms POJlRMU T', U F 2 • RCHO, qTO 3Ta TeopeMa TaKme HBJIHeTCH rrpOCTbIM CJIe.n;CTBHeM Harrren

reopeasr.

OPMAJIH3AQWI HEKOTOPbIX nOHJITHll

213

BAP;:\3HHD, 51. M., 1968, CIIO:HCHOcmb np02paMM, pacOO3HalOUfUX npuuaanexcuocms uamypansuux uucen, He npeeutuatotuux n, pexypcueuo nepetlUC.IIUMOMY MHO:HCeCm6Y,

,UOKJIa;:\bI AH CCCP, T. 182, CTp. 1249-1252 ,UmcrHPb, M. 11.,1969,0 He603MO:HCHOcmu 311UMUHatlUU OOllH020 nepeiiopa npu euuucneuuu rPYHKtlUU OmHOCUme.!lbHO ux 2parPuK06, ,UOKJIa,I:\bI AH CCCP, T. 189, CTp. 748-751 TpAXTEHEPOT, B. A., 1970, 06 asmoceoouuocmu, 'uOKJIaAbI AH CCCP, T. 192, CTp. 1224-1227 XO;::pKAEB, ,UlK., 1969, Hexomopue OtfeHKU cnoxcuocmu 6bltlUC.IIeHUU na uatuunax C opaKYIIOM, BceC0103HaJi KOH Martal(x»). In the work described in JIEt1:KHHA et al. (1966) we have used a language for grammatical conditions that was based on a sort of variable-free logic and made no use at all of anaphoric relations, but I do not claim it to be a general solution. 1.3. The experience in constructing formal grammars for natural languages shows that a 'rigid' phrase-structure (or dependency) grammar even with a number of complicated additional restrictions is not very adequate. Under such a grammar each construction is either fully allowed or fully forbidden and therefore some most common constructions are treated on an equal basis with some rather exceptional ones. Here is an example due to B. M. Leikina. The sentence Six patients recovered completely seems unambiguous to every normal speaker, but a version of formal grammar (one of the tested versions) allows a different parsing, viz., six can be regarded as an object for recovered rather than a modifier for patients ('The patients have recovered six of something'). The grammar can be improved to exclude such 'competitive' constructions by adding a number of restric-

NATURAL LANGUAGES

219

tions, but the resulting grammar is then so complicated that it can be hardly believed that a human brain can do a real-time parsing under such a grammar (note that no real-time parsing procedure is known even for ordinary context-free grammars). There are similar difficulties with the 'rigid' syntax in a language like ALGOL 68. Whoever has followed its history knows how much effort has been spent and how many complications have been introduced to ensure uniqueness of parsing. Moreover, some small amendment may give rise to a new ambiguity and thus necessitate overall changes in the language. Natural languages are far more flexible in this respect. The syntax of natural languages is more liberal in that it allows omitting some parts of grammatical constructions. This tendency is also partly represented in programming languages, especially in the default concept of PL /1. ALGOL 68 also has some default conventions, but it is a remarkable fact that they are introduced mostly by means of extensions, context conditions and standard prelude rather than as a part of the context-free syntax of the strict language. A concept that may contribute to solving these difficulties in description both of natural and programming languages is that of preferential syntax. A preferential description of a language is based on a common (say, contextfree) description, but it also establishes a set of preferences between various rules and constructions, e.g., by means of numerical evaluation. A sentence can have more than one parsing under such a grammar but one of them is usually preferred. The concept of preference applies not only to whole sentences but also to their parts, i.e., an underpreferred parsing of a part of a sentence has very little chance of becoming part of the preferred parsing of the whole sentence (and therefore perhaps may be dropped in the course of parsing before the sentence is accomplished-). I believe that a system of preferences is inherent in natural languages and, unlike some linguists influenced by the set-theoretical approach, I think that preferences pertain to the language itself rather than to statistical characteristics of speech. In programming languages the preferential approach may result in simpler descriptions of powerful languages, first of all because of dropping the unique-parsing requirement. The most striking example is the trick used in the ALGOL 68 Report for identification of operators (see van WJJNGA2 This idea may lead to a real-time parsing procedure for natural languages (see UEllTllH, 1970). Of course in a good parsing algorithm the preference should be influenced by semantic factors and context.

220

G. S. TSEYTIN

ARDEN et al., 1969, 4.3.2.b) where it would just suffice to say that the defining occurrence with the narrower scope is preferred. The preferential approach could perhaps simplify also some coercion rules in ALGOL 68. This approach will possibly lead to new and more flexible programming languages better suited for human users. 2. There are some features of similarity between natural languages and modern programming languages in their semantical concepts. 2.1. Programming languages usually describe operations on data performed step by step. However, a remarkable event in the development of programming languages was the introduction of declarations that involve no operations on data but play an essential part in the program. The 'declarative', or 'fact-stating' approach prevails in natural languages even for messages containing instructions. I think therefore that an ideal programming language should not require specifying all the minor steps. Instead, only the main stages of processing that are significant to the user should be noted and they should be described in terms of general relations between the data and the results (in cyclic processes also in terms of properties of data maintained at each step). This is already achieved for problems or their parts solved by library routines, but in general it is connected with techniques of logical analysis (correctness and termination proving) of programs now being developed by a number of people. A step in this direction would be creation of a debugging-oriented language combining complete operational information with logical description of properties to be verified by an appropriate proof procedure (cf. proof checker in LAUER, 1971, Appendix). The next step would be to allow omitting the operational part in some places and instructing the 'compiler' to supply it in accordance with the logical description" (by means of some heuristic process or as in GREEN, 1969, Chapter 7, B.). 2.2. There are two concepts in programming that have arisen from specific features of computer design and that seem to be alien to natural languages and even to the human way of thinking. These two are assignment and jump, both reflecting an irreversible change of state of some element in the computer. The human memory does not work that way and therefore it is very difficult to explain those notions to a newcomer in programming. Indeed, if we instruct a man to perform some new action we do not expect 3 This approach was outlined by the author in a communication at the Conference on Problems of Theoretical Cybernetics held in Novosibirsk in June 1969.

NATURAL LANGUAGES

221

that he is supplied with precise instructions for the rest of his life but rather that after fulfilling our request he will do what he wants, i.e., resume his normal activities. Thus an order expressed in a natural language is more like a procedure call than a jump. Similarly, when we assign a meaning (a value) to a symbol or a word we do not expect that the effect of this assignment will be valid until the next assignment to the same symbol is made; it is expected to be valid either 'forever' or within a specified portion of text, like an identity declaration in ALGOL 68. To introduce genuine assignments and jumps the concepts of memory state and of time are necessary. Therefore, as the programming languages become less memory- (and time-) oriented they introduce a wide range of means to replace jumps (like procedure calls, conditional clauses, synchronized parallel processes) and assignments (identity declarations, parameter substitution in calls, clauses delivering values). Jumps are also considered inconvenient for logical analysis and in general 'harmful' (DIJKSTRA, 1968). In LISP jumps and assignments enter as an extra feature and in another recursive language, REFAL (Twl.JIDI, 1968) they are totally absent. 2.3. There are a number of other features in which programming languages begin to imitate natural languages. a) The possibility of extending the languages by means of new definitions. b) Including the metalanguage level into the language, as in LISP or the ABC-language of JIABPOB (1971). c) Polysemy, i.e., multiple meaning of the same sequence of symbols depending on context, e.g., in ALGOL 68 through coercions. d) Conversational features. 3. I hope that deliberate imitating of some deep features found in natural languages will show a way to create new, more powerful and convenient programming languages. References DUKSTRA, E. W., 1968, do to statement considered harmful, Communications of the ACM, vol. 11, pp. 147-148 GREEN, C. C., 1969, The application of theorem proving to question-answering systems, Dissertation, Stanford University LAUER, P. E., 1971, Consistent formal theories of the semantics ofprogramming languages, Doctoral thesis, Queen's University of Belfast VAN WIJNGAARDEN, A. (ed.), B. J. MAILLOUX, J. E. L. PECK and C. H. A. KOSTER, 1969, Report on the algorithmic language ALGOL 68 (Mathematisch Centrum, Amsterdam)

222

G. S. TSEYTIN

nABPOB, C. C., 1971, Hsuxoean OCH06a npuMeHeHUU 3BM, )KYPHllJI BbltIHCJIHTeJIbHOii MaTeMaTlUlli R MaTeMaTHtIecI 0. 8. If A ,.." 0 and B s;; A then B ,.." 0. 9. If A,.., 0 and B,.." 0 then AuB - 0. 10. If A s;; B and AlB,.." 01B then A ,.., 0. 11. If A s;; B and A > 0 then AlB> 01B.

Use Axiom 7.

THEOREM

12. If A ,.., 0 then AlB,.., 01B.

The proofs of Theorems 13, 14 and 15 are given in KRANTZ et at. (1971, pp. 230-231). Theorems 16-22 express fundamental elementary properties needed in treating Bernouilli processes and other classical cases that cannot be explicitly discussed here. THEOREM 13. If A s;; B s;; C then AlB unless A ,.., 0 or C-B ,.." 0.

~

AIC. Moreover, AlB> AIC

THEOREM 14. If A s;; B s;; C and B > 0, then BIC BIC> AIC unless B-A - 0.

~

Ale. Moreover,

THEOREM

15. If AuB s;; C s;; D then AIC ~ BIC iff AID ~ BID.

THEO~EM

16. If A = B- C and C s;; B then AlB,.." ICiB.

PROOF:

If C s;; B then (X-C)nB = B-e.

(1)

Thus, we have (hypothesis) AlB - B-C1B - (X-C)nBIB (1) (Axiom 5) ""' X-C1B -ICIB. THEOREM

17. If A s;; BnC then AlB

THEOREM

18. If AnC

THEOREM

19. IfIAnlC = IBnlC = 0 then AlB ~ AnClBnC.

THEOREM

20. If BnC

THEOREM

21. If AlB - A then BIA - B.

= BnC = 0 =0

~

AIC iffC

~

B.

and As;; B then AuCiBuC ~ AlB.

then AlB

~

AIBuC.

522

P. SUPPES

PROOF: Suppose not. Without loss of generality let BIA > B. Then by Axiom 5 we have AIX '" AnBIB and AnBIA > B/X, whence by Axiom 8. AnB/X> AnB/X, which is absurd, since ~ is an equivalence relation THEOREM 22. If A ~ C, B '" D, AlB ~ A and C1D ~ C then AnB ~ CnD. PROOF: From the hypothesis, equivalence properties of "', and Axiom 5 AnBIB ~ CnDID, but we also have by hypothesis B ~ D, whence by Axiom 7, AnB ~ CnD as desired. I turn now to a theorem that does not seem to have been previously stated in the literature of qualitative probability. The reason seems to be that the solvability axioms ordinarily used in subjective theories of probability often do not hold in particular physical situations when the probabilistic considerations are restricted just to that situation. Consequently the familiar methods of proof of the existence of a representing probability measure must be changed. Without this change, the theorem is not needed. The theorem is about standard sequences. Alone, the theorem does not yield a probability measure, but it guarantees the existence of a numerical function that can be extended to all events, and thereby it becomes a measure when the physical hypotheses expressing structural, nonnecessary constraints are sufficiently strong. REPRESENTATION THEOREM FOR STANDARD SEQUENCES. Let be afinite standard sequence, i.e., A j > 0, A j £ Aj+l' and XIX> AdA j+! '" AIIA z . Then there is a function Q such that (i) A j £ A j iffQ(AD ~ Q(A j ) , (ii) if Ai £ A j and A k S;;; Al then

AdAj ~ AklA I iffQ(Aj)/Q(A j) = Q(Ak)/Q(A I ) .

Moreover, for any Q satisfying (i) and (ii) there is a q with 0 < q < 1 and a c > 0 such that Q(A j )

= cq":":',

PROOF: Let 0 < q < 1. Define Q(A j ) as (1)

Q(A j ) =

«":'.

Then obviously (i) is satisfied, since the members of a standard sequence are distinct, otherwise there would be an i such that Ai = Ai+l and thus AjIA I+I ~ XIX, contrary to hypothesis. So we may turn at once to (ii). First, note the following. (2)

OBJECI1VE PROBABILITY

523

The proof of (2) is by induction. For m = 1, it is just the hypothesis that for every i, AilA i+! "" A1IAz . Assume now it holds for m-1; we then have Ail Ai+(m-l) "" AjIA)+(m-l) ,

and also for any standard sequence Ai+(m-dAi+m "" Aj+(m-l)IAj+m,

whence by Axiom 7, AdAi+m ,..., AjIA j+m, as desired. Next, we show that if Ai S;;; A j, A k S;;; Ai and AdAj ,..., AkIAj, then there is an m ~ 0 such that j = i+m and 1= k-i-m. Since Al S;;; Aj and A k S;;; Aj, there must be nonnegative integers m and m' such thatj = i+m and 1= k-i-m', Suppose m =I: m', and without loss of generality suppose m-s-h = m', with h > O. Then obviously In addition, Ai+mIAi+m ,..., X/X> AHmIAk+m+h, and so again by Axiom 7 AdA;+m > AkIAH m" contrary to our hypothesis, and so we must have m = m', With these results we can establish (ii). We have as a condition that Ai S;;; A j and A k S;;; Ai. Assume first that AdAj ,..., AdA i. Then we know that there is an m such that j = i-s-m and 1= k+m, whence Q(AI)jQ(A j )

= q"+1-ijq"+1-i-m

= qn+l-kjq"+l-k-m = Q(Ak)jQ(A i).

Second, we assume that Q(A;)jQ(A j ) = Q(Ak)jQ(A 1) .

From the definition of Q it is a matter of simple algebra to see that there must be an m' such thatj = i-s-m' and I = k-i-m', whence by our previous result, AilA j ,..., AdA l • Finally, we must prove the uniqueness of q as expressed in the theorem. Suppose there is a Q' satisfying (i) and (ii) such that there is no c > 0 and no q such that 0 < q < 1 and for all i Q'(A;)

Let

=

«:':'.

Q'(A n) = ql Q'(A n-1 ) = qz·

524

P. SUPPES

Set q

=!Q. ql

qf

c =-q2

Obviously, Q'(A n ) = cq Q'(A n- 1 ) =

Cq2.

On our supposition about Q', let i be the largest integer (of course i ~ such that

n)

We have whence by (ii) and so Q(A j ) =

C~+1-f,

contrary to hypothesis, and the theorem is proved. 3. Structural axioms for radioactive decay One of the best-known physical examples of a probabilistic phenomenon for which no one pretends to have a deeper underlying deterministic theory is that of radioactive decay. Here I shall consider for simplicity a discretetime version of the theory which leads to a geometric distribution of the probability of decay. Extension to continuous time is straightforward but will not be considered here. In the present context the important point is conceptual, and I want to minimize technical details. Of course, the axioms for decay have radically different interpretations in other domains of science, and some of these will be mentioned later. In a particular application of probability theory, the first step is to characterize the sample space, i.e., the set of possible experimental; outcomes, or as an alternative, the random variables that are numerical-valued functions describing the phenomena at hand. Here I shall use the sample-space approach, but what is said can easily be converted to a random-variable viewpoint.

525

OBJECTIVE PROBABILITY

From a formal standpoint, the sample space X can be taken to be the set of all infinite sequences of O's and L's containing exactly one 1. The single 1 in each sequence occurs as the nth term of the sequence representing the decay of a particle on the nth trial or during the nth time period, with its being understood that every trial or period is of the same duration as every other. Let En be, then, the event of decay on trial n. Let Qn be the event of no decay on the first n trials, so that n

Qn =

,UEi • 1=1

The single structural axiom is embodied in the following definition. The axiom just asserts that the probability of decay on the nth trial given that decay has not yet occurred is equivalent to the probability of decay on the first trial. It thus expresses in a simple way a qualitative principle of constancy or invariance of propensity to decay through time. DEFINITION 3. Let X be the set of all sequences of O's and l's containing exactly one 1, and let \3' be the smallest a-algebra on X which contains the algebra of cylinder sets. A structure fl{ = <X, \3', ~ > is a qualitative waitingtime structure with independence of the past iff fl{ is a qualitative conditional probability structure and in addition the following axiom is satisfied for every n, provided Qn-l > 0: Waiting-time Axiom. EnlQn- l '" E 1 •

The structures characterized by Definition 3 are called waiting-time structures with independence of the past, because this descriptive phrase characterizes their general property abstracted from particular applications like that of decay. For the particular application we are holding in mind, the single structural axiom could just as well have been baptized the Decay Axiom. The great simplicity of this single structural axiom may be contrasted with the rather involved axioms characteristic of subjective theories of probability. In addition, the natural form of the representation theorem is different. The emphasis is on satisfying the structural axioms-in this case, the waiting-time axiom-and having a unique parametric form.

>

REPRESENTATION THEOREM. Let fl{ = <X, \3', ~ be a qualitative waitingtime structure with independence of the past. Then there exists a probability measure on \3' such that the waiting-time axiom is satisfied, i.e.,

(i)

526

P. SUPPES

and there is a number p with 0 < p ~ 1 such that

(ii) Moreover, any probability measure satisfying (i) is of the form (ii). PROOF: I first note that the events En uniquely determine an atom or possible experimental outcome x of X, i.e., for each n, there is an x in X

such that En = {x},

a situation which is quite unusual in sample spaces made up of infinite sequences, for usually the probability of any x is strictly zero. If E l ~ X, then peEl) = 1, and the proof is trivial. On the other hand, if X> E l , then for each n, 0 such that (Let i' be the numbering in the reverse order (n, ... , 1), then i' = n - (i -1), and the exponent n + 1- i in the representation theorem becomes i'.) Starting with a fixed standard sequence of length n, P' can be extended to every i > n in an obvious fashion. The next step is to extend P' as an additive set function to all atoms E i by the equations P'(Ei) = P'(Qi-l -Qi) = P'(Qi-l)-P(Qi)

OBJECTIVE PROBABILITY

527

= C[qi-l_qi]

= cqi-l(l_q). The consistency and uniqueness of this extension is easy to check. Now 00

L: P'(E i=l

i)

= C,

so we set c = 1 to obtain a measure P normed on 1 from P' and let p = 1- q. We then have P(Ei) = p(l-pY-I P(Qi) = (l_p)i. The set function P just defined is equivalent to a discrete density on X, since E, = {x} for some x in X, and thus P may be uniquely extended to the a-algebra of cylinder sets of X by well-known methods. Finally, the proof of (ii) is immediate. If we suppose there is an n such that peEn) "# p(l_p)n-l, where p = peEl), we may use the waiting-time axiom to obtain a contradiction, for by hypothesis P(Qn-l) = (l_p)n-t and P(EnIQn-t) = p, whence peEn) = pP(Qn-l) = p(l_p)n-l. 4. Independence and randomness

Two fundamental aspects of probability as used in science have not been covered by the axioms introduced thus far. One is independence of repetitions of an experiment, and the other is randomness of outcome. Of the two, randomness is the more important and more difficult concept from a philosophical point of view. Some of the reasons for its importance have been stated in the introductory section. The treatment I give here is not complete, and I hope to give a deeper analysis in the near future. On the other hand, independence is of great importance in its own right. Some form of independence assumption seems essential to the theory of any phenomena. If events in a given space-time region were not ordinarily independent of most events in remote regions, systematic scientific theories would be practically impossible. To express these two additional axioms, we need to expand the underlying sample space by taking Cartesian products-a product n times for n repetitions of the experiment. The necessary axioms hold for each dimension-each repetition-and the product space as well. The structural axioms in general hold only for the individual experiment and do not necessarily make sense for arbitrary events in the product space.

528

P. SUPPES

I shall not give a formal definition of the qualitative probability produc. structure. The ideas are familiar in the literature of probability theoryt The product structure f!(11 = (XlI, ty", ~,,> is the n-fold Cartesian product of a structure f!( = (X, ty, ~ as characterized in Definition 1. In formulating the axiom of independent repetitions, which expresses the concept of independence, I use the notation AI) for an event Al on the ph repetition. More importantly,

>

I use the notation in such a way that Ai} and A lk are possibly different events in the product space, but they differ only in the trial (j or k) on which event Al occurs. AXIOM OF INDEPENDENT REPETITIONS.

AI)I

If

n Ai~ > 0 then

k.pj

n AI~ ,..., Air

k.pj

By taking Ai = X as we choose it is easy to show the independence of AI) from any subsequence of repetitions. It follows easily from this axiom that given a probability measure on the product space, then

To formulate an appropriate qualitative axiom of randomness the natural thing is to use DE FINETTI'S beautiful theory of exchangeable events (1937). The qualitative axiom I give expresses exactly his idea of exchangeability, but the interpretation is now that the axiom should hold for empirical phenomena that are held to be random. To express the axiom in a simple way, let a be a permutation of the first n positive integers. Then as a matter of notation aU) is the number j is permuted to. To distinguish events in the structure f!l = (X, ty, ~ I use superscripts, e.g., Apl and Apl.

>

AXIOM OF RANDOMNESS.

For all events AI) and permutations a,

Detailed discussion of whether this axiom captures fully our intuitive sense of randomness must be left for another occasion. It does seem to be a necessary axiom in the sense of following from any of the standard quantitative theories of randomness.

OBJECTIVE PROBABILITY

529

References DE FINETTI, B., 1937, La prevision: Ses lois logiques, ses sources subjectives, Annales de l'Institut Henri Poincare, vol. 7, pp. 1-68. English translation in Studies in subjective probability, ed. H. E. Kyburg, Jr., and H. E. SmokIer (Wiley, New York, 1964) KRANTZ, D. H., R. D. LUCf, P. SUPPES and A. TVERSKY, 1971, Foundations 0/ measurement, vol. 1 (Academic Press, New York) POPPER, K. R., 1959, The propensity interpretation 0/ probability, British Journal for the Philosophy of Science, vol. 10, pp. 25-42

MACROTHEORIES AND MICROTHEORIES P. ACHINSTEIN The Johns Hopkins University, Baltimore, Md., USA

• 1. Introduction The physical sciences are rife with examples of the sorts of theories of concern to me here. Whether these afford philosophical meat for chewing is another matter. But at least let us be clear about examples. JOSEPH BLACK (1803, I, p. 76) in his Lectures on the Elements of Chemistry made many important claims about heat. He noted the "tendency of heat to diffuse itself from any hotter body to the cooler around, until it be distributed among them, in such a manner that none of them are disposed to take any more heat from the rest." He also noted that the quantity of heat in a body should be distinguished from its temperature, and that when two liquids of different temperatures are mixed, heat is neither created nor destroyed during the mixing. It is only after Black has noted these and many other facts about heat that he goes on to raise the question of what sort of thing heat is, e.g., whether it is a fluid or a motion, the two leading theories of his day. Black opts for a version of the fluid theory, although he is quite uncertain about it. At any rate, a distinction can be drawn here between two kinds of statements. There are statements which attempt to cite some of the properties of heat without indicating what the physical composition or mechanism of heat is, for example, 'heat tends to diffuse from hotter to cooler bodies', 'heat is conserved'. And there are statements that purport to say what the physical composition or mechanism of heat is. The postulates of the caloric theory, which Black tended to favor, say that heat is a fluid which is attracted by ordinary matter, which is indestructable and uncreateable, and which can combine with ordinary matter to vaporize a substance. As a second example consider some of Stephen Gray's work on electricity in the eighteenth century. Gray discovered several new materials

534

P. ACHINSTEIN

that could be electrified by rubbing. He discovered that the "electric virtue" can be communicated from rubbed glass to other bodies with which it is in contact and which would then have the attracting power. He also discovered that the "electric virtue" can be communicated from one body to another by means of an intervening body connecting the two. Gray, however, proposed no theory concerning the physical composition of electricity; he did not say whether it consisted of a fluid, a set of particles, or what. This is to be contrasted with the case of physicists in the mideighteenth century who held theories about the physical composition of electricity, for example, the two-fluid theory. Gray's hypotheses about electricity committed him to no such theory concerning its nature. There are numerous examples of theories of these two sorts, and I shall cite just one more. In geometrical optics we learn that light travels in straight lines, that it is reflected by a surface between two media in such a way that the angle of incidence is equal to the angle of reflection, and that it is refracted in accordance with Snell's law. Many facts about images can be explained using these assumptions of geometrical optics without invoking a theory regarding the physical composition of light, e.g., without answering the question of whether light is composed of particles or whether it consists of a wave motion in some sort of matter. ' Let me now draw this distinction somewhat more generally, albeit in a preliminary and tentative way. There are groups of statements which introduce some X -heat, electricity, light-and state a number of its physical properties without indicating or implying anything about its physical composition, mechanism, or structure. The claim that X has the properties attributed to it by such a theory seems to have some more or less direct basis in experiment or observation. I shall call these macrotheories of X. There are other groups of statements whose aim it is to postulate a physical composition, mechanism, or structure for the X introduced in a macro theory, and to do this in such a way as to provide a means for explaining why X's manifest the properties described in the macrotheory. Such theories do not seem to have the same basis in experiment or observation that macrotheories appear to have. I shall call them microtheories of X. Let me emphasize that I do not take such characteristics as constituting definitions for these types of theories. They are far too vague for that. Moreover, by paying further attention to examples we may discover characteristics that are more important than these. Nor is it my claim that the terms 'macrotheory' and 'microtheory' have standard uses in science

MACROTHEORIES AND MlCROTHEORIES

535

or in the philosophy of science which it is our job to expose. My strategy is to begin with examples of the kinds of theories I have in mind, and to appropriate the terms 'macrotheory' and 'microtheory' to refer to these examples. Those I have mentioned, as well as others that I would put in the same categories, are not very much different from ones that are often cited as falling into distinct categories, even though what I have to say about them may turn out to be different from what is usually said. I do not think I am atypical in proposing to put classical thermodynamics, the gas laws, geometrical optics, electrostatics, and classical mechanics into one category, and statistical mechanics, atomic theory, nuclear theory, caloric theory, and the wave and particle theories of light into another. The important point is that I begin with a list of examples of two sorts of theories-a list which I believe can be extended. Whether we use the terms 'macrotheory' and 'microtheory' or some others is of little concern to me. My aim is to explore this distinction by whatever name it is called. 2. Why study this distinction?

Suppose then we try to provide a philosophical clarification of this distinction. What would we expect the payoff to be? What problems within the philosophy of science would be clarified and possibly resolved? Perhaps none at all. Perhaps it is sufficient for the philosopher of science to characterize kinds of theories in science even if no special philosophical problems are thereby raised or solved. Some quite respectable people have defended this descriptive role of the philosopher. On the other hand, I believe there are philosophical issues for which a study of these two types of theories is relevant. E.g., there have been eminent thinkers who have claimed that theories of one type or the other are illegitimate in science. Mach and Ostwald in the ninteenth century and Duhem in the early twentieth century claimed that theories I have called microtheories should be eschewed altogether if such theories are understood in a realistic manner. DUHEM (1914, p. 32), e.g., distinguishes the "representative" part of a theory, by which he means the experimental laws exemplified in observable phenomena, from the "explanatory" part which "proposes to take hold of the reality underlying the phenomena" by postulating an unobserved material realm to explain the experimental laws. This distinction seems to correspond more or less to our distinction between macrotheories and microtheories, or at least macrotheories are included among Duhem's 'representative' part and microtheories among his 'explanatory' part

536

P. AClllNSTEIN

(see Section 9). Duhem's philosophical view is that questions conerning the existence of a "material reality distinct from sensible appearance" (p. 10) have no basis in experimental method, but belong to the realm of metaphysics. The only legitimate way to construe an 'explanatory' theory is not as a description of reality but simply as a group of sentences from which deductions can be made that can be compared with experiments. There was a similar metaphysical fear of microtheories on the part of certain positivistically inclined philosophers later in the twentieth century, and a similar instrumentalist approach was advocated. On the other side of the coin, there are those who claim that macrotheories without microtheories are illegitimate. How can the postulation of an X by a macrotheory explain anything unless the physical nature of X is given, i.e., unless a microtheory of X, at least at some minimal level, is presupposed? Suppose we explain why two bodies are at the same temperature by saying that there is an equilibrium of heat between them. If nothing is said about what sort of item heat is-a fluid, a motion, or what-then does the proposition 'there is a heat equilibrium between these bodies' say any more than the proposition 'these bodies are at the same temperature'? Is not postulating heat to explain equality of temperature like postulating the dormitive virtue to explain why opium is soporific? Either the latter is saying something silly like 'opium puts you to sleep because it puts you to sleep'; or it is saying something mysterious and metaphysical like 'opium puts you to sleep because it has an occult power to produce sleep'. This type of explanation, notorious because of its ridicule by Moliere, is an example of a type of explanation by appeal to a theory of forms that was accepted by many until the early seventeenth century. According to this theory the question of why a substance has a certain property, e.g., why gold is yellow, is to be answered by appeal to the fact that gold contains the form of yellowness implanted in it. Treating heat as a form within matter which is responsible for temperature changes is a version of this theory, which numerous philosophers and scientists from the seventeenth century onward rejected. So one question an examination of macro- and microtheories should help to answer is whether, and if so how, theories of both sorts can be legitimate. Do critics of either type of theory have any justified basis for their criticism? There are other issues such a study should help clarify. Scientists often find a microtheory more satisfying than a macrotheory in the sense that when they have a macrotheory of X they tend to search for a microtheory of X. Why do they do so? Perhaps this feeling of satis-

MACROTHEORIES AND MICROTHEORIES

537

faction is misguided, an expression of some metaphysical bias. There have been scientistis who failed to follow such a procedure. In order to explain the laws of reflection and refraction of light, which constitute a macrotheory of light, Fermat appealed not to a microtheory which would describe the physical nature or composition of light but to a more general macrotheory which would not. He assumed that light, whatever its physical nature or composition, moves in a path for which the time of traversal is a minimum. From this principle it is possible to explain why light obeys the laws of reflection and refraction as well as the principle of linear propagation. Nor did Fermat have or seek a microtheory of light on the basis of which to explain his principle of least time. Rather he based his principle on what he regarded as the fundamental axiom that nature does nothing in vain, that it always takes the simplest course, an axiom that certainly does not constitute a microtheory of light (see SABRA, 1967, Ch. 5). Was Fermat wrong or misguided in doing what he did? Should he have sought a microtheory of light, and if so why? Microtheories figure prominently within a host of problems that have made philosophical headlines recently-problems of theoretical identity, theoretical interpretation, and reductionism. It might be said that a microtheory asserts that some X described in a macrotheory is identical with some physical system described by that microtheory, e.g., heat with caloric fluid. It also asserts that X's having property P-a macro claim-is theoretically interpretable as Y's having Q-a micro claim. What do such assertions amount to? Is theoretical identity strict identity subject to the logician's standard axioms, or something weaker? Must identity claims be implied by a microtheory? Are macrotheories always in principle reducible to microtheories, and what would this mean? Or, as FODOR (1968) and others have argued, are there certain types of macrotheories that are not so reducible, and what would that mean? I do not pretend to have answers to all or even most of these questions, nor am I claiming that a study of macro- and microtheories will automatically provide them. But it might help, and anyway, as with Reichenbach's fisherman, a necessary condition for success is that we cast our net somewhere. So let us consider what can be done to provide some deeper clarification of these types of theories.

538

P. ACHINSTEIN

3. A linguistic characterization It might be thought that one should be able to characterize them according to the types of statements they contain. Those who find the logical positivist viewpoint attractive will probably suggest that statements in a macrotheory are observation statements containing only 'observational' terms while those in a microtheory are theoretical statements containing nonobservational or so-called 'theoretical' terms as well. Even if the positivist's desired distinction between theoretical and observational terms could be drawn-and recently there have been many doubts expressed on this score-this proposal is not promising. Numerous terms in macrotheories are terms that appear on positivists' theoretical lists, terms like 'light ray', 'heat', and 'electricity'. This point needs some emphasizing. The kinds of theories I am classifying as macrotheories do not contain only terms like 'red', 'hard', and 'longer than' Sk)

(2)

to be read: 'the probability that a physical system belonging to the H-class and acted upon by a stochastic state changer of class A makes a transition from state So to the state Sic. Here, H stands for the Hamiltonian or any other property characterizing a class of like or similar systems having at least the same state algebra-a technical term to be explained later. However, since the numerical value of (2) depends only on the two states, we may use the short form

prob, (So -> Sk)

(3)

or-to stress the analogy with (1)(4)

prob 2(So; Sk)'

This of course puts a rather heavy burden on the semantics of 'So': 'So' now represents a virtual ensemble of like or similar systems with at least the same state algebra, all in the same state So. The choice of form is of course largely a matter of convention or rather convenience, but it has far-reaching consequences for the prob, calculus. Indeed, if we choose form (2) we can formulate the correct axiom prob 2(H, A; So

->

Sf)

=0

for A ¥= B

(5)

which cannot be formulated in terms of (3) or (4). Instead of the axiom we must then use a formative rule stipulating that expressions of the form (6)

are inadmissible. This makes prob, a function over a partial Boolean algebra.

609

PROBABILITY IN PHYSICS

The need for partial Boolean algebra in prob, theory arises directly from the existence of two or more complete sets of states, as may be seen from the following. Let LA. = {Sf} and L B = {Sf} be two complete sets of states so that

prob, (So;

V Sf) (i)

= prob, (So; V Sf) = 1. (k)

(7)

If we combine this with the addition theorem prob 2(So; Sf or Sf) = prob 2(So; st)+prob2(So; Sf) we obtain

prob 2(So; V Sf or \/ Sf) (i)

(k)

=

2.

(8) (9)

This paradox can only be avoided if we drop equation (8), and this implies that the compound expression 'Sf or Sf' is an inadmissible argument of the prob, functor. In other words, prob, must be a function over a partial Boolean algebra. To be sure, the paradox (9) does not arise if we use form (2) together with axiom (5). This, however, can not be taken as an argument in favor of (2) since the generalization of standard logic to a logic with partial Boolean algebra is in any case unavoidable as will be explained later on. Still, what we meet here is an instance of a fairly general metalogical theorem which I shall not try to formulate exactly but which would state some sort of equivalence or translatability of different formalizations involving different logics. (Instances of this are known from the field of mathematical logic). Another instance is the translatability of REICHENBACH'S (1935) three-valued logic into a system with standard logic; for the proof I refer to STRAUSS (in press).

4. proh, theory-general formulation In the following I presuppose the short form (4) of the prob; expression and hence partial Boolean algebra for the arguments of the prob, functor. The connective symbols of partial Boolean algebra will be marked by a superimposed dot, e.g., 0, to distinguish them from the corresponding connectives of Boolean algebra. To conform with the modern method of presenting prob, theory, the theory will likewise be built up in three steps. The first step gives a sort of algebraic model of a supposed real structure. In the second step this algebraic structure is represented by a mathematical space. The third step introduces the numerical measure function.

610

M. STRAUSS

To stress the analogy with prob, theory still further I shall quote the current metaprob, terminology, misleading though it is. To start with, instead of 'field of events' C with Boolean a-algebra we have a field of states S with partial Boolean algebra. To every 3 belongs the impossible state with

°

8 0

°=

8

for all 8

E

3

(10)

8 A I = 8

for all 8

E

3.

(11)

and the sure state I with Further, to every state 8 E 3 there exists exactly one complementary state S E 3 satisfying 8 A S = 0, 80S = I. (12) I must warn you that these conditions do not make the dotted bar a proper negation: they would do so only if we had a Boolean algebra. A state 8 ::/= is called a sharp or atomic state iff the equation

°

(13) has only the trivial solutions (i) 8 1 = S2 = S,

(ii) 8 1 = S, S2 = 0,

(iii)

s,

= 0, S2 = S.

(14)

To be sure, for quantum mechanical systems and also for all known games of chance all states actually occurring are sharp states, but this need not be presupposed in the general theory. Even if it were presupposed, other states such as (13) have to be considered in the theory. Finally, we postulate that there exists in 3 at least one n-tuple of sharp states Sl' ... , S; with (i) S, A 8 k =

°

for i ::/= k, i, k = 1, ... , n = I,

(ii) 8 1 0 8 2 0 '" 0 S;

(15)

Such an n-tuple of sharp states will be called a basic set of states. The number n is called the dimension of the field 3. The number f of different basic sets in 3 will be called the freedom number of 3; it corresponds, in a logical though not in a physical sense, to the number of degrees of freedom of a classical system. (For a die we have n = 6 andf = 1.) A field of states 3 statisfying the above axioms is called a state algebra; it corresponds to the 'algebra of events' in prob, theory. Thus, a state algebra can be represented by the symbol Note: the numbers n andf refer to reality, not to mathematical constructs.

3m.

PROBABILITY IN PHYSICS

Sm

611

By definition, a state algebra contains at least nf sharp states; it will be called subnormal iff it contains only nl sharp states; otherwise it will be called normal. Note that the states of a die form a normal state algebra (with 1= 1 and n = 6), the seventh state So being the state of the falling die. The state algebra of a Markoff chain is a subnormal state algebra with 1= 2 and arbitrary n. It is subnormal because the initial state is identical with a state belonging to a basic set so that there are only In sharp states. For a quantum mechanical system we have a state algebra with I equal to aleph, and n equal to aleph. . A state algebra will be called logically degenerate iff the paradox presented above [equation (9)] cannot be construed even when the algebra is supposed to be Boolean. It follows:

Sm

1. A state algebra is logically degenerate iff either (i) 1, or (ii) it is subnormal and f = 2.

THEOREM

1=

Thus, the familiar games of chance and the Markoff chains have logically degenerate state algebras. This explains why their partial Boolean character remained undiscovered. We now come to the second step. In prob, theory the second step is based on a theorem by STONE (1936) stating that to every 'algebra of events' ; there exist an isomorphic set algebra or 'measurable space' [Q, £] such that the elements of ; are related one-to-one to those of £, £ being the system of subsets of Q. Similarly it may be shown or taken from quantum mechanics that to every state algebra sen) there exists an n-dimensional linear vector space v(n) with metric M such that there is a one-one correspondence between the elements of sen) and the elements of a subset of urn), u Zz, ... be an unbounded lineage of machines such that Z, produces Zi+!, i.e., [ZI](X) = c(zz), [zz](x) = C(Z3), ... (cf. Definition 7). Furthemore, we require that the machines differ from each other in some novel way such that the lineage may be considered evolutionary. Now, if this lineage is going to be effectively explicable, in spite of its eventual nonpredictability, there must be a recursive explanation function g(i) such that g(i) = Zi> i.e., [g(y)](x) = c(g(y+ 1». It can be proved that

FORMAUZABILITY OF LEARNING AND EVOLUTION

657

there are recursive g-solutions, i.e., effective explanations, to this functional equation. This is true even if we require some non periodic structural changes in the lineage (see LOFGREN, 1972). Just as we could specify a normal behavior to a self-reproducing organism in Section 4, we can make a similar specification here. Thus the partial evolution equation: [g(y)](x) = i(C(g(y+ 1»), !(Y, x»)

may be interpreted as follows. The function value g(y) is the code number of a Turing machine, Gy , which produces the genetic information c(g(y+ 1») of another Turing machine GY +1 and besides computes the normal function !(Y, x). Again, GY+1 is reproductive and computes a new normal function !(y+ 1, x), and so forth. In Lofgren (1972) we prove that with every partial recursive function (normal behavior)f(y, x) there exists a recursive explanation function g(y), which satisfies the above partial evolution equation, even with the requirement of a nonperiodic structural change of the Gl-machines. This result makes it possible to meet with reasonable novel-criteria, and still have the evolution process explicable (cf. LOFGREN, 1972). For example, the novelty may consist in being able to solve a previously unsolvable problem. This criterion was originally suggested by MYHILL (1964) in connection with an essentially self-reproducing automaton structure. Myhill also suggests the possibility of encoding a potentially infinite number of directions to posterity on a finitely long chromosomal tape. 8. Can hidden mutation rules be found in biological evolution? As we have seen in the last two sections, automaton aspects suggest that evolutionlike phenomena can be programmed in the code that transmits structural information from one generation of automata to the next. Are there biological theories of evolution with explicatory powers such that phenomena of this type can be translated into them? It would seem that recent biological findings concerning repair enzymes in procaryotic cells (see for example HANAWALT & HAYNES, 1967)may account for one such translation. Certain DNA-strands appear to be encoded to direct the production of an enzyme, which to a certain extent repairs DNA-errors. Such a strand can be said to contain a partial program of evolution in the following sense. The repair enzyme cannot possibly repair all errors, but will take care of some, which hence may be classified as

658

L. LOFGREN

'undesirable' errors. Other errors will pass, and among them will be the 'desirable' mutations. Hence, due to the DNA-program, the probability of certain changes (mutations) will be higher than the probability of others, and we may speak of a partially programmed evolution. Although a formal theory of partially programmed evolution processes may explain certain observed phenomena, we can always ask questions beyond the reach of such a theory. For example, how did such programs arise? Are there metalaws or metaprograms for the development of evolutionary programs, or do we have to answer, less explicably, by reverting to randomized variation and a natural selection of evolutionary programs? References AHO, A. and J. ULLMAN, 1968, The theory of languages, Mathematical Systems Theory, vol. 2, pp. 97-125 DAVIS, M., 1958, Computability and unsolvability (McGraw-Hili, New-York) HANAWALT, P. and R. HAYNES, 1967, The repair of DNA, Scientific American, vol. 216, pp. 36--43 HEssE, M., 1972, Models of theory-change, this volume LEE, C., 1963, A Turing machine which prints its own code script, in: Proc. Math. Theory of Automata (polytechnic Press, Brooklyn), pp. 155-163 LOFGREN, L., 1967, Recognition of order and evolutionary systems, in: Computer and Information Sciences II, ed. J. Tou (Academic Press, New York), pp. 165-175 LOFGREN, L., 1968, An axiomatic explanation of complete self-reproduction, Bulletin of Mathematical Biophysics, vol. 30, pp. 415--425 LOFGREN, L., 1969, Relative recursiveness of randomization and law-recognition, Notices of the American Mathematical Society, vol. 16, PP. 685 LOFGREN, L., 1972, Relative explanations of systems, in: Trends in General Systems Theory, ed. G. Klir (Wiley, New York) MAYR, E., 1963, Animal species and evolution (Harvard University Press, Cambridge) MYHILL, J., 1964, The abstract theory of self-reproduction, in: Views on General Systems Theory, ed. M. Mesarovic (Wiley, New York) ROGERS, H., 1967, Theory of recursive functions and effective computability o(McGrawHill, New York) THORPE, W., 1966, Learning and instinct in animals (Methuen, London) Ttrnrxo, A., 1939, Systems of logic based on ordinals, Proceedings of the London Mathematical Society, vol. 45, pp. 161-228 WADDINGTON, C. (ed.), 1968, Towards a theoretical biology 1. Prolegomena (Edinburgh University Press, Edinburgh) WADDfNGTON, C. (ed.), 1969, Towards a theoretical biology 2. Sketches (Edinburgh University Press, Edinburgh) WARREN, J., 1957, The phylogeny of maze learning. I. Theoretical orientation, British Journal of Animal Behaviour, vol. 5, pp. 90 WOODGER., J., 1937, The axiomatic method in biology (Cambridge University Press, Cambridge)

ORGANIZATIONAL PRINCIPLES FOR THEORETICAL EMBRYOLOGyl

M. A. ARBIB University of Massachusetts, Amherst, Mass., USA

One mechanism whereby a tissue may change its form is that of the autonomous change in cell shape. For example it is now well known that various microstructures may be synthesised within cells during characteristic changes of shape, and that their destruction impairs such changes. Thus cells seem able to elongate themselves by producing microtubules aligned parallel to the axis of elongation. Again, cells seem able to constrict a portion of themselves by producing microfilaments which can then contract to provide the constriction by a sort of 'purse-string effect' (BAKER & SCHROEDER, 1967). SCHROEDER (1970) has combined such mechanisms to provide an elegant model of neurulation-the process whereby a plate of cells on the back of the embryo is formed into a trough which then rolls up into a tube running the length of the embryo to then disappear beneath the surface of the back and form the rudiments of the spinal cord and brain. A crude caricature of the mechanism is shown in Figure I-the reader will find a more subtle and careful treatment in Schroeder's paper. Another mechanism whereby a tissue may change its form involves the combined effects of cellular adhesiveness and cellular motility. Such 1 This paper presents a number of points that were delivered at the International Congress on Logic, Philosophy and Methodology of Science, Bucharest, Rumania, August 30, 1971. A full version has appeared under the title "Organizational Principles for Theoretical Neurophysiology" in "Towards a Theoretical Biology; 4 Essays" (C. H. Waddington, Ed.) Edinburgh Univ, Press, 1972, pp. 146-168, and will be expanded upon in the book "Patterns and Models in Developmental Biology", by D. A. Ede and M. A. Arbib, Academic Press (to appear). Preparation of this report was supported in part by the United States Public Health Service under grant No. NS 09755-01 COM from the National Institute of Neurological Diseases and Stroke, and in part by the University of Massachusetts under Faculty Research Grant SI2-7I(1).

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a mechanism helps us understand situations in which the attachments of cells change over time, but where there seem to be important specificities in the ensuing pattern of cellular attachments. GUSTAFSON and WOLPERT (1967; for an exposition see also WOLPERT & GUSTAFSON, 1967) have given a masterly analysis of cellular movement and contact in sea urchin

" '--y---J II I I I I I

j

constriction of apices of these cells yields a

sh~ugh

II

kQw5~

~ elongation of these cells yields deepening of the trough

III

~

r(ff~:~mn

lspreading of these tissues pushes lips of groove together to yield a tube

IV

'

6 ~

Fig. 1. Dangerously oversimplified schematic of neurulation. The 4 stages are not chronological. Rather, each of the 3 transitions schematises a mechanism (there are. others) found by SCHROEDER (1970) to playa role in forming the neural tube. All views are in crosssection.

morphogenesis. EDE and AGERBAK (1968) have been able to correlate changes in adhesiveness of cells (and the consequent change in their motility) in normal and talpid" mutant chick embryos with changes in the developing limit pattern in these embryos, while EDE and LAW (1969) have expressed this correlation in the specific form of a computer simulation of limb development. While elegantly showing how changes in cell shape, motility or adhesiveness can provide mechanisms for morphogenesis-both in the nervous

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system and elsewhere-the above schemes do not make explicit how a cell 'knows' what contribution it is to make in the overall pattern. It is for this reason that other workers have developed the idea of 'positional information'. Here, the line of argument runs 'If the cell is to change appropriately it must have information about its position within the embryo (and perhaps it will need to consult a clock, too)'. An early approach to such positional information was in gradient theory (e.g., CHILD, 1941)-ifa source of some metabolite were located at one end of the axis and a sink at the other, with a uniform gradient in between, then the concentration of metabolite in any cell would signal its position on the axis. WOLPERT (1969) has suggested ways in which such a model needs refinement and elaboration, and GOODWIN and Courx (1969) have instantiated Wolpert's ideas in a model in which position is signalled by the phase differences between families of pulses propagating with different delays from cell to cell. By contrast, automaton theorists have shown how cells may be formed into complex arrays without explicit 'addressing'. Rather, each cell is capable of a finite number of states, and at any time the cell changes state in a way dependent upon its previous state and that of its neighbors. For example, VON NEUMANN (1966) exhibited a self-reproducing array with tens of thousands of components, but the cells were only capable of 29 states, and so could not 'know where they were'. ARBIB (1967) has attempted to place this approach in a more biological context. The work of APTER (1966) should also be mentioned here. Other authors have compared the change of state rules used by von Neumann and others to the rewriting rules employed by linguists to 'grow' a sentence from its grammatical description, and are now exploring the applicability of formal linguistics to theoretical embryology (LAING, 1969; LINDENMAYER, 1968). In considering the specificity of cellular connections, we must not be misled by estimates that the amount of information in DNA is far less than that contained in the connections of the brain, which some have taken to imply that connections in the brain must be random. To see this, consider the following computer program which comprises four instructions: 1. Set n equal to zero. 2. Print out n. 3. Replace n by n+ 1 4. Return to the second instruction. If you observe a computer executing this program, it will emit a stream of numbers which is endless-at least till you have exhausted the capacity of the computer. Arguing that a comparison of the number of DNA

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bases with the number of connections in the brain shows that the brain must be a random network is as naive as comparing the four instructions of the above program with the number of positive integers and concluding that the sequence of positive integers is a random sequence! In other words, one of the things we know from our study of programming computers to do clever things is that our programs have loops within them which are hierarchically structured to provide for a great deal of economy in the way we specify processes. As a biological example of a plausible 'use' of such 'nested subroutines', we may cite the retina of the frog. The connections between the interneurons of the second layer in the retina and the ganglion cells which send their output down the optic tract to the brain have been schematized by LETTVIN and MATURANA (LETTVIN, MATURANA, MCCULLOCH & PITTS, 1959) as falling into two or three segregated layers. A plausible wiring scheme would then prescribe that certain types ofaxons from the interneurons terminated in one layer and so are highly likely to connect one level of the dendrites of the ganglion cells while other types ofaxons bearing different transforms of the visual input would terminate in the other layer thus hitting other parts of the ganglion cell dendrites. By this means, one can very simply specify how to get a retina that would function perfectly for the frog trying to snap flies in its world, without having to specify point-by-point interconnections. Hence, a sort of 'nested subroutine' approach could probably explain a great deal of the specificity of the nervous system without requiring an immense investment in genetic material. References APTER, M. J., 1966, Cybernetics and development (pergamon Press, Oxford) ARBIB, M. A., 1967, Automata theory and development, I, Journal of Theoretical Biology, vol. 14, pp. 131-156 BAKER, P. C. and T. E. SCHROEDER, 1967, Cytoplasmic filaments and morphogenetic movement in the amphibian neural tube, Developmental Biology, vol. 15, pp. 432-450 CHILD, C. M., 1941, Patterns and problems of development (University of Chicago Press, Chicago) EDE, D. A. and G. S. AGERBAK, 1968, Cell adhesion and movement in relation to the developing limb pattern in normal and talpid" mutant chick embryos, J. Embryol. Exp. Morph., vol. 20, pp. 81-100 ED!!, D. A. and J. T. LAW, 1969, Computer simulation of vertebrate limb morphogenesis, Nature, vol. 221, pp. 244-248 GooDWIN, B. C. and M. H. COHEN, 1969, A phase-shift model for the spatial and temporal organization of developing systems, Journal of Theoretical Biology, vol. 25, p. 49 GUSTAFSON, T. and L. WOLPERT, 1967, Cellular movement and contact in sea urchin morphogenesis, Biological Reviews, vol. 42, pp. 442-498

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JACOBSON, M., 1970, Developmental neurobiology (Holt, Reinhart and Winston, New York) LAING, R., 1969, Formalisms for living systems, University of Michigan, Report 08226-8-T LETTVIN, J. Y., H. R. MATURANA, W. S. McCULLOCH and W. H. PITrs, 1959, What the frog's eye tells the frog's brain, Proceedings of the Institute of Radio Engineers, vo\. 47, pp. 1950-1959 LINDENMAYER, A., 1968, Mathematical models for cellular interactions in development, Journal of Theoretical Biology, vol. 18, pp. 280-299, pp. 300-315 SCHROEDER, T. E., 1970, Neurulation in Xenopus laevis. An analysis and model based upon light and electron microscopy, J. Embryol. Exp. Morphol., vol. 23, pp, 427-462 TRlNKAus, J. P., 1969, Cells into organs: The forces that shape the embryo (prentice-Hall, Englewood-Cliffs, New Jersey) VON NEUMANN, J., 1966, Theory ofself-reproducing automata (U. of Illinois Press, Urbana) (edited and completed by A. W. Burks) WOLPERT, L., 1969, Positional information and the spatial pattern of cellular differentiation, Journal of Theoretical Biology, vol. 25, pp. 1-47 WOLPERT, L. and T. GUSTAFSON, 1967, Cell movement and cell contact in sea urchin morphogenesis, Endeavour, vol. 26, pp. 85-90

POLAR ORGANISMS WITH APOLAR INDIVIDUAL CELLS

G. T. HERMAN State University of New York at Buffalo, Amherst, N.Y., USA

1. Introduction

This paper is intended to demonstrate how a particular automatontheoretical model for cellular interactions in development, the one originally proposed by LINDENMAYER (1968), can be used to answer a particular problem in biology. The problem that we shall be concerned with is whether polarity of individual cells is a necessary condition for the regulatory polar development of organisms. We quote SINNOT (1960) to explain the notion of polarity in biology. "A notable feature of these bodily forms of plants (and animals) is the presence in them of an axis which establishes a longitudinal dimension for organ or organism. Along this axis, and symmetrically with reference to it, the lateral structures develop. The two ends or poles of the axis are usually different both as to structure and physiological activity" . "This characteristic orientation of organisms, which is typically bipolar and axiate, is termed polarity". "Polarity is simply the specific orientation of activity in space. It refers to the fact that a given biological event, such as the transfer of material through an organ or the plane in which a cell divides, is not a random process but tends to be oriented in a given direction. If this were not so, an organism would grow into a spherical mass of cells, like tissue in a shaken culture". An example given by SINNOT (1960) refers to some original experiments of Yachting. "YaCHTING (1878), cut twigs of willow and kept them under moist conditions. Some he left in their normal, upright orientation and others were inverted. Regardless of orientation, however, roots tended to be regenerated more vigorously from the morphologically basal end and

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shoots from buds at the original apical end. This is the classical example of polarity. If such a shoot were cut into two or more parts transversely, each part regenerated roots and shoots in the same polar fashion. Even very short pieces of stem showed this polar character. Yachting removed a ring of bark in the middle of a shoot and confirmed earlier observations that roots were formed above the ring and shoots below, just as if the stem had been cut in two. From these and similar experiments he concluded that polarity was a fixed and irreversible characteristic of the plant axis and that probably the individual cells of which the axis was formed themselves possessed a polar character". Our major concern will be, whether such a conclusion about the polar character of the individual cells is justified on the basis of the experiments. In other words, we shall investigate whether the kind of global behavior observed in the experiments can be achieved with individual cells which are apolar. What we shall describe now is a summary of HERMAN (1971, 1972). The interested reader will find a more detailed discussion in those papers. For the individual cell, polarity can have two kinds of manifestation: 1. External polarity, e.g., transportation of certain substances may take place in one direction and not in the other. 2. Internal polarity, e.g., unequal division, the daughter cells of a cell may have different characters. In this paper, we shall incorporate the notions of external and internal polarity into the automaton-theoretical model of Lindenmayer, and then investigate whether the kind of behavior, which led people to believe that there is polarization of individual cells, can be achieved with cells which are apolar. Our major result will be that complicated polar regulatory behavior can be achieved with apolar individual cells, and so the argument, which concludes the polarity of individual cells from certain kind of polar behavior of the organism as a whole, is false. This is not to say that we claim that cellular polarity never plays a role in achieving global polarity of the organism. This mayor may not be the case. What we shall show is that one cannot argue that cellular polarity is necessarily present from observations of the kind described by Sinnot. 2. Definition of Lindenmayer models If G is a nonempty finite set, G* denotes the set of all strings of elements of G. G* includes the empty string, which is denoted bye.

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Given a string p E G*, pR denotes the string which we obtain by writing elements of p in reverse order. A Lindenmayer model L is a quadruple ,

3 :::;;; r :::;;; x,

, 1:::;;; s < s+2:::;;; x.

Assuming that a cut changes the first component of the state of the cell on either side of it to i, the intended meaning of the definition above is the following. A filament is a legitimately altered subfilament, if it is at least of length 3, and it has been obtained by a series of cuts from an originally disturbed filament, the only restriction being that the first cut can only be performed after a time equal to the length of the filament.

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Using this terminology, we can now state the final condition which L must satisfy. (D) If p is a legitimately altered subfilament of length x, then there exists a to such that, for all t ~ to, )..f(p)

= (r,0,m t>(r,0,m2> ... (r,O,m u>(w,O,n t>(w,O,n 2> ... (w, 0, nv> cb, 0, II> (b, 0,12) .. , (b, 0, Iw>,

where 13u-xl ~ 2, 13v-xl ~ 2 and 13w-xl ~ 2. This says that any legitimately altered subfilament will turn into a French flag with the same orientation as the original filament would have done if no cutting took place. (Note again that the symmetry of the model, and of the definition of a legitimately altered subfilament, guarantees that a property analogous to (D), with the right-hand end initially disturbed, will be automatically satisfied.) HERMAN (1972) gives a symmetric propagating Lindenmayer model L, which has the properties (A)-(D). Thus, regulatory polar behavior can be achieved with apolar individual cells. For lack of space we cannot reproduce here the details of the solution. 5. Conclusion We have offered two pieces of evidence of the power of symmetric Lindenmayer models: universal computing ability and solvability of the French flag problem. Other evidence can also be given, e.g., the solvability of the firing squad synchronization problem (see HERMAN. 1972). This demonstrates that an argument for the polarity of individual cells on the basis of regulatory polar behavior of the organism as a whole is unacceptable. In a larger context, we have demonstrated the applicability of cellular automata to an honest-to-goodness biological problem. This is not an isolated instance. In HERMAN (1972), the relevance of the solution to the French flag problem given there is discussed, considering such biological notions as mosaic development, positional information, gradients, etc. We wish to conclude with the words of LONGUET-HIGGINS (1969), the truth of which is further supported by this paper. "We are beginning to realize that the interest of an organism lies, not in what it is made of, but in how it works". "The most fruitful way of thinking about biological problems is in terms of design, construction and function, which are the concrete

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problems of the engineer and the abstract logical problems of the automata theorist and computer scientist". "If-as I believe-physics and chemistry are conceptually inadequate as a theoretical framework for biology, it is because they lack the concept of function, and hence of organization ... This conceptual deficiency of physics and chemistry is not, however, shared by the engineering sciences; perhaps, therefore, we should give the engineers, and in particular the computer scientists, more of a say in the formulation of Theoretical Biology". References ARBrB, M. A, 1969, Self-reproducing automata-e-some implications for theoretical biology, in: Towards a Theoretical Biology, vol. 2, Sketches, ed, C. H. Waddington (Aldine, Chicago), pp. 204-216 ARlnB, M. A, 1972, Automata theory in the context of theoretical embryology, in: Foundations of Mathematical Biology, vol. 2, ed, R. Rosen (Academic Press, New York, N.Y.), pp. 141-215 HERMAN, G. T., 1969, The computing ability of a mathematiial model for filamentous organisms, Journal of Theoretical Biology, vol. 25, pp. 421-435 HERMAN, G. T., 1970, The role of environment in developmental models, Journal of Theoretical Biology, vol. 29, pp. 329-341 HERMAN, G. T., 1971, Models for cellular interactions in development without polarity of individual cells, Part I: General description and the problem of universal computing ability, International Journal of System Sciences, vol. 2, pp. 271-289 HERMAN, G. T., 1.972, Models for cellular interactions in development without polarity of individual cells, Part II: Problems of synchronization and regulation, International Journal of System Sciences, vol. 3, pp. 149-175 LrNDENMAYER, A, 1968, Mathematical models for cellular interactions in development, Part I: Filaments with one-sided input, Part II: Simple and branching filaments with two-sided inputs, Journal of Theoretical Biology, vo!' 18, pp. 280-299; pp, 300-315 LoNGUET-HrGGINs, C., 1969, What biology is about, in: Towards a Theoretical Biology, vol. 2, Sketches, ed. C. H. Waddington (Aldine, Chicago), pp. 227-232 MINSKY, M. I., 1967, Computation: Finite and infinite machines (prentice Hall, Englewood Cliffs, N. J.) ROSEN, R., 1970, Dynamical system theory in biology, Vol. 1: Stability theory and its applications (Wiley-Interscience, New York) ROSEN, R., 1971, Some comments on the concepts of regulation and positional information in morphogenesis, International Journal of System Sciences, vol. 2, pp, 325-335 SINNOT, E. W., 1960, Plant Morphogenesis (McGraw-Hili, New York) VOcHrING, H., 1878, Uber Organbildung und Pfianzenreicb (Cohen, Bonn) WEBSTER, G., 1971, Morphogenesis and pattern formation in hydroids, Biological Reviews, vol. 46, pp. 1-46 WOLPERT, L., 1968, The French jfag problem: a contribution to the discussion on pattern development and regulation, in: Towards a Theoretical Biology, Part I: Prolegomena, ed. C. H. Waddington (Edinburgh University Press, Edinburgh), pp. 125-133

CELLULAR AUTOMATA, FORMAL LANGUAGES AND DEVELOPMENTAL SYSTEMS A. LINDENMAYER University 0/ Utrecht, Utrecht, The Netherlands 1. 'Cellular automata' are formal constructs which in the last decades came into increasing use as biological models. The first two, and still important, cellular-automaton constructs with biological motivation were the nerve-net models of MCCULLOCH and PITTS (1943) and tge self-reproducing cellular automata of von Neumann (published posthumously by BURKS, 1966). Nerve-net models have been constructed to exhibit various kinds of simple behavior like reflex loops or the looming response of frogs. Selfreproducing cellular automata are meant to demonstrate how very complex structures can be constructed from relatively simple components and be able to replicate themselves. The term 'cellular' refers in this case to the subunits with which this construction is carried out, and does not imply an analogy of these subunits with cells of living organisms. In fact, the whole self-reproducing structure might be considered to be analogous to a single living cell. Both of these models consist of a number of interconnected simple units. These units are switching elements in the model of McCulloch and Pitts, and input-output devices with a small number of internal states, but without any other additional memory, in the case of von Neumann's system. Both of these kinds of units are examples of what are now called 'finite automata', i.e., machines with finite number of states, and sequential generation of outputs as determined by the sequence of inputs and the configuration of states. The interconnections of the units in the first model can be arbitrary in form and richness, while in the second one the units are placed in a rigid square grid, and are connected in uniform, time-invariant way with neighboring units, e.g., to exactly four neighbors, in the von Neumann model. While McCulloch's and Pitts' constructs were subsequently shown (KLEENE, 1956) to be not any more powerful computing machines than their own

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subunits the finite automata, the construction of von Neumann turned out to be equivalent in computing ability to Turing machines, i.e., to the most general and powerful computing devices. This difference in computing power between the two kinds of models is a consequence of the fact that the first ones are of stable size, they consist of a constant number of units and interconnections, while the latter ones can expand, and computation can be carried out with increasingly larger numbers of subunits. Cellular automata of both kinds have been further investigated and their biological usefulness has been commented upon by several workers (ARBIB, 1969a, 1969b, on the use of cellular ways in simulating regeneration and on self-reproducing systems in which the neighbors of a cell are allowed to change; CODD, 1968, on simplified models of self-reproducing automata; VITANYI, forthcoming, on sexually reproducing automata; several of these and others being also mentioned in BURKS, 1968). 2. Cellular-automaton models have also been introduced (LINDENMAYER, 1968) with reference to the development of organisms, rather than their self-reproduction, or their nervous behavior.' In this case, the cells of the model are assumed to correspond to individual cells of growing multicellular organisms. Living cells are well established as autonomous units of metabolism, heredity and physiological function. In these models cells are, in fact, construed to be finite automata, that, in any moment may be present in one of finite number of states, can receive one of a finite number of inputs from neighboring cells and give rise to one of finite number of outputs, the outputs in turn serving as inputs to other cells. As in all such models, time is to pass in discrete steps, and whether a cell changes its state, divides, or dies at the next moment is determined by its present state and by the present inputs it receives. The outputs of cells at each moment are also determined by their states and inputs. Since cells can divide or die, the number of cells may increase or decrease from moment to moment. Thus one can obtain a series of cellular arrays, each corresponding to momentary developmental stage of an organism, beginning with a single cell, the fertilized egg, and increasing or decreasing in size. Furthermore, morphogenesis is simulated by the production of stable patterns in the array, provided that certain cellular states are such that they change no further under any input combinations.

a

1 I would like to add here that W. Ross ASHBY (1956) was among the first to enunciate clearly the method of construction for interacting automata and lowe his book a debt for learning the basic principle of this method. I am also indebted to John R. Gregg for his having introduced me to automata theory in the first place.

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It may well be asked whether these kinds of automaton-theoretical models can at all be interpreted in terms of the biochemical mechanisms supposed to underlie development and morphogenesis. GOODWIN (1970) posed this question in the following way: "The implication of the computer analogy is that the cell computes its own state, looks at the DNA program for further instructions, and then changes state accordingly. This is not in fact what a cell does, although formal analogy can be made between the biochemical behavior of a cell and the operation of an automaton following a program. It may seem elementary to insist that all the operations of the automaton must at some point be interpreted in biochemical and physiological terms, when discussing such a process as epigenesis," but I have been somewhat dismayed at the amount of confusion that has arisen because of the failure of those using the computer analogy to illustrate the operation of algorithmic instructions at the biochemical level". In fact, the concept of finite automaton is general enough t~ give it the desired interpretation in biochemical and cell-physiological terms, one needs only to elaborate it somewhat. We may safely assume first of all that the same genes (or operons) are present in every cell of a certain organism, and that each gene may be present in an active or a passive form (masked or unmasked) and thus at each moment in each cell there is a particular combination of active and inactive genes. We can assume furthermore that in each cell the active genes produce specific enzymes, which under the proper conditions are able to convert various metabolites into other compounds, some of which accumulate in the cell, resulting in its differentiation. The change of particular genes from inactive to active form, or vice versa, is presumably determined by the product (an inducer) of some enzyme, together with a repressor protein. Whether the inducer -comes from the same or another cell, we observe the induction of the synthesis of a new enzyme, which then might result in differentiation, or in the initiation of a new cell line. In a very short summary, these are the currently accepted hypotheses concerning differentiation and induction, and they represent an important portion of the mechanisms underlying development (leaving aside for the time being the spatial orientation of cell division and enlargement which represent another important aspect). We recognize of course that many different control mechanisms may be operating on the synthesis and activity of various enzymes (as, e.g., discussed by WADDINGTON, 1968); 2 Epigenesis is meant to include both morphogenesis and differentiation, for further explanation see, e.g., WADDINGTON, 1970.

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nevertheless, all of these controls can be expressed as either turning on and off the genes, or as affecting the activities of the enzymes and thus the appearance of their products. Now as for the automaton-theoretical connections of all these statements. Let us first consider all metabolites and other cellular components, except for the active proteins and nucleic acids (informational macromolecules), and call the set of all the species of such components C. At each moment in a given cell there would be present (in significant amounts) a particular combination of the elements of C, i.e., the cell can be associated with a particular subset of C. Let us further consider each gene and the enzyme it gives rise to as a transformation rule, transforming certain combinations of cell components (elements of C) into other components, and call the set of all such transformation (or production) rules P. In each cell at a given time there will be a certain combination of genes active, concomitantly there will also be the corresponding enzymes active, i.e., the cell will possess productions representing a subset of P. To each cell at each moment one can thus assign a state, consisting of a particular subset of C and a particular subset of P. The inputs of a cell consist of the compounds (elements of C) which have entered the cell during the last time interval, and the outputs of the cell consist of the compounds which have left it (in significant amounts). 3 The next state of the cell will depend entirely on its present state and the inputs it receives, according to some specifiable functions. First, one needs a function which determines the next subset of P, i.e., the combination of active and inactive genes (and their enzymes) at the next moment, including also the possibility that two new subsets of Pare to be specified (the cell divides), or no new subset of P is specified (the cell dies). The combination of cell components at the next moment will be determined by the present combination of components, by the input to the cell, and by the subset of P applicable to the cell at the time (the elements of P being rules which specify the interconversion of components). If the cell divides, it has to be further specified whether the cell components are 3 WALTER R. STAHL (1966) has introduced the concept of 'gene-enzyme automata' in his computer model of a self-reproducing cell, and I would like to acknowledge the influence of his paper in construing here genes and enzymes as production rules. I had many interesting discussions with him, and I felt his early death a great loss. A theoretical biologist of very broad scope, he was also among the first to be concerned with computational complexities of cells and organisms, as compared for instance with Turing machines.

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to be divided equally or unequally between the daughter cells (one more function is needed in the latter case). The output of a cell, i.e., the compounds which diffuse or are transported from a cell into the neighboring cells, will depend only on the components present in the cell (assuming that it is known which cell components are transportable). If an inducible permease is involved in the transport of a compound, then the production of the permease is controlled as that of an enzyme. It can be seen that both the next-state and the output functions can in principle be constructed for a given population of cells, thus the cells can be validly represented by finite automata. I cannot agree, therefore, with Goodwin's objections. What he calls the 'DNA program: is the set of production rules we introduced, and according to our short discussion above it can be said in a completely natural way that the next state of a cell is computed by using these production rules." The only serious objection against these sort of mathematical constructs, that I can envision, is an objection against their discreteness. It is quite true that not only time and space appear here in discrete units (the time steps and naturally the cells themselves are these units), but the enzymes, metabolites and inducers must also be considered as discrete entities, being either present in concentrations above certain thresholds or not. The genes are of course always present in integer units anyway. Having to deal with discrete temporal and concentration parameters is clearly a disadvantage of this approach one must recognize. This is counterbalanced, however, in my view, by great conceptual and practical computational advantages, some of which will presently become more apparent, I hope. 3. As for our developmental models, up to now we have investigated primarily one-dimensional cellular arrays. Each cell in the array is conDr. B. Goodwin has commented on this passage in the sense that his remarks were primarily directed against the position expressed by C. LONGUET-HIGGINS (1969, p. 231) as follows: "Not until we can interpret the DNA of a new species without actually growing an individual from it will we be able to claim a full understanding of epigenesis". The term 'DNA program' is also from the article of Longuet-Higgins. Goodwin has objected to its use, and wished to state that this view is misleading and erroneous, while he considered my exposition in the present paper an acceptable formulation. In spite of this willingness of Goodwin to agree with me, there still seems to persist a gap between our positions, since I do not find the views of Longuet-Higgins at all erroneous, only in need of amplification. In fact, I attempted to provide in this paper just that. 4

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strued to be a finite automaton with uniform sets of states and next-state functions throughout the array. For the sake of mathematical simplicity, we have considered states and outputs to be identical and consequently the inputs to be the states of neighboring cells. Under these simplifying assumptions, a filamentous developmental system (called a Lindenmayer model by HERMAN, 1969, 1970, or an L-system by VAN DALEN, 1971) is a construct consisting of a finite set of states, a next-state function, and an initial array. This construct is formally somewhat analogous to grammars, as the term is used in formal language theory (see, e.g., AHa & ULLMANN, 1968). There the set of states is called an alphabet, the next-state function a set of productions, and the initial array an axiom. Just as one can obtain new cellular arrays from previous ones by using the next-state function, so in language theory one obtains new strings of symbols by the use of production rules. But the difference between our constructs and grammars lies in the fact that (a) in our case all cells in the array must simultaneously be transformed at each step; in other words, all symbols in the string must simultaneously undergo substitution, according to the production rules; and (b) in our constructs no distinction is made between terminal and nonterminal symbols as is done in the case of grammars. In formal language theory a language is defined as the set of all terminal strings, generated from the axiom by the production rules (terminal strings are string-composed only of terminal symbols). Analogously, we may define developmental languages (or L-Ianguages) as the set of all strings generated from the axiom by simultaneous application of production rules to all symbols in each string (no restriction needed concerning terminal or nonterminal symbols). An L-system may either be deterministic or nondeterministic, depending on whether for every state symbol there is exactly one next-state "transition rule (production) or not. If a system is deterministic, then there will be a single sequence of arrays (strings) generated by the rules beginning with the initial array (axiom); in nondeterministic systems many such sequences may be produced, possibly diverging and converging at various points. Each such sequence of strings is supposed to correspond to the entire life of an organism, each string being a momentary description of it. A deterministic system permits only one such developmental sequence to occur, while a nondeterministic one yields many possible life histories. A further division of L-systems is that into propagating and nonpropagating ones, meaning thereby systems, respectively, without production

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rules resulting in cell deaths, and with such rules. Such rules correspond to erasing in language theory, i.e., to productions which erase particular symbols. The presence of erasing rules in grammars is well known to give rise to much more complex languages. Similarly, preprogrammed cell death has been recognized by developmental biologists (e.g., SAUNDERS, 1966) to be an important mechanism in many morphogenetic processes. A third, and equally important distinguishing notion for L-systems is the size of the neighborhood of the cells. One can namely consider systems in which each cell acts independently of all other cells, i.e., systems with context-free productions, and others in which each cell receives inputs from a certain number of its neighbors on either side in the filament, in other words, systems having productions with varying lengths of context. In general one may speak of L-systems with k left and I right inputs, meaning inputs received by each cell from k left and I right neighbor cells. The biological interpretation of such contexts was already given above, i.e., the transport of metabolites or other biologically active molecules from cell to cell. If one wishes to allow the transport to take place in one direction only, as in the case of auxin being transported from apex to base in stems of higher plants, then one needs only one-sided (undirectional) context. The higher the number of neighbors contributing to the context of a cell in one direction, the faster is the transport of the active substance along the filament (relative to the length of the time steps). 4. The simplest of the L-systems in terms of context are the ones without any interaction among the cells, called 0 L-systems, and these have been the subject of most work so far. Some of the results obtained as well as formal definitions of OL-systems and languages are given in the Abstracts of this Congress by DOUCET (1971) and VAN LEEUWEN (1971). Two simple examples for OL-Ianguages are: (a) The set of strings consisting of the letter a of lengths 2ft , for all nonnegative integers n. This language is generated by the single production a ~ 00, from axiom a. (b) The set of strings, for all integers n ~ 1, consisting of n-times the symbol a, followed by the symbol c or nothing, followed by n-times the symbol b. This language is produced by the productions: a ~ a, b ~ b, c ~ acb, c ~ empty string; from axiom acb. It is interesting to note that the language consisting of n-times a followed by n-times b symbols, a well-known context-free language, cannot be generated by OL-systems. It can be generated by L-systems with one-sided inputs. Various theorems (due to DOUCET, 1972; HERMAN, forthcoming; LIN-

684

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DENMAYER, 1971; ROZENBERG, 1970, 1971a; ROZENBERG & DOUCET, 1971; SALOMAA, manuscript; VAN DALEN, 1971; VAN LEEUWEN, 1971) deal with the relationship of 0 L-languages to the languages of the Chomsky hierarchy, of which the main classes are in order of increasing complexity: regular, context-free, context-sensitive, recursive, and recursively enumerable languages (see, e.g., AHO & ULLMAN, 1968). It can be proven that all OL-Ianguages are context-sensitive. In fact, the set of 0 L-Ianguages intersects the set of regular, context-free and contextsensitive languages, but does not exhaust any of them (thus, there are regular languages which are not OL-Ianguages). The usual closure properties (under set-theoretical union, intersection, etc.) do not hold for OL-Ianguages. OL-systems which have identity productions for all symbols in their alphabet (thus for every symbol a, there is a production a --+ a) produce context-free languages only. Among the above-mentioned formal results perhaps only the last one has biological significance. It shows, namely, that the fact that certain OL-systems are able to produce non-context-free languages (for instance the language simulating the development of the red alga Callithamnion roseum, LINDENMAYER, 1971) is due primarily to the synchronous cell divisions and changes of state imposed by the production rules of these systems. As soon as the synchrony requirement is relaxed by adding identity production rules for all symbols in these systems, the resulting language becomes context-free. Another biologically interesting theorem was found by VAN LEEUWEN (1971) stating that for every OL-system G, such that every symbol in its alphabet either leads to the empty string or to a cycle but not both, there exists a propagating 0 L-system G', and a nonerasing homomorphism h such that the language generated by G is identical to the h transform of the language generated by G' (with the addition of the empty string to the latter, should the empty string be in the language of G). For example, the language (the set of all strings) consisting of ab repeated all possible finite numbers of times can be generated by an OL-system, but cannot be directly generated by a propagating 0 L-system. A nonpropagating OL-system generating this language is the one with the productions: a --+ ab, a --+ abab, b --+ empty string; with the axiom: abo The same language can be obtained by the use of a nonerasing homomorphism h, such that h(a) = ab, from a language produced by a propagating OL-system, namely, the one with productions: a -+ a, and a -+ aa, and with axiom: a. Thus developmental patterns which are produced by systems without cellular

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interactions and for which preprogrammed cell death appears to be necessary, may also be produced by a system without cell death, provided that a biologically interpretable homomorphism can be found. Still another biologically promising aspect of OL-systems has been the study of 'locally catenative' systems (ROZENBERG & LINDENMAYER, forthcoming). In this case we are concerned with propagating, deterministic OL-systems which give rise to development such that certain previous developmental stages appear as part of the whole organism at a later time. One frequently sees such cases in the development of compound leaves and branching structures. Formally, we are dealing here with sequences of strings such that (after an initial period) each string consists of the concatenation of strings which were previously encountered in the sequence. It was possible to find a certain property (called dependence) of the set of productions of propagating, deterministic OL-systems which entails the generation of such locally catenative sequences. This property has to do with the existence of certain types of cycles among the states of the system. On the other hand, given any locally catenative formula, one can always find a propagating deterministic OL-system which gives rise to a sequence satisfying that formula. Thus the observation of a locally catenative developmental pattern in nature ensures the existence of a propagating deterministic system without cellular interactions which can give rise to such a pattern. Of course, the fact that such a no-interaction system could exist does not mean that in the actual case no interactions are taking place, since every pattern which can be produced without interactions can also be produced by a system with interactions. Nevertheless, a biologist might find it useful to know that such seemingly very complex patterns can arise by development without cellular interactions and without cell death, and thus he does not need to search for interactions or occurrences of cell death to explain the underlying developmental mechanisms. From a biological point of view an important result would be an explicit characterization of OL-Ianguages or OL-sequences. Algebraic characterizations are available at present among the Chomsky languages for the class of regular languages only (the class which can be recognized by finite automata or by the McCulloch-Pitts nerve-nets). Partial characterizations are available for the set of context-free languages, in the sense, that given a context-free language one knows certain properties it must have, but given an arbitrary language with those properties it does not follow that it is context-free. Once even a partial characterization (or algorithmic procedure) would be available for the recognition of OL-Ianguages, one

686

A. LINDENMAYER

would be able to tell from the developmental description of an organism or of an organ whether it is necessary that cellular interactions occur in the course of its development. This is so, because the partial characterization would provide us with a formal property for which the developmental history of the organism could be tested (just as mentioned above, in the case of the locally catenative property for certain sequences), and if the organism does not fulfill the required property, then it would follow that its development could not have taken place in the absence of cellular interactions. At the present time characterization is available only for OL-Ianguages over one-latter alphabets (ROZENBERG & DOUCET, 1971; VAN LEEUWEN, ROZENBERG & HERMAN, in preparation), but further work is in progress on characterizations of deterministic OL-Ianguages. An extension of OL-systems has been proposed by ROZENBERG (forthcoming), in which the productions are organized into tables, each table containing at least one production for each symbol, and requiring that in anyone string only productions of a single table may be used. These kinds of developmental systems, which still have no interactions among cells, but in which the entire organism may be subjected to sudden changes of programs, are particularly suited to simulate developmental processes controlled by some critical value of an environmental parameter. We may think of the sudden switch from vegetative to flowering condition in higher plants as the length of daylight (or length of night) exceeds a certain threshold. A simple table OL-model for such a change in a filamentous fungus from vegetative to reproductive state under the control of light has been presented before (SURAPIPITH & LINDENMAYER, 1969). Although more powerful than regular OL-systems, table OL-systems were shown to be properly included in the set of context-sensitive languages. 5. As to L-systems with inputs, the first point to be mentioned is that they exhibit a dramatic rise in complexity. The results obtained by HERMAN (1969), ROZENBERG (1971b) and VAN DALEN (1971) show that L-systems with inputs from single cells from one side only (called lL-systems) can generate nonrecursive languages, i.e., languages of the highest complexity class in the Chomsky hierarchy. With proper coding lL-systems can simulate any Turing machines. Nevertheless, if one considers the languages which can be directly generated by lL-systems, i.e., without any coding or canonical extensions, then it turns out that there are many regular languages which cannot be gener-

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ated by them. Thus, the set of 1L-languages intersects all the major Chomsky-classes of languages, but does not include any of them. The set of 1L-languages includes, of cours~, the set of OL-languages, and it also includes the set of all finite languages. Beginning with OL- and 1L- languages, one can build an entire hierarchy of L-languages, according to the number of inputs coming to a cell from either side, each set of L-languages with a higher total number of inputs (from both sides) containing the sets with lower total numbers of inputs. But no matter how high numbers k of left inputs and l of right inputs we choose, there always remain certain regular languages which cannot be generated by a (k+/) L-system. The generation of finite languages is of particular biological interest, since it corresponds to development of organisms or organs with a final (adult) stage. This is what is called determinate growth in plants, of which leaf growth is a good example. Among animals the occurrence of an adult stage with a final size and form is widespread. It is important, therefore, to ask what kind of developmental programs would be necessary to yield such terminating growth and development, i.e., a sequence in which strings would eventually occur repeatedly. Curiously enough, while finite languages are considered to be of the lowest complexity type in formal language theory, the production of many of these languages by propagating Lsystems requires a high degree of interactions (large contexts). An attempt has been made to find general conditions for the production of symmetric terminating growth patterns with reference to the development of leaf marginal meristems (LINDENMAYER, paper in preparation), and it was shown that such a pattern can be generated by a propagating L-system with two-sided inputs which is completely apolar. (In a polar system the productions distinguish one side from the other either by giving rise to different states under mirror images of inputs, or by giving rise to two cells in unequal states under any combination of inputs. An apolar system is one which is not polar.) The most interesting aspect of developmental systems with cell interactions is their ability to simulate regeneration. To regenerate a complete organism of the original size after amputating a part at a previous stage, or to reorganize a cut off part to form a complete organism reduced in size, are processes which definitely require interactions among cells. An abstract statement of the latter case is the 'French flag problem' proposed by WOLPERT (1969). Systems with two-sided inputs, capable of accomplishing this kind of regeneration have been constructed by us as well as by

688

A. LINDENMAYER

ARBIB (1969a) and HERMAN (1971, 1972). The latter investigator has even given an apolar L-system which exhibits this kind of regeneration (this work is described elsewhere in these proceedings). A general theory of developmental systems with regeneration properties is, however, lacking. In the previous discussions -(except for that of table OL-systems) it was always assumed that the environment remained constant during the course of development, and we asked what kinds of patterns can be produced by what class of systems. However, if the environment does not remain constant but fluctuates in some unknown way, then one is faced by an entirely different sort of construction and classification problem. HERMAN (1970) has begun to work on this topic, as well as on the problem of constructing L-systems to fit incomplete developmental data (see also FELICIANGELI & HERMAN, forthcoming). 6. There have been other developmental models based on cellular automata published recently, which differed from: the ones discussed here either (a)jby being more than one dimensional, or (b) by using continuous variables (concentration, age, distance) as well as discrete variables, or by both respects (a) and (b). For instance, BAKER and HERMAN (1972) studied the distribution of heterocysts along filaments of blue-green algae by a one-dimensional model in which the concentration of an inhibitor (which can diffuse from cell to cell) and the age of the cell determined cell division and the transition from vegetative to heterocyst condition. Another kind of process which can be considered in a somewhat extended one-dimensional framework is the development of branching structures. Models giving rise to branching patterns and based on L-systems have been studied by HOGEWEG and HESPER (1971); and probabilistic cellular models for the production of branching patterns were given by COHEN (1967), who also considered the interaction of branches in terms of exhaustion of nutrients in their surroundings. Cellular-automaton models for two- or three-dimensional development have also been investigated, such as the model of RAVEN (1968), and RAVEN and BEZEM (1971), for the early embryonic development of the pond snail Limnaea stagnalis, in which the sizes and positions of the cells are algorithmically computed after each cleavage; or the cellular modelfor avian limb development by EOE and LAW (1969), in which cell movements, in addition to cell divisions, are programmed in order to simulate in two dimensions the morphogenesis of limb initials; or a cellular model for the origin of

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regularly spaced leaf primordia (phyllotaxis) on a cylinder surface, controlled by the diffusion of an inhibitor (LINDENMAYER & VEEN, unpublished manuscript). There is, at the present time, no general theoretical framework within which we could handle the various cellular-automaton models, some of which are of more than one dimension, and some of which use continuous parameters as well. Clearly, the morphogenetic role of spatially directed cell divisions, cell enlargement, and cell movements can only be investigated if such models are further developed. At the present time there is also a lack of clearly elaborated connection between the biochemical and physiological mechanisms of development which we mentioned earlier and the automaton models described afterwards. Although we maintained that in principle such a connection is possible to find, detailed interpretations have not yet been attempted, and their testable consequences have not been obtained. These shortcomings of cellular-automaton theories of development are both serious. But the results obtained so far could provide certain automaton-theoretical and formal linguistic interpretations for a number of fundamental concepts of developmental biology, such as differentiation, induction, regeneration, cellular interactions, polarity, and the roles of synchrony, cell death, and environment in development. There clearly remains a great deal more of the theoretical framework to be constructed before we can speak of a 'theory of development'. References AHa, A. V. and J. D. ULLMAN, 1968, The theory of languages, Mathematical Systems Theory, vol. 2, pp. 97-125 Aaars, M. A., 1969a, Self-reproducing automata-Some implications for theoretical biology, in: Towards a Theoretical Biology, vol. 2, ed. C. H. Waddington (Edinburgh University Press), pp. 204-226 Anura, M. A., 1969b, Theories of abstract automata (Prentice-Hall, Englowocd Cliffs, New Jersey) AsHBY, W. R., 1956, An introduction to cybernetics (Chapman and Hall, London) BAKER, R. and G. T. lIERMAN, 1972, Simulation of organisms using a developmental model, International Journal of Biomedical Computing, vol. 3, pp. 201-215, 251-267 BURKS, A. W. (ed.) 1968, Essays on cellularautomata (University of Illinois Press, Urbana) CODD, E. F., 1968, Cellular automata (Academic Press, New York) COHEN, D., 1967, Computer simulation of biologicalpattern generation processes, Nature, vol. 216, pp. 246-248 DOUCET, P. G., 1971, Some results on Ol.-languages, Abstracts of IVth International Congress on Logic, Methodology and Philosophy of Science, Bucharest, pp. 87-88

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DOUCET, P. G., 1972, On the membership question in some Lindenmayer systems, Indagationes Mathematicae, vol. 34, pp. 45-52 EDE, D. A. and J. T. LAW, 1969, Computer simulation of vertebrate limb morphogenesis, Nature, vol. 221, pp. 244-248 FELICIANGELI, H. and G. T. HERMAN, Algorithms for producing grammars from sample derivations: A common problem of formal language theory and developmental biology, Journal of Computer and System Sciences (forthcoming) pBIS.

16 We assume for the sake of simplicity that D and A are mutually exclusive. If they are not exclusive, but S, A and Dare mutualIy statisticalIy independent according to the scheme:

we will also have a positive association between Sand B, because here: pBIS =pD+pA-pD'pA pBIS = pA

and because (PD-pD' pA) > 0

then (pD+pA-pD' pA) > pA

therefore we will always have here pBIS > pBIS.

CONDmONAL CAUSAL RELATIONS

777

We can summarize it by saying that if all components of all alternative sufficient conditions of our effect are mutually statistically independent, this will result in positive association between any of the essential components of any of sufficient conditions and their effect. This seems to be the necessary assumption in the everyday research practice, which takes for granted that the causal relations between variables can be assessed by the observation of positive associations and correlations between them. But this assumption should be clearly formulated in any inference from statistical to causal relations, especially because it is by no means universally valid, as we could see above. Let us discuss finally the last problem of this section of the paper. Until now we were using the term 'probability' as equivalent to the term 'relative frequency'. It is of course not always justified. Suppose that the relative frequency of D when 8 occurs is equal to p in the whole population of 8'S,!7 but the time-space area occupied by the events of type 8 may be divided into two subareas with clear time-space boundaries. In one of these subareas all 8's always occur jointly with D, whereas in the other subarea D never occurs jointly with 8. In this case we would still say that the relative frequency of D in relation to 8 is p in the whole population of 8. But we would not say that there is a certain probability p of D when 8 occurs.l" because the notion of probability usually refers to such situations in which the distribution of D is relatively even throughout the whole population of S's in all the time-space area occupied by the events 8. We would not say either that there is a certain probability of D when 8 occurs even if it is in the case when there is a certain 'even distribution' of D's throughout the all time-space area of occurrence of S's, but where we can discover a definite 'systematic pattern' of relations between S and D. Suppose that each second S, in space or time order, is accompanied by D. In this case relative frequency of B/8 is exactly 0.5, but since we know exactly which of the S'« produces B and which of them does not produce B, we would not use the term 'probability' here because this term 17 In order to make this argument meaningful we must assume that the population of S's is finite, even if a very numerous one. 18 But we could use quite meaningfully the term 'probability' for the situation in which we draw a random sample of 5's from the whole population of 5's. Here we can predict that the probability p, that 5 will occur jointly with D is equal to p. The same applies to the systematic pattern of distribution of D in relation to 5 discussed in the next paragraph. See S. NOWAK (1972).

778

S. NOWAK

refers to the situations in which we are basically uncertain of the occurrence of B in relation to singular cases of S. This uncertainty is usually denoted by the term randomness. Therefore we would say that to justify the use of the term 'probability' for the pattern of joint occurrence of Sand D, the occurrence of D in relation to S must be both relatively evenly distributed and random in its character. Only then will we observe such sequence of events Sand B which may be characterized as strictly probabilistic in terms of frequency theory of probabilityr? i.e., B will occur randomly in relation to S with pDjS as the limiting value of probability of occurrence of B when S has occurred. Therefore only if different elements of the same sufficient conditions and different alternative causes occur randomly with approximately equal relative frequency in relation to each other, they will produce statistical relations, characterized above, which we will call the probabilistic laws of sciences. We can end this section by saying that the statistical-or, more strictly, probabilistic-character of most of the regularities we observe in social reality is by no means incompatible with the notion of causality. As I tried to show above, the idea of conditional causal relations can help us to understand better the causal mechanisms underlying the statistical relations between the phenomena. 4. Historical generalizations as approximations to universal causal laws Until now we have discussed situations in which the conditions on which the generality of the causal relation between Sand B depends exist or occur in relation to Sin such a way that could be characterized as a statisticalor probabilistic pattern of occurrence, and to which the notions of relative frequency or probability could be meaningfully applied. If we exemplify, e.g., our D as a 'disposition to a given behavioral response' one can suppose that a disposition such as 'intelligence' (understood as 19 Von MISES postulates in his frequency theory of probability another necessary condition for the relations which are 'probabilistic'. They should be random not only in the sense that the occurrence of B when S has occurred is indeterminate with p as limiting value of their frequency, but also in the sense that there is no way of changing this frequency limit by taking into account any additional conditions. This requirement is not fulfilled of course in the situations where we will have different alternative causes and different components of one sufficient condition, because taking any additional 'predictor' changes here the probability of occurrence of B when S occurred, which was discussed above. For some solution of this problem see S. NOWAK (1972).

CONDmONAL CAUSAL RELATIONS

779

a genetically determined predisposition to learning and problem solving) exists probably throughout the human race in a way that corresponds to the notion of probability, i.e., which may be characterized as random with relatively equal frequency of occurrence. But this is not the only way in which our D, or alternative sufficient condition A, can occur. I mentioned already in the Section 3 such a situation, when D is the characteristic of all human beings but only within some definite time-space limit H, when outside these limits D does not occur at all. Sociologists could give many examples of such dispositions with a definite 'historical' localization. It might be, e.g., a disposition which is shaped by the cultural heritage of a given nation If and its actual sociopolitical system. Therefore it is common to all, or almost all, members of the nation If in the giventime and does not exist outside these 'historical boundaries' H. Assume now that the causal relation which could be used to explain the occurrence of B has the form S n D = B. 2 0 This relation is of course theoretically general and universal. It applies anywhere and to any time-moment. But its actual operation is limited to time-space area H, because D occurs only there even if S could occur anywhere. Therefore B can occur only within the historical boundaries H. Suppose now that we do not know the importance of the property D for the relation between Sand B, and try to formulate a generalization about the relation of Sand B only. In this situation we will not be able to formulate and to test a universal causal law describing the sequence of occurrences of Sand B, because, as it was mentioned above S may be followed by both B and non-B.21 But, even without discovering the importance of D we can formulate a general historical proposition (NOWAK, 1961) which says that within the time-space boundaries H (in the group population, nation, culture), S is always followed by B. After formulating it we can conduct a series of controlled experiments testing the causal relation S = B, and within the time-space 'coordinates' H, these experiments will fully confirm our causal historical generalization. On the other hand such experiments will fail outside the extension of H, because the condition D does not exist there. The historical coordinates H playa substitutive role in such historical generalizations: they substitute the unknown conditions D of a universal Le., that (S n D) is a necessary and sufficient condition for B, when S is a necessary but not sufficient condition for B. 21 We couId formulate here a universal proposition of a negative kind, because S --+ ii, but this is not point of the discussion here. 20

780

S. NOWAK

conditional causal relation, permitting us to formulate a true general historical proposition limited in its time-space validity. They also inform us where'? our unknown conditions exist, and where they do not exist. This plays an essential role in the strategy of verificational studies aimed toward discovery of these unknown conditions. Suppose now that the causal regularity has such a form that (SliD) uA = B, i.e., that S is only an essential component of one of many alternative conditions, but D still occurs always and only within time-space limits H. Here again we will have a causal generalization S ~ B valid for the population H only. But we will discover here that also outside the boundaries of the population H, S is sometimes followed by B. This occurs of course when S is accompanied by A, which makes this relation-? a spurious one. We may discover its spuriousness by trying to make an experimental test of hypothesis S ~ B outside the time-space area H. The results of experiments will be negative in this situation. This approach permits us to understand better the fact that so many findings in social and behavioral sciences have their validity only for some specific populations, cultures or periods of history. 24 This also explains why so many replications of experimental studies in behavioral sciences fail in other cultures or nations. We can regard the hypotheses tested by them as approximations to some conditional causal laws, the unknown conditions of which are the characteristics of these areas for which they are valid and do not exist outside these boundaries. If we could discover them, i.e., if we substituted the historical coordinates H by a generally defined variable D, we would have a general causal law, the theoretical validity of which would be universal, even if its factual operation was still limited to the population H. Discovery of these conditions is important not only for reasons of 'theoretical elegance', it may also have practical importance, because when we know which conditions are neccessary for the causal sequence S -+ B, we may be sometimes able to create them outside the area H,25 22 I would like to note here that in probabilistic universal laws the value of probability p informs us how often these unknown conditions occur. 23 Which may have approximately even and random, i.e., a strictly probabilistic character if A occurs at random in relation to S. 24 One could be reminded here of behavioral scientists who called modem social psychology 'general theory of verbal behavior of the contemporary American undergraduate'. 25 E.g., by extending the impact of cultural patterns which create D to the new populations.

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781

where they did not exist until now. In this way we can also apply the stimulus S to the stimulation of behavior B in these areas, where until now B would not be produced. But how could the nature of our D be discovered? The answer is very simple, and we can find it in J. St. Mill's canons of induction. We should apply the method of only agreement and try to discover the only common feature of the persons, who are the members of the population H and therefore which is the condition of the sequence S -+ B. We should also apply the method of only differences" and try to find only this feature which makes population H different from all others, and therefore is responsible for the failure of the generalization S -+ B in the populations non-H. But these very simple rules cannot be usually applied in inductive analysis of social phenomena. The cause of that lies in what might be called the syndromatic character of variation of social phenomena. The traits according to which the populations, cultures and epochs in history may differ usually occur and change in 'nonrandom clusters'. When we study one population we find its members similar in many aspects in the same time. We see, for example, that all members of population H have the full syndrome of properties D j , D2 , ... , Dn • When they differ from the members of other populations, they normally differ in many features at once, because the members of other populations may be characterized by the absence of the whole syndrome, i.e., by D1 , D2 , ••• , Dn • The effect of this is that when we are making inductive comparisons of instances within the population H, instead of 'only agreement' we meet 'too many agreements', when we compare the cases from different historical populations, we see too many differences between them. Does it mean that the situation is hopeless for a prospective social theory which would be composed of universal laws? Not necessarily. The idea of conditionality of social regularities may be the source of some important rules of strategy of research oriented toward the discovery of laws of universal validity in our sciences. At first, the syndromes mentioned above are seldom characterized by perfect, universally valid correlation of all features, which are characteristic of the population H. With the increase of number of populations accessible for inductive comparisons and the increase of number of 26 The meaning of these rules is here a little different from the classical formulation of Mill's canons of induction, because the factor D, which they should help us to discover, is not the condition for the occurrence of D, but the condition of validity of regularity S -> D.

782

S. NOWAK

experimental replications in different cultures we can find populations that will be similar to our H on most of the features except one, e.g., D 3 • Knowing that regularity 8 ~ B also fails where D 3 is absent, we may discover the nature of this disposition D 3 and accept it as the condition of our regularity. We may also find populations that differ from on most features except D 3 , and then discover that our regularity 'works' there. To increase the effectiveness of our search and to decrease its costs, the choice of populations for comparative inductive research should be guided, if it is possible, both by some hypotheses on the nature of conditions essential in our regularity, or some alternatives of such possible conditions, and by the knowledge of empirical occurrence of different combinations of these alternative conditions in different populations. In such situations the test of alternative hypotheses would be especially fruitful and efficient. If this method fails, and we are still faced by too many agreements and too many differences, we can try to create the combinations of independent variables necessary for the test of alternative hypotheses by the technique of experimental manipulation. This method is extremely useful in other sciences which also face the problem of syndromatic character of variations of events and properties, especially in biology. The biologist who would like to describe in terms of general properties the conditions that determine a certain regularity 8 ~ B characteristic for all objects of the same species perceives too many features in common for all objects of one species, and too many differences between the species, to evaluate which of them is important as the 'modifier' D ofa given regularity '8 ~ B'. In biology this problem is usually solved by experimental manipulations in vivo, which permit to discover, which features of organisms are essential for the given regularity, and which are irrelevant for it. There is no doubt that this method can be also applied to much greater extent than now in the study of social phenomena. Finally, if we are unable to find proper combinations of independent variables in natural conditions or are unable to create them by experimental manipulations, there is still another way of distinguishing among all variables D 1 , D 2 , ... , D n which jointly characterize the members of the population H, this one (or more than one) which is a necessary condition of the regularity 8 ~ B. We can do it sometimes by trying to explain our conditional regularity 8 ~ B by some more general laws of human behavior. In other words, we can try to reduce it to more general laws and theories.

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783

Such reduction usually permits us to see which of the accompanying factors ••• , D; are essential for the causal sequence S --+ B from the point of view of a more general theory and the existence of which seems to be irrelevant for this regularity. Let us take as an example the propositions which say that the members of the contemporary American working class have a rather strong tendency to react in the 'authoritarian way' in different social situations (UPSET, 1963). This is only a historical generalization of an obviously conditional character and we do not know which of a great number of characteristics of the contemporary American working class are conditions codetermining it. But if we are able to reduce this regularity to a more general psychological theory, e.g., theory of frustration or aggression, we may come to the conclusion (as in LIPSET, 1963) that only these variables are essential in our regularity, which are responsible for especially strong feelings of social frustrations in the American working class today, when all the others are here irrelevant. The reductive systematization of laws and theories permits us not only to explain less general laws by more general ones but also to identify essential variables of less general causal laws in such situations where we have no possibility of discovering their essential conditions by comparative inductive study. In the above analysis we assumed that 'our' D existed as a characteristic of all members of one historical population H. As a result of that we were able to discover a historical generalization S --+ B, even if we did not know the importance of D for the sequence S --+ B. But it may well happen that D exists or occurs in relation to S in a way that corresponds to the pattern of a statistical random •distribution with probability p, but D occurs in that way only in one population H. If this is the case, even without discovering the nature of these conditions, our observations-? permit us to formulate a proposition about the sequence Sand B, which will be both historical and statistical (probabilistic) at the same time. This proposition may be again transformed into a universal law of science if we identify D as a condition which codetermines this regularity. This we can do by testing the hypotheses about its importance either by multivariable statistical analysis or by experimental manipulations, or finally by in reductive explanation. D1 , D2 ,

27 Under assumption that (S n D) = B and we do not know the role of D in this regularity.

784

S. NOWAK

5. Patterns of 'conditional objectivity' of the laws of social sciences If the conditions D, necessary for the causal sequence of events S ~ B have a definite time-space localization H, this results in 'historical' localization of this sequence: it operates only within the time-space coordinates H. Until now we were considering rather the space dimension of this localization and the methods of discovery of the unknown conditions of its validity by cross-cultural comparative induction. If we look closer at the time dimension it now has some additional implications for the character of regularities we observe in the area of social phenomena. The main point is that factor D, necessary for the 'operation' of the conditional causal sequence S ~ B, may occur at a certain point of time and from this moment on D may become a general characteristic of all beings, or at least a general characteristic of large civilizational areas. As OSSOWSKI (1967) pointed oue s important historical events such as the rise of Christianity or the outbreak of the French Revolution may create by its impact some basically new conditions, which are the essential factors of some regularity of human behavior. As one could add here, their impact may be also of a negative character-they may create conditions which 'cancel' factual operation of some regularities of human behavior, which were 'valid' before their occurrence. It means that some regularities of social behavior have such localization of their factual operation that we are able to show a certain moment before which they did not operate or from which they cease to operate. The existence of a certain 'terminus ab quo' or 'terminus ad quem' would be therefore one of the effects of historical localization of conditions, which codetermine the regularities of social behavior or man. There is no reason to expect that all possible 'modifiers' of regularities of human behavior did already occur in the past or do occur in contemporary social situations. We have to face the possibility that some basically new conditions will occur in the future. Remembering that their effects may result either in codetermining some regularities of social behavior, or in 'cancelling' their operations, we must be prepared that: 1. Some of the regularities that, according to our theoretical and em-

pirical knowledge, could not operate until now might still be possible 28 The interpretation of D as 'disposition' presented and discussed above was only one of many different possible 'qualifiers' or 'modifiers' of regularities of human behavior.

CONDITIONAL CAUSAL RELATIONS

785

in the future, if in the course of historical transformations of societies the necessary conditions of their operations will occur. 2. On the other hand, we must also admit the possibility, that the historical change of creating some basically new conditions will 'eliminate' some regularities of social behavior believed until now to be 'iron laws' or social and behavioral sciences. For an observer of factual sequences of events of the type S ~ B it might create the impression of occurrence of a 'basically new' regularity of human behavior, or elimination of a certain 'old' one. When he discovers that the occurrence of these new conditions D, which codetermine the regularity S ~ B is due to (purposeful or not purposeful) action of human beings and whole societies, a philosophically naive observer might come to the false conclusion that man can 'create' laws of social sciences by his purposeful action or behavior. This would be incompatible with the notion of 'objectivity' or 'necessity' of such laws. This is of course not a justified conclusion. Nevertheless we have to admit that some hypotheses about the regularities of human behavior, which we could formulate on the basis of some general theories, e.g., by deriving them from more general laws, may be basically unverifiable or basically unfalsifiable by the known, past and present, empirical data. They must 'wait' for their confirmation or rejection until the course of historical transformations (or purposeful human action) brings to the existence these conditions for which these generalizations are valid. The conditions which determine the validity of certain generalization may be of a different kind. They may be the elements of an external-socialinstitutional or cultural context in which the behavior takes place. They may also be the properties of the behaving persons themselves-their 'objective' or 'psychological' characteristics. Here we should mention one special, but methodologically important, category of such properties as our factor D, which is the condition of validity of certain regularities, or their disturbing 'cancelling' factor, namely, the knowledge about the social reality. Sociologists of ideology and knowledge, e.g., Karl Marx or Karl Mannheim, have shown that knowledge about the society by the members of that society is in itself an acting social force. This problem was also discussed by R. K. Merton in his essay on the 'self-fulfilling prophecy'. By the self-fulfilling prophecy he means that a prediction must be known to the social group if the prediction is to be 'true' for that group.

786

S. NOWAK

Since each prediction implies certain regularity of events, we could present the paradigm of 'self-fulfilling prophecy' in the following way: 1. There is a certain generalization in social science or common-sense knowledge which says '8 -+ B'. This generalization is not true in that form. It is conditional and depends on the occurrence of a certain D, therefore, a 'true' generalization should be written as follows: D

-+

(8

-+

B).

2. The condition D is of a special nature, namely, D means that the members of the group for which the proposition '8 -+ B' has been formulated know the propositions '8 -+ B' and believe that it is true. If they do not know it, or do not believe in its truth, 8 is not followed by B. Similarly we could define a 'self-destroying' prophecy as one in which the fact that people believe in the truth of propositions'8 -+ B' (we denote this fact by D) makes this generalization false. It is true only when they do not believe in it, or do not know it. Symbolically: D -+ (8 -+ B). Sometimes this condition D is of a different kind, namely, D means that the members of a certain group believe that they should behave in a certain way-that they should react as B reacts to stimulus S. The car drivers stop before the red light and start when it turns to green only when they have been efficiently taught that they should behave in this particular way. A certain kind of psychological condition, which we could call 'acceptance of a certain social norm', is here the condition for the causal connection between the given stimulus and given behavior. In general we can say that enactment of normative laws and other rules of social behavior is not an act of 'creation of social laws'-if the law means a description of an 'objective regularity', but it means the creation of some conditions without which some regularities of behavior would not operate.:" It means also that the enactment and operation of normative standards of human behavior can be well understood in terms of a strictly causal approach, and that it is compatible with the notion of conditional causal relations. The last question which I would like to emphasize is the problem of generality of consequences of different mechanisms of conditionality of causal relations discussed above, and their applicability to other sciences. 29

For a much more detailed analysis of this problem see S.

NOWAK

(1966).

CONDmONAL CAUSAL RELATIONS

787

It seems to be rather obvious that their validity is by no means limited to the area of social phenomena and social behavior of human beings. Although in our disciplines their consequences are more visible-most of them, although sometimes in slightly different form-ean also be observed in the sciences which study the 'natural' phenomena, but the analysis of them in these sciences would be beyond the scope of this paper.

References KOTARBINSKI, T., 1955, Traktat 0 dobrej robocie (Treatise on Good Work) (Warszawa), p.23 LAZARSFELD, P. F., 1955, Interpretation 0/ statistical relationships as a research operation, in: The Language of Social Research, eds. P. F. Lazarsfeld and M. Rosenberg (Free Press, Glencoe, Ill.), pp. 115-125 LIpSET, S. M., 1963, Political man (Doubleday, Garden City, New York) MERTON, R. K., 1968, Social theory and social structure (Free Press, New York), pp. 475-490 NOWAK, S., 1960, Some problems 0/ causal interpretation 0/ statistical relationships, Philosophy of Science, vol. 27, pp, 23-38 NOWAK, S., 1961, General laws and historical generalizations in the social sciences, The Polish Sociological Bulletin, No. 1-2, pp. 21-32 NOWAK, S., 1966, Cultural norms as elements 0/ prognostic and explanatory models in social theory, Polish Sociological Bulletin No.2, pp. 40-57 NOWAK, S., 1967, Causal interpretation ofstatistical relationships in social science, Quality and Quantity, vol. 1, pp. 53-89 NOWAK, S., Inductive inconsistencies and conditional laws 0/ science, Synthese, vol. 23, pp. 357-313 OSSOWSKI, S., 1967, Two kinds 0/ historical generalizations, Works, vol. V (Warszawa) OSSOWSKI, S., Dwie koncepcje historycznych uogolnien, Dziela, t, IV (Two notions 0/ historical generalizations, Works, vol. IV), pp. 319-328

CAUSE-EFFECT RELATIONSHIPS: OPERATIVE ASPECfS

H. WOLD University of Gothenburg, Gothenburg, Sweden

1. Three dichotomies of relationships This is an introductory section, reviewing some few fundamental distinctions among causal relations and other types of relationships. Explicit relations between observed variables are in focus. The symbols follow current usage. Illustrations and cases in point are briefly indicated. For a more detailed exposition, see WOLD (1967b). 1.1. Deterministic v. stochastic relations. Deterministic relations are functionally exact (they involve no residuals). 1.1.1. Formal presentation. a. Deterministic relation: (1) y =f(x) b. Stochastic relation: Y =f(x)+e } E(Ylx) = f(x) .

(2a-b)

In words, it is assumed that y is formed by a systematic part f(x), corresponding to the right-hand member of the deterministic relation, and a residual, e. The systematic part is defined as E(Ylx), the conditional expectation of y for given x. To emphasize its operative aspects, E(Ylx) is called a predictor (of y in terms of x) (WOLD, 1961). 1.1.2. The special case of linear relations. a. Deterministic relation: y = a+f3x

(3)

b. Stochastic relation: y = a+ f3x+e} E(ylx) = a+f3x .

(4a-b)

790

H. WOLD

1.2. Predictive v. causal (cause-effect) relations (WOLD, 1967a, 1967b). 1.2.1. Predictive relations. Relations 1.1.1a-b are predictive as they stand: predy =f(x) (5) writing predy for the predicted value of y when x is known. a. Deterministic relations 1.1.1a. The prediction (5) is exact; there is no prediction error 8. b. Stochastic relations 1.1.1b. The prediction (5) has prediction error B. The following theorem is an immediate implication of 1.1.1b (WOLD, 1961, 1963). THEOREM.

The prediction error has zero conditional expectation

E(elx)

= 0,

(6)

0.

(7)

and zero overall expectation,

E(e)

=

1.2.2. Causal (cause-effect) relations. The following definitions a-b are preliminary and incomplete. We shall revert to this matter in Section 2. a. Deterministic relations 1.1.1a. Suppose that on a change from x to x-l-zl the value ofychanges fromy =f(x) to y+L1y, with zlj =f(x+L1) -f(x); in symbols,

Ylcp(x

-+

x+L1) -+ y+L1y } L1y = f(x+L1)-f(x)

(8a-b)

where 'cp' denotes the ceteris paribus clause that when x varies, other factors of relevance are kept constant. Then 1.1.1a is a causal relationship, with x for cause and y for effect. b. Stochastic relations 1.1.1b. Using the same notations as in 1.2.2a, suppose that on a change from x to x+L1 the expected value of y changes from f(x) to f+L1f; in symbols, E(Ylcpx

-+

x+L1) -+ f(x)+L1f } L1f = f(x+L1)-f(x) .

(9a-b)

Then 1.1.1b is a causal relation, with x for cause and y for effect. c. We note an immediate corollary of 1.2.2a-b: A causal relation is predictive, but a predictive relation mayor may not be causal. 1.2.3. Illustrations. Two simple examples follow, and many more can be provided by the reader.

CAUSE-EFFECT RELATIONSHIPS

191

a. Hooke's law for the stretch of a wire. This is a deterministic relation,

y =j(x)

(10)

where y is the length of a wire that is subject to stretch by a weight x, Experiments show that the function f(x) and its derivative f'ex) increase with x. Hooke's law is predictive and causal, with x for cause and y for effect. b. The fertilizer-yield relationship in agricultural field experiments. This is a stochastic relation, y

= f(x) + 8;

E(ylx)

= f(x)

(11a-b)

where x is the amount of fertilizer spread over a plot which at harvest time gives the yield y. The shape of function f(x) depends upon the soil characteristics, the irrigation and other features of the experimental field, and its system of minor plots. The residual 8 sums up the effect of secondary factors which influence y and are not specified or otherwise taken into account in the experiment. The fertilizer-yield relation (11 a-b) is predictive and causal, with fertilizer x for cause and yield y for effect. 1.2.4. Vector notation for multivariate cases. a. Without formal change, using vector notation

x =

(Xl' ... ,

x m)

(12)

the notions 1.1.1a-b extend immediately to relations that involve several right-hand variables Xl' ... , x m • Similarly, using vector notation (13) the same notions allow straightforward extension to systems of relations with one relation for each of the left-hand variables. For example, spelling out relation (1) when y and x are vectors (12)-(13), we obtain the system

~~ -: (~: .: :~~) J.

(14)

Y. - f,,(XI, ... , x m)

In what follows, we shall mark the distinction between ordinary and vector variables only when this is needed for the argument. b. In accordance with (12)-(14), the notions of predictive relations and cause-effect relations stated in 1.2 allow direct extension to vector

792

H. WOLD

variables x, y. For example, spelling out the causal relation (9a-b) in the case of vectors (12)-(13):

1.2.5. Illustration: Boyle-Gay Lussac's law for ideal gases. This is a deterministic relation, usually written in the form PV= cT. (16) As applied to a cylindric gas container with movable top, V is the volume of the cylinder as read off along its side; P is the pressure of the gas as read off on a manometer; T is the absolute temperature of the gas; c is Planck's constant. For the experimental facts referred to below, see, e.g., Zemansky (1937). The variables P, V, T can be subjected to experimental variation, namely, (A) temperature T can be regulated by heating the container by a burner; (B) volume V and pressure P vary at the same time if the top of the cylinder is moved up or down (the movement can be performed either by pushing a handle at the top, or by placing or removing weights at the top). In situation (B) we note two alternatives to perform the experiment, namely, (B l ) moving the top of the cylinder so as to regulate volume V, while P is passively registered; or (B 2 ) moving the top of the cylinder so as to regulate pressure P, while volume V is passively registered. The experimental evidence is in accordance with the following relations, obtained from (16) by solving in turn for the three variables, P = cT. V'

V=cT. p'

1 T=-PV. c

(17a-c)

Each of these relations is of type (1), with one variable in the left-hand member, and two variables to the right. a. Each of relations (17a-e) is predictive in the sense of (5), allowing us to predict the left-hand variable when the two right-hand variables are known. b. Relation (17a) is a cause-effect relationship in the sense of (8a-b). It is an experimental fact that temperature T and volume V can be regulated by procedures (A) and (Bl ) , respectively, while P is passively registered, and that in conjunction with such regulation of T and V the pressure P varies in accordance with (17a). c. Similarly, (17b) is a cause-effect relation. In fact, temperature T

CAUSE-EFFECT RELATIONSInPS

793

and pressure P can be regulated by procedures (A) and (B 2 ) while volume V is passively registered; it is an experimental fact that V then varies in accordance with (17b). d. Relation (17c), on the other hand, is not causal in the sense of (8a-b). It is an experimental fact that volume V and pressure P cannot be regulated by procedures (B l)-(B2 ) so as to make temperature T vary in accordance with (17 c). 1.3. Formal v. causal reversibility. 1.3.1. Example: A deterministic relation 1.1.1a. a. Under general conditions of functional regularity, the relation y = f(x) is formally reversible, giving X

=f-l(y),

(18)

where f- l (.) is the inverse of f(.). b. A cause-effect relation y = f(x) mayor may not be causally reversible. If causally reversible, x = f-l(y) is causal, with y for cause and x for effect; in symbols, xltp(y

--+

y+Ll) --+ x+Llx } Llx =f-l(y+L1)-f-I(y) .

(19a-b)

1.3.2. Illustration: Boyle-Gay Lussac's law (16). By the experimental evidence referred to in 1.2.5a-d, relation (17a) is causal and causally reversible with respect to pressure P and volume V, but not with respect to pressure P and temperature T. Similarly, (17b) is causal and causally reversible with respect to volume V and pressure P, but not with respect to volume V and temperature T. 1.3.3. Operative v. inferential reversibility. a. A stochastic relationship is never reversible with respect to prediction. Thus if we consider relation 1.1.1b, E(xIY) is not identical to f-l(y), E(xly)

~f-l(y),

(20)

except when e is identically absent. b. The direction of predictive or causal inference is always reversible, whether or not the relationship itself is reversible. Thus if we consider a causal relation' 1.1.1b which is not causally reversible, it is perfectly legitimate to ask: Given y = Yo, for which value x can we expect y to equal Yo? The answer is given by the inverse inference (21)

794

H. WOLD

2. Definition of the notion causal (cause-effect) relation The following definition has been given by the author (WOLD, 1966); also see MOSBAEK and WOLD (1970, Sec. 1.4.7), and WOLD (1959, 1969a). 2.1. The relation between stimulus (cause) and response (effect) in a genuine or fictitious stimulus-response experiment is, by definition, a causal relation. Elaboration: 2.1.1. The definition 2.1 takes the notion of stimulus-response relationship as the prototype for causal relations. In stimulus-response experiments, the stimulus (x) is subject to controlled variation, and response (y) is passively registered, while other relevant factors are kept constant. The experiment mayor may not be subject to random disturbance; this dualism corresponds to the distinction in 1.1.1a-b between deterministic and stochastic relations. 2.1.2. In nonexperimental situations, a relationship 1.1.1a or 1.1.1b mayor may not involve the hypothesis that if x were subjected to controlled variation, and y passively registered, while other relevant factors were kept constant, then y would vary in accordance with 1.1.1. In case this hypothesis is adopted, relation 1.1.1 is a causal relation, with x for cause and y for effect. 2.1.3. In accordance with 1.2.4 and without verbal change, the definition 2.1 extends to causal relations where the cause variable x and the effect variable yare vector variables. 2.2. Illustrations. Again, the following short list can be extended by the reader. 2.2.1. Hooke's law (10) is a deterministic case of experimental causality; 2.1.1 with 1.1.1a. 2.2.2. The fertilizer-yield relation (11) is a stochastic case of experimental causality; 2.1.1 with 1.1.1b. 2.2.3. Boyle-Gay Lussac's law (16). Relation (17a) is a deterministic case of experimental causality with two cause variables (T, V) and one effect variable, P. Similarly, (17b) is a causal experimental relation with two cause variables (T, P) and one effect variable (V). Relation (17c), however, is not causal. 2.2.4. Expenditure as dependent upon income (WOLD, 1967a, 1967b, 1969a, 1969b). In survey data on family income and expenditures, let y

= f(x)+e;

E(Ylx)

= f(x)

(22a-b)

CAUSE-EFFEct RELATIONSIDPS

795

stand for the hypothetical relation between family income per consumer unit (y) and family expenditure on food per consumer unit (x). Then (22) is a predictive relation for y in terms of x, while e sums up the influence upon y of other factors than x. Imposing the hypothesis of 2.1.2, relation (22) becomes a causal relation with x for cause variable and y for effect variable. Assuming that f(x) takes the form f(x) = cx E

(23)

and maintaining the causal hypothesis 2.1.2, E will be the elasticity of demand for food with respect to income. We note in passing that the notion of demand elasticity thus is a causal concept. 2.2.5. In a group of university students of age 20, let x be the body weight, and y the length. The data on x, y give a relation 1.1.1b that is predictive: when x is known, f(x) gives the expected value of y. But the relation is not causal. If a student puts on weight, deliberately or not, or reduces his weight, the length will remain practically constant, and will not vary in accordance with f(x). 2.3. The definition 2.1 of causal relations makes use of a number of more fundamental concepts which are supposed to be known and established, notably 'experiment', 'controlled variation', 'hypothesis', and 'stimulusresponse experiment'. This is in accordance with the hierarchic structure of human knowledge; see CARNAP (1928) and, from the present points of view WOLD (1969a). 3. Causality: Model building aspects The theoretical definition of cause-effect relations is one thing; their empirical assessment is quite another matter. Speaking broadly, the empirical assessment is relatively simple in experimental situations, whereas it is much more difficult, as a rule, on the basis of nonexperimental data. The dualism between theoretical and empirical notions goes to the core of the much-debated problems of causality. The limited aim of this brief section is to link up the dualism and the debate with the basic principles of model building. 3.1. Cognitive models (WOLD, 1967a, 1967b, 1969a). A model, or a cognitive model as we shall say when we want to emphasize the epistemological aspects of the notion, is a joint theoretical-empirical construct about phe-

796

H. WOLD

nomena in the world around us. In general symbols, a cognitive model, say M, allows the representation (24) The rectangle denotes the frame of reference of the model; that is, the frame specifies the group of phenomena which the model serves to describe or explain: T represents the theoretical content of the model; E represents the empirical content of the model; the two-sided arrow ~ denotes a relationship of matching, the aim of the model being that its theoretical and empirical content should in essential respects agree with each other. 3.2. In the epistemological literature, models are usually dealt with as theoretical notions, as purely theoretical constructs designed with a view to empirical applications; see, e.g., CARNAP (1928), FEIGL (1953), NAGEL (1961). Such a model corresponds to the theoretical content T of a cognitive model (24). In point of principle, the difference between the two types of model is formal rather than real, inasmuch as the empirical applications are in both cases the raison d' etre of the model. 3.2.1. The representation (24) spells out explicitly that cognitive models are joint theoretical-empirical constructs. The explicit display of the synthesis has considerable advantages. For one thing, it makes for more clarity in the model construction, inasmuch as the notions that make the building material are.a mixed bag, some notions being theoretical while other ones are primarily of empirical nature. The notion of causal (cause-effect) relation provides an illustration of this last point. As emphasized in the definition 2.1, the relation between stimulus and response in stimulus-response experiments is a fundamental type case of cause-effect relations; stimulus-response experiments are a wellrecognized category of procedures in scientific workshops; hence the notion of cause-effect relation is essentially of empirical nature. 3.2.2. At the same time, and again in conformity with everyday procedures in scientificworkshops, the representation (24) allows us to distinguish between a cause-effect relationship as a theoretical hypothesis on the one hand, and as an empirical fact on the other. The systematic distinction between theoretical and corresponding empirical concepts is a basic principle in model building; see, FEIGL (1953), NAGEL (1967), WOLD (1956, 1959, 1961, 1963). The representation (24) is in complete harmony with this principle.

CAUSE-EFFECT RELATIONSHIPS

797

3.3. To repeat, most of the epistemological literature deals with models as purely theoretical constructs. It is a general weakness of this approach that concepts with an empirical foundation are difficult to incorporate in the model construction. The notion of causal (cause-effect) relations is a case in point. Reference is made to the classical definition of causality, which is very narrow on two scores: (a) it is deterministic, and (b) it runs in terms of binary variables A, A and B, ii; see, e.g., FEIGL (1953), NAGEL (1961). Evidently, such a definition is by far too narrow to cover illustrations 2.2.1-2.2.4 and other causal relationships in everyday use in scientific workshops. 3.3.1. In his invited address to this Bucharest Congress, G. H. VON WRIGHT (this volume) deals with causality on the basis of the purely theoretical definition of models. His approach builds up a theoretical world in terms of binary observations, in which he develops a formal-mathematical definition of causality; then with regard to the applications von Wright emphasizes that 'action' is a key word to characterize situations where the notion of causality applies. Clearly, von Wright's approach is much more ambitious than the classical definition in terms of binary variables, but it is safe to say that it is by far too theoretical to incorporate the operative features that are typical for the causal usage that is current in scientific workshops. To put it otherwise, his reference to 'action' and related operative features should in some way be incorporated in the very definition of cause-effect relations; to wait with this reference until dealing with the applications is like throwing in the yeast when the bread is baked. 3.3.2. Again referring to the classical definition of causality, this is confined to the special case of deterministic relationships. In his recent monograph, PATRICK SUPPES (1970) has explored the probabilistic aspects of causality. Time has long been overdue for a push in this direction, and I am full of admiration for the many brilliant facets of Suppes' approach. At the same time I must confess that I am not happy about his basic notions. Much of his analysis revolves around the notion of prima facie causality, defined in terms of conditional probability, with deterministic causality as a limiting case. It is my understanding that this way of looking at things has led Suppes into a collision course with established physical facts. a. To bring home this last statement, let Pi> Vi, T,

(i

= 1, ... , n)

(25)

be data from n observa tions on a gas container as governed by Boyle-Gay

798

H. WOLD

Lussac's law (16). The data will give plots in the coordinate system (P, V, T) that form a statistical scatter, and the scatter will be in agreement with 1.2.5b within observational accuracy. Thus if the scatter is interpreted in terms of conditional probability, the situation in 1.2.5b will be in accordance with Suppes' notion of prima facie causality, and the limiting deterministic relation is causal in the sense of physical experiments. b. The same scatter will be in agreement with 1.2.5d, again within observational accuracy. Interpreted in terms of conditional probability, this situation, too, will be in accordance with Suppes' notion of prima facie causality, and the limiting deterministic relation (17c) will be causal in Suppes' sense, but not in the sense of physical experiments. c. The illustration 3.3.2a-b is of general scope. As I see it, Suppes' notion of prima facie causality is essentially the same as the notion of predictability, defined in terms of conditional probability. It is a classical tenet that predictability is not the same as causality. Illustration 3.3.2.a-b shows that this tenet is true also in the limiting case of deterministic relations. 4. The empirical assessment of causal relations. The reader is assumed to have a working knowledge of the extensive literature; see, e.g., Box and JENKINS (1970), DEMPSTER (1971), KERLINGER (1964). The approach to be adopted is very much dependent upon the amount of prior information which is available and can be utilized in the empirical analysis. The following situations will be briefly reviewed; the exposition leans heavily on WOLD (1956) and WOLD (1967a). 1. High degree of information: Controlled experiments. 2. Nonexperimental situations: High information about suitable models. 3. Nonexperimental situations: Low information about suitable models. 4. In 4.1-4.3 the variables in play are directly observed. In psychological and educational model building-on the basis of experimental or nonexperimental data-it is typical that the variables in play are only indirectly observed (for a recent review, see DEMPSTER, 1971). 4.1. High information: Controlled experiments. The causal variable x (stimulus) is subject to controlled variation, the effect variable y (response) is passively registered, while other relevant factors are kept constant. The experiment can be repeated. The experimenter tries to make the replications mutually independent. The independence can be reinforced by randomization (that is, the various levels of x are allocated at random among the experimental objects). 4.1.1. OLS (Ordinary Least Squares) regression of effect variable y

CAUSE-EFFECT RELATIONSHIPS

799

on cause variable x will provide consistent estimation of the function f(x) that describesy as dependent onx (WOLD, 1963). On the null hypothesis the estimation is optimal in the sense of predictive inference of y in terms of x (smallest standard deviation for the prediction error). 4.1.2. On the further assumption thatf(x) is linear, and that the residuals are normally distributed, OLS will be equivalent with ML (Maximum Likelihood) estimation. In this special case the aspirations of the estimation and the hypothesis testing can be raised still higher: Optimal efficiency, optimal power, and much more. 4.2. Nonexperimental situations: High information about suitable models. The researcher has information (theoretical or empirical) about which variable is to be specified as effect variable y, and which variables x are the causal variables of primary importance. As a rule, there is further an array of causal factors x of secondary or minor relevance. 4.2.1. OLS regression of effect variable yon cause variables x will under mild conditions of stochastic regularity provide consistent estimation (WOLD, 1963). On the null hypothesis the estimation is optimal in the sense of predictive inference of y in terms of x. As compared with controlled experiments, there is a shift of emphasis in the statistical problems (WOLD 1967a, 1969b). The selection of variables x to be included in the regression comes to the foreground. The selection and the ensuing regression is often marred by collinearity. The high aspirations of 4.1.2 cannot be maintained; the main problem is to obtain consistent estimates. Hypothesis testing according to the principle of the self-contained experiment has to be supplemented by predictive testing (using the estimated model for forecasting, and comparing the forecasts with fresh evidence). 4.3. Nonexperimental situations: Low information about suitable models. Here the analysis is tentative not only with regard to the selection of causal variables xthat are of primary importance, but also with regard to the distinction between cause variables and effect variables (WOLD 1956). The high aspirations of 4.1.2 become even more remote. The main problem is to obtain meaningful relationships. Predictive testing gains in importance. Only when a relation has been (tentatively) established as meaningful is it relevant to speak about consistency in the estimation. The fundamental importance of OLS in situations 4.2 and 4.3 is reflected in recent developments of statistical methods. Specific reference is made to the various strategies that have been developed for the selection of causal

800

H.

WOLD

variables x. This is a key topic in modern textbooks of statistics; see Box and JENKINS (1970), KERLINGER (1964) and others.

In situations 4.2 and 4.3 the problem at issue often is in the nature of a system of relationships rather than one fundamental relation. This brief review deals with unirelational models, and I can only give general reference to the growing literature on multirelational model building; see, for example, MOSBAEK and WOLD (1970). 4.4. Directly v. indirectly observed variables. The limited scope of the above review must be emphasized, and in particular it is restricted to models built in terms of directly observed variables. Models in terms of indirectly observed variables are of great importance in psychology, education, and several other fields. The applications at issue may be experimental or nonexperimental. In the present panel discussion, Professor Wiley devotes his contribution to model building in terms of indirectly observed variables. References Box, G. E. P. and G. M. JENKINS, 1970, Time series analysis, forecasting and control (Holden-Day, San Francisco) CARNAP, R., 1928, Der logische Aufbau der Welt (Meiner, Hamburg) DEMPSTER, A. P.,1971, An overview of multivariate data analysis, Journal of Multivariate Analysis, vol. 1, pp. 316-346 FEIGL, H., 1953, Notes on causality, in: Readings in the Philosophy of Science, eds. H. Feigl and M. Brodbeck (Appleton-Century-Crofts, New York), pp. 408-418 KERLINGER, F. N., 1964, Foundations ofbehavioral research (Holt, Reinhart and Winston, New York) MOSBAEK, E. J. and H. WOLD, 1970, Interdependent systems. Structure and estimation (North-Holland, Amsterdam) NAGEL, E. 1961, The structure of science (Routledge and K. Paul, London) SUPPES, P., 1970, A probabilistic theory of causality, Acta Philosophica Fennica, vol. 24 VON WRIGHT, G. H., On the logic and epistemology of the causal relation, this volume WOLD, H., 1956, Causal inference from observational data. A review of ends and means, Journal of the Royal Statistical Society, Ser. A, vol. 119, pp. 28--60 WOLD, H., 1959, A case study of interdependent versus causal chain systems, Review of the International Statistical Institute, vol. 5, pp. 5-25 WOLD, H., 1961, Unbiased predictors, in: Fourth Berkeley Symposium of Mathematical Statistical and Probability Theory, vol. 1, ed. J. Neyman (University Press, Berkeley, California), pp, 719-761 WOLD, H., 1963, On the consistency of least squares regressions, Sankhya, Ser. A, vol. 25, pp. 211-215 WOLD, H., 1966, On the definition and meaning of causal concepts, in: La Technique des Modeles en Sciences Humaines, eds. R. Peltier and H. Wold (Union Europeenne d'Editions, Monaco), pp. 265-295

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Wow, H., 1967a, Forecasting and scientific method, in: Forecasting on a Scientific Basis, by H. Wold, G. H. Orcutt, E. A. Robinson, D. Suits and P. de Wolff (Gulbenkian Institute of Science, Lisbon), pp. 1-65 Wow, H. 1967b, Nonexperimental statistical analysis from the general point of view of scientific method, Bulletin of the International Statistical Institute, vol. 52, pp. 391-424 Wow, H., 1969a, Mergers of economics and philosophy of science. A cruise in deep seas and shallow waters, Synthese, vol. 20, pp, 427-482 Wow, H., 1969b, E. P. Mackeprang's question concerning the choice of regression. A key problem in the evolution of econometrics, in: Economic Models, Estimation and Risk Programming: Essays in Honor of Gerhard Tintner, eds. M. Beckman and H. P. Kiinzi (Springer, New York), pp, 325-341 ZEMANSKY, M. V., 1937, Heat and thermodynamics (McGraw-Hill, New York, 2nd edition 1943)

PROBLEMS OF THEORETICAL LINGUISTICS H. SCHNELLE Technische Universitdt Berlin, West Berlin

1. Introduction

In discussing problems of a science one has to be selective, by necessity. The selection is, furthermore, always biased by subjective evaluation and by the historical and social factors determining the status of that science. I shall select four problem areas and give arguments why I exclude a fifth area from discussion, though its problems have received much attention during recent years. Should some reader think that some other problems are still more important I should not be surprised and I hope he will not be too much suprised either in not finding his problem discussed in my article. My first problem relates to the general methodological status of linguistics and the form of its theories. Both aspects strongly depend on the answer one is inclined to give to the question: Is there an essential difference between natural languages and interpreted calculi constructed for scientific purposes? Depending on the answer one is likely to consider either linguistics as a methodologically basic discipline or as just a particular science among many others. The answer affects at the same time the form one is likely to give to systematic results of linguistic analyses: Either they are rules we conform to or should conform to in our intellectual activities or else they are sets of statements about linguistic objects, their properties and relations forming a deductive nomological theory.' Since I shall argue for an essential difference between natural languages and constructed languages, linguistics as the science of natural languages is not to be given a special status among 1 It is quite possible that in the last instance we shall have to consider also deductivestatistical or even inductive-statistical theories in linguistics. For hints to this effect. cf. Y. BAR-Hu.LEL (1970, p. 205) with reference to A. MARGALIT'S (1971) thesis. From a methodological point of view it seems, however, at least at present preferable to consider primarily deductive nomological theories.

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the sciences, and the form of its theories should be comparable to the form of theories in other empirical disciplines. My second problem relates to the field of clarification and abstraction in linguistics. In tackling this problem one will have to decide which explicanda one is going to choose as basic in linguistics, those of linguistic communication or those of a competence-knowledge of an abstract syntactic or semantic system. Connected with this is the problem of how much abstraction is tolerable under criteria of adequacy such that the description is still manageable from a formal point of view. The treatment of this problem will be the central concern of my contribution. My third problem relates to the field of the empirical evaluation of linguistic theories. There are two questions: What is the status of standard observations and descriptions in linguistics, and what is the status of introspective grammatical judgments? My fourth problem relates to the internal structure of linguistic theory treating the question whether the statements of syntax or those of semantics are basic. I shall discuss this problem since it is currently much debated among linguists, though I personally consider it to be a pseudoissue. I refrain from reiterating the discussion on the innateness hypothesis since I completely agree with Chomsky in that "I cannot see how one can resolutely insist on one or the other conclusion in the light of the evidence now available to us [CHOMSKY, 1969, p. 80]". Moreover, I do not even believe that Chomsky has stated the problem in a satisfactory way. 2. The status of linguistics and the logical form of linguistic theories 2.1. Linguistics, as the science of languages (and acts in languages), is looked upon as a fundamental science by many philosophers and scientists. Sometimes this position is characterized by the dictum that the logic (and methodology) of science is based on the analysis of language. This, then, establishes a special status for the science of languages since, apparently, it would not make sense to subsume it under the logic of science in the same way as any other science because of circularity: The science of language based on the logic of science based on the science of language. It follows from this point of view that the analysis of language is prior to any other methodological analysis. I cannot help feeling that the central status of the science of language (in general) as expressed by this dictum is unwarranted in the same sense as was the central status of psychology toward the end of the nineteenth

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century. The error lies in having excessively generalized on an originally justified assumption, namely, according to Carnap: The logic of science is the analysis and theory of the language of science. This thesis would only be equivalent to the first one if the analysis and theory of the language of science are essentially the same as the analysis and the theory of languages in general (comprising natural languages in particular). This assumption is indeed made by a number of scientists. In contrast to this, I believe that a fundamental difference between natural languages on the one hand and 'regimented and sterilized' interpreted calculi for scientific purposes on the other can be shown (though a genetic dependency of the latter on the former cannot be denied)." It is obvious that this position affects the evaluation of the first thesis. 2.2. Let me briefly indicate what I conceive to be the essential difference between ordinary spoken languages and constructed calculi for scientific purposes." A natural language may be compared to a complex system of instruments serving their different purposes appropriately only in vivo, i.e., in the context of their normal cultural and social environment. This system of instruments together with its quasi-ecological environments is flexible: it may be modified, developed and adapted to particular tasks and by such adaptations special changes of the language may be brought about. One such development is in the direction of greater independence from the nonlinguistic environment. The introduction of writing presented historically the basis for a development in the direction of greater independence from situative context, but introduced at the same time a potentially greater dependence on contextual factors of knowledge and experience. One such 2 Carnap seems to deny in his reply to Strawson's contribution (ScmLPP, 1963, p. 934) this fundamental distinction. I think, however, that a closer analysis would reveal that this denial mainly concerns the genetic aspects and the aspect of mutual 'contact' and influence. But from a systematic point of view, he distinguishes ordinary language in everyday use (and contexts) (p. 936), (more exact parts) of ordinary language "chiefly used in technical, scientific contexts" (ibid.) and constructed language systems. It seems, that he would consider the language of science as belonging to the second or third type and, since being determined by principles manifesting themselves in systems of the third type, these are not parts of a natural language. "The meta language in which he [the linguist] formulates the description of a natural language, is itself not a part of natural language but rather a part of the language of science; therefore this meta language should be-s-and usually is-much more exact than the language he describes. I have no doubt, that the best procedure for this meta language would be to use the sharp dichotomies of two-valued logic" (p. 942) and of the methodology of science, as we may add. 3 A more detailed understanding will be possible on the basis of what will be said in Section 3.

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process of modification and development is the adaptation of languages to the purposes of sciences which-during the centuries-produced technical languages on the one hand (via terminologies, the introduction of variables, formulas etc.) and finally the logical analysis of the pure aims and forms of such systems for scientific purposes. In the last instance, we get instruments functioning in a predictable way in vitro, i.e., in a determinate context. The original plasticity of the system of a natural language is thereby lost, the instruments are sharpened and hardened for their scientific use: A scientific sublanguage or variety of a language is created. This modification of the natural language system changes its qualities essentially and an analysis of the scientific sublanguage does not allow one to draw conclusions as to the properties of the languages in general, just as on the other hand an adequate description of the complex and plastic system is without interest to the analysis of the scientifically determined sublanguages. This does not mean, however, that the special purpose sublanguage is specialized with respect to the range of facts onto which it may be applied. In particular, the subsystem grown out of the complete system of natural language might be bent back to its own origin: The language of science may be applied in setting up a theory oflanguage and of the languages in general. Let me therefore stress the following two points: 1. The language of science is a-possibly radical-modification of certain aspects and parts of natural languages (or, in some cases, it may even be constructed without a genetic relation to any natural language). 2. I reject the assumption made by many logicians and theoreticians that the languages of science (in particular the constructed ones) may serve adequately as first approximations to the objects of theoretical linguistics in general, though I should not exclude that this may be the case for certain subtheories such as syntax and semantics. I said that the language of science may be bent back to its genetic origins: it may be applied to the analysis of natural languages. This application is, however, not essentially different from any application of the logic of science. If so, we may express this by another thesis: The science of language is based on the logic of science. From the conjunction of this thesis and Carnap's we get: The science of language is based on the logic of science and the logic of science is based on the analysis of the language of science. In taking the language of the science of linguistics as a specialization of the general form of the language of science as analyzed in logic and meth-

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odology we derive from this conjunction the special case by substituting the Jast three occurrences of 'science' in the last thesis by 'the science of linguistics'. This is exactly what was meant by our metaphorical expression of bending the language of science back to its origins. In other words: Theoretical linguistics will have to start with the analysis of the language of linguistics. 2.3. As a particular proposal we might take as the language of linguistics a language out of the reservoir of formalized languages analysed by logic (such as interpreted calculi of first-,second-, etc., order, 'languages' of intensional logic). The descriptive determination of this language is achieved by the introduction and explication of a set of descriptive constants. These descriptive constants should be obtained as explicata in a process of clarification, the result being precise concepts partially determined by meaning rules and partially by the theories in which they are used or by precise specification of their formal designata. A linguistic theory, then, is to be a set of statements in this language of the science of linguistics presenting explanatory schemata (say in the sense of the Hempel-Oppenheim scheme) for linguistic facts. It contains an analytic part and a complex system of empirical subtheories. The analytic part is the set of those statements that may be proven on the basis of logic and the meaning rules for basic concepts alone. The analytic part delimits the set of communicative phenomena that can be described as a language, a family of languages or a language variety. The empirical statements of linguistics are to be partitioned into a number of subtheories such that the partitioning reflects the multiplicity of linguistic phenomena: the fact that there are many languages which are historically and culturally interrelated and that each language is a complex of elementary varieties of languages, dialectally and diatypically interrelated." Accordingly, we shall have to present a linguistic theory for each factual elementary variety of each language and theories for each factual interrelation of varieties within a language or of the languages among each other. For certain linguists the varieties of languages may be neglected in the first approximation and substituted by just one homogeneous theory for each language. 2.4. Though, ultimately, a standard linguistic theory should be presented in this form, formalizations of this type are not common in linguistics. 4 Following GREGORY (1967) and LIEB (1970, p. 54) we refer to the distinction between dialectal varieties and diatypical varieties. In an attempt of clarification (see below) the primary characteristic of the former is that it represents a linguistic norm whereas the latter is mainly related to the purpose of communication.

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Instead, one usually introduces, for convenience of research, notations presenting in a compact and intuitive way complexes of theoretical statements on" certain linguistic entities (such as sentences), and algebraic or relational systems in terms of which the appropriate notations are generated." Particularly outstanding examples are generative grammars. Formally the algebraic systems and systems of notations may themselves be presented in an axiomatic form. Then the resulting presentation differs from the one envisaged above by having in addition to the linguistically descriptive constants descriptive constants of the mathematical theory applied. Under the general assumption made above that the science of linguistics may be formulated in terms of a specialized interpreted calculus developed by logic, the formal descriptive constants should be either interpretable as descriptive constants of linguistics also or they should be eliminable from the description." There is no doubt that an algebraic description may be very convenient for practical purposes and may provide a fruitful basis for general research on the properties of linguistic subtheories (such as grammars). For example, the analysis of the general properties of linguistic subtheories (in particular of syntax) for all natural languages is of central importance in linguistics, at least for many linguists, such as Chomsky. It seems that his analysis has only been possible on the basis of algebraic or relational systems of mathematics. His work on general properties ofgrammars proceeded on the hypoth, Some linguists (Halle, Chomsky, and others) think: that "when particular notational devices are incorporated into a linguistic theory ... , a certain empirical claim is made, implicitly, concerning natural language [CHOMSKY, 1965, p. 45]". This statement, as it stands, is obviously quite confusing. It is, however, conditional on Chomsky's assumptions relating notations and evaluation measure of explanatory adequacy (cf. CHOMSKY, 1965, p.42). In eliminating Chomsky's rather vague notion of significant generalization one might say that the introduction of a new symbol proposed for purposes of abbreviating linguistic descriptions is related to the claim that (a) it will receive application in many descriptions of various phenomena in natural languages and that (b) its application achieves in each case a sensible reduction of the number of occurrence of symbols necessary to state all expressions belonging to a certain type. Though these aspects do playa role in the development of the descriptive linguistic apparatus, I consider their role to be of rather heuristic nature, in contrast to Chomsky. Moreover, I should think: that length ofexpression is a rather crude measure of evaluation What is really aimed at is intuitivity admitting though that this notion is not easily amen able to adequate numerical evaluation. 6 E.g., the relations expressed by the arrow expressing a rule of a grammar as well as the arrows expressing direct and indirect derivability in a grammar formalized in terms of a semi-Thue system.

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esis that (a) the syntactic structures of natural languages are only of certain types, other possible types of structuring beingexclu ded, (b) an effective 'knowledge' ofthis restriction is a constant of human beings which is (c)genetically transmitted (innateness hypotheses), and that (d) within this restriction, an evaluation function on pairs of a finite set of texts and the set of structuring grammars is given for which (b) and (c) hold correspondingly. He tried to state both the restrictive properties and the evaluation function. An outcome of the first attempt is his famous classification of constituentstructure grammars and corresponding languages which was extremely fruitful in the theory of programming languages and automata. Though the fruitfulness of theoretical linguistics in terms of such algebraic and relational systems has been demonstrated by this it neither proves Chomsky's working hypotheses nor the superiority of the formal descriptive apparatus used. Though this should be obvious the work of Chomsky and his followers had the practical result that it tends to be regarded not as a particular proposal but as the only way offormalizing linguistic description and thereby imposes restrictions on the theoretical and practical power of formalization which are-in my opinion--excessive, unnecessary and unjustified. Since Chomsky's assumptions (a) to (c) are independent from each other (just as its correlates in (d) Chomsky's arguments derived from the speed of language acquisition by children does not affect (a). Therefore, even if it is rather doubtful that children acquire languages because of their innate 'knowledge' with respect to 'language (in general)' it does not follow that linguistic science whose task is the explicating of this very problem should enjoy the same privilege. Therefore, in order to balance the unjustified restrictions which follow practically from the assumptions of algebraic linguistics for many linguists, the eliminability of the mathematical constants and the translatability into a standard form of metalanguage provided by logic should be shown in each case of algebraic linguistics." At the same time one should try to write grammars and the general linguistic theory immediately in the standard version. 3. Clarification and explication in linguistics 3.1. After having decided on the general form of presentation for the results of linguistic research, we shall have to turn to the introduction of 7 For the class of constituent-structure grammars as well as for transformational grammars this has been done by JVN-TIN WANG (1971a, 1971b).

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descriptive constants and their interrelation (usually mediated through their use in the sentences of the language-variety of the science of language). We must, however, watch our first steps. It is customary to take it for granted that linguistics will have to describe mainly words, sentences and texts as well as their meaning. From this point of view our explicanda might be taken to be such phrases as 'word (of a language L)', 'sentence (of a language L)', 'meaning (of some word of language L)', 'meaning (of some sentence of language L)' and some additional explicanda such as '(formational) part (of word, sentence, meaning ... )' possibly specialized to '(inflexional or derivational) ending' 'prefix'. But these explicanda referring to the entities of some language yield a much too restricted view of the task of language description. There are other words equally to be taken as explicanda that do not refer to entities of some language but to actions in a language, e.g., 'saying something (to somebody) in language L', 'understanding what was said (by somebody) in language L'. Both sets of explicanda may apparently be related by certain other acts, e.g., somebody says something (to somebody) in a language L in or by uttering a word or sentence of the language L and thereby expressing a meaning of the language L. These latter actions relate the former to linguistic entities, showing that the entities of a language are used for the actions in that language. 3.2. The class of actions in a language is obviously a special case from the class of actions in general. From this it follows that the explication of actions in a language has to be executed in the general framework of a logical theory of action. In particular a number of distinctions made for actions and systems of actions in general may and should be applied in theoretical linguistics of actions in languages. Among these we have the relation between action, process, event and state of affairs, the distinction between action type and action event (the generic-individual distinction), the distinction between the complete transformation and the final result brought about in an action, the various sorts of interdependency of actions (inherent, i.e., language determined, causal, combinatorial), actions with singular or collective agents, sorts of the agent's having command of an action, and the acquisition of the command, the types of acquisition and the conditions for the acquisition, such as a capacity for the action in the agent. 8 3.3. On the other hand, the actions in a language are to be characterized as special cases in the general class of possible actions. The specialization 8 For clarification and explication of these terms I refer to G. H. 1968).

VON WRIGHT

(1963,

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comes about in considering the following pragmatic dimensions of a system of communicative actions (a) intentions in a hierarchy of means-end nexus, (b) contextual functioning of the means, (c) systematic organization of the set of means, (d) conditions of having command of a language and its systematic organization and conditions of acquisition and change of this command, (e) status of normativity of this system, (f) organization of the system with respect to a complex of purposes, (g) processing restrictions (awareness, storage space, time for processing required) related to the physical basis in the actor. With respect to almost all of these factors a natural language is different from a constructed language. The intentions of use of the latter are restricted to those of conveying truth, of convincing by argumentation, etc., contextuality is confined to the explicitly formulated methodological or theoretical context, the systematic organization comprises just one directly interpreted semantic system (in contrast to the more complicated systems presented in Section 4.2), command and acquisition usually has a rather straightforward and mechanical or quasimechanical nature, the normativity completely depends on the purpose which is methodological or empirical representation, processing restrictions usually are completely neglected since written communication is assumed. Let me add a few remarks on each of these factors with respect to natural languages. A primary specification is the one that the actions in a language stand in a means-end nexus which is intentionally controlled: The speaker usually intends a complex of goals (or ends) the primary and essential one being that the hearer whom he addresses understands what he says (at least partially). The second is that the hearer, usually, in consequence of his having understood, will take certain (internal or external) actions. Moreover, his doing so will possibly be only one element of a complex activity the speaker wants to effect in the addressee. This chain of further aims might lead to (taking part in) organizing a society or a personality. All of this is effected, in the last analysis, by uttering entities of a language and by expressing thereby certain meanings that may be expressed in that language. The entities of the language used thereby acquire the function of a means. The goals of an action in a language are, however, not obtained automatically and independently of the situation. Every speaker is aware of a system of conditions for successful execution of an action (contexts) whose presence he tests before performing that action. Among these are: (a) The assumption that his addressee has command of the variety of language the speech be-

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longs to, that the variety is a variety that mayor may not be used under the given circumstances and that his partner shares his evaluation of this status of normativity of the variety. (b) The situation, its systematic significance and its evaluation as well as the possible difference in its perceptual presence, significance and evaluation for the speaker and hearer. (c) The knowledge about the structuring of situations, the plans, wishes, etc., of situations aimed at, etc. (d) Thefeelings present in the speaker and assumed in the hearer. (e) Their respective attitudes, characterization (such as belonging to a certain class or type of human being) and valuation, actual as well as habitual. (f) The aims the speaker has and assumes with respect to the hearer (e.g., doing something for the hearer). Every hearer decodes an action on the basis of what he would have said if he (instead of the speaker) had been the speaker (taking account, obviously, of the difference of personality). This decoding proceeds under corresponding contextual factors and assumptions as those present to the speaker. As a consequence of this essentially context-determined functioning of action in a language, actual linguistic communication usually is not just linguistic communication of linguistic entities but communicative action with utterances of linguistic entities embedded as necessary parts. Measures of redundancy operate on the full complex of communication as well as coding and decoding mechanisms. In terms of cybernetics: The communication proceeds in several channels simultaneously (including channels of stored information of various characters only one of which is the linguistic one); coding and decoding take the interdependency of parallel linguistic and nonlinguistic actions as well as their relation to stored material in the sender and receiver into account. However, there is a notable difference between spoken and written actions in language. Written language has to be much less dependent on situative context than spoken language. But the other side of the coin is that very often, nonlinguistic context has to be substituted by linguistic context: In the extreme this means, as Buhler has put it, sympractical (deictic) functioning is transformed into synsemantical functioning. Actions in a language are not just invented on the spot; the agents of these actions have the ability to produce and reproduce these actions under certain circumstances, they have command of these actions. However, no human being has command of any particular action of any language in the beginning of his existence, the command for any set of actions is acquired. But, each human being has the capacity to acquire, in principle,

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any set of actions in any language. The practical application of this capacity undergoes characteristic changes in the course of the development of each human individual. The process of acquisition of a set of actions in a language is a process of several years passing through a large number of intermediate stages. The full sense of this statement becomes clear only in the next paragraph. The process of language acquisition requires by necessity (a) communication with competent speakers of the language to be acquired and (b) existence of perceptual and nonlinguistic communicative abilities and intellectual capacities before the process of acquisition is initiated. In other words, in addition to being executed intentionally by contextually functioning means, systems of actions in any language variant are such that human beings as agents do not have them by necessity but have the capacity to acquire them. 3.4. Up to this point, the presentation made the impression that the actions of a language which speaker and hearer have command of are just sets of actions. This is not the case: The essential building blocks of a language are systems of actions organized by a complex of structuring principles; an action is, in a sense, only determined when shown to be an element of such a system. A natural language is, however, not just one system of actions of this type but a reservoir of, usually, a large number of such systems. With respect to the language such a system belongs to, one says that it is a variety of that language. Besides being structurally different the varieties represent different norms and different purposecomplexes of structuring expressions and meanings. The norms and purposecomplexes exist only insofar as they are internalized in the linguistic agents having command of the corresponding variety. A language variety representing a certain (internalized) norm is called a dialect (or sociolect) and a language variety adapted to a certain internalized purpose-complex is called a register. The unit of a language variety is a dialect-register, i.e., a language variety according to some socially effective norm and to some purpose-complex. The different dialect-registers may be structurally interrelated in various ways. Every linguistic agent has command of only a subset of the dialectregisters or varieties of a language. Moreover, even the set of speakers existing in a certain temporal interval do not exhaust the varieties of the language; a language has a historical development represented in a historico-genetic relation over the varieties. Though no speaker has command of all varieties of the language to say of somebody that he has command of a language, it is sufficientthat he has command of a dialect of the language.

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A detailed analysis of this system of factors should show precisely how natural languages and actions in natural languages to be described in theoretical linguistics differ from other systems of actions, in particular from other representational or communicative systems. As a special case it must show the difference between natural languages and constructed languages. In order to do this we shall have to inquire how the systematic linguistic principles organizing actions in a variety of a language, contextually functioning or not, are distributed over the linguistic agents of that language, whether conditions of normativity and of purpose complexes produce a number of varieties (dialect-registers) allowing for the adaptation and transformation of the language to a multitude of systems thereby producing historical change of languages. It is important to note that the system of organization usually represents an internalized norm together with certain ways of licence one may take from the norm (and on some occasions even should take). The latter indication shows that the kind of norm associated with a system of a language variety is not of the kind of a rule system defining behavior-as in certain games, where one cannot take licence of the rule without ceasing to play the game, nor of the kind of a custom where there is no point in taking licence from the regularity of the custom. Usually, it seems to be the case that a licence taken is tolerated if a motive for doing so can be made plausible and if is otherwise shown that one has command of the system primed as the norm. Let it be stressed, however, that there is not just one norm in a language but quite a number of norms, each norm being applicable when the conditions for the applicability of the norm are given, i.e., when the partner apparently belongs to a particular social group or has a particular social role. The concept behind a constructed language, to emphasize the difference once again, is usually such that the norm status is one of a rule system. Because of its character as a rule system, there is no licence possible, because violating the rules means leaving the system of actions defined by the rules. It is, furthermore, homogeneous for all users, with one principle of organization only, a principle not depending on context except in the rules for the substitution of variables which, linguistically spoken, have an anaphoric character. In other words, a constructed language is extremely poor in structure compared with a natural language. The analysis of actions in a language and of entities in a language may be related to spatiotemporal parts of the linguistic agents and the states

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of the channels between them. In this case it is appropriate that the analysis uses the languages of the corresponding sciences, the languages of acoustics and optics, of physiology and of neurophysiology and psychology. The statements on actions in a language executed by linguistic agents are translated into statements on processes within or between organisms. With respect to the facts related to acoustic or optic signals, linguistics provides indeed, in its modern versions, translations into the language of acoustics (and optics). These translations are very often considered to be one form of relating the facts of linguistics to observable facts. However, in most cases, the translations provided are just a number of essential acoustic features of standard phenomena. In actual use a much wider variety of phenomena occurs in utterances of sentences. We shall return to the discussion of these facts in our analysis of the empirical evaluation of linguistic theories. The situation seems to be much more complicated for the relation of the processes of linguistic analysis on the one hand and neurophysiological processes on the other. With respect to a realization of linguistic actions in computers the situation is obviously again different though in this case the linguistic varieties that could be programmed are still very poor. I shall not enter further into these details.

4. Abstractions in linguistics 4.1. At the end of the last century and during the first decades of this century the movement in history of science against simple positivist psychologism and physicalism in the analysis of language (and of logic) produced a scientific attitude based on the assumption that scientific progress requires a more radical abstraction from conditions of use in the field of basic sciences such as logic and linguistics. Essential aspects of these sciences were to be sought rather in the structural (or even combinatorial) conditions of the system of their objects than in the actual processes of their use; these processes were considered to be determined by many additional factors which could not be described by logic and linguistics alone. In spite of this common tendency, the differences between logic and linguistics were never overlooked, the latter being clearly conceived as an empirical discipline whereas the former as an aprioristic one. This implied that the linguistic structures had to be constituted from, or at least evaluated with respect to, data forming the empirical basis of this science. I shall return to this in the next section.

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Independently of the ways of empirical constitution, reduction or evaluation of linguistic structures the following conception of the use of a structural theory seems to underly its abstraction: The relation between the articulated sound produced in a speech act and what is meant by the speaker in the act of producing this sound is to be taken as a result of the activation of different systematic factors of action. Among these there are certain particularly basic ones the treatment of which is the genuine task of linguistics: The construction underlying the complex of articulated sound, and what this construction signifies or designates. The linguistic form of linguistic entities-expressions and their structure on the one hand and content or sense on the other-and the relation of signifying or designating are the necessary and primary factors in each instantiation of sound articulation and meaning. The systematic description of the form of the constructions resp. of meanings belongs to the linguistic subfields phonology (or graphemics) and syntax on the one hand and semantics on the other. The description of these fields may be undertaken without taking into account the other factors determining language use. Explanation of language use may, according to this view, take the following form: Whenever A produces a certain sound complex and this can be taken (on certain additional evidence) to be a speech act, then he thereby uttered an expression with a certain structure determined by phonology and syntax and he may be taken to mean what is determined by the semantic representation which the semantics assigns to the expression. Accordingly, the prevalent conception of abstract linguistic structuralism? may be stated in the form that the complete theory T c of actions in a language contains a subtheory TB , which is independent and basic with respect to the complete theory Tc in the following sense: No statement and no explanatory argument in TB involve a statement of the rest of the complete theory, i.e., a statement or argument from Tc - TB but most statements and explanatory arguments of Tc - TB involve statements and explanatory arguments of TB • Furthermore, the objects of T B are to be taken as the forms of the means with which linguisticactions are performed, namely, the words or morphemes' 9 As against behavioristic linguistic structuralism prevalent in America. Though, in its earlier form (Bloomfield and immediate followers) American structuralism does not abstract a priori from any observable fact (though, radically, from unobservable facts) later approaches (Harris, Chomsky in his earlier phase) abstract from all factors determining meaning altogether and therewith from all semantic functions and pragmatic dimensions. They consider phonology resp. syntax as basic subtheories.

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sentences and texts and their functionally distinctive parts, such as phonemes or distinctive phonetic features. The theory TB states the properties, relations and systematic constructions of these linguistic entities. This, combined with the postulate of its independence and basicality with respect to the complete theory, implies that (a) the actions in a language depend on the standard systematic form of the means with which they are performed and (b) the structure of the means can be presented (though possibly not heuristically ascertained) without reference to the multidimensional system of pragmatic functions as described above (see Section 3.3). It is not inadequate at present to assume that the theory TB in its intended final form is a deductive nomological theory (though, usually, this is not explicitly stated)."? In most descriptions presented so far the deductive nomological theory aimed at is a special case of an empirically interpreted algebraic or relational system, i.e., it is assumed that it can be represented in terms of some such mathematical system. (In this case, the advent of mathematical or algebraic linguistics as the formal basis of theoretical linguistics is obvious.) This aspect has already been discussed above. I think that it is even possible to show that the basic assumptions of structuralist linguistics correspond, with respect to abstraction, rather closely to language descriptions developed in connection with formal logic. The assumption, with respect to the form of their description, that there is no essential difference between natural and constructed languages (MONTAGUE, 1970a, b) might be quite acceptable from the standpoint of abstract linguistic structuralism. On the basis of this assumption the basic and independent theory TB on linguistic objects may take one of three different forms: 1. T B may take the form of an abstract syntax enumerating the set of linguistic objects that may be uttered and assigning to each object its construction from its parts and from underlying, more abstract, objects (such as word forms from words and sentences from a combination of underlying sentences). In addition to these properties of formation of the objects, derivations between sentences (or underlying sentences) may be defined expressing semantic relations such as synonymy, paraphrase, consequence, semantic change, etc. (e.g., HIZ, 1968, 1969). 2. T B may take the form of a direct semantical system (either a purely 10

Compare, however, footnote 1.

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truth-conditional one as proposed, e.g., by D. DAVIDSON, 1969, 1970, or a model-theoretic one as proposed, e.g., by R. MONTAGUE, 1970a, 1970b and 1973) where, in addition to the construction of each linguistic object its truth condition or interpretation is specified (usually as a recursive function of the syntactic construction of the object). 3. T B may take the form of an indirect semantical system consisting of a pair of systems, the first being an abstract syntax (as in (1) the second being a. direct semantical system (as in (2) together with a translation from the first to the second system. An example is the universal grammatical description proposed by R. MONTAGUE (1971b and 1973). We call such a system an indirect semantical system, since the linguistic objects described receive their interpretation not directly (as in (2) but indirectly through translation into a semantical system whose set of expressions is different from the set of expressions of the syntax of the language though both sets are translatable intoe ach other with the help of their constructions. An indirect semantical system is apparently aimed at in HJELMSLEV (1961, p. 112) where it is argued that all semiotic systems are characterized by the fact that they have to be described on two planes whose (purely combinatorial) forms are not 'conformal' (or isomorphous or even homomorphous), and it seems also aimed at by generative semanticists. Let us summarize the assumptions of abstract linguistic structuralism (corresponding apparently to nonpragmatic constructionism in analytic philosophy and philosophical logic): The complete theory Te describing all phenomena of language behavior contains a subtheory TB which is independent from the rest of the theory, basic to it, about linguistic objects (the means of linguistic actions) and their structure only, may be brought into the form of deductive nomological theory or even a manifestation of some special mathematical structure or a system of such structures generating compact and intuitive representations of statement complexes. Some structuralists even hold that insofar as action in a language is determined by the structure of its means the processes of acquisition and of historical change are also independent of the pragmatic factors of use mentioned above. In other words, the broader theory of these phenomena T~ contains equally a subtheory T~ having the same relation to T~ as TB has to Te . 4. A large number of linguists (probably the vast majority) is working on the basis of the assumptions outlined. There is a growing number of linguists and philosophers of language who feel that the incorporation of theories worked out on the basis of the abstract structuralist assump-

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tion without making explicit the pragmatic conditions of use referred to above will present at least enormous difficulties and that the abstract theories are therefore relatively unrevealing with respect to the task of describing systems of communicative actions in a language taking account of the details of pragmatic dimensions. However, the arguments presented so far l l did not succeed in shaking the position of the abstract structuralists. The main reason seems to be that the detailed description of the contextually determined actions in a language seem to be so complicated that most linguists are easily discouraged, in particular, since, besides informal classifications of the problems, not even an outline of a theory of linguistic action exists. Therefore one feels justified in radical abstraction because of the complications apparently inherent in each attempt at less abstract but precise description.'? But this conclusion is by no means the only way out of the dilemma. On the one hand. the complications to be expected from a detailed analysis should not prevent at all the development of a precise methodological framework for such an analysis. On the other hand, there is substantiated hope that the complications encountered in fully developed systems of actions in languages might be extremely reduced in certain restricted systems of language use. As an example one might consider models of language learning (under abstraction, for the time being, from intricate psychological factors of intellectual development in the child-d. SCHNELLE (1971)). With respect to the first task, there are two ways open: Either one modifies the abstract structuralist theory by indexing the statements in some or all parts of the theory of the language system with respect to factors of the (pragmatic) functionality of language, i.e., with respect to the dimensions mentioned above, or else one rejects a theory on the form and function of linguistic objects (words, sentences and texts) altogether referring in the theory directly to actions with linguistic objects (or even to correlated processes over parts of organisms, e.g., in terms of automaton theory). I should propose a moderate version of a nonstructuralist theory: A theory constructed analogous to an indirect semantical system (with detached phonetics, however) in the following way: There are three subtheories: phonetics, syntactics and semantics. Each of these is represented 11 One might refer to Y. BAR-HILLEL (1970, Chs. 15, 16, 17; 1971), T. G. BEVER (1970) and to C. F. HocKE'IT (1968); P. F. STRAWSON (1970), the arguments of Hockett are unfortunately not always sufficiently precise. 12 Arguments to this effect are presented, e.g., by J. J. KATZ and J. A. FODOR (1963).

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by the set of true statements containing grammatical rules that allow theorems to be derived from a subset of statements, the axioms. The statements of the lexical entries are among the axioms. The subtheories taken in isolation are conceived in the comprehensive and nonrestrictive sense; i.e., they exclude only the statements patently false according to the phonetic, syntactic or semantic regularities considered in isolation (with respect to the syntactics this resembles Chomsky's position of '57). If taken in this comprehensive sense it is possible to neglect on the levels of phonetics, syntactics and semantics pragmatic factors altogether, i.e., for each phonetic, for each syntactic and for each semantic representation there is at least one circumstance under which it may occur in some fully successful speech act. In phonetics, syntactics and semantics one may abstract from pragmatic factors. But as soon as one begins to inquire whether, by producing a sound complex, an individual thereby uttered a certain sentence of a certain syntactic structure and whether, by uttering that sentence, he thereby expressed a sense and a reference representable by a certain semantic representation and, if so, which one it was, one no longer can abstract from the pragmatic factors. In other words the translation between phonetic, syntactic and semantic representations and the interpretation of the semantic representations is dependent on context. One might moreover try to restrict the contextual factors relevant in the semantic representation to the data necessarily connected to linguistic utterances, namely, the communication frame (speaker, hearer(s), utterance, time of utterance, place of utterance). This would allow a pragmatic language for semantic representation containing a number of deictic features such as persons and tense and being thus closer to natural languages than constructed language in logic. For a more detailed presentation of this theory I must refer to my book (SCHNELLE, forthcoming). Another way of evading the dilemma of abstraction mentioned above is the analysis of restricted language varieties. In this field analysis of the learning process of languages seems to offer the most fruitful approach. Such an analysis might be undertaken from the point of view of rational reconstruction abstracting completely from empirically given situations and inquiring only into necessary conditions of learning languages.P It is, however, not yet clear whether the factors determining the initial processes are sufficient for the development of the complex varieties of natural languages we encounter empirically. An alternative to this is an 13

A particularly fruitful model for such procedures has been presented by K.

(1970).

LORENZ

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analysis of the learning process as it is reflected in the assumptions of adults in their communication with children. The model of the adult seems to represent the language of a child as a subsystem of his own. Accordingly he uses this same subsystem in communicating with the child. Together with some colleagues I have begun to analyze language learning and states of learning processes by taking account of all contextual factors considered to be necessary.!" 5. Empirical evaluation of linguistic theories 5.1. After having become confident about their abstractions, structuralist linguists (with the exception of the high times of American structuralism and certain Prague-school linguists) no longer seemed to worry particularly about the empirical evaluation of their proposals being content with predicting standard observations. According to their approach linguistic descriptions will have to provide a phonetic representation designating a sound complex on the one hand and a semantic representation (if the sentence is unambigous) designating state-of-affairs on the other. In the case of a language variety referring only to observable state of affairs both representations designate observable phenomena by use of the descriptive constants of the observation language for linguistic description. But it is not the case that all observations that actually are made on actions in a language in uttering a sentence occur as statements of the theory on that language. In fact the set of statements that are derivable from the theory intersects with the set of statements on facts actually observable; there are statements of the theory that might not occur at all under normal circumstances among the observed ones whereas other actual observations 'deviate' from the observations postulated by the theory on some sense. Because of this the observation statements postulated by the theory should be called standard observation statements (with respect to the theory). In contrast to other empirical sciences, such as physics and mathematical economy, but in correspondence to certain parts of the theory of automata and information processing, the statements of the theory concerning standard observations are themselves statements of the observation language. Therefore, they are theoretical as well as observational. Instead of the correspondence rules of the methodology of physics relating statements 14

For a more detailed exposition I refer to

SCHNELLE

(1971).

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of the theoretical part of the language with those of the observational part, we shall have to specify rules interrelating standard with actual observations.t? However, the standard observations are usually not completely but only partially defined. Particularly perspicuous examples are the sounds having the feature plosive. This feature is not completely determined by the acoustic properties of the sound it characterizes: The immediately following vowels and their acoustic properties have also to be taken into account. 5.2. Usually, the observable properties of the sound continuum and of the state of affairs a speaker might intend to refer to in acts of producing such a sound continuum are not sufficient as an empirical basis. In addition to observations of this kind judgments of certain types are taken as an additional empirical basis. These judgments are particularly important for morphology and syntax. They are usually of the following forms 'x is grammatical' (or 'x is (syntactically or semantically) acceptable', 'x is (syntactically) well formed)' 'x is synonymous to y' 'x implies y' 'x presupposes y', etc., (with x and y as variables over expressions). Recently, there has been a discussion on such matters that is relevant here (cf. Y. BARHILLEL (1971, pp. 403, 4) to which I should like to refer now. Be (1) a name for the string of English words "Harry reminded me of myself". Paul Postal says that (1) is ungrammatical (to the first degree). John Kimball says that (1) is not ungrammatical in his (Kimball's) dialect and in the dialect of most of his informants, though he admits that it might be so in Paul Postal's dialect. Yehoshua Bar-Hillel says that he does not accept the content of what Paul Postal said or what John Kimball admits, namely, that (1) is indeed ungrammatical in Paul Postal's dialect. To prove his case BarHillel is ready to concoct for a reasonable fee a situation in which Postal will show by his conduct that he perfectly understands what Bar-Hillel would be about to state by uttering (1) in this situation. He says that he would even be able, for a much higher fee, to create a situation in which Postal would utter (1). Bar-Hillel proposes that linguists should argue in such a way on facts, i.e., whether people on certain occasions would utter a certain string of words and how easily such an occasion might present 15 In a very detailed and precise analysis A. KASHER (forthcoming) presented practically the same structure. He distinguished between clear spots (standard observations) and unclear spots (nonstandard observations) and functions relating the former to strings and the latter to sets of clear spots (to which they may be equivalent). Setting up the first function is according to Kasher the task of certain linguists, setting up the second, the joint task of linguists and psycholinguists,

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itself or might be produced. The outcome of arguing on such facts is considered to be empirically relevant to the question whether Postal is justified in stating the existence of certain grammatical constraints as part of his system of explanation of ordinary English (resp. of his dialect). Postal and Kimball, as well as many generative grammarians, seem on the other hand perfectly satisfied with judgments as an empirical basis. They seem ready to accept the statement (particularly by a linguist) that a certain string of words is grammatical (or ungrammatical), and if the statement seems to mean what is usually thereby expressed, i.e., it is not a joke or a lie or it does not cite a false example, etc., then it can be taken to be true - (at least in his ideolect or dialect). I think that both positions are unsatisfactory mainly because certain distinctions are not made explicit. First, Postal's and Kimball's position, as well as that of the generativists in general, might be stated in a very weak sense, as I did myself in (1970). According to this the judgments do have only heuristic value in deriving a theory to be taken as a complex empirical hypothesis. Only after the completed setup of this complex will it be empirically evaluated. The judgments only serve as preliminary material for producing shortcuts toward the theory. But on the other hand, one might also try to take a stronger position, namely, to take judgments as empirical data. In this case they should, however, be made more precise. Postal might argue against Bar-Hillel that even if Bar-Hillel should succeed in showing that he (Postal) understands and even sometimes uses utterances of (1) it is not relevant, since he (Postal) admits that indeed he sometimes makes errors in speaking or that he sometimes accepts and even expresses himself in odd ways though he would rather have used another way of expressing what he wanted to say. He might also argue that normally he would have used another way of expressing himself, more conforming to the norm, but that in this situation a deviation from the norm had a nice effect. Or again he might hold that the situation in which he or the speaker used the utterance was so odd that it was beyond the task of linguistics proper (linguistics of competence) to take account of language use in situations of such oddity. But the very fact that there are several ways to evade Bar-Hillel's conclusion shows that the judgments taken as empirical basis are not sufficiently precise in their status. In rejecting the last counterargument I would propose that the empirically basic judgments should have the following form: 'x' may be used (or not) under a circumstance of type C as according to some internalized (dialectal) norm N applicable in circumstance type C

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or alternatively-as a licence leN) with respect to some internalized (dialectal) norm N applicable in circumstance Type C. If at the place of 'x' we not only allow names of expressions but also certain aspects of structural and other conditions, then there are indeed cases where there are no circumstances imaginable under which such an expression will occur. As an example we might have the empirical hypothesis (reformulated cf. BAR-HILLEL, 1971, p. 405): The utterance of the sentence "Harry reminded me of himself" is under no circumstance in accordance with an internalized norm (or a licence to that norm) of the better-known variants of English, when in uttering this sentence one intends to refer to one and the same temporal sector of a person named Harry. I think that empirical hypotheses of this type would be acceptable as observations in linguistics, though it must be admitted that the situation is not yet sufficiently clarified. 6. Generative semantics vs. interpretative semantics - A pseudo-issue? 6.1. For severalyears a lively little war has been going on between generative grammarians about the relation of syntax and semantics. Due to the sometimes obscure terminology it is difficult to evaluate the issue. I shall try to give a short indication of the problems in my terminology. Generative grammar concentrated on the analysis of sentences. Due to the abstraction discussed in Section 4, linguistic actions using sentences are primarily determined by the structure of the sentences: the phonological, morphological and syntactical structure determining the structure of the utterance, the semantic structure and interpretation determining what is expressed in uttering the sentence, i.e., what it is about or what it refers to. The metalanguage for the theoretical description of the entities occurring in the structure of sentences must accordingly provide representations for the expressions (morphemes, formatives, words, sentences) on the one hand, and for the contents expressed on the other and it must show how the contents expressed are related to expressions by eventually relating the corresponding metalinguistic expressions. A representation of an expression is a metalinguistic sentence (of the syntax language) consisting of a complex predicate (the structural description) and an individual expression designating the expression of the object language. The sentence expresses a statement true of the expression designated by the individual expression according to the theory of the language. The complex predicate usually is a combination of representations of parts of the expression or other expres-

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sions 'underlying' the given expression or its parts. In generative transformational grammar alternative representations are used in which the complex predicate true of the sentence either is a tree structure or a sequence or half-ordering of such tree structures. A representation of a content is equally a metalinguistic sentence (of the semantics-language) consisting of a complex predicate (the structural description of the combination of primitive symbols of the semantics language) and an individual expression designating a content (state-of-affairs, event process, action, a modal aspect of these, etc.). In addition to complex one-place predicates (structural descriptions) one might consider two- and more-place predicates on expressions resp. contents. Among these one might have such relations as 'x is a syntactic variant of y (where the selection of x or y is contextually determined)' (as an example consider word-order variants for main clauses and dependent clauses in German) or 'x is a (partial or full) paraphrase of y', etc., or else relations between elements co-occurring in a sentence such as 'x is the object of the verbal phrase y' 'x is a modifier of y'. Some of these relations might only be statable when the complex predicates true of x and yare at least partially given. Finally, it is necessary to specify the pairings of semantic representations and syntactic representations expressing the relation between expressions of the object language and what they express. In a systematic description of a given language, we are interested in the derivation or proof of all representations and pairings true for that language (according to the theory presented). Since they form a subset of all possible representations and pairings, they might be considered to express constraints (on possible combinations of the constants of the metalanguages) to use a favorite term of the generative semanticists. Now, in a first analysis, the essential difference between generative semanticists and interpretative semanticists seems to be whether semantic representations should be derived (by interpretative rules) from syntactic representations which in turn were derived from the axiomatic statements expressed by the syntactic rules of the grammar and the lexicon or whether the semantic representations were (generatively) determined by a set of statements of semantic constraints on the constants of semantic representation and whether syntactic derivations could be derived from them by derivation rules making use at certain places of lexical statements? 6.2. The direction of the derivations (syntax --+ semantics or vice versa) no longer seems to be strictly at issue. It is generally admitted that since

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the rules determine just relations on certain representations and not functions their direction of use might be easily inverted. The discussion concentrates on the other hand on the ways and positions, in the derivations from semantics to syntax, in which the lexicon is used. Chomsky argues (1972) that the insertion of lexical material should be executed en bloc before the underlying structures are transformed into surface structures, beginning with the innermost embedded sentences proceeding to larger and larger stretches (in a so-called cyclic way). The arguments for the insertion en bloc are: that certain restrictions on their simultanous occurrence in a sentence need to be stated only once thereby avoiding unnecessary repetitions of portions of the grammar. Generative semanticists argue against this assumption, because in dropping this restriction one is able to give an analysis of lexical items occurring in sentences into the semantic notions contained in them during the derivation process from semantics to syntax.!" In other words, generative semanticists seem to favor semantic representations, that do not contain for each word and morpheme a corresponding primitive semantic constant, but certain words are directly analyzed into certain combinations of primitive constants. Since such an analysis might involve primitive semantic constants which are in the semantic representation on different levels of embedding, the requirements of Chomsky cannot be met in this case. As an example of an analysis in generative semantics one might cite the example (due to Postal) "Harry reminds me of Fred Astaire". According to Postal's very problematic analysis the word 'reminds' in this sentence is to be analyzed into [(The speaker) (perceive) [(Harry) (similar) (Fred Astaire)]]; 'reminds' incorporates (perceive) and (similar), i.e.; elements on different levels. Another example is the analysis of 'kill' into (cause) (become) (not) (alive) or of 'seek' as (try) (find) in appropriate structures. I think that the arguments, though possibly of empirical interest, are of no basic importance at all. Chomsky on the one hand claims that the demonstration of the empirical necessity of the insertion of lexical material en bloc would show the existence of an intermediate level of description. f think, however, that the merely formal argument that the derivation process from semantics to syntax has a certain formal property is not sufficient to establish the existence of a level of description. There might be a number of other formal properties which would allow us to postulate 16

For recent discussions see J. J.

KATZ

(1971), G.

LAKOFF

(1970),

P. POSTAL

(1970).

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829

other 'levels'. To show the existence of a level one should show that it corresponds to an explicandum. Chomsky and the other interpretative semanticists do not succeed in this or, at least, argue in the wrong direction. As a convincing argument (at least to me) I should point to the necessity of explaining syntactic relations within syntax. In order to do this, there are the alternatives either to state them directly on surface structures (as in CHOMSKY, 1957) or to relate the corresponding surface structures to some underlying structure. This obvious task of stating syntactic relations is not to be confounded with the possibility of deriving these relations from a semantic account. This confusion is often made by generative semanticists. On the other hand, the generative semanticists do not prove that lexical elements necessarily have to be analyzed in the derivation from semantics to syntax. The alternative to analyze these elements or rather their semantic counterpart in the inference part of 'natural logic' (as Lakoff puts it) together with the other semantic aspects is not refuted. If one should argue this would multiply the number of primitive descriptive constants in semantics (namely, introducing for each lexical item one such constant which is an explicatum corresponding to the lexical element) as well as the rules I would not be much impressed. I should valuate the methodological advantage of distinguishing semantic analysis from translation of semantics into syntax higher than formal simplicity of the descriptive apparatus. The apparatus will be complicated anyway. Therefore, its organization into intuitively perspicuous subtheories is of predominant importance. In view of the important methodological problems outlined in previous sections the heated debates on the issues analyzed in this section seem to be a waste of intellectual energies to be stopped as soon as possible. I would like to finish by stressing a moral Bar-Hillel recommended for linguists: "Be more careful with forcing bits and pieces you find in the pragmatic waste basket into your favourite syntactico-semantic theory. It would perhaps be preferable to first bring some order into the contents of this waste basket as is, to clarify somewhat better the explicandum-to use Carnap's undeservedly neglected slogan-before embarking on the explication [BAR-HILLEL, 1971, p. 405]". 7. Concluding remarks I think that linguistics is once again in a position where a new start is required, and it is quite possible that this new start is mediated and initiated by developments not in the center of linguistics but from its periphery, from

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philosophical logic or psycholinguistics and sociolinguistics, for instance. The center of linguistic research is unfortunately much absorbed by the discussion of details of an apparatus which has shown its fruitfulness in syntax but seems to be a straitjacket for semantics. But, as indicated, there is some hope that a new turn in linguistics may get started, less abstract, more adequate, but still formally precise, even if the prize of more limited covering would have to be paid. References BAR-HILLEL, Yo, 1970, Aspects of language (North-Holland, Amsterdam) BAR-HILLEL, Yo, 1971, Out of the pragmatic wastebasket, Linguistic Inquiry, vol. 11, pp. 401-407 BEVER, T. G., 1971, The cognitive basis for linguistic structure, in: Cognition and the Development of Language, ed. J. R. Hayes (Wiley, New York), pp. 279-362 CHOMSKY, N., 1957, Syntactic structures (Mouton' The Hague CHOMSKY, N., 1969, Linguistics and philosophy, in: Language and Philosophy, ed. S. Hook (New York University Press, New York), ppo 51-94 CHOMSKY, N., 1972, Some empirical issues in the theory of transformational grammars in: Goals of Linguistic Theory, ed. S. Peters (Prentice Hall, Englewood Cliffs), ppo 63-~30 DAVIDSON, D., 1969, Truth and meaning, in: Philosophical Logic, ed. J. W. Davis, D. J. Hockney, Wo K. Wilson (Reidel, Dordrecht), pp. 1-20 DAVIDSON, Do, 1970, Semantics for natural languages, in: Linguaggi nella societa e nella tecnica, ed. B. Visentini (Edizioni di Communita, Milano), pp. 177-188 GREGORY, M., 1967, Aspects of varieties differentiation, Journal of Linguistics, vol. 3, pp. 177-198 HIZ, He, 1968, Computable and uncomputable elements ofsyntax, in: Logic, Methodology and Philosophy of Science, vol. 3, eds. B. van Rootselaar and J. F. Staal (NorthHolland, Amsterdam), pp. 239-254 HrnLMSLEV, L., 1961, Prolegomena to a theory of language (University of Wisconsin Press, Madison), HOCKETT, C. F., 1968, The state of the art (Mouton, The Hague) KAsHER, A., 1972, Sentences and utterances reconsidered, Foundations of Language 8, pp. 313-345 KATZ, J. J., 1971, Generative semantics in interpretative semantics, Linguistic Inquiry, vol. 11, ppo 313-331 KATZ, J. J. and J. A. FODOR, 1963, The structure of a semantic theory, Language, vol. 39, pp. 170-210; reprinted in: The structure of language, ed. J. A. Fodor and J. J. Katz (prentice Hall, Englewood Cliffs, New Jersey, 1964), pp. 479-518 LAKOFF, G., 1970, Linguistics and natural logic, Synthese, vol. 221/2, pp. 151-271 LIEB, H. H., 1970, Sprachstadium und Sprachsystem (Kohlhammer, Stuttgart) LORENZ, K., 1970, Elemente der Sprachkritik (Suhrkamp, Frankfurt) MARGALlf, A., 1971 The cognitive status of metaphors (The Hebrew University, Jerusalem) (mimeographed paper as an abstract cf. Ph.D. thesis in Hebrew)

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MONTAGUE, R., 1970a, English as a formal language, in: Linguaggi nella societa e nella tecnica, ed. B. Visentini (Edizioni di Communita, Milano), pp. 189-233 MONTAGUE, R., 1970b, Universal grammar, Theoria, vol. 36, pp. 373-398 MONTAGUE, R., 1973, The proper treatment of quantification in ordinary English, in: Approaches to Natural Languages, eds.: J. K. Hintikka, J. Moravcsik and P. Suppes (Reidel, Dordrecht) POSTAL, P., 1970, On the surface verb 'remind', Linguistic Inquiry, vol. 1, pp. 37-120 SCffiLPP, P. A., ed., 1963, The philosophy of Rudolf Carnap (Open Court, La Salle) SCHNELLE, H., 1970, Theorie und Empirie in der Sprachwissenschaft, Bibl. Phonet., vol. 9, pp. 51-65 SCHNELLE, H., 1971, Communication with children-Toward a theory of language use, in: Pragmatics in Natural Languages, ed. Y. Bar-Hillel (Reidel, Dordrecht), SCHNELLE, H., Introduction to Theoretical Linguistics (Springer, Heidelberg, New York) (forthcoming) STRAWSON, P. F., 1970, Meaning and truth (Clarendon Press, Oxford) VON WRIGHT, G. H., 1963, Norm and action (Routledge and Kegan Paul, London) VON WRIGHT, G. H., 1968, An essay in deontic logic on the general theory of action, Acta Philosophica Fennica, Fasc. 21 (North-Holland, Amsterdam) WANG, J., 1971a, On the representation of context-free grammars as first-order theories, in: Proceedings of the 4th Hawaii International Conference on System Sciences (Western Periodicals, Honolulu), WANG, J., 1971b, On the representation of generative grammars as first-order theories, Contributed paper at the IV International Congress on Logic, Methodology and Philosophy of Science, Bucharest

SOME REMARKS ON THE NOTION 'UNIVERSAL SEMANTICS,l

S. C. DIK University of Amsterdam, Amsterdam, The Netherlands

Two of the most important trends in recent linguistic thinking are the growing concern with semantics as a fundamental component in the system of language and the rehabilitation of 'universal grammar' as a respectable topic of linguistic investigation. In this paper, I should like to make some remarks about the intersection of these two trends, embodied in the notion universal semantics. My intention is, on the one hand, to inform philosophers about linguistic developments in a field which many of them will, not without some reason, regard as their own, and, on the other hand, to bring together and critically evaluate some of the alternative approaches to the problems of universal semantics which can be found in the literature. My remarks will be divided over the following four topics: 1. The notion 'semantic content of a linguistic expression'; 2. The theoretical status of 'universal semantics'; 3. The possible empirical content of universal semantics; 4. The possible explanations of the existence of a universal semantics. 1. The notion 'semantic content of a linguistic expression' It is generally agreed among linguists that a description of a natural language should, for each well-formed expression of that language, specify at least a phonological structure, representing the make-up of the expression in terms of basic sound elements and their interrelations, a syntactic structure specifying the syntactic categories and relations in terms of which the expression is organized, and a semantic structure representing, in terms of basic semantic components and relations, the semantic content of the ext I am grateful to Norval S. H. Smith for having corrected the English of this abstract,

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pression. Furthermore, the description should explicitly interrelate these three types of structure, thus explaining, in the last analysis, how certain meanings are expressed in certain forms, and how certain forms can be interpreted as carriers of certain meanings. It is thus evident that, no matter how the relations between phonological, syntactic, and semantic structure are conceived of, any theory about the structure and the description of natural languages will have to contain a subtheory about the properties of the semantic content of linguistic expressions and about the way these should be represented. This subtheory is generally called semantic theory. The objects to be accounted for by semantic theory are the semantic contents of linguistic expressions." In order to get a better picture of what should be understood by semantic content, it is first of all essential to realize that linguistic expressions are typically used as communicative instruments in concrete situations defined by a particular speaker, particular addressees, a particular time and a particular place. Through the use of a linguistic expression in a specific situation the speaker causes it to happen that the addressee(s) arrive(s) at a certain amount of information. Let us refer to this amount of information as the final interpretation of the linguistic expression involved, and to the information contained in the linguistic expression as such, as the semantic content of that expression. Given these conventions it must be stressed that final interpretation and semantic content are never identical; that the final interpretation of a linguistic expression always contains more information than its semantic content; that different addressees will arrive at different final interpretations in the same situation; and that the same addressee will arrive at different final interpretations in different situations. Let me explain this by means of a simple example. Suppose I say, in some particular situation: (1) I am very fond of marihuana. Then some of the addressees may take this as a disinterested expression on the part of the speaker of genuine appreciation for this particular kind of soft drug. Many will immediately evaluate this according to their personal stand on these matters. Others may think I am wanting to get a share of their personal supplies. Others again will think I am telling a lie or trying to make a joke which again they will either evaluate positively or, more probably, classify as being in poor taste. Some, however, will not interpret 2 The view of semantic content outlined here is inspired by Anton Reichling. See, e.g., Doc (1968, Ch. 12) and REICHLING (1963).

UNIVERSAL SEMANTICS

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my utterance as a factual statement at all, but simply as an example in a linguistic discussion. As such it can again be evaluated differently because of its fashionable character. All these interpretative differences are heavily underdetermined by the semantic content of my linguistic expression which, in fact, is the same in all cases. This is possible because, in interpreting anyone utterance, the addressee not only uses his knowledge of the language involved, but also his knowledge of the context and situation in which the utterance is embedded and his general, long-range knowledge (including his beliefs, preconceptions, etc.) with respect to any facts relevant for the interpretation of the utterance. We can now define the semantic content of a linguistic expression as: the information which, given the role of context and of extralinguistic knowledge, it is necessary and sufficient to assign to a linguistic expression in order to explain its different final interpretations. The semantic content of a linguistic expression is a structured entity. It can be conceived of as consisting of basic semantic elements belonging to different categories, fulfilling different functions in the total content, and contracting various types of relation with other semantic elements. On the basis of their internal structure semantic contents stand in various relations to each other, such as synonymy and paraphrase, presupposition, implication. It is the task of semantic descriptions to define the internal properties and external relations of semantic contents without over- or under specifying them with respect to the possible final interpretations. And it is the task of semantic theory to define the general framework within which such descriptions can be formulated. 2. The theoretical status of 'universal semantics' According to some basic requirements put on theories in general, semantic theory should be as general and as simple as possible. It should preferably specify the properties of semantic structure in a way relevant for sets of languages, and even more preferably in a way relevant for all languages. We can speak of universalsemanticsto the extent that semantic theory is relevant for all languages. Seen in this perspective, it is evident that the notion of 'universal semantics' is a methodologically necessary notion. It is necessary to define the framework in which semantics would achieve the required maximum of generality and the optimal degree of explanatory power.

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Universal semantics will have to specify those properties of semantic structure which are invariant over the set of all natural languages. What those properties are may, even at this quite abstract level, be specified in different ways. BAR-HILLEL (1969, pp. 9-10) has proposed to restrict the term 'universal' to those properties which are inherent in the definition of 'natural language'. GREENBERG, OSGOOD, and JENKINS (1966, p. 14), on the other hand, define universals as: "characteristics possessed by all languages which are not merely definitional; that is, they are such that if a symbolic system did not possess them, we would still call it a language". I believe it is best to use the term 'universal' for any property common to all natural languages, and to follow MORAVCSIK (1967, p. 224) in distinguishing among: 1. Analytic universals: properties which all languages have by definition; 2. Accidental universals: properties which all languages just happen to have by coincidence (e.g., suppose that English, Chinese, and Maori are the only languages having a certain property. Then, if all languages except these three were to become extinct, this property would thereby become an accidental universal of natural language. (Cf. HOCKETT, 1966)).

3. Synthetic nonaccidental universals: properties which all languages have, though not by definition or coincidence, and which thus represent substantive restrictions on what could logically count as a natural language. As CHOMSKY (1968, p. 51) has stressed, it is this last type of universal which perhaps is most interesting, since it can tell us something about the types of language which human beings are capable of handling. Though this threefold distinction of universals is theoretically useful, it will no doubt prove difficult to apply in particular instances, since there is no a priori insight into what properties should be reckoned to belong to the 'defining set' for natural language (cf. HOCKETT, 1966), or into what factors determine the substantive restrictions on the class of logically possible natural languages. Much more must be known about what can count as universal in the first place, before these questions can be answered (if they can be answered at all). There are also different opinions about the relationship between universal semantics on the one hand, and semantic descriptions of particular languages on the other. I shall refer to three different points of view on

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this matter as to the union interpretation, the intersection interpretation, and the metainterpretation of universal semantics. The union interpretation holds that there is a fixed and finite set of semantic primitives from which each natural language makes its own selection. Universal semantics would thus consist of the union of the sets of semantic primitives to be found in particular languages. The intersection interpretation considers universal semantics to consist of the intersection of the sets of rules, categories, and elements relevant to the semantics of particular languages. The metainterpretation says that universal semantics also contains a set of metarules specifying certain universal properties of semantic content without themselves figuring in the description of particular languages. There are several weaknesses inherent in the union interpretation of universal semantics. In the first place, it has yet to be demonstrated that the semantic primitives of all languages can be considered to be drawn from a fixed and finite universal set of such primitives in any nontrivial way. In the second place, there is a danger that nonlinguistic, e.g., perceptual or cognitive, categories may be taken to be linguistic ones. Let us illustrate this with an example. In a fascinating' cross-linguistic study of color terminology BERLIN & KAY (1969, p. 2) have demonstrated that: " ... although different languages encode in their vocabularies different numbers of basic colors, a total inventory of exactly eleven color categories exists from which the eleven or fewer basic color terms of any given language are always drawn." They take this as a proof that 'semantic universals do exist in the domain of color vocabulary'. What they have shown in fact, however, is that the famous semantic relativity in color terminology is rather heavily constrained by the existence of a limited number of perceptual categories (cf. what they say on p. 109), which are more fully and more precisely given expression in one language than in another. Such categories define limiting conditions for the semantic structuring of particular languages. They do not define semantic universals. The intersection interpretation and the meta interpretation, on the other hand, seem to constitute necessary components of universal semantics lacking the methodological weaknesses of the union interpretation. Some element denoting the speaker without naming him will probably belong to the intersection of the semantic structuring of all languages. Languageindependent characterizations of such relations as synonymy or antonymy and universal conditions on the format of semantic structures and semantic rules will be part of the metatheory of semantics (cf. CHOMSKY, 1968,

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p. 57). If we combine these two components, we can say that universal semantics will consist of two distinct parts: viz.: (i) a set of rules, definitions, and conditions defining universal properties of semantic contents and their descriptions; (ii) a set of rules and primitive semantic elements figuring in the semantic contents of all natural languages. 3. The possible empirical content of universal semantics About the possible empirical content of universal semantics few definitive statements can be made at this moment, although the situation is less gloomy than just a few years ago, when CHOMSKY (1968, p. 50) could refer to "a still-to-be-developed universal semantics" and mentioned the "tangle of confused issues and murky problems" one was likely to get into when trying to solve general semantic problems. Indeed, several different though only partly incompatible approaches to the analysis of semantic structure have been developed in the past few years, the major results of which can be summarized in the following way. In spite of differences in formulation and elaboration, many semantic theorists agree in that the basic form of semantic representations should look like the structure given below: prop

_//;~ (2)

pred

I

GIVE

/

/

arg,

I

JOHN

"'~ arg ,

I

arg;

I

MARY BOOK

where 'prop', 'pred', and 'arg' are variables ranging over propositions, predicates, and arguments, respectively. The basic structure is thus closely parallel to the form of expressions in predicate calculus, with the important difference that the constants assigned to the variables in (2), i.e., the elements printed in capitals, represent the basic content elements of natural language. In a certain sense, these basic content elements can again be regarded as predicates, with the variables to which they are assigned as arguments. For this reason, these elements are often referred to as 'semantic predicates'. Structures of type (2) must be made recursive by allowing propositions in argument positions. Thus, we can get structures like the following: (3) a. John believes that Bill thinks that Mary is beautiful b. BELIEVE(JOHN, THINK(BILL, BEAUTIFUL(MARY»)

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UNIVERSAL SEMANTICS

(4) a. John seems to begin to realize that he has made a mistake b. SEEM(BEGIN(REALIZE(JOHN, MAKE(JOHN, MISTAKE»» It is a matter of debate whether, in addition to predicates, a specific class of operators must be distinguished, or whether the semantic information often assigned to such a class, i.e., negation, tense, modality, quantification, can be expressed in structures like (2), (3b), (4b) without further adjustment. In the case of quantifying expressions in particular, this issue (which in other cases may be purely terminological) seems far from settled. 3 The primitive 'semantic predicates' referred to in the last paragraph should be sharply distinguished from the lexical items figuring in particular languages. This is evident in the case of lexical elements like father and mother, in which at least the relational predicate PARENT OF (x, y) and the nonrelational predicates MALE/FEMALE (x) should be recognized. But it also applies in the case of a lexical element like female which seems to coincide with the predicate FEMALE. The word female, however, is part of the object language. The predicate FEMALE is part of the metalanguage used for describing the meanings of words in the object language and is, as such, part of the meaning of many other lexical elements. This analysis in terms of primitive predicates is necessary in order to account for quite a few properties of semantic contents, e.g., the contradictory (or at least out-of-the-way) character of:

(5)

My female father kissed me,

and the tautologous character of: (6)

My female mother kissed me.

The idea is, of course, to determine, for each particular language, the set of primitive predicates necessary and sufficient for describing the semantic content of its linguistic expressions. This set of primitive predicates, which may be termed the semanticon of the language, is hoped to be substantially smaller in size than the set of lexical elements, the lexicon of that language. It is, in general, not known what the primitive predicates of anyone particular language are. It is a fortiori unknown what the intersection of all the semantics of particular languages might contain, i.e., what the set of universal primitive predicates might be. It is clear, however, that primitive predicates have a much better chance of being relevant to more 3 On these various points, see BACH (1968), (1970).

FILLMORE

(1968),

McCAWLEY

(1970), LAKOFF

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S. C. DIK

than one particular language than lexical elements, which are typically language-dependent. It is also probable that a certain number of primitive predicates, e.g., those connected with the speaker, the addressee, the time and place of speaking, the various speech acts which can be performed, the human body, the basic coordinates of the environment in which we live, basic relationships like identity, similarity, and so on will prove to be very generally if not universally valid. Primitive predicates should not be thought of as independent entities. There is, e.g., an obvious relationship between the predicates MALE and FEMALE which can be expressed by presenting these predicates as mutually exclusive values of a semantic category SEX. In a similar way, other predicates may be grouped into semantic categories. In analyzing verbs of motion, e.g., we can make use of categories like the MEDIUM through or over which the movement takes place, the TYPE OF INSTRUMENT used, the SPEED, the MANNER, the SPATIAL ORIENTATION of the movement, and the like. Since these categories are more general than the predicates which specify their values, they have again a better chance of being generally valid. It is difficult to conceive of a language in which the semantic category of COLOR would not in some way or other be relevant, though languages differ in the particular values they can specify for this category. Other types of relationship between predicates may be defined by general metarules. The relation between PARENT OF (x, y) and CHILD OF (x, y) could be laid down in an equivalence defining the one as the converse of the other. Similar conventions have been proposed by BAR-HILLEL (1969), and will probably prove to be necessary ingredients of universal semantics. Universal validity is also claimed by FILLMORE (1968) for what he calls the semantic 'cases', i.e., roles or functions, of the participants required by anyone verb, in the sense in which a verb like break requires an Agent responsible for the action described, an Object affected by the action, and an Instrument used by the Agent in affecting the Object; similarly, a verb like accuse requires a Judge pronouncing the verdict, a Defendant whose responsibility is asserted, a Situation for which the Defendant is said to be responsible, and an Addressee to which the assertion is directed. Such semantic roles, which can be shown to be largely independent of the syntactic position of the expressions fulfilling them, playa part in semantic analyses of different provenance. Various relationships between semantic contents will also find their

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way into universal semantics, e.g., relations like presupposition, implication, contradiction, and the like. In particular, the notion of presupposition, first developed by philosophers like Straws on and Austin, proves increasingly useful in the semantic analysis of natural language. It is not yet clear, however, to what extent the information presupposed by a certain semantic content should figure in the specification of this content and if so, how this could be done. Space prevents me from giving a more detailed survey of possible candidates for inclusion in universal semantics. It may have become clear, however, that there is ample reason to believe that the two components of universal semantics, i.e., both the intersection of the semantic descriptions of particular languages and the meta component containing general rules, definitions, and conventions, will finally prove to be far from empty. 4. The possible explanations of the existence of a universal semantics I should like to finish this survey with some remarks about how the existence of universal semantics could be explained. One explanation, which is currently very much in vogue, is embodied in Chomsky's Innateness hypothesis which, in its strongest form, states that linguistic universals reflect so many properties of the human mind which are genetically determined. The strongest argument for the correctness of this hypothesis is that it is very difficult to understand how the child, given the fragmentary and degenerate data which it is able to perceive, could arrive in so short a period at a fully fledged competence in its native language, unless it is presumed that the child is preprogrammed as it were with highly structured and specific expectations about what a natural language can be. The best candidate for this innate information on the possible form of natural languages is precisely the set of linguistic universals. I find it difficult to argue about this hypothesis in abstracto. Judging from the number of recent contributions for and against the Innateness hypothesis, this is probably my fault. It appears to me, however, that advanced research into the nature of language acquisition may well show that the data on which the child has to go in learning its language are less fragmentary and degenerate, the time of learning less short, and the competence finally acquired less fully fledged than they are so often asserted to be. If this is true, then the need for postulating substantive initial equipment would diminish proportionately. More important, it seems to me, is the fact that the proponents of the

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Innateness hypothesis almost exclusively concentrate on the individual psychological factors involved in language acquisition and in linguistic competence and performance, to the neglect of their social aspects. It has been rightly stressed that what we want to explain first of all is communicative competence, i.e., the ability of human beings to communicate with each other by means of language (CAMPBELL and WALES, 1970, pp. 249 ff.). This is a social phenomenon not only determined by the biological and psychological nature of man, but also by the fundamental features of the sociosituational environment in which he lives. If, following anthropologists like Murdock and Kluckhohn who have concerned themselves with the problem of cultural universals in general, we take this sociosituational factor into account, it is immediately evident that any explanation in terms of innate predispositions alone is unduly restrictive. Let us propose a less restrictive hypothesis for explaining linguistic universals and call it the Means-Ends hypothesis. In this hypothesis, language is regarded as (1) an instrument for general communication, (2) to be used by human beings, (3) in social situations, (4) availing themselves of vocal sounds as a communicative channel. The conditions (1)-(4) each place heavy constraints on any system which is to qualify as such a communicative instrument. KLUCKHOHN'S (1962, p. 317) statement: "All cultures constitute so many distinct answers to the same questions posed by human biology and by the generalities of the human situation" may then well be read with 'languages' in the place of 'cultures'. Due to the constraints mentioned the answers which natural languages provide, though infinitely many, are not infinitely diverse. References BACH, E., 1968, Nouns and noun phrases, in: Universals in Linguistic Theory, eds. E. Bach and R. T. Harms (Holt, Rinehart and Winston, New York), pp. 91-122 BACH, E. and T. HARMS (eds.), 1968, Universals in linguistic theory (Holt, Rinehart and Winston, New York) BAR-HILLEL, Y., 1969, Universal semantics and philosophy of language: quandaries and prospects, in: Substance and Structure of Language, ed. J. Puhvel (University of California Press, Berkeley), pp. 1-21 BERLIN, B. and P. KAY, 1969, Basic color terms: their universality and evolution (University of California Press, Berkeley) CAMPBELL, R. and R. WALES, 1970, The study of language acquisition, in: New Horizons in Linguistics, ed. J. Lyons (Harmondsworth, Eng.), pp. 242-260 CHOMSKY, N. 1968, Language and mind (Harcourt, Brace and World, New York) DIK, S. C., 1968, Coordination; its implications for the theory ofgeneral linguistics (NorthHolland, Amsterdam)

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FILLMORE, J., 1968, The case for case, in: Universal in Linguistic Theory, eds. E. Bach and R. Harms (Holt, Rinehart and Winston, New York), pp. 1-88 GREENBERG, J. H., 1966, Universals of language (M. I. T. Press, Cambridge, Mass.) GREENBERG, J. H., C. E. OSGOOD and J. J. JENKlNS, 1966, Memorandum concerning language universals, in: Universals of Language, ed. J. H. Greenberg (M. I. T. Press, Cambridge, Mass.), pp. 15-27 HOCKE1T, C. F., 1966, The problem of universals in language, in: Universals of Language, ed. J. H. Greenberg (M. I. T. Press, Cambridge, Mass.), pp. 1-29 KLUCKHOHN, C., 1962, Universal categories ofculture, in: Anthropology Today: Selections, ed. S. Tax (Chicago University Press, Chicago), pp, 304-320 LAKOFF, G., 1970, Linguistics and natural logic, in: Semantics of natural language, ed. D. Davidson and G. Harman (Reidel, Dordrecht), pp. 545-665 McCAWLEY, J. D., 1970, Where do noun phrases come from? in: Readings in English Transformational Grammar, eds. R. A. Jacobs and P. S. Rosenbaum (Ginn, Waltham), pp. 166-183 MORA VCSIK, J. M. E., 1967, Linguistic theory and the philosophy of language, Foundations of Language, vol. 3, pp. 209-233 REICHLING, A., 1963, Das Problem der Bedeutung in der Sprachwissenschaft (Auslieferung durch das SprachwissenschaftIiche Institut der Leopold-Franzens-Universitat Innsbruck)

UNCONTAINED RULES OF MEANING

J. F. STAAL University of California, Berkeley, Calif., USA

Many seperate treatises on different branches of general grammar are very properly considered as appertaining to the science of logic. COLEBROOKE. On the Sanscrit and Pracrit Languages (1803)

Since Chomsky began to turn linguistics from a respectable branch of scholarship into a fast-developing scientific discipline, there has been increasing awareness of the close relationship between certain linguistic and certain logical problems, and accordingly a slow convergence between research in linguistics and in logic. The area of this convergence is largely the area of semantics and reflects the growing importance of semantics within linguistic theories of language. Yet, there are differences between logic and linguistics. Logicians deal with regularities that mayor may not correspond to regularities and universal features of natural languages. Linguists also have to consider irregularities which are by definition not universal but occur in specific natural languages. Where in a language do such irregularities occur, and where and how do have they to be incorporated in the theory of language? Linguists, traditional structuralists as well as contemporary generativists, generally answer that irregularity and arbitrariness lie in the vocabularies of particular languages and have to be treated in the lexicon. BLOOMFIELD (1933, p. 274) for example writes: "The lexicon is really an appendix to the grammar, a list of basic irregularities". And BACH (1968, p. 117): "In positing a universal base component it is, of course, necessary to exclude the lexical component. It is precisely in the set of lexical elements that the famous 'arbitrariness' of the linguistic sign becomes most obvious". The beginning of 'transformational generative grammar' in the late

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50's was marked by Chomsky's discovery of deep-lying regularities in syntactic structure. This led to a picture of language as consisting of a combination of syntactic regularity and lexical irregularity. But there are regular processes by which lexical items are inserted into syntactic structures, and the meaning of lexical items contributes to the meaning of the syntactic structures into which they are inserted. Semantics thus came to be added to syntax and was conceived of as consisting of two parts: a lexicon, dealing with irregularities, and a set of projection rules and (later) rules of semantic interpretation, dealing with regularities. This view led to the position nowadays referred to as 'interpretative semantics'. In the 60's an alternative view within the general perspective of transformational generative grammar made its appearance. Syntactic structures were no longer regarded as pivotal, new semantic regularities were discovered, and increasing attention was focused on semantic structures and subsequently on logic. Though this approach, leading to what is nowadays called 'generative semantics', was .in some respects nothing but a notational variant of interpretative semantics, it also developed notions and ideas which are genuinely different from those of the rival view. In generative semantics, grammar is conceived as a continuum of rules describing and explicating semantic and syntactic regularities, with lexical irregularities inserted at different places along the way. Interpretative and generative semantics met with semantic problems that did not fit into the postulated components of grammar. It is now beginning to be realized that some of these problems occupy in fact a vast and important area of semantics which is so far uncontained within the existing theories. I will try to show that this area of semantics requires a theory of its own (for which the way is paved to some extent by logical work) which will have to be explicitly related to the other components of grammar before it is possible to evolve an adequate general theory of language. I shall do this mainly by evaluating some of the ideas propounded in interpretative and generative semantics, not because I believe that the fashionable controversy between these two theories runs as deep as it is sometimes made to appear, but because this controversy tends to obscure the fact that there are many profound insights, as well as profound shortcomings, which are shared (though sometimes unwittingly) by both camps. A settlement will not be reached by providing counterexamples to the opponent's theories (for counterexamples do not refute theories: only theories do) and not unless the area of semantics I have in mind receives its full due.

UNCONTAINED RULES OF MEANING

847

The area of semantics I wish to consider comprises numerous wellknown semantic relationships between concepts that may be expressed by lexical items, e.g., the relationships between the senses (which I shall write in capitals) PRECEDE and FOLLOW of the English lexical items (which I shall write in italics) precede and follow, respectively. Such a relationship could be expressed by a semantic expression of the form: FOR ALL X AND Y, IF X PRECEDES Y, Y FOLLOWS X.

(1)

The semantic relation expressed by (1) is connected not only with the English lexical items precede and follow, but also with any other two lexical items or composite expressions in English, e.g., come before and come after, or in any other language, e.g., French preceder and suivre, which connote the two senses PRECEDE and FOLLOW. (1) itself is not a sentence of English, though it is obviously related to sentences of English, e.g.:

If John's birth preceded his mother's wedding, then her wedding followed h~b~~.

~)

The meaning of (2) cannot be accounted for by a compositional function of the kind proposed by Katz, deriving it from the syntactic structure of (2) and the meanings of its parts, without invoking (1) or an equivalent statement. For (1) itself is not a compositional function. Hence, the required meaning could only be derived if at least one of the lexical items, for example precede, was accompanied by a meaning specification equivalent to (1). Attempts to decompose the lexical items, e.g., by explicating precede in terms of before and follow in terms of after, would not come any closer to (1) and only shift the problem to another couple of lexical items. And such a process of decomposition cannot go on forever. It is possible to attach (1) to a lexical item, but not without changing the character of the lexicon fundamentally: the lexicon would fall apart into two separate domains, one incorporating the lexical entries, the other, expressions such as (1). Is it possible that (1) is not just attached to a lexical entry, but actually part of it? The lexical entries listed in the lexicon are each of the form: pew) is '"

(3)

where P is a one-place syntactic or semantic predicate (like 'the-meaning-of'), w is a lexical item, and a semantic notion, e.g., a 'meaning' or 'reading', has to be substituted to the right of "is". The requirement that the lexicon must be constrained in this way is analytic: (3) gives expression to the notion of lexicon. This is presupposed by all grammatical theories

848

J. F. STAAL

that have considered this matter in any detail, in particular, interpretative semantics and generative semantics. In interpretative semantics, the lexicon must provide the meanings of lexical items that are inserted into preterminal strings so that the projection rules or rules of semantic interpretation can apply. In generative semantics, the lexicon must provide at least the meanings of the smallest units into which transformationally derived lexical items are decomposed. Assuming that precede is not a composite lexical item (in which case the argument would shift to the units into which it should be decomposed), its lexical entry must be of the form: P(precede) is ... .

(4)

It is conceivable that (4) would be of the form:

P(precede) is PRECEDE.

But it is inconceivable that (4) is of the form: P(precede) is 1(1)1,

(5)

(6)

for what is required on the right of "is" in (6) is a semantic notion, not a relation between semantic notions. Hence expressions such as (1) cannot be part of lexical entries. Even if these arguments were incorrect, and it were possible for expressions such as (1) to be part of, or attached to, lexical entries, the question would arise why (1) should be attached to or incorporated in precede rather than follow. Any decision taken in this respect seems arbitrary. If on the other hand (1) were related to both lexical entries, this would introduce an obvious redundancy. Lastly, how could a presumably universal relationship like the one that is expressed by (1) be tucked away in the lexicon of a specific language such as English? All these considerations show that indispensable semantic expressions such as (l) are not part of the lexicon, but must be accommodated elsewhere in the' semantic system. I shall tentatively assume that such rules of meaning include at any rate expressions that are formulated in terms of semantic concepts (hence written in capitals) and relate the meanings of at least two lexical items, or of the elements into which they are decomposed, to each other. It is impossible to decide where and how expressions like (1) have to be incorporated into the grammar without considering how lexical items themselves are related to other parts of the grammar. I shall now review some of the solutions proposed with regard to two related problems: how to deal with expressions such as (1), and how to formulate lexical insertion.

UNCONTAINED RULES OF MEANING

849

Logicians have recognized the kind of meaning relationship expressed by (1) and have recognized (1) as a meaning rule or meaning postulate. These notions were originally proposed by Carnap in order to explicate the concept of analyticity, or truth based upon meaning, for an artificial language or "semantical language system" (CARNAP, 1947, especially pp. 222-229). Carnap's idea was simply to explain the analyticity of such statements as:

If Jack is a bachelor, then he is not married

(7)

by a meaning postulate: (x)(Bx

~ ~

Mx),

(8)

which expresses the incompatibility of the properties Band M without requiringthat BandM designate the properties BACHELOR and MARRIED, respectively. The concept of analyticity for a language L is then explicated in terms of the conjunction of the meaning postulates of L. Quine has criticized this explication of analyticity by pointing out that the notion of analyticity is not clarified by a list of truths called meaning postulates: these truths themselves might as well be called analytic truths, which would not explicate the notion of analyticity either (QUINE, 1953, 32-37). Meaning postulates such as (8) can also be interpreted within a different perspective. When they are provided with the further specifications that Band M designate the properties BACHELOR and MARRIED, respectively, such meaning postulates provide information about the semantic relation between any two lexical items or composite expressions of any language (say, bachelor and married in English, or expressions such as 'man who has never had a wife' and 'legally united to a person of the opposite sex for the purpose ofprocreation') which refer to the properties BACHELOR and MARRIED, respectively. This second notion of a meaning postulate or meaning rule was rediscovered by linguists only much later, and its importance is beginning to be appreciated.' Linguists were originally occupied with the statement of lexical insertion, in logical terms the substitution of a constant for a variable. The first detailed proposals 1 At the Bucharest Congress C. G. Hempel argued that meaning rules are no longer needed. So it seems that linguists are beginning to demand what philosophers at long last have discovered to be redundant. But the contradiction is only apparent. It is quite possible that the sciences (including linguistics) do not need separate meaning rules specifying the meanings of key terms and concepts occurring in scientific theories; but linguistics has meaning among the objects it must account for, and is therefore in quite a different position.

850

J. F. STAAL

for lexical insertion occur in Chomsky (1965, p. 84 and following). Here a grammatical base consists of Phrase-markers or trees, each generated by a system of context-free rewriting rules. While most nodes are occupied by symbols representing grammatical categories, e.g., S, NP, N, V, certain P-markers terminate in so-called preterminal strings, which consist of grammatical formatives and complex symbols. Complex symbols are sets of syntactic features introduced into the tree by special rewriting rules. The syntactic features are unspecified, positively specified or negatively specified with respect to each lexical item. For example, boy will have the syntactic features [+ Common), [+ Human), among others. Neglecting further complications, a part of a tree terminating in a preterminal string may look like this: 5

~ .r ANIMATE X), which would express the required redundancy perfectly well. Chomsky cannot use such an expression, since he does not in this context make use of variables over terms. Lexical categories in preterminal strings may also dominate a variable of a special sort, introduced in a different statement of the theory (1965, p. 122 and following) and called "a fixed dummy symbol", Lt. zl specifies 2 I think it is incorrect to say that a transitive verb is a verb followed by a NP. Rather, a transitive verb is a verb that requires an object. For further discussion see STAAL (1967a, p. 83 and following).

852

J. F. STAAL

the posinon of a lexical category, which determines a specific structure to which a transformation (the operation of which is conditioned by this structure) mayor must apply. Actually, this L1 is redundant, for it is not difficult to determine whether the structure of the terminal string of a Pmarker meets the structural condition of a transformation; such a structural condition is formulated precisely in order to enable us to see whether or not it is met. I mention L1 because it was recently extracted from its oblivion by KATZ (1971, p. 322), who admits that Chomsky introduced this symbol to trigger such transformations as permutation transformations, but who observes that its occurrence can also trigger a lexical insertion rule, which is after all a transformation. This would allow the insertion of lexical items at any stage of a derivation. The validity of this assertion depends on whether L1 is allowed to occur only in preterminal strings in deep structure, or also elsewhere. CHOMSKY (1965, p. 22) thought only of occurrences of L1 in preterminal strings, for occurrences of L1 are replaced by lexical items and lexical insertion is only defined with respect to the nodes occurring in preterminal strings. Chomsky in his later work has adhered to the mains outlines of this theory of lexical insertion. There have been several refinements and extensions, e.g., CHOMSKY (1970a), and CHOMSKY and HALLE (1968, pp. 373-380). But the general position with regard to the lexicon remains that the vocabulary is introduced into the grammar with the help of the lexical insertion of lexical items specified by sets of syntactic features, into positions marked by a variable in a preterminal string which is specified by the same set of syntactic features. It is not clear where the list of syntactic features is supposed to end and whether there are semantic features that are basically different and are also required. But it is clear that in addition to these syntactic features and the lexical items themselves, rules are required which relate them to each other, which are basically equivalent to the meaning rules discussed earlier and which are not properly accommodated in any specific part of this theory of language. Chomsky's ideas provide the basic semantic notions treated in such works as KATZ and FODOR (1963) and KATZ and POSTAL (1964). Chomsky sometimes improves upon these, e.g., by replacing the syntactic feature trees, untenable because of cross-classifications, by sets of features: Aspects, p. 79 and following. The only important respect in which the Fodor-KatzPostal theory goes beyond Aspects is in the theory of projection rules: compositional functions which are postulated in order to derive the meaning

UNCONTAINED RULES OF MEANING

853

of a higher constituent in a tree (and ultimately therefore the meaning of the top-node marked S) from the meanings of the constituents immediately dominated by it. The examples given for such projection rules are rather crude and ad hoc in the early publications; but they have been constantly refined by Katz in later publications. In the course of these refinements, new ad hoc solutions have appeared: e.g., Katz' formulation of the lexical item Neg, ridiculed by Bar-Hillel (in PUHVEL, 1969, p. 6; reprinted in BARHILLEL, 1970, p. 187). The projection rules of the Fodor-Katz-Postal theory were defined for the deepest underlying Phrase-marker, or Deep Structure, of a derivation. But as soon as the formulation of such rules for more complicated deep structures was attempted, e.g., for structures incorporating quantifiers, serious difficulties began to appear. In order to deal with these, Chomsky gave up the postulate that transformations do not change meaning (the main thrust of KATZ & POSTAL, 1964) and introduced semantic interpretation rules which operate on the P-markers of the Surface Structure, derived from the Deep Structure of the base via transformations. These rules of semantic interpretation give interpretative semantics its name. Like projection rules, they are different from the meaning rules considered earlier, and cannot handle the problems the latter are destined to solve. Chomsky had a premonition of these problems in Aspects (1965, pp. 162-163) when he listed puzzling semantic facts, e.g., the meaning relations between the members of the following sentence pairs: John strikes me as pompous/I regard John as pompous I liked the play/the play pleased me John bought the book from Bill/Bill sold the book to John John struck Bill/Bill received a blow at the hands of John.

(15)

Most or all such meaning relationships could be accounted for by meaning rules. Attention has been drawn to such meaning relations by logicians and philosophers who worked in the Carnap tradition, e.g., Bar-Hillel in his criticisms of Katz (in PUHVEL, 1969, pp. 1-21; reprinted in BAR-HILLEL, 1970, pp. 182-201, and elsewhere). Also linguistic semanticists considered them (especially GRUBER, 1962; LYONS 1963, Ch. 4, and 1968, Ch. 10; cf. also WEINREICH, 1966). No one succeeded, however, in incorporating such observations and suggestions into a full-fledged theory of language, as has always been the principal concern of Chomsky and his followers. Clear presentations of puzzling meaning relations between verbs, involving shifts in case of the accompanying nouns, were given by FILLMORE (l968a,

854

J. F. STAAL

1968b), who showed that such semantic relations cannot be dealt with within a Chomskian framework (actually, neither within the perspective of interpretative semantics, nor within that of generative semantics), but he did not propose a semantic theory. I tried to treat some cases within a slightly extended Chomskian framework (STAAL, 1967b), an attempt criticized (together with Katz') by Bar-Hillel, who stressed the need for meaning rules in addition to lexical entries (BAR-HILLEL, 1967). I acknowledged the need for meaning rules but found Bar-Hillel insufficiently concerned with the need to specify the incorporation of a theory of meaning rules within a full-fledged theory of language (STAAL, 1968). Bar-Hillel's article of 1967 clearly indicated the direction in which the solution of these problems has to be sought. In generative semantics there is no separate level of deep structure. Transformations relate successive members of a sequence of P-markers, each defined by a well-formedness condition on the configuration of its nodes. This necessitates a specification of all the relations Dominates (x, y) and Precedes (x, y) which determine a tree (the variables ranging over nodes). Each transformation affects one of these relations (which is reminiscent of what Chomsky had called "elementary transformations"). According to Lakoff there are also global rules, which place constraints on sequences of more than two P-markers. The P-markers exhibit semantic as well as syntactic structures. The initial P-marker of each sequence is equivalent to an expression of the Predicate calculus. For example, the tree: S

//x/1\ y z

(16)

Pred

corresponds to the logical expression Pred(x, y, z). While the symbol Pred may dominate verbs, adverbs, quantifiers, negatives, etc., the variables x, y, z , ... (also called arguments) are ultimately replaced by nouns. In generative semantics lexical insertion is not confined to what Chomsky had called preterminal strings. Lexical items may be substitued for any part of a tree, provided it is dominated by a single node (this constitutes a claim about the notion 'possible lexical item', to which no exceptions have been found; actually this statement may be analytic, since an apparent counterexample would suggest a different tree structure). The fruitfulness of this generalization of the lexical insertion rule (whether interpreted as such, or in the opposite direction as a lexical decomposition rule) constitutes one of the best arguments for generative semantics. The combination of various

8SS

UNCONTAINED RULES OF MEANING

nodes under a single one, which the theory requires, necessitates certain transformations. These are confirmed in a variety of cases of lexical insertion, and required in other areas of the syntax as well. T shall illustrate this with the help of three examples. McCawley has repeatedly argued that the lexical item kill has to be transformationally derived (via cause to die) from cause to become not alive in the following manner (simplified):" So

~

Pred

/

cause

I

I

S,

x

/\

Pred N

/

r:

Pred

/

not

So

~

Pred

/

cause

I

become

(17)

N

N

I 53

/: Pred N / yI alive

N

I

I

S,

x

~

So

~

Pred N

N

~N /\ \ \ /\ / I Prect** N => cause Pred y x => become S2 r. become/ \Pred\y become/\Pred Pred N /\ /\\ /\ not Pred not Pred not Pred y I I I alive alive /\

Pred N

S2

N

N

/

cause

I

S,

\

x

Pred* N

alive

The predicates are combined under a single node by applying the same transformation (called Predicate-raising) on each cycle. Once all are combined under the node marked Pred*, the lexical insertion of kill takes place. Note that die could have been inserted at the stage of the derivation where Pred** appears. (For criticism, see CHOMSKY, 1970b; FODOR, 1970). A different kind of illustration is provided by POSTAL (1970), who has argued that one sense of remind is derived from strike as similar by a combination of lexical insertion and other transformations. (For criticism, see KIMBALL, 1970). A third example is taken from LAKOFF (1970), who has proposed that one sense of persuade y to, e.g., hit z, be derived from cause 3 One respect in which the following derivations are simplified is that little attention is paid to the order of constituents. I think that there are good reasons for this (cf, STAAL, 1967a).

856

J. F. STAAL

to come about that y intends to (hit z). Analogously, persuade y that is derived from cause to come about that y believes that. Lakoff observed that these relationships could not only be explained by a lexical decomposition of the kind he suggested, but also by meaning rules: "Ix, y, z(PERSUADE 1 (x , y, z) ;: CAUSE(x, (COME ABOUT (BELIEVE(y,

z»»)

"Ix, y, z(PERSUADE 2(x, y, z) ;: CAUSE(x, (COME ABOUT (INTEND(y,

z»»)

(18)

(LAKOFF 1970, p. 213). The relationship between kill and cause to die, the relationships discussed by Postal, and all similar relationships could also be formulated in terms of meaning rules. The question arises which of these two very different possibilities is correct (they are different, e.g., because they utilize very different kinds of rules; because meaning rules refer only to semantic notions and are presumably universal, while derivations, also in generative semantics, refer to semantic notions as well as phonological spellings of lexical items). Lakoff has argued, I think convincingly, that the regularities stated here should be explicated in terms of grammatical derivations rather than in terms of meaning rules. One of Lakoff's arguments is a consequence of the Fillmore-Binnick proposal to regard bring as the causative of come. This proposal, among other things, would explain the regular pattern illustrated by: come come come come

about/bring about up [bring up (for discussion) to (awaken)/bring to together/bring together,

(19)

etc., (LAKOFF, 1970, p. 215). These regularities cannot be simply explained if come and bring are only related by a meaning rule, stated in terms of COME

and BRING. For there would have to be additional meaning rules positing the same relationship for COME ABOUT/BRING ABOUT, COME UP/BRING UP, COME TO/BRING TO, etc: each of these expressions symbolizes an atomic semantic predicate. But such symbols for atomic predicates have nothing in common and do not yield a simple generalization. If come and bring are transformationally related, however, the linguistic regularities illustrated by (19) can be simply explained. This suggests, moreover, that the regularities exhibited in (19) are language specific and do not belong to universal semantics.

UNCONTAlNED RULES OF MEANING

857

Lakoff's observation suggests that all meaning rules cannot be reduced to transformational derivations. The fact that in special cases a meaning rule may be constructed which could perform the function of a grammatical derivation, is not surprising and does not affect the general claim that meaning rules have to be accommodated in a separate area of the semantic system. Lakoff acknowledges the need for certain meaning rules, e.g., those which he writes as: REQUIRE (x, y, S) ::::> PERMIT (x, y, S) CERTAIN (S) ::::> POSSIBLE (S)

(20)

(LAKOFF 1970, p. 213). It is obvious that meaning rules of entailement could not in general correspond to grammatical derivations; for in interpretative semantics, transformations do not change meaning, whereas in generative semantics, transformations may affect meaning, but only as the case requires and not in any systematic way. Lakoff has therefore included a 'natural logic' in the theory of generative semantics, where such relations are treated. McCawley (in lectures given in 1970 in Tokyo) reiterated the position of generative semantics that lexical insertions may be interspersed among transformations, but also suggested an alternative mechanism of lexical insertion about which, however, he only said the following: "I leave open the question of whether there is a separate lexical insertion transformation for each lexical item (i.e. that each lexical item essentially is a transformation that attaches the morpheme or morphemes in question to the semantic material in question) or whether there is a single 'lexical insertion transformation' which reaches into a 'dictionary' for the appropriate lexical item" (McCAWLEY, 1970, pp. 3-4,5). This new suggestion (if it is more than a notational variant) does not affect my earlier argument, that meaning rules cannot occur in the dictionary. This brief survey of generative semantics shows that there is greater awareness of the need for meaning rules among generative semanticists than among interpretative semanticists (which is not surprising in view of the semantic bias of the former). Still, neither in generative, nor in interpretative semantics are there investigations into the specific form, the function and the limitations of meaning rules on a level with Chomsky's early investigations into the form, function and limitations of context-free, context-sensitive and (to a smaller extent) transformational rules, e.g., in Chomsky, 1961). Lakoff has gone furthest in postulating 'natural logic'

858

J. F. STAAL

as a separate component of universal grammar which incorporates meaning rules. What we now need is detailed discussions of the interaction between meaning rules and other rules and structures of language, and of the position of the area of meaning rules in relation to other areas of grammar. I have no specific solutions to any of these problems, which appear to be wide open. I have been primarily concerned to show that meaning rules are required for an adequate theory of language and that they have to be accommodated in a separate area of the grammar. I do not know arguments which show that they have to be formulated in a certain manner. However, it is possible to make certain assumptions about meaning rules on the basis of the present discussion. (1) Meaning rules relate the meaning of lexical items, or the semantic elements into which such meanings are decomposed, to each other. (2) Meaning rules do not refer to semantic properties that are languagespecific, but are formulated in terms of notions of universal semantics. It is likely that such notions will become more abstract than they are at present and that the set of smallest semantic elements will gradually decrease. For example, the lexical items precede and follow may be first reduced to other items including before and after, then further decomposed into elements characterized in terms of variables ranging over time sequences, and ultimately formulated in terms of semantic relations similar to the arithmetic relations and). Whatever direction this process will take, it cannot continue indefinitely, if only because language is not merely a "systeme ou tout se tient", but also an instrument that enables us to talk about the world. Meaning rules, will always be required to state relationships between the ultimate semantic elements. (3) There is no need for an explicit listing of the features introduced by Chomsky: these are implicitly defined by the system of meaning rules (cf. LyONS, 1963, p. 57 and following). (4) Meaning rules include not only statements about relations of paraphrase and synonymy between semantic elements, but also of entailment between meanings, which are in fact much more numerous, much less controversial, and which also belong to the competence of the users of a language. It is likely that entailment rules should express the presuppositional relations recently discussed, e.g., in connection with "factive verbs": KIPARSKY and KIPARSKY (1970) or "implicative verbs": KARTTUNEN (1971). (5) Many logical notions and relations also belong to the competence of the users of a language. Many, e.g., logical connectives, cannot be explicated


: ___________.

NP-----S-----------.VP ·1 Pro V

II

Something

was

NP~

Li/s~ in spider

something frightened Miss Muffet

And the analysis of sentences like 76. A spider frightened Miss Muffet. NP

S

VP

T/2~ /~

Something

something was in spider

frightened Miss Mullet

The virtue of these analyses is that they account both for the obvious similarity of these sentences-by deriving them from the same pair of underlying sentences-and for the (less obvious) differences between them-by making the existential sentence the main clause in Sentence 75 but the subordinate clause in Sentence 76, there was a spider being the surface form of something was spider when it is the main clause, a spider being all that survives in the surface after the deletions and consequent tree pruning that take place when it is a subordinate clause; these results being guaranteed by the fact that the relative clause transformation occurs early in the transformation cycle, certainly earlier than the 'there-copying' rule-if indeed the latter is not a postcyclical rule-so that it will operate only in the case of those sentences occupying a nonsubordinate position. Several of the claims that have been advanced here on linguistic grounds bear a striking resemblance to views held by many logicians since about the turn of the century. The claim that noun phrases derive from structures in which the noun forms part of a predicate is consistent with the view that common nouns are to be treated as functions rather than as arguments.'? 10 It is interesting that C. S. PEIRCE (1933, p. 290), one of the first modem logicians to adopt this kind of formulation should choose to justify it in the following way: "Our European languages are peculiar in their marked differentiation of common nouns from verbs. Proper nouns must exist in all languages, and so must such 'pronouns, or indicative words as this, that, something, anything. But it is probably true that in the great majority of tongues of men, distinctive common nouns are akin to participles, as being mere inflexions of verbs. If a language has a verb meaning 'is a man', a noun

876

J.P. THORNE

The analysis proposed for existential sentences is reminiscent of the claim made by some logicians that sentences like Sentence 1 are to be interpreted as 'Something is God'. The hypothesis that existential sentences form part of the structure of noun phrases is an essential part of the theory of descriptions. However, it is precisely with regard to this point that STRAWSON (1950; 1952, pp. 173-194; 1954) has argued that the theory of descriptions fails to constitute an accurate description of natural language structure. Strawson's position is that anyone uttering Sentence 76 does not assert that a spider existed but presupposes that it did. He would deny, therefore, that an existential sentence actually forms part of the structure of Sentence 76}1 But even if one accepts the analysis of Sentence 76 given above it remains a nice question whether or not one really wants to say that in asserting this proposition one asserted that there was a spider. However this problem arises with regard to any proposition contained in a restrictive relative clause. Consider the sentence 77. The pretty girl is the best tennis player. NP

S-------VP

D~l~s /~ I I /~

The

girl

who is pretty

is the best tennis player

If I say this to you in all probability I expect you to agree or disagree with the claim that the girl in question is the best tennis player. It is this that I regard, so to speak, as news. But, of course, it is also open to you to agree or disagree with the proposition contained in the subordinate clause, for you to say "Yes she is pretty" or "But she isn't pretty". Now in uttering Sentence 77 did I assert that the girl was pretty? Well yes and man' becomes a superfluity... The best treatment of the logic of relatives, as I contend, will dispense altogether with class names and only use such verbs". The point that, according to this analysis, noun phrases derive from complete sentences except when they occur in predicate position may explain why many logicians, notably Frege, have felt that subjects are complete ('saturated') in a way in which predicates, including nominal predicates, are not. 11 STRAWSON (1952) defines the relation'S presupposes S' as follows "The truth of S, is a necessary condition of the truth or falsity of S". Notice that presupposition as it is used in this paper is a much weaker notion.

GRAMMAR OF EXISTENTIAL SENTENCES

877

no. Certainly it seems that some kind of distinction is needed to reflect the difference in the degree of commitment that I have to the proposition contained in the subordinate clause compared with that which I have to the proposition contained in the main clause. It is just this difference that is captured by the distinction between asserting and (positive) presupposing. That is to say in uttering Sentence 77 I assert the proposition contained in the main clause, that the girl is the best tennis player, but presuppose that she is pretty, the proposition contained in the relative clause. (Presumably this is why I would choose this proposition to act as an identifying description in the first place.) Similarly in the case of Sentence 76 the speaker presupposes the existence of a big spider, which is the presupposition contained in the relative clause, but asserts that it frightened Miss Muffet, which is the proposition contained in the main clause. On the other hand in the case of Sentence 75 where the main and subordinate clauses are switched so are the assertions and presuppositions. This is assuming that the relative clause in Sentence 75 is a restrictive relative clause. If it is taken as a nonrestrictive clause then there seems to be no difference in the kind of commitment the speaker of the sentence will have to the proposition expressed by this clause and to that expressed by the main clause. It seems, that both of them will be assertions. This can, perhaps, be taken as evidence in support of STAAL'S (1970) suggestion that nonrestrictive relative clauses represent a form of conjunction in which a sentence and its performative verb are conjoined to another sentence and its performative verb. In other words, in which, as it were, two speech acts, rather than just two sentences, are conjoined. In all the sentences discussed below it is assumed that the relative clauses are to be taken as restrictive relative clauses. In this discussion of the relationship between Sentences 75 and 76 great emphasis has been placed on the point that they are not synonymous. Other linguists have assumed that sentences of this kind are completely synonymous and, therefore, that they have a common underlying structure. Some have taken this to be that underlying Sentence 75, others have taken it to be that underlying Sentence 76. There are arguments against both these hypotheses. Against the hypothesis that sentences like Sentence 75 derive from sentences like Sentence 76 there is the objection that this would mean extending the domain of the transformation which produces there at the beginning of the sentence to structures in which the main verb is not be-a particularly unlikely proposal in view of the fact that the full effect of this

878

J. P. THORNE

transformation would now be to produce a sentence (like the first clause in Sentence 75) in which the main verb is be. The fact that Sentence 76 has the surface structure of a simplex sentence and Sentence 75 the surface structure of a complex sentence may have a further significance. Whereas there are many well-established cases in which a surface structure which is a simplex sentence is produced from an underlying structure that is a complex sentence, as the result of the application of deletion rules and subsequent tree pruning, there are only two cases, this and the usual account of cleft sentences, where rules have been proposed that have the opposite effect. In both cases the transformations proposed are optional and in both cases there are reasonably well-motivated accounts which derive these sentences in different ways. At the moment it is impossible to be certain on this point but it might well be the case that rules that generate complex sentences from the underlying structure of simplex sentences are in defiance of a restriction on the output of a transformational grammar. It could be that linguists who accept the hypothesis that Sentence 75 is derived from Sentence 76 are being misled by the fact that in some dialects the relative pronoun occurring in Sentence 75 can be omitted, as in 78. There was a spider frightened Miss Muffet. so that the presence of the relative clause is somewhat obscured. But the fact that the speakers of these dialects can find no other difference between Sentence 75 and Sentence 78 makes it plain that the latter is merely a variant of the former, deriving from it by a rule which optionally deletes relative pronouns in these constructions. On the other hand, like all English speakers, those who speak these dialects regularly make a distinction between sentences like 79. I noticed that there was a policeman watching me. and 80. I noticed that there was a policeman who was watching me. This is accounted for by the hypothesis that there in the complement of Sentence 79 is the residue of a copy of the present participle watching, whereas in the case of Sentence 80 it derives from a copy of the noun phrase a policeman; the present participle in this case forming part of a different underlying sentence. Another explanation of why it is so many linguists have been ready to take sentences like Sentence 75 and 76 as synonymous might be sought in what is more or less a standard practice amongst logicians. Asked to

GRAMMAR OF EXISTENTIAL SENTENCES

879

produce a representation of the predicate calculus type for Sentence 761 they will, almost invariably, produce an expression with an existentia, quantifier which they will then proceed to verbalize as 'there was a spider etc.', in this way covertly substituting Sentence 75 for Sentence 76 (cf. REICHENBACH, 1966, p. 264; BACH, 1968; DAHL, 1970). Now let us consider the suggestion that Sentences 75 and 76 derive from a common underlying structure which is that underlying Sentence 75. This would also mean that a sentence like 81. Alex found a book. should derive from the underlying structure of 82. There was a book that Alex found. But now consider the relationship between the sentences 83. There was a book that Alex was looking for. and 84. Alex was ooking for a book. If it really were the case that Sentence 84 derived from the underlying structure of Sentence 83 then anyone uttering Sentence 84 must necessarily be asserting the existence of a book. Obviously this is not so. One is not being inconsistent if one utters Sentence 84 and then adds. 85. But there was no book that he was looking for. This also suggests that the correct analysis of Sentences 83 and 84 should provide a basis for distinguishing what they assert from what they presuppose. Let us say that a verb like know has positive presupposition; that is to say, anyone uttering a sentence like 86. The ancient Greeks knew that the world is round. assumes the truth of the proposition contained in the subordinate clause forming the complement of knew. Indeed it would be inconsistent to utter Sentence 86 and then add But it's not. (This, of course, is to employ a much weaker notion of positive presupposition than that employed by Strawson (See Footnote 10». Let us say that a verb like pretend has negative presupposition since anyone uttering a sentence like 87. John pretended to be rich. assumes that, in fact, John is not rich. And let us say that a verb like believe has zero presupposition since anyone uttering a sentence like 88. The ancient Greeks believed that the world is round. does not commit himself as to whether or not it is the case that the world is round. Hence it is possible to utter Sentence 88 and to add either And it is the case the world is round or But it is not the case that the world is

880

J. P. THORNE

round, without being inconsistent. (Compare (and contrast) 1970).

LAKOFF,

One of the advantages of adopting Bach's proposal to derive noun phrases from sentences is that it makes it possible for the differences in the interpretations of Sentences 81 and 84 noted above to be explained through the use of the concepts positive and zero presupposition. Following Bach's proposal, if we take Sentence 81 as equivalent to Alex found something that was a book. and Sentence 84 as equivalent to Alex was looking for something that was a book. it seems clear enough that the differences between the way in which these are interpreted relate to find being a verb with positive presupposition and look for a verb with zero presupposition. The relationship between Sentence 81 and Sentence 82 is exactly that which one would expect to obtain between sentences in which the individual clauses are the same but where the main and subordinate clauses are switched. One asserts what the other presupposes and vice versa. But this relationship (it has already been shown) cannot be stated in terms of transformational rules. These only express relationships between sentences having exactly the same underlying structure, that is not just sentences made up of the same set of clauses, but sentences in which the main clause is the same and the subordinate clauses are the same, and which, therefore, make the same assertions and presuppositions.

Bibliography BACH, E., 1968, Nouns andnoun phrases, in: Universals in Linguistic Theory, eds. E. Bach and R. T. Harms (Holt, Rinehart and Winston, New York), pp. 91-122 BOLINGER, D., 1971, The nominal in the progressive, Linguistic Inquiry, vol. 2, pp. 246-250 BRAATEN, B., 1967, Notes on continuous tenses in English, Norsk Tidsskrift for Sprogvidenskap, vol. 21, pp. 167-80 CHOMSKY, N., 1970, Remarks on nominalizations, in: Readings in English Transformational Grammar, eds. R. Jacobs and P. Rosenbaum (Ginn, Waltham, Massachusetts), pp. 184-221 CHRISTIE, J., 1970, Locative, possessive and existential in Swahili, Foundations of Language, vol. 6, pp. 166-178 DAHL, 0., 1970, Some notes on indefinites, Language, vol. 46, pp. 33-42 ELIOT, C., 1890, A Finnish grammar (Clarendon Press, Oxford) EMONDs, J., 1969, A structure-preserving constraint on NP movement transformations, in: Papers from the Fifth Regional Meeting of the Chicago Linguistic Society April

GRAMMAR OF

EXISTENTIAL

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18-19, 1969, eds. R. Binnick, A. Davidson, G. Green, J. Morgan (Department of Linguistics, University of Chicago, Chicago), pp. 60-65 FILLMORE, C., 1968, The case for case, in: Universals in Linguistic Theory, eds. E. Bach and R. T. Harms (Holt, Rinehart and Winston, New York), pp. 1-88 GARDINER, A., 1950, Eygptian grammar (Oxford University Press, London) KAHN, C., 1966, The Greek verb 'to be' and the concept of being, Foundations of Language, vol. 2, pp. 245-266 KATZ, J., and P. POSTAL, 1964, An integrated theory of linguistic descriptions (M.LT. Press, Cambridge, Massachusetts) KUNO, S., 1971, The position of locatives in existential sentences, Linguistic Inquiry, vol. 2, pp. 333-378 KURYLEWIcz, J., 1964, The inflectional categories of Indo-European (Carl Winter, Universitatsverlag, Heidelberg) LAKOFF, G., 1970, Linguistics and natural logic, Synthese, vol. 22, pp. 151-271 LYONS, J., 1968, Existence, location, possession and transitivity, in: Proceedings of the 3rd International Congress for Logic, Methodology and Philosophy of Science, eds. B. van Rootselaar and J. F. Staal (North-Holland, Amsterdam), pp. 496-504 MILLER, J., 1970, Stative verbs in Russian, Foundations of Language, vol. 6, pp. 488-505 PEIRCE, C. S., 1933, Exact logic, Vol. III, ColIected Papers, eds. C. Hartshorne and P. Weiss (Harvard University Press, Cambridge, Massachusetts) PERLMUTTER, D., 1970, On the article in English, in: Progress in Linguistics, eds, M. Bierwisch and K. Heidolph (Mouton, The Hague), pp. 233-248 POSTAL, P., 1966, On so-called pronouns in English, in: Report of the Seventeenth Annual Round Table Meeting on Linguistics and Language Studies, ed, F. Dineen (Georgetown University Press, Washington D.C.), pp. 176-206 REICHENBACH, H., 1966, Elements of symbolic logic (The Free Press, New York) Ross, J., 1967, Auxiliaries as main verbs (M.LT. Xerox) STAAL, J., 1970, Performatives and token-reflexives, Linguistic Inquiry, vol. 1, pp, 371-381 STRAWSON, P., 1950, On referring, Mind, vol. 54, pp. 320-344 STRAWSON, P., 1952, Introduction to logical theory (Methuen, London) STRAWSON, P., 1954, A reply to Mr. Sellars, Philosophical Review, vol. 63, pp. 216-231 WRIGHT, J., 1905, The English dialect dictionary, vol. 6 (Henry Frowde, London)

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S. TOULMIN University of California, Santa Cruz, Calif., USA

1. Anyone who sets out to survey the area covered by Section 12 of this Congress is confronted by a basic ambiguity. Its title can be seen as pointing in either of two quite different directions. So the task might be, first, to consider the historical development of men's ideas about the intellectual and practical status of formal logic, about the structure and methods of argument by which scientific theories can be subjected to logical scrutiny, and about the philosophical significance and implications of the sciences. Or, alternatively, it might be to look and see how the course of history has changed the formal procedures, research methods and philosophical presuppositions which have guided the development of men's ideas within the different sciences-whether practical, natural or exact. Maybe these two topics ought ideally to be one and the same. Maybe, that is, scientists ought-ideally-to be guided in their work by methodological maxims derived from the work of formal logicians and philosophers of science; and maybe philosophically minded logicians ought-ideallyto pay more direct attention than they commonly do to the research procedures and methodological considerations that influence working scientists in actual practice. But though there have been periods in history when there really was such a convergence-as when Aristotle, the marine biologist and taxonomist, argued for a theory of explanatory ideas as 'inherent essences', while Plato, the advocate of ideas as 'independent quasigeometrical entities', made planetary kinematics the focus of scientific work in his Academy-we find ourselves now, in the last third of the twentieth century, looking back, in both science and philosophy, on fifty years of work during which the divergence between philosopher's theories about science and the working philosophies of scientists, has been extreme-at times even absolute. Since 1920, writers on 'inductive logic' have only rarely paused to consider

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whether the formal patterns of argument they are discussing have any genuine counterparts in (say) the Physical Review, the Proceedings of the Royal Society of London, or the scientific publications of the Academia Romana. For their part, most working scientists have meanwhile taken to dismissing the philosophy of science as irrelevant and intellectually sloppy, if not worse. Max Born used to speak about the 'iridescent fancies' of philosophers. Peter Medawar spends a fair amount of time correcting the scientific errors of philosophers, more in sorrow than in anger. As for Richard Feynman's views on the subject: these are not strictly printable. All this being so, it may be more profitable if this lecture avoids any historical surveyor catalogue of a strictly chronological kind, and concentrates instead on the historical sources of this divergence between philosophers' views of science and the working philosophies of scientists. Has this professional divorce between working scientists and philosophers of science-between formal logicians, and the practical reasoners whose arguments they theorize about-gone too far? What, if anything, could be done to restore the lost contacts between them? Or is there, after all, some intrinsic and insuperable division between the general and invariant standards of formal validity in argument, which are the business of logicians, and the local and variable conceptions, with an eye to which practitioners in different fields of scientific inquiry have chosen their intellectual goals and procedures in one historical period or another? These are the questions from which I shall be beginning today. 2. It would be nice if we could just dispose of these questions instantly and crisply, to the agreement of all concerned. It would be nice (that is) if we could wave a magic wand-say, one of those splendid epigrams which so appealed to our nineteenth-century forerunners, such as "The Science of Logic is the Logic of Science"-and so make them disappearin a puff of smoke. Unfortunately, the problems at issue here are real and serious ones, which do not respond to a philosophical coup de theatre. For a while, we may have been blinded to them by overreliance on mottos like 'There is no Logic of Discovery'; but, now that the smoke screen is lifting, they are still there, large as life and unsolved as ever. Nor should this fact surprise us. For the central difficulties involved are-equally-topics of active and inconclusive debate today, both among historically minded philosophers in the European Marxist tradition, and among their more formally and empirically minded colleagues in Britain and the United States. They include (for instance) those difficulties on

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which our lamented colleague George Lukacs used to focus, when he recalled the forgotten subtleties of Karl Marx's early philosophical writings: I have especially in mind Lukacs' discussions about the nature of 'consciousness', and its role in cultural history. For is there any single universal and essential aim around which scientific inquiry is-and must necessarily be-organized, in all cultures and at all historical times? Or do we, rather, have to recognize that the very nature of 'science' must be defined with an eye to the changing aims in terms of which men working in different scientific domains, within different historico-cultural milieux, have for the time being identified their intellectual tasks? How far (in short) has Man's historically developing 'consciousness' of the goals and possibilities of scientific explanation itself been an inescapable element in any adequate definition of the rational enterprise we call 'science'? As to this question, one highly influential twentieth-century philosopher in particular had no doubt. In the introduction to his Foundations of Arithmetic, Gottlob Frege made it clear that-in his view-it would be entirely disastrous to permit references to human attitudes, or 'consciousness', to appear in the definitions of our intellectual enterprises. At the very beginning of his argument, he drew a sharp distinction between two kinds of inquiry: first, rigorous logical analysis of "concepts in their pure form", and, second, historical or psychological studies of all the differing conceptions with which men have experimented, at one time or another, while fighting their way towards a clear recognition of those "pure" concepts. As to these latter, historical and/or psychological issues (Frege argued), they raise purely empirical questions, and are none of the philosopher's business: the philosopher must concentrate his attention exclusively on the logical analysis of "pure concepts". Like Descartes and Plato before him, Frege apparently believed that, while the varieties of intellectual error are unlimited, there exists-in any particular domain-one and only one set of concepts in terms of which a man can operate rationally. Thus, he says: Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, by stripping off the irrelevant accretions which veil it from the eye of the mind. But the history of all that intermediate effort is-from Frege's point of view-irrelevant to the "conceptual analysis" which alone concerns the philosopher of mathematics.

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For Frege's immediate associates and successors, this doctrine carried an evident charm and conviction: especially, in relation to arithmetic and similar branches of pure mathematics. This was so for two reasons. In the first place, it was indeed plausible to argue that the concepts of modern arithmetic, as analyzed by Peano, Russell and himself, have a genuine intellectual superiority over their historical precursors, so that they deserve a quite particular attention from the philosopher of mathematics. In the second place-and more importantly-it was indeed possible, within pure mathematics as so conceived, to demand formal criteria both for judging the validity of arguments stated in terms of the currently established concepts of the subject, and also for judging which among the conceptual changes now under debate would constitute improvements. During the twentieth century, Frege's admirers have extended the doctrines and distinctions which he defended originally in an arithmetical context, and have applied them immediately in the philosophy of natural science also. And this (I believe) is the point at which we must ask some searching questions. For this step involves us in some very substantial assumptions: e.g., that it is possible, in the natural sciences, to distinguish clearly and cleanly between, on the one hand, eternal and logically definable 'concepts in their pure form' and, on the other, the changeable and merely empirical 'conceptions' of different historico-cultural milieux. Yet, whatever case we might make out for treating earlier number conceptions as no more than a clouded vision of our own modern arithmetical concepts, it is far less plausible to advance the corresponding argument about (say) electrodynamics or genetics. Was Maxwell's electromagnetic field concept the single rational ideal towards which all earlier physicists of light, electricity and magnetism had struggled? Is DNA, in Frege's sense, the "pure" concept arrived at by "stripping away accretions" from Mendel's "factors" and Johannsen's "genes"? Or is not the relationship between earlier and later explanatory concepts in natural science more complex and subtle than it is in arithmetic? Correspondingly, philosophers of science have taken Frege's distinctions as justifying them in distinguishing, equally clearly and cleanly, between two things: (a) the entirely general standards of validity in scientific argument, which apply in all domains and milieux alike-standards which can arguably be analyzed by a straightforward extension of existing formal logic, and (b) the more specific and historically changing methodological conceptions accepted by practitioners in one scientific domain or another, and in one milieu or another. They have regarded the first, entirely general

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set of standards alone as being their proper philosophical concern. They have been inclined to set aside questions about the changes and chances of history, psychology and sociology-all those empirical details about Kepler and Galileo, about mercantile economy and the secularization of learning, about Mendel and Michurin-as beside all legitimate philosophical points; and to dismiss all such empirical questions as irrelevant to their own quite general analysis of the 'logical structure' of any valid scientific theory. Frege's example seemed to provide an unanswerable justification for concentrating exclusively on the internal articulation of scientific arguments, considered as formal networks of propositions. But did it really do so? Even assuming the absolute validity of Frege's approach within the philosophy of pure mathematics, did he really provide an example which philosophers of natural science can take as authoritative? In retrospect (I believe) we can now see that his example has in fact been quite misleading. For his own procedure in the philosophy of mathematics was to ignore all that numbering, counting and calculating behavior, by abstraction from which men have, over the centuries, gradually worked their way towards modern arithmetic; and so to confine himself to the systematic network of theoretical propositions finally arrived at as the supposedly ideal, ultimate and definitive outcome of all earlier practical activities in the numerical domain. And the corresponding procedure can succeed fully in the case of natural science, only to the extent that we can assume that the historically developing explanatory behavior of scientists over the centuries is directed towards similar idealized and definitive propositional networks, and so lends itself to a similar formal analysis. Whether or not that is so, is a matter of debate, and the question is far too uncertain to be begged at this stage. Meanwhile, by cutting the proper and necessary links between philosophical theories about science and the history of methodological practice-I nearly said 'praxis'-this extension of Frege's program has impoverished our philosophical understanding of the scientific enterprise. And I shall be happy, today, if my lecture does no more than to restate the case for recognizing the history of methodology as having a central and indispensable part to play in any adequate philosophy of science. 3. Up to this point, I have spoken about only one of the two topics covered in my title. I have contrasted an essentialist view of science-a view that credits all the natural sciences, in every milieu and domain, with a single universal aim and set of intellectual values, which we may hope

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to define and analyze in purely formal terms-with a more historical view-one that recognizes some variation in aims and methods between different domains and different milieux as not merely inevitable in fact, but necessary in principle. And I have suggested that, after fifty years during which the philosophy of science has been dominated by essentialism, it is time to pay more serious attention to 'the changing aims of scientific inquiry', and so to the historical aspects of methodology. At this point, it is convenient to introduce the second of my two themes: namely, the notion of 'rationality'. For, once we begin to take the historical variability in the aims, methods and fundamental concepts of the science more seriously, we have to reconsider a whole string of neglected philosophical questions about the rationality of scientific ideas and procedures. And, as to these issues, I want to argue straight away that their neglect has been no accident. On the contrary, Frege's concentration on the logical analysis of idealized "pure" concepts-extended from arithmetic to natural science-made it inevitable. Within a logistic approach to the analysis of science, philosophers have been interested only in the formal relations linking the propositions of science; and these formal relations can-in the nature of the case-hold only between propositions stated in terms of a given, accepted or assumed set of concepts. Questions about the manner in which, and the reasons for which, scientists are led to abandon one set of theoretical concepts in favor of another are extremely difficult to analyze in these terms. As for questions about the manner in which, and the reasons for which, scientists have revised their very conceptions about the explanatory tasks facing one domain of science or another: these are ruled out of philosophy entirely. They may perhaps be open to some sort of historical study and documentation; but, being evidently unformalizable, such considerations are clearly not ones about which questions of strict 'validity' or 'invalidity' can properly arise. As to this, we must remark: "Maybe not". Maybe, fundamental changes in the concepts and methods of the various natural sciences involve considerations about which no questions of 'formal validity' can properly be raised. Yet, certainly, such changes give rise to questions of 'rationality' none the less. And here the shortcomings of Frege's approach are evident, even in relation to arithmetic itself. In his enthusiasm to establish rigorous connections between pure mathematics and formal logic, Frege cut away all those things that he regarded as "historical and psychological accretions", so as to bring to light the essential formal structure of modern

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arithmetic; and, in doing so, he ignored the great mass of empirical questions about those 'centuries of immense intellectual effort' after which humanity had at last succeeded in arriving at that system of formal arithmetic. Yet this involved him in setting aside, also, all questions about the rationality or intellectual merit, of all the successive steps comprised in this historical sequence. Or rather: it involved him in assuming that all those intellectual merits reflected solely the extent to which those successive steps helped mathematicians approximate more closely to the modem arithmetic which he and Peano were themselves analyzing. On closer examination, this was a very curious assumption. When Western mathematicians took over the Arabic system of numerals in place of the earlier Greek and Roman notations, for example, was this a 'rational' thing to do solely because it brought arithmetic nearer to the Fregean ideal? And when the 'zero' was introduced into arithmetic, were the intellectual merits of that step, once again, solely of a Fregean kind? And when decimals were invented; and ways of operating with negative numbers; and so on, and so on ... ? Are we, in all these cases, to regard these inventions as having been, at the time, 'rationally meritorious achievements'-as innovations to be accepted 'for good reasons'-solely with an eye to the eventual, Fregean system? Such a suggestion puts the whole business of mathematical invention and judgment in a very odd light. For it implies that the 'rationality' of intellectual innovation in early mathematics depended on a clairvoyant capacity to move towards an intellectual ideal whose very possibility-let alone, desirability and intellectual authority-the mathematicians in question had as yet had no opportunity to imagine. In dealing with questions about the 'rationality' of conceptual changes in s?ience (I am suggesting) the logistic approach to the philosophy of science has involved one long ignoratio elenchi. Having confined themselves to questions about the formal relations between the propositions of science, the philosophers concerned have had no alternative except to identify 'rationality' with 'logicality', and to look for some index of rationality within the formal properties of scientific arguments. The effect of this has been to distract their attention from the very class of cases and situations about which 'rational' questions arise most urgently and significantly. These are the cases, and the situations, in which more or less far-reaching changes are introduced into the basic concepts and presuppositions of a science; with the result that strictly formal relations no longer obtain between the older theoretical propositions of a science and the new. An

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overpreoccupation with logical coherence and systematicity has thus led to an overemphasis on the validity or invalidity of intellectual steps taken within the framework of a science at some one particular stage in its development; it has correspondingly swept aside questions about the steps by which scientists move from one set of explanatory concepts to another later and logically incongruous set of concepts. It is over precisely these latter questions (as I shall now go on to argue) that 'rationality' is inextricably connected with the 'collective consciousness' of the various scientific professions, and with the changing aims of inquiry in the various domains of the scientific enterprise. 4. To come straight to the heart of the matter: in the natural sciences, we have from the outset to drive a wedge between (a) the formal criteria by appeal to which we judge arguments framed in terms of the currently established concepts of any science, and (b) the substantive criteria by appeal to which we judge whether some proposed conceptual innovation is or is not an improvement. The plausibility and charm of Frege's method sprang precisely from the fact that, in pure mathematics, both these sets of criteria can be specified in purely formal terms. But in the natural sciences, that possibility is no longer open to us. The criteria for judging the validity of arguments within any given phase in the historical development of a science may be formalizable; but as for the criteria by which we judge whether some proposed conceptual change is or is not an improvement-whether, as a result of this change, the science in question will be enabled to 'do a better explanatory job'-these criteria rest, in the nature of the case, on substantive considerations. The character of these conceptual selection criteria, or standards for judging conceptual improvements, depend directly on what, in the particular domain of science, is conceived of at the particular time as 'doing a better explanatory job'. For what (we may ask) was 'the proper explanatory job' for scientists working in (say) the field of electrodynamics in the year 1870, or 1910, or 1950? And how did this task resemble, or differ from, the 'proper explanatory jobs' of taxonomists, or cell biologists, at those same times? So long as we accept the logistic model for scientific theory uncritically, we may confine ourselves to generalities about 'science as such' and so succeed in holding history at bay; but, once we start examining in detail the ways in which, in different sciences at different times, considerations of different kinds have borne on the adoption or rejection of conceptual innovations, we shall find (I suspect) that no entirely general or

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universal account of 'the aim of science' has historical relevance or plausibility. The intellectual merits of conceptual innovations in science have differed both between one domain of science and another at any given time, and also from one time to another within the development of any given science. And the guiding factor by reference to which these judgments have been made is-precisely-the 'collective consensus' by which the scientists in question have jointly identified their outstanding problems in that domain, have recognized possible solutions to those problems, and have shaped the methodological or theoretical strategies for their future inquiries. It is in this sense that the history of (say) 'physics' can with advantage be spoken of as a history of the consciousness of physicists. To the extent that the current mainspring of any science lies in its outstanding problems, indeed, it will have such a mainspring at all, only so long as the scientists involved recognize certain issues as being 'problematic'. And they will have substantial reasons for recognizing problematic issues, in tum, only to the extent that they share some agreed conception of the current theoretical goals of their science, by comparison with which the understanding provided by the current aggregate of concepts falls short of what they are now entitled to demand. If we define the current domain of any particular science in terms of the objects, properties and events which pose 'problems' for the science, and so contribute to its its recognized 'phenomena', our definition will accordingly have to refer not just to Nature, but also to the intellectual attitudes with which men currently approach Nature. In saying this, I do not of course mean the individual psychological attitudes of scientists, but rather the professional expectations which have been born of all the experience which past and present scientists in the domain concerned have collectively brought to bear on their understanding of the systems in question. Just what kinds of happenings are 'problematic' for science thus tells us (in Kant's terms) not about Nature as a 'thing in itself', but only about the current 'representation' of Nature: specifically, it tells us how the scope of our current explanatory methods of representation in this field of science falls short of our reasonable expectations. The gap in any science between these two things-the difference between our reasonable explanatory expectations and our actual explanatory capacities-is a measure of the distance that this particular science still has to go. More exactly, it is a measure of the distance this science still has to go, in order to fulfill its current intellectual ambitions. Meanwhile,

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everyone concerned will admit that further horizons, involving fresh explanatory hopes and possibilities, will in due course become apparent beyond the goals which form the immediate destination of conceptual development in this field; and that the future intellectual hopes of the science will be both more ambitious and more specific than those which at present direct its development. As a result, features of the world which the scientists of one generation find unmysterious can become puzzling and 'problematic' for the men of the next generation, simply because their intellectual ambitions have expanded. To quote one revealing example: the classical nineteenthcentury theory of matter took as its ultimate level of analysis the ninety-odd so-called 'elements'. Each of these possessed properties supposedly conferred on it in the original Creation, and was thought of as comprising solid, indivisible atoms having eternally fixed shapes and sizes. Within these limits, one was free to inquire what precise color, or electrical conductivity, any particular element possessed, in point of observed fact; but no scientific opportunity existed for inquiring, in point of theory, why the vapor of sodium (say) when excited electrically should emit radiation in the yellow part of the spectrum, rather than elsewhere. The only available way of answering this question was to say, quite literally-as Newton himself had done-"God alone knows". (Incidentally, this was Maxwell's answer, too.) By conceiving the possibility of a subatomic physics, J. J. Thomson, Ernest Rutherford and their students removed this particular limitation, and carried the ultimate level of analysis below the atomic level. By finding an effective way of conceiving of the elementary atoms as composed of common smaller particles, that is, they made it conceptually possible to treat as 'phenomena' and so as 'problematic'-properties of matter that had hitherto been accepted,faute de mieux, as arbitrary features of Nature. From now on, facts like the existence of twin yellow lines in the spectrum of sodium vapor ceased to be arbitrary (or theological) and became genuine scientific problems, to be resolved in term of the configurations of subatomic components within the sodium atom. As a result of this feat of intellectual imagination-as a result of this deepening of the physicist's consciousness-the scope and nature of 'physics' itself had been transformed. In a significant sense of the phrase, after Rutherford and Bohr, physics was no longer what it had been hitherto. 5. Now: anyone who sets out to make explicit the part played in the

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conceptual development of any science by the scientists' own collective conception of their proper intellectual tasks is of course exposed to certain familiar objections. For surely (it may be said) any philosophical account in which the crucial test of what is or is not 'physical' (say) depends on the current opinions of particular groups of physicists is bound to end in some kind of subjectivism or historical relativism? And, surely, any references to the psychological attitudes or ideals of the men engaged in scientific inquiry lays one open to the charge of idealism; for does this not misrepresent the essentially 'objective' character of the issues facing scientists as concerned merely with the satisfaction of scientists' own mental preferences? If the changing intellectual strategies of the different sciences were decided solely by the arbitrary choices and preferences of the scientists involved, it might well be claimed that all 'objective external constraints' had been removed from science. And if that were the inevitable consequence of following my present advice-that is, that it is time to turn the philosophical analysis of science away from questions about the truth of theoretical propositions to questions about the adequacy of theoretical concepts-then we should indeed have opened a Pandora's Box. But this alarm would be premature. In arguing that the selection of theoretical problems, and the approval of conceptual innovations in science, can be rationally judged only in relation to the intellectual goals and methods currently recognized by the actual practitioners of the science in question, I am in no way suggesting that these conceptions are arbitrary, subjective, or matters of personal taste; nor am I implying that the resulting scientific developments are the products of human idiosyncrasy, uncontrolled by external requirements or constraints. The force of these objections rests, in fact, on misunderstandings which the logistic model of science only serves to reinforce. For, of course, both scientists and philosophers are justified in claiming that the development of the natural sciences is governed-above all-by 'objective, external constraints'. Yet it is essential to be clear about the variety of different ways in which such external constraints can operate in a science, and not to confuse the issue by over-simplifying the empirical burden of science: that is to say, the relations which hold between the theoretical concepts of any science, on the one hand, and the experiences which those concepts enable us to understand, on the other. All of the judgments on which the historical development of a science depends are, in one way or another, the outcome of mankind's accumulated

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experience in dealing with the succession of problems raised by studying the corresponding aspect of the 'external' world. But the key phrase here is the phrase 'in one way or another'. There is no single way in which, no single point at which, empirical experience must in every case be brought to bear on the judgment of our scientific ideas. Whereas the logistic model of science implies that the point of connection between theory and experience lies exclusively in the 'empirical observation statements' which it is the business of theoretical generalizations to entail (and thereby 'explain'), an account of scientific judgment given in terms of the adequacy of its concepts distributes the empirical burden much more widely. Certainly, enunciating true empirical propositions about relationships found in the world around us is an 'objective task' of one kind; and, insofar as the discovery of such empirical relationships brings grist to our explanatory mill, this will represent one way in which the tasks of sciencecan be subjected to external objective constraints. But it is not by any means the only way, let alone the most powerful or significant one. The task of formulating well-founded concepts, in terms of which it is possible to state further true propositions about the world around us, is an 'objective task' of another, different kind. By introducing some new concept-inertia (say) or optical dispersion, or nucleotide-we do not give ourselves the means of verifying or falsifying, confirming or probabilifying, refuting or corroborating propositions which we had earlier stated, but about whose truth, falsity or probability we had been uncertain. Rather, we put ourselves in a position to identify and state new problems, and so to find new ways of 'putting Nature to the question'. In this respect, the empirical burden of a new theory is not borne by questions about 'truth', 'falsity' and 'entailment' alone. What we now have to consider is how far all these new questions turn out to be 'fruitful', and what sort of 'light' they throw on those aspects of Nature which they were introduced to explain. Nor is this the end of the matter. Going even further, beyond the task of formulating particular new concepts, scientists may find themselves having to devise more comprehensive methods or strategies of inquiry, capable of yielding entirely fresh sets of well-founded concepts and explanatory procedures-and so, at a remove, of generating true propositions. And this, too, is an 'objective' task, subject to the 'external' constraints of scientific experience, in no less demanding a sense of the terms. (On this point, let me refer you to the classic exchange between Max Planck and Ernst Mach in the Physikalische Zeitschrift for 1910-11.) The unfortunate effect of traditional empiricism in the philosophy of

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science has been to suggest that there is one-and-only-one way of mobilizing our accumulated experience of 'objective' problems, as a constraint on the intellectual development of an empirically based natural science: namely, by matching 'particular propositions' against 'individual facts'. When contrasted with this direct mapping of empirical propositions against observed relations, by contrast, the ways in which we judge the adequacy of our concepts inevitably appear 'meta-empirical'. When Huygens proposed his novel concept of double refraction, for instance, the question that arose was not primarily one of 'truth' or 'falsity' -concepts are not' true or false-but one about its 'scope of application', and its power, in turn, to make sense of optical phenomena which had hitherto appeared quite simply 'anomalous': that is, optical phenomena about which the previously established set of concepts-so far from generating false empirical propositions-gave no way of saying anything coherent at all, whether true or false. As for statements about scientific strategies, these represent general policies for the production and judgment of conceptual changes, and they are even further removed from the elementary 'fact matching' on which empiricists have placed such weight. Yet this in no way entails that the problems of conceptual change and strategio redirection in science are any the less 'objective', or any the less subject to 'external' constraints, than questions about particular empirical phenomena. It means only that our conceptual and strategic decisions are exposed to criticism in the light of experience in very different ways from propositions about particular empirical correlations. To some extent-but only to some extent-we can verify scientific propositions one at a time. To some extent-but only to some extent-we can condense the accumulated experience of mankind in making sense of perplexing natural phenomena into well-defined rules and procedures-> 'laws' and methods of representation-and reach a point at which we possess methods of explanation for handling certain types of systems and phenomena having a proven scope and reliability. To some extent-but only to some extent-it turns out that we can in fact group the resulting rules, procedures and methods of representation into compact and coherent scientific disciplines, whose conceptual development is itself governed by common, more-or-Iess agreed strategies. Yet even where, for once in a while, none of these conditions holds, so that the best way of advancing scientific understanding in a particular domain remains unclear: even there, the issues confronting scientists are none the less 'objective'.

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In response to the current dilemmas about the ultimate adequacy of quantum mechanics, for instance, two equally distinguished and authoritative physicists may propose different strategic directions for the future development of their science; and each of them will base himself on his individual reading of the current problem situation, as viewed in the light of all the historical experience of his theoretical predecessors in physics. As it stands, neither of these two strategic proposals needs directly entail any 'true' empirical propositions, nor at once serves to establish any 'wellfounded' concepts: all it can do (in fact) is to give us a new set of considerations for judging future conceptual innovations. Yet, in due course, these alternative intellectual policies will nonetheless prove to have been objectively 'sound' or 'unsound', 'fruitful' or 'unfruitful', in their own ways: for, if their claims are to be recognized in retrospect as firmly based, they will have to make it possible, in due course, to recognize and establish new and more powerful sets of concepts and explanatory procedures. Initially, then, two such proposals may be the products of individual judgment; but, when it comes to deciding retrospectively which of the two proposals was the 'better judged', we shall not take into account any personal considerations about the men who put them forward, but rather the actual, practical sequels to which the two proposals gave rise. Such strategic proposals are no more directed than anything else in science at satisfying the individual tastes or prejudices of the scientists concerned. All of them are directed, rather, at the general, objective and external task, of suggesting how our intellectual grasp of Nature can best be improved. Nor is there any greater difficulty in dealing with the other objection: namely, the charge that a 'conceptual' account of scientific theory lands us in a vicious idealism. For, although the question of what kinds of events and processes are genuinely problematic, for a given science at a given time, may depend in part on the explanatory expectations that scientists have formed in the domain concerned, the actual character of the resulting problems-once identified-can and must once again be specified in entirely 'objective' terms. Such and such an explanatory procedure-for instance, 'ray-tracing' in optics or 'renormalization' in quantum electrodynamics, or the use of numerical methods in biological systematics-has been discovered to have reliable application over such and such a range of cases, with such and such a degree of accuracy; and the problem is then to see, for instance, how this same technique can be extended or modified, so as to embrace a further class of cases, which might in principle be expected to

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fall under the same general theory, but for which no satisfactory treatment yet exists. The fact that some particular question is currently 'problematic', as seen from the point of view of physics, as we now know it, is indeed a fact about 'the point of view of physics, as we now know it'; and so about the collective attitudes of physicists. But the problem itself-the problem of finding a new way of drawing ray diagrams, or a new kind of computational procedure for dealing with problems in quantum electrodynamics or the like-that is an objective question of explanatory technique, involving no essential reference to human attitudes or expectations. This is not to say that such a problem can be specified in terms of some 'empirical generalization' which simply reports observed correlations in Nature. Not at all: it remains a problem about the techniques which can be used in giving an explanatory account of Nature-and used, we may add, not just by human scientists, but equally by electronic computers, super-intelligent apes, or anything else that would qualify for Kant's title of a "rational-thinkeras-such". . 6. I have been able to touch here on only a few of the topics that demand our attention, if we set out to give a properly historical analysis of the development of methodology in the different natural sciences. In doing so, I have deliberately picked on topics which help to narrow the divide from which I began-between philosophical ideas about science, and the working philosophies of scientists. This means that I have had to omit many topics of deep philosophical interest. For instance, I have said nothing today about the distinguishing characteristics of 'conceptual' problems in scienceabout how they are marked off from purely empirical problems, on the one hand, and purely formal problems on the other. (Contemporary dissatisfaction with the synthetic/analytic distinctions, I believe, springs from a healthy recognition that conceptual problems are adequately analyzed neither as purely empirical nor as purely formal, nor as a mixed of both.) But there is one topic about which I must say a little more in conclusion, if it is to become clear why I speak of the 'history of methodology' as having a central part to play in any future philosophy of science. Instead of making the initial abstraction of treating scientific theories as 'propositional systems', whose logical articulation and formal validity are the sole matter of direct interest to the philosopher (I have argued) an historical approach must begin, rather, by stepping back and viewing the arguments of science within a larger explanatory context. This means,

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among other things, viewing the very activity of 'explaining' itself in a new light. As things stand, most philosophers of science probably take it for granted that the term 'explanation' necessarily refers to a propositional argument; yet even this seems to me to be a mistake. An explanation is not necessarily a propositional argument; and a propositional argument is not necessarily an explanation. On certain conditions and with certain qualifications, a propositional argument can serve as an explanation, and an explanation can take the form of a propositional argument. But explaining is an intellectual function: something we do, using propositional arguments or any other appropriate explanatory procedures. Propositional arguments can thus serve a genuinely 'explanatory' purpose, only when given the right kind of application in the right kind of context. Only where the existing problematic needs of a scientific discipline call for such an explanation (that is) can an argument be put to use as an 'explanation'. Conversely, if it is a 'scientific explanation' we want, we do not necessarily have to produce a propositional argument, either. There are many situations in which scientists offer 'explanations' by the use of quite other sorts of explanatory technique-by drawing graphs, by programming computers, by producing ray diagrams, or in half-a-dozen other ways. And the basic empirical discoveries in which the 'collective experience' of any. scientific profession is properly expressed are themselves discoveries about the circumstances in which, the conditions on which, the range of cases to which and the degree of accuracy with which any particular technique-whether argumentative, computational, taxonomic, graphical, diagramatic -or whatever-can be put to an authentical explanatory use, within this scientific domain or that. The possibility of writing the history of scientific methodology in this way, so that it became of direct relevance to the philosophy of science, was recognized quite clearly by Ernst Mach, in his own 'historical and critical' accounts of the conceptual development of mechanics, heat theory, and the physics of light. If Mach himself was unable to reap the full harvest of this philosophical insight, this was not because his program for making the philosophy of science 'historical and critical' was in any way at fault. It was, rather, because, in Mach's own mind, that sound and valuable project was associated with an empiricist-specifically, with a sensationalist-view about the basic data of science, according to which all fundamental 'scientific observations' had to be equated with 'sense impressions'. Once we free ourselves from Mach's picture of 'experience' as a collection of 'sense data', and from the assumption that the entire empirical burden

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of a science is comprised in straightforward factual observation statements and generalizations-whether about sensory data or about physical observations-we shall be in a position to return to Mach's historical and critical program for philosophy of science, but from a new direction. For we can now study the way in which the explanatory techniques of the science have developed down the years, and the discoveries by which scientists have brought to light the scope and range of application of their different techniques, in ways that throw direct light on the changing character of the intellectual strategies by which conceptual change has been governed within this scienceor that, at this particular historical period or that, as well as the reasons for which those strategies changed as they did, and when they did. Within such an account, however, we shall have to speak of the 'experience' of scientists, neither in a 'sensationalist' way, nor in a 'physicalistic' way, but rather in a way that brings it into line with the 'experiences' of other professional men: for example, of engineers, lawyers, or airline pilots. All of these men accumulate 'experience' by finding out what can or cannot be achieved, using the different items in their repertory of professional equipment. So here: a typical item in the experience of a physical scientist will be, neither a 'sense datum' nor an 'empirical correlation', but a fresh discovery about the scope or relevance of some 'technique of representation': e.g., the discovery that the straightforward techniques of geometrical ray tracing can (or cannot) be extended to cover cases of 'double refraction' also, by using such an such and additional modification. With all respect to Galileo and Descartes, then, Nature has no language in which she can speak to us on her own behalf, and it is up to us as scientists to frame concepts in which we can 'make something' out of our experience of nature. Whether this can be done at all, and what kinds of intellectual construction will prove-methodologically speaking-most effective: these are things we can find out only as we go along. So it is indeed true that any genuine science must have about it both something empirical and something logical. But the logic resides within the articulation of the explanatory procedures associated with the use of some given set of -theoretical concepts. Whether this overall theory is applicable at all-and if so on what conditions, in what sorts of situations, and with what degree of accuracy-that is an empirical question: the most fundamental and general empirical question, indeed, that arises about the theoretical concepts. But the fundamental empirical task of a natural science is not to discover a super-major premise from which all else in this science can be formally deduced. It is to establish progressively more adequate and comprehensive

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concepts and methods of representation with the help of which we can make the phenomena in any domain that much the more intelligible. The choice of a particular theoretical method of representation as an intellectual strategy in (say) biological systematics or atomic physics is, thus, not the result of any straightforward empirical generalization, but rather a prospective estimate of the likely conceptual consequences of alternative intellectual policies. As such, it amounts to a 'rational bet'; and it is concerned, less with uninterpreted Nature-regard as the world of neutral objects, coexisting with the human race-than with the possibility of making the natural world itself a more intelligible object of understanding. So it should be no more surprising if twentieth-century writers like Planck and Mach, Einstein and Heisenberg, have found the fundamental questions of physical theory merging into those of epistemology than it is that their seventeenth-century forerunners found 'natural' philosophy inseparable from 'metaphysical' philosophy. The ultimate question for science, now as then, is in what terms-by the application of what general concepts and explanatory procedures-we can make the world of nature more fully intelligible to ourselves. And on this level of theoretical sophistication, as on every other, the question of what concepts we use when framing our scientific questions about the world is-and is necessarily-prior to all questions about the verification or falsification, confirmation, corroboration or probabilification, of the propositions we can subsequently state in terms of those concepts. One last word will bring the various threads of this lecture together. The methods of theory construction and the selection criteria for judging conceptual innovations in the natural sciences cannot usefully be analyzed by reapplying philosophical techniques-however admirable-originally developed for the purposes of the philosophy of arithmetic, for one clear and fundamental reason. The intellectual goals at which the entire rational enterprise of natural science is directed are quite other than those at which pure mathematics rightly aims. Within pure mathematics, formal systematicity is indeed a necessary virtue: within natural science, it is at most a source of contingent advantages. So (as I have argued elsewhere) what Hilbert was doing when he set out to axiomatize pure mathematics was essentially different from what Hertz did when he reanalyzed the axiomatic structure of Newtonian mechanics. And the formal rigor which makes 'rational mechanics' rational-when regarded as a branch of pure mathematics-is for this same reason different from the substantive considerations which make the procedures of conceptual change in a natural science

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'rational'. And the central contention of my whole argument is this: that the true character of the natural sciences, regarded as the developing outcome of a continuing effort to make progressively more intelligible sense of the world in which we find ourselves-in a phrase, the true character of the natural science is as 'rational enterprises'-is something of which we can hope to give a philosophically adequate account, only if we are prepared to pay a properly historical attention to the changing intellectual strategies and principles of conceptual change by which the evolving intellectual content of those sciences has been selectively perpetuated.

LA DOCTRINE DE VUNIVERSEL CHEZ ARISTOTE A. JOJA Academie Roumaine, Bucarest, Roumanie

1. La querelle des universaux est aussi vieiIIe que la philosophie et la logique, puisqu'elle opposait deja Platon au nominaliste Antisthene (David Proleg. ad Porph. Isag. b. Brand, p. 20 a 2), ainsi qu'a Aristote, partisan d'un realisme modere, i. e., immanentiste (Anal. Post. A 11, 77 a 5; Metaph. A 9, 999 a 33; K 1, 1059 b 1; - M 4, 1078 b 7; M 5, M 6, 7, 8, 9, 10; N 1, 1087 a 29; N 2, N 3, N 4, N 5, N 6).

Ensuite, elle devait opposer les Stoiciens nominalistes (Simplicius, Categ. 26) a Aristote et a Platon. Au Moyen Age, avec les platoniciens de Chartres, avec le conceptualiste Abelard, l'aristotelisant Thomas d'Aquin et Ie nominaliste Occam, elle passionne et divise philosophes et logiciens (BREHIER, 1937; GILSON, 1944; VON PRANTL, 1927). Dans les temps modernes, la fameuse querelle subit une eclipse, mais de nos jours et, paradoxalement, avec l'avenement de la logique mathematique et notamment de la semantique, elle connait un regain d'interet et d'actualite. Uber Sinn und Bedeutung de Frege (1892) et On Denoting (1905) de Russell semblent avoir stimule la discussion du statut logique et ontologique des universaux. Platonisants ou nominalistes, Husserl, Wittgenstein, Lesnievski, Carnap, Quine, Church et Goodman (STEGMULLER, 1970) ont contribue a 1'approfondissement du probleme des universaux. Le professeur Quine a formule le critere suivant pour juger du statut ontologique d'une theorie: "An entity is assumed by a theory if and only if it must be counted among the values of the variables in order that the statements affirmed in the theory be true" (QUINE, 1953, p. 103). "To be is to be the value of a bound variable", dit encore Quine. Dans la configuration actuelle, il semble qu'il n'y ait plus que deux conceptions adverses, les platonisants et les nominalistes; on ne parle

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plus de conceptualistes, ni meme d'aristoteliciens, Ie realisme modere du Stagirite n'apparaissant que comme une variete du platonisme. Or, cette variante ne sacrifie ni les droits de l'individuel, comme Ie fait Ie platonisme, ni les prerogatives de I'universel, comme Ie nominalisme. Done, ce n'est pas une simple variante. Le realisme modere (ou immanent) d' Aristote semble fournir les moyens de trouver la veritable solution du probleme des universaux. C'est la raison pour laquelle nous no us proposons de degager sommairement la doctrine aristotelicienne de I'universel, que resume la celebre antinomie: existentia est singularium (Aristote, Categ. 5, 2 b 5: !L~ oOa'wv ouv 'rwv 7tpw'rwv OOO'LWV liouvoc'rov 'rwv (f.MWV 'rL EtvOCL ergo nisi primae substantiae sint, caeterarum rerum nulla esse potest), scientia est de universalibus (Anal. Post. I 33, 88 b 30: ~ !LEV EmO''r~!J.."1J xoc86AOU xoct ilt &'VOCyxocLwv scientia vero est universalis et ex necessariis). II ressort des textes du Stagirite, que les caracteres des universels sont les suivants: a) l'objectivite b) l'idealite c) la predicabilite d) la necessite e) l'intelligibilite f) l'anteriorite logique. II ressort egalement des textes d' Aristote que les caracteres des singuliers sont: a) la substantialite b) la concretude c) l'impredicabilite d) l'intelligibilite e) la contingence f) l'anteriorite chronologique. a) Objectivite. - "Il est evident, dit Aristote, que les universels appartiennent necessairement aux choses" (Anal. Post. I 5, 73 b 27). En general, les formes intelligibles sont contenues dans les formes sensibles: in ipsis sensibilibus formis ipsae sunt intelligibiles formae (De Anima III 8, 432 a 5). L'universel est non pas separe, mais il est dans les choses singulieres et multiples, (mais non comme une partie) dont il ne peut etre separe que sola cogitatione.

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Seules les choses singulieres existent, mais elles ne peuvent acceder type de structure.

a I'existence qu'en se subordonnant a un universel, a un

b) Idealite. - Or, si I'universel est objectif, en tOIS pragmasi, en tant qu'universel ontologique il se presente toujours sub specie individui et durationis. Ce n'est que dans l'intellect qu'il existe sub specie universalitatis et aeterni, concentre et pur, plus authentique que son homologue ontologique. En effet, Aristote dit: " ... tout universel en repos dans I'ame, eomme une unite en dehors des multiples" (Anal. Post. II 19, 100 a 5), ex omni universali quiescente in anima, uno praeter multa ... In re, l'universel est entraine dans Ie flux du devenir; in mente, il se tient ealme, il est eomme en repos. Objectif in re, il est ideel in mente. Ideel, parce que les faits de l'intellect sont tels: "Car ce n'est pas la pierre qui est dans l'ame, mais sa forme (De Anima III 8,431 b 29), to eidos. L'intellect est "forme des formes" (De Anima III 8, 432 a 2), "espaee des formes" (De Anima III 4, 429 a); or, l'universel est une telle forme, caracterisee par l'idealite. L'intellect est Ie monde des intelligibles et l'universel est un intelligible, done ideel. L'objectivite et l'idealite de l'universel ne sont nullement contradictoires; ee sont, au contraire, deux aspects complementaires: l'aspeet ontologique, immerge dans Ie reel objectif, et l'aspect logique, emergeant concentre et pur dans I'intellect. c) Predicabilite. - "En effet, il est evident que rien de ce qui existe comme universel n'est une substance et qu'auqun des predicats communs ne signifie cette chose-ci (tode ti), mais telle qualite (toionde)" (Metaph. Z 13, 1038 b 35). La substance n'est predicat d'aucun sujet, tandis que l'universel est toujours predicat de quelque sujet. C'est en raison de sa nature propre que l'universel est predicat commun. L'universalite engendre spontanement la predicabilite, comme l'individualite, l'impredicabilite. "Parmi les etants, les uns sont predicats d'un sujet, mais ne sont dans aucun sujet; ainsi homme s'affirme d'un certain homme determine, eomme d'un sujet, mais n'est dans aueun sujet" (Categ. Z, 1 a 20). d) Necessite. - "J'appelle universel ce qui appartient a tout sujet, par soi et en tant que soi. II s'ensuit clairement que les universels appartiennent necessairement aux objets" (Anal. Post. I 4, 73 b 26).

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L'universel appartient necessairement au sujet, puisque celui-ci n'existe qu'a la condition de se conformer aux universaux, de se mouler dans l'universel. Aristote identifie l'universel au necessaire, it. ce qui ne peut pas etre autrement qu'il n'est. Dans la conception d' Aristote, l'universel est l'hypostase la plus exterieure d'une trinite: essence - necessite - universalite. C'est parce qu'un attribut appartient it. l'esse de la classe respective, que les singuliers, que cette classe subsume, ne peuvent pas etre, esse, sans posseder cet attribut (essentiel) - c'est done pour cette raison que cet attribut est necessaire (ex necessitate inest rebus). Or, etant necessaire, il se distribue universellement aux singuliers subsumes it. la classe respective. L'universalite est une consequence de la necessite et celle-ci derive de l'essence meme, ousia, to ti eii einai, essentia quidditas. e) Intelligibilite. - En effet, tandis que Ie singulier, to kath hekaston, est ineffable et, en soi, inintelligible, l'universel est eminemment intelligible. "II est evident que des substances singulieres il ne saurait y avoir ni definition, ni demonstration. Car les etants corruptibles deviennent obscurs pour ceux qui sont pourtant capables de connaitre et, bien que leurs notions se maintiennent dans l'ame, il n'y aura ni definition, ni demonstration. C'est pourquoi il faut, it. l'egard de telles definitions des choses individuelles, ne pas ignorer qu'une telle definition est toujours precaire" (Metaph. Z, 15, 104 a 1). Omne individuum ineffabile: ineffable et indefinissable, parce que Ie singulier est objet, tandis que l'universel est objet de connaissance: "necessairement nous percevons Ie singulier, tandis que la science consiste dans la connaissance de l'universel" (Anal. Post. I 31, 87 b 37). "Les choses les plus universelles sont les plus eloignees des sens et les choses individuelles en sont les plus proches" (Anal. Post. I 1, 72 a 4). Si Ie singulier, en tant que tel, est indefinissable, indemontrable et ineffable, it peut etre connu et concu par l'intermediaire de l'universalite qui y reside: sa forme, eidos. L'universel est intelligible, parce qu'il reduit les multiples singuliers it. des classes relativement peu nombreuses: infinita perlustrare impossibile est (Phys. 8, 263 a 6). L'universel est intelligible en vertu de sa stabilite et de son ubiquite:

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"il est impossible de percevoir l'universel, qui vaut pour tous les cas, car ce n'est ni cette chose-ci, ni ce moment, puisqu'alors il ne serait pas universel: nous disons, en effet, que l'universel est partout et toujours. Done, puisque les demonstrations sont universelles et qu'elles ne peuvent pas etre percues, il est evident qu'il n'y a pas de science par la sensation. Mais il est aussi evident que, meme s'il nous etait possible de percevoir que Ie triangle a ses angles egaux a deux droits, nous en chercherions, pourtant, une demonstration et nous n'en aurions pas, comme certains Ie disent, une connaissance rationnelle, car necessairement on a la sensation de l'individuel, tandis que la science reside dans la connaissance de l'individuel" (Anal. Post. I 31, 87 b 28). 11 faut, pourtant, noter que l'universel surgit deja au niveau meme de la sensation, car la sensation est une certaine connaissance, gniisls tis. Lorsque je vois Callias, je vois en meme temps l'homme en Callias; je ne puis percevoir Ie singulier, sans l'enchasser spontanement dans un universel. L'intellect est de meme nature, syggenes, que les intelligibles; c'est pourquoi il peut saisir les intelligibles. "L'intellect est en puissance les intelligibles memes, mais il n'est en entelechie aucun d'eux, avant d'avoir pense", prius antequam intelligat (De Anima III 4, 429 b 31). L'universel est un intelligible par excellence, car tous les intelligibles sont des universaux. L'intellect saisit d'emblee l'identite; or, Ie fondement de l'universel, c'est I'identite: "11 n'est pas necessaire qu'il y ait des idees (eide), ou bien une unite (hen ti) en dehors des multiples, pour qu'il y ait demonstration, mais il est necessaire de dire avec verite une unite - pertinente - aux multiples (hen kata pollon). En effet, il n'y aura pas d'universel si cette unite n'existe pas; or, si l'universel n'est pas, il n'y aura pas de moyen terme, par suite, pas de demonstration. II faut done qu'il y ait quelque chose d'un et d'identique et non homonyme dans une pluralite " (Anal. Post. A 11, 77 a 5). L'universel ontologique se rattache organiquement a l'identite en tant que telle, en tant que phenomene de base du monde des etants: l'universel logique se rattache au principe d'identite, principium a quo cogitatio et logice. Sans identite, pas d'unite; sans unite, pas d'universel; sans universel, pas d'intellection et pas d'intelligibilite, Ainsi, l'universel est eminemment intelligible. f) Anteriorite logique. - La primordialite existertielle appartient a la substance: "Done, si les premieres substances n'e xistent pas. il est im-

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possible que rien d'autre existe, car toutes les autres choses ou bien se prediquent de celle-ci comme de leurs sujets, ou bien sont dans ces sujets eux-memes; il en resulte que, si les substances premieres n'existent pas, il est impossible que quelque chose d'autre existe" (Categ. 5, 2 b 4). Chronologiquement, la substance est anterieure a l'universel, mais logiquement -logo -l'universel, le genre, I'espece semblent anterieurs a la substance. Logiquement, puisque l'individu se definit par Ie genre et la difference specifique, que l'individu n'existe et ne saurait exister qu'en se moulant dans Ie type de structure que lui offre l'universeI. L'universel - type de structure, forme, qualite, relation - se repete indefiniment a travers les series d'individus et, par cette repetition structurante, precede, en quelque sorte, secundum quid, l'individuel. II semble qu'on ne puisse, finalement, eviter Ie realisme absolu des essences. 2. Et "puisqu'il y a des choses universelles et d'autres singulieres, j'appelle universel ce qui, de par sa nature, peut se prediquer tkategoreisthai) de plusieurs sujets, et singulier ce qui ne Ie peut; ainsi, homme est un terme universel, Callias un terme individuel - il est necessaire d'enoncer (apophainesthai) que quelque chose appartient ou non tantot a un universel, tantot a un singulier" (De Interpr. 7, 17 a 38. Pacius (Aristoteles Latine, interpretibus variis) traduit ainsi: Quoniam autem rerum aliae sunt universales, aliae singulares [universale apello, quod suapte natura multis attribuitur, singulare quod non attribuitur, ut homo est res universalis, Callias singularis], necesse est utique enuntiare aliquid inesse aut non inesse, interdum rei universali, interdum rei singulari). C'est parce que, dans Ie plan ontologique, il y a des choses singulieres (effectivement n'existent que les singuliers) et des entites universelles (qui sont, en fait, des tota attributiva) c'est - parait-il- pour cette raison, sur la base de son realisme modere, qu'Aristote opere Ie passage ali plan logique. La division des onta en singuliers et universels (immanents) justifie la division logique des termes universels et singuliers. Le rapport de l'universel et du singulier, leur intrication, in re, semble justifier I'enonciation, l'ap6phansis, Yapophainesthai qu'un attribut appartient (ou n'appartient pas) a un universel ou a un singulier, que l'attribut s'applique a l'universalite du sujet ou seulement a une partie de celui-ci.

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Mais on peut enoncer universellement ou non-universellement d'un sujet, praedicari de universali universaliter vel non universaliter. Aristote observe que, si l'on enonce universellement d'un universel qu'une chose lui appartient ou non, il y aura des propositions contraires (enantfai apophdnseis) (De Interpr. 7, 17 b 3). Ainsi se profile l'opposition de contradiction (antikeisthai antiphatikosi et celIe de contrariete (antikeisthai enantiiisi. Exemple de de universali universaliter enuntiare: tout homme est blanc - nul homme n'est blanc. Propositions enoncant l'universel, mais non universellement : l'homme est blanc, l'homme n'est pas blanc. Homme est un universel, mais iI n'est pas utilise universellement dans la proposition, car Ie mot tout (pas, omnis) ne signifie pas l'universel, mais seulement que Ie sujet universel est utilise comme universel (sed quod universale universaliter). lei intervient Ia celebre theorie aristotelicienne de la non-quantification du predicat, Aristote la formule en ces termes: "Mais si au predicat universel on attribue l'universel, ce n'est pas vrai: car aucune affirmation ne sera vraie dans laquelle l'universel est attribue au predicat universel, par exemple "tout homme est tout animal" (De Interpr. 7, 17 b 12). Ammonius et, a sa suite, Boece, ainsi que Thomas d' Aquin, au Moyen Age, adherent a la these d' Aristote. Ammonius pense qu'il est impossible de dire la verite tout en attribuant l'universalite au predicat, Ammonius distingue seize propositions formees par la specification (prosdiorism6s) du sujet et du predicat. II observe que si omnis homo omnis animal est evldemment fausse a jamais, taei pseude], en revanche la proposition non omnis homo omne animal est "toujours vraie"; de meme est vraie la proposition omnis homo non est omne animal. La quantification, meme vraie, du predicat est a rejeter, puisqu'elle n'ajoute rien a la clarte du predicat, dont la fonction n'est pas d'indiquer une quantite, mais une qualite, C'est pourquoi les Scolastiques disaient: subiectum se habet materialiter, praedicatum se habit formaliter. Le predicat peut bien etre pris universellement, mais ce n'est pas Ia sa fonction: au contraire, sa mission est d'indiquer une qualite, une forme qui s'applique a un sujet universel, particulier ou singulier. Le predicat est universe! par destination: iI ne designe pas un tode ti, mais un toionde, une certaine qualite d'une chose universelle ou singuliere. Si je dis: "tout homme est quelque animal", je dis une chose exacte, mais la proposition, dont Ie predicat est ainsi specifie en plus (prosdiorismosi n'ajoute rien a la clarte de la proposition "tout homme est un

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animal". Car il ne s'agit pas de denombrer Ie genre animal, mais de situer exactement l'homme dans son genre. "Tout et nul ne signifient rien autre chose si ce n'est que l'affirmation ou la negation du nom (onoma) est prise universellement"; par consequent, il faut adjoindre les autres parties de la proposition comme identiques" (De Interpr. 20 a 12). Le veritable sujet est une substance premiere, tode ti, puisque seule elle existe effectivement et est porteuse d'attributs. En effet, "tout Ie reste est ou bien predique des substances premieres sujets, ou bien est dans ces sujete eux-memes, Car si les substances premieres n'existaient pas, il serait impossible que quelque chose d'autre existat" (Categ. 5, 2 b 4). Les substances individuelles sont des partes subiectivas in toto attributivo, des particuliers ou des singuliers qui sont inherents, resident (hypdrchousin) dans un tout d'attribution. Les especes et les genres sont done des tota attributiva, des touts d'attribution, lesquels ne peuvent etre qu'en s'attribuant et se distribuant, s'incorporant aux substances individuelles. Les substances secondes ne peuvent etre que par attribution. II s'ensuit que le veritable sujet est le singulier determine et concret, hoc aliquid, ce cheval-ci ou cet homme que voila. "Ce qui est signifie est un individu numeriquement un", individuum enim et unum numero est id est quod significatur " (Categ. 5, 3 b 13). Homme ou animal ne signifient pas un etre determine, tode ti, mais plutot un toionde, un quale quid, une qualite, car le sujet n'est pas un, comme Ia substance premiere, mais homme et animal se disent d'une pluralite, kata pol/on, de multis. Cependant, homme ou animal ne signifient pas la qualite absoIument, comme, par exempIe, Ie blanc (car Ie blanc ne signifie rien d'autre que Ia qualite), L'espece et le genre determinent Ia qualite par rapport a Ia substance, precisement ils signifient: quelque substance de telle qualite (Categ. 5,3 b 18). Callias ou Socrate sont de veritables sujets, des sujets-types; homme et animal, du fait qu'ils sont des substances secondes et qu'ils designent des qualites et non des individus numeriquement un, ne sont pas de veritables sujets. Le propre d'homme ou d'animal est d'avoir une extension qui depasse indefiniment Ie hos aliquid, Yindividuum unum numero; par consequent, l'espece et le genre, et a plus forte raison Ies qualites comme Ie blanc, sont suiets au second degre,

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913

L'homme n'est pas mortel et Yanimal non plus. Ce qui est mortel, c'est Yindividuum, to dtonom. Seul I'individuel est vraiment subjicibile: praedicato potest subjici, L'universel est predicat, kategorotanenon.

L'universelle "L'homme (tout homme) est morteI" n'a de valeur existentielle que si on substitue Callias et Socrate a l'universel "homme". Existentia est singularium, ou en langage de semantique moderne, to be is to be a value of a bound variable. Les universels homme ou animal ne sont un sujet que par extrapolation,

puisqu'ils ne sont pas capables d'avoir des attributs, mais qu'ils sont euxmemes attributs. Ce sont des concepts, or, les concepts ne peuvent etre porteurs d'attributs que par assimilation aux singuliers multiples. "En outre, les substances premieres, parce qu'elles sont Ie fondement (hypokeisthai) de tout Ie reste et que tout Ie reste est predique d'elles ou est en elles, sont pour cela appelees au plus haut degre substances. Et de meme que Ies substances premieres se rapportent a tout Ie reste, de meme l'espece se rapporte au genre; car l'espece est un fondement pour Ie genre. En effet, les genres sont prediques des especes, mais les especes ne sont pas prediquees des genres, en sorte que de tout cela resulte que l'espece est plus substance que Ie genre" (Categ. 5, 2 b 15). Ainsi, les substances individuelles sont Ie fondement de tout ce qui peut pretendre avoir une realite: tout Ie reste, tout ce qui presente un certain degre de realite est ou bien predique des substances individuelles, ou bien est en elles (en tautais einai, in ipsis sunt}. C'est pourquoi les substances individuelles sont les substances mdlista, Ies veritables substances. Elles sont Ie sujet ontoIogique et, par suite, Ie prototype, Ie paradigme du sujet logique. Mais Ia position des substances individuelles a l'egard de tout Ie reste, se repercute dans la position de l'espece a l'egard du genre et, par suite, d'un universel d'extension moindre a un universel d'extension superieure, Ainsi, par assimilation des universels aux singuliers, par un transfert de fonctions, les concepts (universels par definition) assument Ie role d'un sujet Iogique dans Ia proposition, bien que, en eux-memes, ce soient des predicats. En effet, suivant Ia definition aristotelicienne, les universels sont des predicats communs (koine kategoroumenai et qui ne sont jamais substance. Par exempIe, dans la proposition "I'homme est animal", homme aussi bien qu'animal sont des universels, mais homme est Ie sujet que Ie predicat animal determine, mais Ie sujet homme devient predicat dans Ia propo-

914

A. JOJA

sition "Le Grec est homme", tandis qu'il est impossible que les singuliers Socrate ou Callias se metamorphosent jamais en predicat. Socrates est albus (De Interpr. 7, 17 b 28) est une proposition a sujet singulier. En revanche, omnis homo est albus (De Interpr. 7, 17 b 18) est une proposition a sujet universe! et quantifie universellement, comme quidam homo est albus est quantifie particulierement (De Interpr. 7, 17 b 19). Socrate est un sujet authentique, puisque seul le singulier est porteur d'attributs, tandis que homme ou tout homme ne peut etre blanc que si on sousentend: chaque individu subsume a homme. L'homme n'est ni blanc, ni beau, ni juste, parce que l'homme est un universel qui, comme tel, n'est pas subjicibile; ce qui ales caracteres d'un sujet, ce sont Socrate, Callias et Coriscus. Cependant, homme est, quand meme, une substance seconde; comme la substance individuelle, elle n'est pas dans un sujet, nul/am in subiecto esse. En effet, homme est predique du sujet "un certain homme", mais n'est pas dans le sujet; au contraire, c'est l'individuel qui se trouve contenu dans l'homme comme une partie subjective dans un tout attributif. C'est en raison de son caractere d'universalite attributive, de sa disponibilite, que la substance seconde est prediquee, legetai, praedicatur, de la substance individuelle, essentiellement impredicable, Or, non seulement les substances secondes sont prediquees des substances individuelles, mais les accidents universels, qui sont dans Ie sujet, sont prediques du sujet. Les entites qui sont dans un sujet - ta d'en hypokeimenii onta -la plupart du temps ni leur nom, ni leur definition ne sont attribues au sujet. Dans certains cas, rien n'empsche que le nom ne soit attribue au sujet, mais quant a la definition, c'est impossible. Ainsi, Ie blanc etant dans un corps comme dans un sujet est predique de ce sujet, car Ie corps est dit blanc, mais la definition du blanc ne sera jamais prediquee du corps (Categ. 5, 2 a 27). En effet, l'homme ne peut pas etre defini comme une couleur. On voit, cependant, Ie gIissement qui s'opere de la substance individuelle, seule veritable sujet porteur d'attributs, a la substance seconde (espece et genre) et, ensuite, a I'accident universeI. Le resultat sera que tout universel peut remplir, par rapport a un autre, Ie role de sujet. Ainsi, Ie sujet devient ce dont on affirme quelque chose et Ie predicat ce qui s'affirme de quelque chose. Le sujet est id de quo dicitur, kath' OU kategoreitai, et Ie predicat id quod dicitur.

LA DOCrRINE DE L'UNIVERSEL CHEZ ARISTOTE

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De cette facon, sujet et predicat sont vides de tout contenu ontologique et ne gardent plus qu'une signification logique et souvent grammaticale. La logique est essentiellement formelle et iI etait dans sa nature d'evoluer toujours dans la direction du formalisme. Le progres dans la voie du formalisme est normal, pourtant Arendt Heyting rappelle a bon droit que des qu'on veut appliquer la logique, on doit bien se poser la question de la signification du mot "vrai" et des autres termes logiques, et on est alors conduit a des enonces comme Ie suivant: "Une proposition est vraie si l'etat des choses qu'elle exprime existe dans Ie monde reel. La definition varie selon Ie point de vue du philosophe, mais elle presuppose toujours une conception du reel; cela revient a dire que la logique a besoin, pour son interpretation, d'une ontologie... La logique des propositions, sous sa forme traditionnelle, n'est done independante que lorsqu'on la traite comme un calcul purement formel. Des qu'on essaie de l'interpreter, on doit avoir recours a une metaphysique" (HEYTING, 1956, p. 226). Les Premiers Analytiques traitent du syllogisme d'un point de vue purement et strictement formel, mais ils sont ordonnes en vue des Seconds Analytiques, ou Aristote expose la theorie de la science. Au premier abord, iI semble que tout pont entre les Premiers Analytiques et le reel soit rompu et, pourtant, l'arriere-plan ontologique se dessine. De meme, dans les notions de sujet et de predicat, l'arriere-plan ontologique semble se derober, alors qu'en realite elles se rattachaient, initialement, a ce plan. Ce rattachement est involontairement marque par Aristote a la fin du premier chapitre des Anal. Pr.: In toto esse alterum altero et alterum de altero omni praedicari idem est. Dicimus autem de omni praedicari quando nihil subiecti sumi potest, de quo alterum non dicatur; de nullo (praedicari didmus) similiter, etre contenu dans la totalite d'un autre (terme) et etre predique universellement d'un autre (terme) c'est la meme chose. Or, nous disons qu'un terme est predique universellement d'un autre terme quand on ne peut prendre aucune partie du sujet dont I'autre ne se dise pas; et nous disons que n'€tre predique d'aucun se comporte de la meme facon (Anal. Pro I, I 24 b 26). Ainsi, en holii einai (etre dans la totalite) et kata pantos kategoreisthai (etre predique universellement) sont deux expressions identiques. En Mia einai garde encore la resonance ontologique, mais elle finit par se perdre avec kata pantos kategoreisthai. II faut entendre ici par totum l'attributif, par exemple le genre. In toto esse et did de omni sont la meme chose en fait, mais different par leur

916

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fondement, comme l'ascension et la descente. Quand l'attribut est affirme universellement du sujet, comme animal de tout homme, on dit que Ie sujet est dans la totalite, c'est-a-dire qu'il est contenu dans l'attribut comme sa partie subjective. Et lorsque no us descendons de l'attribut au sujet, no us disons de omni, mais lorsque nous remontons du sujet a l'attribut, nous employons l'expression esse in toto (Julii Pacii a Beriga in Porphyrii Isagogen et Aristotelis Organon commentarius, p. 116). Dans Ie passage des Categories, 3, 1 b 10, on voit Ie sujet et Ie predicat s'opposer comme les deux constituants de la proposition: in toto autem esse alterum (8) altero (P) et alterum (P) de altero (8) praedicari idem est. On y surprend aussi les resonances ontologiques dans les formules 10-giques. De meme, dans De Interpretatione 3, 1 b 10, nous lisons: "quand une chose est prediquee d'une autre comme de son sujet, tout ce qui est dit du predicat sera dit aussi du sujet; par exemple, homme est predique de cet homme-ci, et animal est predique de I'homme. Done animal se prediquera aussi de cet homme-ci, car cet homme-ci est a la fois homme et animal" (Categ. 3, 1 b 10). Le passage ci-dessus indique d'une maniere suggestive l'interrelation du singulier, du particulier et de l'universel, tant sur Ie plan ontologique que sur Ie plan logique. Pour ce qui est des genres subordonnes les uns aux autres (hypdllela), rien n'empeche que les differences soient les memes; car les genres superieurs (ta epdno) sont predicats des genres subordonnes, de sorte que toutes les differences du predicat seront aussi differences du sujet (Categ. 3, 1 b 20). Ainsi, les genres inferieurs sont sujets par rapport aux genres superieurs et, par suite, les differences du predicat (genre superieur) seront egalement les differences du sujet (genre inferieur), II se forme ainsi une hierarchic des predicats, les predicats inferieurs devenant les sujets des predicats superieurs. Le predicat est predicat parce qu'il affirme quelque chose d'une autre chose, en general de moindre extension. II arrive, pourtant, que Ie predicat soit un singulier, par exemple "nous disons parfois que ce blanc-la est Socrate et que ce qui approche, c'est Callias (Anal." Pr. I, 27, 43 a 25). Mais Aristote, qui n'admet pas semble-t-illa valeur logique du singulier et qui a edifie une syllogistique , des termes universels, a soin de preciser "que certains etants ne puissent, de par leur nature, se prediquer d'aucun, c'est

LA DOCTRINE DE L'UNlVERSEL CHEZ ARISTOTE

917

manifeste, car presque chacun des sensibles est de nature telle qu'il ne puisse se prediquer d'aucun, si ce n'est par accident" (Anal. Pro I 27, 43 a). "Categorein kata symbebekos, observe Bonitz, se referant a la proposition "ce blanc-la est un homme", ou la substance est predicat de l'accident, categoric para physin, contre nature - s'oppose a kategorein hapliis, prediquer absolument" (BONITZ, 1955, p. 714 b 38). C'est une predication contre nature, parce que Ie blanc, qui est un accident de la substance, devient, lui, substance, et que celle-ci devient predicat de l'accident "Ie blanc". 3. Lukasiewicz reproche a Aristote d'avoir dit que des termes individuels ou singuliers, comme Callias, ne peuvent pas etre prediques avec verite de rien autre chose. Car Aristote Iui-meme donne des exemples de propositions vraies avec un predicat singulier, come "ce blanc est Socrate" ou "ce qui approche, c'est Callias", disant que ces propositions sont "accidentellement vraies. II y a d'autres exemples de ce genre qui ne sont pas tout simplement par accident vraies, comme "Socrate est Socrate" ou "Sophronisque etait Ie pere de Socrate". Une troisieme inexactitude, c'est la conclusion que tire Aristote de la classification. "II est evident que les termes individuels sont aussi importants que les universels, non seulement dans la vie quotidienne, mais dans les recherches scientifiques. Le plus grand defaut de la logique aristotelique, c'est que les propositions et termes singuliers n'y trouvent place" [LUKASIEWICZ, 1951, p. 6]. La cause de cette omission, explique Lukasiewicz, ne doit pas etre attribuee a l'influence platonicienne sur la logique du Stagirite, car les Premiers Analytiques sont exempts de toute contamination philosophique (LUKASIEWICZ, 1951, p. 7). Aristote pretend qu'un terme singulier est impropre a figurer comme predicat dans une proposition vraie, de meme qu'un terme de grande extension ne peut convenablement figurer dans une telle proposition. La veritable raison de ces deux assertions d'Aristote, c'est qu'il est essentiel pour la syllogistique aristotelicienne que Ie meme terme puisse etre employe une fois comme sujet et une autre fois comme predicat, Dans les trois figures syllogistiques, connues par Aristote, il existe un terme qui se presente une fois comme sujet et ensuite comme predicat, Dans la premiere figure, c'est Ie moyen terme; dans la seconde, c'est Ie majeur et, dans la troisieme, Ie mineur (ibid., p. 7). La syllogistique, telle qu'elle dt concue par Aristote, exige l'homogeneite des termes quant a leur position possible comme sujet et predicat. C'est

918

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la, semble-t-il, la raison pour laqueIIe les termes singuliers sont omis par Ie Stagirite (ibid., p. 7). Certes, il n'y a aucune confirmation de l'influence directe de Platon dans les Premiers Analytiques. Aristote a cree cet ouvrage dans un esprit de parfaite positivite, mais l'omission des termes singuliers releve du meme ideal scientifique que celui de Platon: scienta est universalis et per necessaria. L'esprit de positivite n'exclut guere, mais suppose une ontologie OU Ie singulier devient intelligible par son insertion dans I'universeI. Aux yeux d'Aristote, seul Ie singulier existe effectivement, c'est-a-direen tant que substance separee, mais seul I'universel est objet de science. La syllogistique aristotelicienne exige que les termes soient homogenes en ce qui conceme leur position possible comme sujet et predicat. Le moyen terme (sub-prae; prae-prae; sub-sub) est Ie pivot du syIIogisme: medium, per quod fit syllogismusi. Or, pour remplir son office mediateur, Ie moyen doit etre un concept universel, designant une nature universeIIe, une essence. Si Ie moyen n'etait pas un universel, mais un singulier, il ne saurait communiquer sa nature et, par suite, it serait instable et desordonne, dtakton. C'est ce caractere d'universalite qui garantit la demarche syIIogistique. La syUogistique aristotelicienne conclut necessairement (necessitas concludendi), meme si c'est a partir de premisses contingentes qu'eIIe precede. La syIIogistique est edifice en vue de demontrer la nature generale et necessaire des phenomenes; Ie syllogisme approprie est Ie syllogisme categorique, Or, Ie syIIogisme provient de termes universels: syllogism us ex universis (Eth. Nic. Z 3, 11 39 b 29). La syIIogistique d' Aristote a ete concue pour exprimer des relations eidetiques - done intemporeIIes. EIIe baigne dans I'universalisme, tandis que Ie synemmenon stoicien s'inspire du nominalisme pro pre aux Stoiciens: "lIs disent que Ie general n'est rien ... en effet, I'Homme n'est pas quelqu'un, car la generalite n'est pas quelque chose" (Simplicius, Categ. 26). La logique stoicienne baigne dans Ie nominalisme; elle traite des termes singuliers, dont elIe etablit la liaison necessaire dans Ie temps. L'universel est, par definition, supratemporel: semper et unique. Au contraire, Ie singulier se temporalise: il est hoc aliquid et nunc, La logique stolcienne est axee sur Ie singuIier, non sur l'essence, mais sur l'evenement, dont il faut determiner les rapports necessaires avec d'autres evenements; c'est une logique de la consequence, tie la connexion necessaire. La vision sensualiste et nominaliste des choses est cause de la substitution du synemmenon au syIIogisme categorique d' Aristote. L'enchainement dynamique des evenements singuliers produit une logique de la connexion conditionneIIe, OU

LA DOCTRINE DE L'UNlVERSEL CHEZ ARISTOTE

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les modes et les figures disparaissent. Les termes hypdrchei, enhypdrchei et enesti sont remplaces par des termes typiquement stoiciens: hepetai et akolouthei. Le renversement de perspective provoque par l'utilisation des termes singuliers dans la logique stoicienne est visible. Nous avons affaire a un autre horizon du Logos, que la logique aristotelicienne n'epuise pas, mais dont elle traduit les aspects essentiels. La difference de la logique aristotelicienne et de la logique stoicienne reside, peut-etre, dans la predilection de l'une pour les termes universels, et de l'autre pour les termes singuliers. De la formule qui resume la doctrine du Stagirite: existentia est singularium, scientia est de universalibus - les Stoiciens n'ont retenu que la premiere partie. Aristote est aussi convaincu que les Stoiciens que, dans Ie plan ontologique, seul Ie singulier existe effectivement, mais il croit que, dans Ie plan logique, seul l'universel, intelligible par definition, est objet de science en general, de science logique en particulier. La science moderne se fonde sur Ie meme principe d'intelligibilite de l'universe1 enonce par Aristote, a la suite de son maitre Platon. En effet, lois et structures valent par leur caractere d'universalite, ou les phenomenes singuliers viennent se mouler, afin d'acquerir l'intelligibilite. References BoNITZ, R., 1955, Index aristotelicus (second edition, Akademische Druck- u. Verlagsanstalt, Graz) BREHIER, E., 1937, La philosophie du Moyen Age (Michel, Paris) GILSON, E. R., 1944, La philosophie au Moyen Age (Payot, Paris) REYTING, A., 1956, La conception intuitionniste de la logique, Les Etudes Philosophiques, No.2, p. 226 LUKASIEWICZ, J., 1951, Aristotle's syllogistic (second edition, 1957; Clarendon Press, Oxford) . QUINI!, W., 1953, From a logical point of view (Harvard University Press, Cambridge) STEGMULLE'R, W., 1970, Main currents in contemporary German, British and American philosophy, translated from German by A. E. Blumberg (Indiana University Press, Bloomington) VON PRANTL, c., 1927, Geschichte der Logik im Abendlande (Fock, Leipzig)

"

ARISTOTLE, LUKASIEWICZ AND THE ORIGlNS OF MANY-VALUED LOGIC G. PATZIG Georg-August-Universitdt, GiJttingen, German Federal Republic

Many-valued logic may be defined by the thesis that we should dismiss the principle according to which every proposition must be either false or true. This principle, often having been taken to be just another formulation of the law of excluded middle, has been distinguished from it, in modern times, by Lukasiewicz. The law of excluded middle just says, that for any proposition p we have 'p V non-p'. This principle implies that, of a statement and its negation, at least one must be true. But it does not already guarantee, for any given p, that p must be true or false. The difference becomes of practical importance in the famous case of future contingents. Even for events which will, or will not, happen in the future we may be convinced that, for every p, it will be the case that p, or it will be the case that non-p, but we may have our reservations if we also can say that the proposition 'It will be the case that p' is false or true at any given moment. We may be sure that a certain event, say the event that 1 shall receive a letter from Berlin, will happen tomorrow or will not happen tomorrow. But we may be less sure whether we might also say that the sentence 'I shall receive a letter from Berlin tomorrow' is true or not true as things stand today. Lukasiewicz indeed took this line of argumentation. He thought that Aristotle in the much-debated Chapter 9 of "De Interpretatione" had also tried, if not very successfully, to argue for the same position. So the introduction of what has been called non-Aristotelian logic, namely, a logic with more than two truth-values, was historically influenced by a famous view Aristotle had put forward. Aristotle's example is the case of the sea battle which mayor may not occur tomorrow. Aristotle probably thought of the situation on the evening of the 27th of September 480 B.c., the evening before the battle of Salamis. If we try, in imagination, to place us in the situation of the Greeks on the eve of that battle, we can say with

922

G. PATZIG

them that it is (logically) necessary that a sea battle will be fought tomorrow or that a sea battle will not be fought tomorrow. But it is not logically necessary that one of these sentences be true at the present time. That there are just two truth-values to be divided among these two propositions we know for certain; therefore we know that one of them will be true, the other false; but we do not know now which of the propositions is the one which will be true and which is the one that will be false; and very probably neither of these propositions is true nor false already. They are jointly candidates for the two truth-values that must be divided between them. Lukasiewicz has taken Aristotle to develop the position, in this passage, that we may accept the "Tertium non datur" even for statements that are concerned with contingent events or facts in the future, but that we need not and should not accept also the principle of bivalence for such statements. DOROTHEA FREDE (1970) has shown that it is impossible to derive such an interpretation from Aristotle's Greek text. Aristotle has nowhere, not even in any text outside "De Interpretatione" ascribed truth to molecular propositions such as 'If p, then q', 'p or q', or 'p or non-q'. He could not, therefore, say that 'p or non-p' (the law of excluded middle) might be true even in the case when p is a statement about a future contingent fact such that p cannot meaningfully be regarded to be true or false. Therefore, the interpretation that makes Aristotle accept the law of excluded middle in all cases, but reject the law of bivalence for statements concerning future contingent events is not compatible with what Aristotle says. Another nonstandard interpretation that has been put forward, for instance, by Miss Anscombe, is based on another hypothesis. According to Miss ANSCOMBE (1956) Aristotle wanted to state a difference between the logical necessity of the truth of a statement, the factual" necessity of the truth of a statement and, finally, the simple truth or falsity of propositions without any necessity attached to them. That there will be- a sea battle tomorrow or that there will not be a sea battle tomorrow, is, according to this interpretation, 'necessarily true' for Aristotle. In the case of statements concerning past or present events, their truth has what might be called a factual necessity, while in the case of statements about future contingent events no such necessity, neither logical nor factual, can be admitted. Such statements are just true or false without any modal functor (as necessity) attached to them. This interpretation, however, and some others discussed by modern authors on similar lines, are shown to be incompatible with Aristotle's text by the observation, also due to D. Frege, that Aristotle nowhere in his writings introduces modal functors to de-

923

ORIGINS OF MANY-VALUED LOGIC

termine the predicates of truth or falsity. Aristotle of course sometimes says that it is necessary that if the premises of a syllogism are true, the conclusion must be true also. But he never uses expressions like 'necessarily true' or 'necessarily false' in his writings. J. HINTIKKA (1957 and 1964) has put forward the view that in Aristotle to be always true and to be necessary are equivalent predicates. And if you accept this equivalence, you may find in Chapter 9 of "De Interpretatione" some phrases which may seem to show that Aristotle has actually modalized truth or falsity. He says for instance (18 b 10): "

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rl

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CJ.l(l"'t'e: ,xe:L OCI\1)ve:c; 'IV e:L7t'e:LV O't'LOUV 't'UlV ye:VOfLe:VUlV O't'L e:(l"'t'OCL

"Hence it was always true to say of whatever happened that it would happen". But this phrase should be taken in the sense that "At all times the situation was such that it was true to say that something which has happened was going to happen", the temporal functor modifying the simple fact that truth applies to a statement, not truth itself. The difference may become more obvious by some other example: We may assert that someone is always healthy. This does not mean that he possesses what might be called eternal health. Only the latter interpretation has the overtones of immutability and inevitability which authors like Hintikka have read into Aristotle's rather harmless expression. In Aristotle we find neither an established usage of the predicates 'true' and 'false' for propositions other than categorical ones, nor do we find an established practice of applying modal functors to the predicates of truth and falsity. What we have in Aristotle is what has been well characterized by ACKRILL (1963, p. 140) as a "rather crude realistic correspondence theory of truth". Wherever there is no fact yet, there can be no true proposition about it. And we find in Aristotle a (mistaken) notion that, if propositions concerning future events that seem contingent would be true or false even before that event, deliberation would be useless and a rigid determinism would be the consequence, ruling out the possibility that some future events, especially those that depend on human decisions, are in fact contingent. So we find the interpretation best in tune with Aristotle's text (and not just with some single sentences or phrases taken out of context) is what might be called the 'traditional' interpretation. This traditional interpretation we find (roughly) in the ancient commentators Boethius and Ammonios: According to this interpretation, Aristotle expressed the view that for all statements concerning future contingent events we have the rule that either the proposition or its negation must be true, but that it is not settled be-

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forehand which will be the true one and which will be the false one. This view seems odd and is actually logically highly questionable: In what way could we justify an assertion that a proposition of the form 'p or non-p' may be true, if we are not prepared to accept truth or falsity of at least one of its subordinate propositions? What Aristotle may have intended to say would be just this: We can even now be sure that in all cases of future contingents either p or non-p will be true; but we cannot be sure which of the two propositions is true and which is false. I think this view of Aristotle's is acceptable, but not acceptable is the way he put it, namely, that p or non-p is true even before the event, but that neither of its components is true. After stating, basing our interpretation On the investigations of D. Frede, what Aristotle in Chapter 9 of "De Interpretatione" must have had in mind, we now turn to Lukasiewicz's arguments, which may be considered apart from their relation to Aristotle's theory, even if they seem to be influenced by it. LUKASIEWICZ (1922) has developed his argument in a paper "On determinism" which is based on his inaugural speech as Rector at the University of Warsaw. A final version of this paper was drawn up by Lukasiewicz in 1946. At that time, he said expressly that he was still in agreement with the line of argument in this paper. Lukasiewicz first discusses the following example: John met Paul yesterday at noon on the Old Town Market in Warsaw. This event, having taken place yesterday, is not actually existent today. But it has become, by its effects, part of the reality which we must reckon with. Therefore we may say "It is true today that John met Paul yesterday at noon on the Old Town Market". Generally, we may say: "If an object A has the property b at time t, it is true at any moment later than t that A had the property b at time t", The question arises: Has it been true at every time earlier than yesterday noon, too, that John would meet Paul on the Old Town Market yesterday at noon? Is everything true today that has happened once or will happen in some future time? A determinist would answer 'yes', an indeterminist 'no'. Determinism, according to Lukasiewicz, may be defined as the conviction that the following principle is valid: "If an object A has the property b at time t, it is true at any time earlier than t that A has the property b at time t". Lukasiewicz says: "Whoever accepts this view, cannot think the future to be fundamentally different from the past. If everything which will happen some time, and at some time will be true, is true already today and has

ORIGINS OF MANY-VALUED LOGIC

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been true from eternity, the future must be as fixed as the past is and can be different only in this, that it has not yet taken place." For determinism, according to Lukasiewicz, two main arguments may be adduced: The one is taken from logic, the other from ontology. The logical argument is derived from the 'Law of excluded middle'; the ontological argument for determinism is connected with what might be called the 'Principle of causality'. We begin with the logical argument: The law of excluded middle says that of two contradictory propositions one must be true. To formulate the logical argument for determinism, Lukasiewicz introduces the following premises: (a) It is either true at time t, that John will be at home tomorrow at noon, or it is true at time t, that John will not be at home tomorrow at noon. This premise seems plausible. A second premise, also plausible, is the following: (b) If it is true at time t that John will not be at home tomorrow at noon, then John will not be at home tomorrow at noon. The second premise (b) has the form of a conditional 'If p, then non-q'. If we have 'If p, then non-q', we have also (by contraposition) 'If q then non-p'. We can, therefore, derive from (b) the sentence: (c) If John will be at home tomorrow at noon, then it is not true at time t, that John will not be at home tomorrow at noon. The first premise (a) has the form of an alternation 'r or p'. Now 'r or p' implies, according to the logic of propositions 'If not-p, then r', So we may derive from (a) the new proposition (d) If it is not true at time t, that John will be not at home tomorrow at noon, it is true at time t that John will be at home tomorrow at noon. The propositions (c) and (d) are both conditionals, the consequence of (c) being identical with the antecedent of (d). Therefore, by the law of the hypothetical syllogism, we may derive from (c) and (d) the proposition (e) If John will be at home tomorrow at noon, it is true at time t that John will be at home tomorrow at noon. Since t is a free variable ranging over moments of time, we may replace it by the name of any moment of time we wish. Therefore we may say that, if John will in fact be at home tomorrow at noon, it is true at any time t that John will be at home tomorrow at noon. We have shown, then, that the principle of determinism is valid. If an object A has the property b at time t', it is true, at any time t you please, that A has the property b at time t',

926

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The second argument, derived from the 'Principle of causality' might be formulated, according to Lukasiewicz, in the following manner: An event F is cause of an event G, if F precedes G and if F and G are so connected that if we know that F occurs, we may infer by accepted laws that G will occur also. The causal relation is taken by Lukasiewicz to be transitive, i.e., if F is cause of G and G is cause of H, F will also be cause of H. This is obviously a very special way to define the causal relation. According to our everyday use of 'cause', the causal relation is not transitive: If F is cause of G and G cause of H, F is not necessarily cause of H too. If we accept Lukasiewicz's transitivity condition for the causal relation, we cannot of course maintain that there is, for every event, just one cause of it. If we keep to the one-many relation of causes and effects, we cannot, consistently, say that the relation is transitive. Lukasiewicz takes the first course. (Note that in the English translation of his paper we find in these places, pp. 27, 28, the expression "Fis the cause of Goo. The Polish language not having a definite article, this translation would be grammatically defensible, but is philosophically certainly not correct.) Since for every event G we have, according to the principle of causality, another event that is causing it, the series of causes for every G must be infinite. If it is true today that John will be at home tomorrow at noon, causes for this event exist even now and in every moment of time that precedes tomorrow noon. Therefore, it is settled and determined today and has been always settled and determined that John will be at home tomorrow at noon. In order to avoid this conclusion, which he abhors, Lukasiewicz develops the following idea: We may accept the principle that for every event there is an event preceding it and causing it. But this does not necessarily imply that all strings of causes must reach back indefinitely into the past. Since time is a continuum, we may have an infinite number of events in a limited stretch of time. The string of causes may have no first member, but there may be a moment in which it has not yet started. The string of causes leading up to John's staying at home tomorrow at noon need not begin earlier than, say, tomorrow morning. There are events the causal strings of which reach far into the past, as is the case, probably, with the movement of stars and planets and the development of our galaxy. On the other hand there may be no cause yet for the fact that on the 1st of June 1974 in the garden of my neighbor a fly which does not exist today will be landing on the nose of a baby the parents of which do not even know

ORIGINS OF MANY-VALUED LOGIC

927

each other today. Therefore, an event like this is not yet determined; it mayor may not happen. The argument for determinism derived from the law of excluded middle, is, as Lukasiewicz thinks, dependent on the law of causality. The proposition: "It is true now that John will be at home tomorrow at noon" cannot have a definite import if there is at present no cause at all from which it might be inferred that this proposition is true. If there were a cause already, this would be the "objective correlate" to the proposition that it is true now that John will be at home tomorrow at noon. The same would hold of the negation: the proposition "It is true now, that John will not be at home tomorrow at noon" presupposes that there are causes which guarantee John's absence from home tomorrow at noon. The two propositions "It is true now, that John will be at home tomorrow at noon" and "It is true now that John will not be at home tomorrow at noon" may both be false, since they are not contradictory propositions. Only the proposition: "It is not true now, that John will be at home tomorrow at noon" would be incompatible with and contradictory to our first proposition. Therefore, the principle of determinism must be invalid, since it postulates that, if G happens at time t, it must be true at any time before t that G will happen in t. With this way out, however, Lukasiewicz is not yet satisfied. He thinks it does not fit in well with our everyday convictions to say that the two propositions "It is true now, that John will be at home tomorrow at noon" and "It is true now, that John will not be at home tomorrow at noon" may both be false in the case that no causes exist at present for the one event or the other. We cannot maintain that a proposition is false, if we have no positive reasons which may be adduced against the statement expressed by the proposition. It is for getting out of this uncomfortable situation that Lukasiewicz suggests his revolution of logic consisting in the introduction of additional truth-values besides the True and the False. If we do not find it acceptable to say that such statements are false, there must be other truth-values besides the True and the False; at least we must have a third value, which would apply to such propositions that are concerned with contingent events in the future. By introducing this third value, we reject the principle of bivalence. Lukasiewicz asserts, as we have seen already, that this is also the answer Aristotle found to the deterministic argument. As we have shown, Lukasiewicz's interpretation of Aristotle's text cannot be accepted. Moreover, it is obvious that Aris-

928

G. PATZIG

totle says: a proposition that refers to a future contingent event and forms a disjunction 'p or non-p' with another proposition has no fixed truthvalue before the time at which the event in question is bound to happen (or not to happen). Aristotle does not say, and never meant to say, that such propositions have a third value, which is different from the true or the false. He asserted only that it is not yet determined which of the two truth-values such a proposition might have. But this is another point of Aristotelian exegesis, and we may continue, after clearing up this point of history, with Lukasiewicz's argument. If we introduce a third value, which we may call the 'neutral', the principle of determinism: "If an object A has the property b at time t', it is true at any time t before t', that the object has the property b at time t'" must be regarded as a conditional, the consequent of which has the value "neutral" in all cases in which we cannot produce a cause from which to infer that the event is going to happen at time t' or going not to happen at time t'. A conditional with a true antecedent and a neutral consequent cannot be accepted as true. The consequent of a true conditional with a true antecedent must be true. Therefore the principle of determinism must itself be regarded as neutral in truth-value. Lukasiewicz concludes that the two main arguments for determinism have been refuted. This does not show that determinism is false, but shows that we may assert that the arguments for determinism are not stronger than those that may be brought up in support of indeterminism. Therefore we are free to accept indeterminism. But is it really necessary to adopt "many-valued logic in order to evade the unpleasant consequences of determinism? Aristotle sacrificed the general validity of the "Tertium non datur", Lukasiewicz wanted to keep it but sacrificed the law of bivalence. Both, however, were sharing a certain presupposition which I feel is at the root of their troubles and which I think is entirely mistaken: They thought there must be an 'objective correlate' of a proposition in order to make it possible to call that proposition true. If the event itself is not there to play the role of this 'objective correlate', as Aristotle required, at least a cause of this event, according to Lukasiewicz, must take its place. But if there are no causes yet, the proposition stating that the event in question will happen could never be true. According to Lukasiewicz, the principle of determinism might be expressed by the proposition: "If p is a fact at time t', then it is true at any time t earlier than t' that p holds in t'", But this proposition does not, I submit, express the thesis of determinism. To define determinism, we

ORIGINS OF

MANY-VALUED LOGIC

929

must not only state that, for every fact p at time t', it is true before t' that p at t', but also that it can be known at any time t before t', that p at t', The relation between a proposition and the state of affairs it asserts is a logical relation. The necessity of the state of affairs obtaining if the proposition is true is a logical necessity, no causal or factual necessity, If I shall go swimming in Gottingen next weekend, the statement made by me now: "Next weekend I shall go swimming in Gottingen" is, necessarily, true. But nothing can keep me from changing my plans, and there may anyway be events which prevent me from doing what I had planned. Some people, who know me well, may even say they know how I shall vote at the next election. But this does not imply the absurd thesis that these elections will not be free as far as I am concerned. To 'know', in this sense, how some person will act, is just the kind of knowledge we may have of free decisions and actions of other people, or, for that matter, of our own future actions. This knowledge is always open to revision and does not imply any necessity that would limit our choice of actions. One may be an indeterminist, as Aristotle and Lukasiewicz have been (and I myself feel inclined to be), and one can at the same time accept, consistently, the "Tertium non datur" and the principle of bivalence even for future contingents. We may have other good reasons to prefer a manyvalued logic to the classical system, but fear of determinism should not be one of them. References ARISTOTLE, De Interpretatione 9, 18 a 28-19 b 4 ACKRILL, J. L., 1963, Aristotle's categories and De Interpretatione, translated with notes (Clarendon Aristotle Series, Clarendon Press, Oxford) ANSCOMBE, G. E. M., 1956, Aristotle and the sea-battle, Mind, vol. 65, pp. 1-15 FREDE, D., 1970, Aristoteles und die Seeschlacht. Das Problem der Contingentia Futura in De Interpretatione 9, Hypomnemata, Heft 21 (Vandenhoeck & Ruprecht, Gottingen) HINTlKKA, J. K., 1957, Necessity, universality and time in Aristotle, Ajatus, vol. 20, pp. 522-531 HINTlKKA, J. K., 1964, The once and future sea-fight, Philosophical Review, vol. 64, pp. 461-492 LUICASlEWICZ, J., 1922, 0 determinizmie, in: J. Lukasiewicz, Z zagadnien logiki i filozofii, ed. J. Siupecki (Warsaw, 1961). English translation by Z. Jordan, On Determinism, in: Polish Logic 1920-1939, ed. Storrs McCall (Clarendon Press, Oxford 1967), pp. 19-39

THE APPROXIMATIVE EXPLANATION AND THE DEVELOPMENT OF PHYSICS

E. SCHEIBE University of Gilttingen, German Federal Republic

In this lecture I shall be interested in one particular problem within the field of intertheory relations in physics: the problem of the possibility of approximative explanations. The concept of an approximative explanation is occasionally dealt with in the literature on the foundations of the empirical sciences, especially physics, of the last ten or fifteen years. In the papers I have in mind you can find it exemplified verbally by alluding to cases in which one physical theory is a limiting case of another such theory, as for instance Kepler's theory with respect to Newton's theory of gravitation, or the latter with respect to Einstein's, or beam optics with respect to wave optics, or classical mechanics with respect to quantum mechanics or what not. To the best of my knowledge there is, however, no thoroughgoing and rigorous treatment of the matter. By now we neither have any clear general concept of an approximative explanation, comparable, e.g., with the concept of a deductive-nomological (D-N-) explanation, nor do we have any detailed case study in this field.' The principal aim of my lecture is to give such a case study and to argue that the elaboration of a general concept of an approximative explanation may be of some help for our understanding the development of physics." My case study is devoted to the relation between Newton's law of gravitation and Kepler's laws on the planetary motions. Regarding this relation there ought to be mentioned first what may be called the physical folk view on the matter. According to this view Kepler's laws can somehow be explained by Newton's law whereas the latter cannot be explaine2 , 1 Refening to intertheory relations in general this situation has been deplored by BUNGE (1970) three years ago. But the situation has not changed since then. 2 Concerning the philosophical background of my approach I may refer to SCHEIBE (1971).

932

E. SCHEIBE

by the former. Thus this view would take the relation (a) as an explanation, and (b) as a one-sided explanation, expressing by this asymmetry the superiority of Newton's law over Kepler's laws. Beside the physical folk view you can find in the literature about the foundations of physics occasional remarks and even deeper considerations on the relation in question. A case in point is DUREM'S (1954, p. 193) well-known analysis which he summarized in the statement: "The principle of universal gravity, very far from being derivable by generalization and induction from the observational laws of Kepler, formally contradicts these laws. If Newton's theory is correct, Kepler's laws are necessarily false." Duhem's analysis was not intended to elucidate the explanatory relation between Newton's and Kepler's theories. But the quoted result has influenced later work going in just this direction. Trying to exemplify the concept of a D-Nexplanation by historically interesting cases the Newton-Kepler case could not escape notice. In papers written during the last fifteen years several authors, among them Popper, Nagel, Feyerabend and Hempel, have pointed out that Kepler's theory cannot be deductively explained by Newton's. Some of these authors explicitly refer to Duhem's incompatibility thesis and get their result as an immediate consequence of it. In view of the fact that Kepler's laws, being strictly refuted by Newton's law, nevertheless do hold to some degree of approximation, Hempel has offered his conception of approximative deductive explanation to get Kepler's laws at least approximately explained by Newton's theory. A similar view seems to be taken by Popper, whereas Feyerabend has refused this way out. For all this see FEYERABEND (1963, p. 13; 1965, pp. 228-230), HEMPEL (1962, pp. 101, 108; 1965, p. 344), NAGEL (1961, pp. 57-58), POPPER (1949, p. 57; 1957-58, pp. 28-30). The principal aim of my own analysis is to show that approximative explanation is possible in the case before us. As a by-product we shall see that all the views just mentioned have to be corrected in this or that respect. Preparing the field, I shall have to assimilate the two theories to make them comparable in a convenient way, thereby cutting down Newton's and blowing up Kepler's. Second, I shall show the existence of exact D-N-explanations which, however, cover only a negligible small part of the field. This will lead us in a final part to the approximative explanations. Beginning my preparation and assuming the usual formulations of Newton's gravitational theory and Kepler's laws to be known, I am anxious to remark that at first sight it is not quite clear how a comparison, and especially a purely logical comparison, of the two theories could be per-

APPROXIMATIVE EXPLANATION

933

formed. In order to refute, e.g., the statement that Kepler's laws are logical consequences of Newton's theory one must have a precise formulation of this statement. To achieve this it is first of all necessary to have a common basis of comparison for the two theories. Now, so much is clear from the outset that both theories are theories about the motions of bodies. Therefore, as the common basis of comparison we are looking for, a most general kinematical theory suggests itself. Let us try to take for such a theory the theory of space and time which is characterized by the notion of Galilean frame of reference. Idealizing the position of a body as a point ~e can represent the motions in a system of N bodies by N vector functions 1k with suitable assumptions of differentiability. I shall speak of these functions as the possible spatiotemporal state descriptions of the system in question and of the set of all these functions-N given-as the state space J of the system. Having thus established the common basis of comparison for the two theories, we now have to consider these theories separately. As regards Newton's theory, for a comparison with Kepler's theory it is convenient to eliminate that part of it which consists of the general dynamics and with it the concept of force. We are then left with the gravitational masses mk of the bodies of our system which together with the spatiotemporal state description 1k make up the Newtonian state description of the system if the mk and the 1k obey Newton's gravitational equations'

a

(1)

Newton's equations are invariant under all Galileo-transformations and therefore can have the aforementioned general kinematical theory as their basis. The equations are furthermore of a deterministic character. I shall refer to a system of bodies represented in the way just described as a Newtonian system. My next aim is to get a corresponding concept of a Keplerian system which has the same theory of space and time as its kinematical background. To this end one has first of all to disregard the fact that Kepler's laws speak of the solar system. Second one has to gain a Galileo-invariant formulation of these laws. In their usual formulation the laws are not invariant under the special Galileo-transformations, the reason being that 3 In formula (1) the gravitational constant is suppressed because this constant is inessential as long as one considers only one type of force.

934

E. SCHEIBE

the sun is assumed to be at rest. A formulation of Kepler's laws which is invariant under the full Galileo group is given by

Vk =

VI = 0 -,u('l'k-'l'I)I'l'k-'l'11-3

21 I''l'k-'l'1. 12 -,u 1'l'k-'l'1 1- 1

< 0

(2 (2

~ k ~ N) 1 ~

k

~

(2)

N)

with the positive constant ,u and the 'l'k making up a Keplerian state description of a system of bodies. Though one of the formulas (2) is an inequality I shall call them the Kepler equations and the constant ,u the Kepler constant of the system. In the subclass of Galilean reference frames in which the body distinguished by the first Kepler equation is at rest the Kepler equations turn out to be logically equivalent to the three usual Kepler laws, the Kepler constant appearing, apart from a numerical factor, as the well-known ratio mentioned in Kepler's third law (BO'\N, 1949, pp. 129-133). Like the Newton equations the Kepler equations are deterministic. Thus we have in a rather strict formal analogy to the concept of a Newtonian system the concept of a Kepler system being a system of bodies represented by its Kepler constant and its spatiotemporal state description obeying the Kepler equations. Having at hand these two concepts we were already in the position to begin our comparison if we would subscribe to an instrumentalistic view of the cognitive status of physical theories. For in this case we would have to wait only for this or that real system of bodies to which both concepts could be applied. If, however, we prefer a descriptive view of physical theories, we would still have to formulate two empirical assertions, one for each of the two theories. Using the previously defined concepts, we would have to say in the Newtonian case that the system of all bodies in the universe is a Newtonian system and in the Keplerian case that the solar system is a Keplerian system. I shall now perform the comparison for the two views in turn, first the purely logical comparison and afterwards the approximative one. According to the first view we shall have to consider one system of N bodies and its state space f. Now the masses and the Kepler constant appearing in the description of a Newtonian and a Keplerian system, respectively, are one last obstacle to our comparison which, however, can easily be removed by simply taking the set-theoretical projections of the respective sets of full state descriptions into the space f of the

APPROXIMATIVE EXPLANATION

935

purely spatiotemporal state descriptions. We thus get two subsets of f as expressed by my formula (3)

the Kepler set and the Newton set, as I shall call them. Everything now depends upon the question of how these two sets are related to each other. My first thesis about this relation is expressed by formula fk;J n fk~1 'I: 0 (4) in which the sign 1.. appearing in the brackets signifies the set-theoretical complement in f, the brackets themselves indicating that this sign may occur or not occur. Thus in (4) we really have four formulas which, taken together, express the statement that the concepts of a Newtonian and a Keplerian system in application to one and the same system are entirely logically independent. In particular the formulas with only one sign for the complement occurring express the statements that these concepts are not logical consequences of each other. The formula with no complement sign in it expresses the statement that the concepts likewise do not contradict each other, this statement being true only for systems with more than two bodies. The proof consists merely in showing that there are possible state descriptions in f which can be completed by masses to make up solutions of Newton's equations as well as by a Kepler constant to make up solutions of Kepler's equations. Perhaps the simplest examples of such systems are sorts of merry-go-round systems. In a suitable Galilean reference frame they can be described as follows: N -1 bodies arranged in a regular polygon rotate with constant angular velocity on a circle with the N-th body at rest at the center of the circle. Choosing suitable masses and Kepler constant such a system can prove to be a Newtonian and a Keplerian system at the same time (SCHEIBE, 1973). The compatibility of Newton's and Kepler's theories has consequences with regard to the possibility of applying the concept of an exact D-Nexplanation to them. Since the theories are not logical consequences of each other, an unconditional D-N-explanation of one by the other is, of course, out of the question. But you can very well have conditional D-N-explanations in both directions as symbolized in formula

f{ Kep New IIAdd} IIAdd'

f;,

f{Add 1 Add" f{NeWAAdd} Kep x Add'

f{ Kep I New

p} $ f{Ke New

'I: 0

(5)

936

E. SCHEIBE

As the additional premises Add and Add' in this formula one can take any set of initial conditions which together with Newton's or Kepler's equations lead up to a unique set of solutions of Kepler's or Newton's equations, respectively. Apart from the shown compatibility which guarantees the existence of such conditions the procedure is made possible essentially by the already mentioned deterministic character of both sets of equations. Thus we have seen that Duhem's incompatibility thesis and its consequence of the nonexistence of D-N-explanations in the Kepler-Newton field is false if it is not relativized to this or that range of application, in which case however you would leave a pure comparison between the theories in question. On the other hand, the demonstrated possibility ofD-Nexplanations is of rather limited importance for at least three reasons. First, it makes the explanatory relation of the two theories appear as entirely symmetrical, contrary to our intuition of an existing asymmetry. Second, the individual ranges of application of the explanations symbolized in formula (5) are obviously confined to the intersection of the Kepler set with the Newton set. And though this set is not empty, it is of a very limited extent. Third, in the context of the instrumentalistic view of our two theories, which is still our epistemological background, the investigation of exact as opposed to approximative relations is of a somewhat academical nature. For even on this view you can hardly escape the argument that as long as there is still one body outside the system you are considering, the application of Newton's equations to that system can only be approximative. Let me therefore turn quickly to the descriptive view of theories. There we have to consider one big system of N bodies, in fact the system of all bodies in the universe, as a Newtonian system and a small subsystem of M bodies, in the original case the solar system, as a Keplerian system:. On lines similar to the previously treated case one can prove the very same results: complete logical independence of the two concepts of the big Newton system and the small Kepler system, in particular their mutual compatibility, i.e., the existence of Newton systems containing Kepler subsystems, and, as a consequence thereof, the existence of exact D-Nexplanations of the Kepler theory concerning the small system by Newton's theory applied to the big system (SCHEIBE, forthcoming). Of course, this time there are no such explanations in the opposite direction. Nevertheless we have still not reached a genuine asymmetry between the two theories. For the assumption that the Kepler system is a subsystem of the Newton

APPROXIMATIVE EXPLANATION

937

system is completely arbitrary on purely logical grounds and can very well be reversed, however absurd this may appear from an empirical point of view. And if you reverse it, then you get mutatis mutandis the same results for the third time: logical independence, compatibility and D-Nexplanations (SCHEIBE, forthcoming). But let us get back to the empirically reasonable case of the big system being Newtonian. Apart from too small ranges of applicability for the existing D-N-explanations I want to mention another feature of its inapplicability. Though there is, from the standpoint of an objective descriptivism, no ontological argument against the assumption that the system of all bodies in the world is exactly, not only approximately, a Newtonian system, there is an epistemological argument against this assumption. As I have not been able to prove but am convinced to be provable, the following theorem holds: no subsystem of a Newtonian system is itself a Newtonian system. Now we are obviously and on grounds of principle not in the position to establish the whole universe to be a Newtonian system even if it should really be such a system. Rather we have to confine ourselves to pretty small systems in that matter. But these subsystems according to the theorem just mentioned are not Newtonian systems in the very case that the whole system is Newtonian. Thus we are faced with the situation that, provided the whole universe is a Newtonian system, we cannot establish it as such and that the systems for which we could establish the Newtonian character simply do not have it. Of course, they can have it approximately. So let us turn finally to this case. Perhaps the most important thing to have in mind if one is about to make precise the concept of an approximative explanation is that one needs a topology, for a topology is the very mathematical tool to settle all matters in which our intuition works with the idea of an approximation. Thus every individual explanation of the desired sort will depend on a topology, and this topology has to be mentioned explicitly. In the case before us it is obviously the state space J which has to be equipped with a suitable topology. In my formula

(6) I have indicated a complete system of neighborhoods for the topology I want to choose. Intuitively it corresponds to the idea of an approximation of one system of orbits in an N-body-system by another such system of orbits for all time. The case of one big system and one small subsystem of it is easily reducible to the case in which the concepts of a Newton system

938

E. SCHEillE

and a Kepler system are applied to one and the same system, so I take up only the latter. In this case the following theorem, expressed by formula J Kep s; J New

«.: $

(7)

.9Kep

is likely to hold: For any given N the Kepler set is contained in the closure of the Newton set, whereas at the same time, as we already know, it is not contained in the Newton set itself. On the other side, the Newton set is just as little contained in the closure of the Kepler set as it is contained in this set itself. The supposed result expressed by formula (7) brings out an obvious asymmetry between Newton's and Kepler's theories. Does it already solve the problem to gain a unidirectional explanation of the latter theory by the former? Unfortunately it does not. Strangely enough, the first line of formulas (7) rather provides us with an approximative D-Nexplanation in the opposite direction, as can perhaps better be read from the next formula .9Kep s; (.9 New). (8) though it is logically equivalent to the first line of formula (7). Here the set on the right side is the set of all state descriptions of .9 which are in an s-neighborhood of a state description belonging to the Newtonian set.9New' Now let me oppose to this explanation the explanatory possibilities in the direction we want to have, i.e., Newton's theory explaining Kepler's. This is done in formula .9New AAddA Add.

S;

(.9Kep).

J

.9AddAAdd. $ (.9Kep)• .

(9)

.9NewAAddAAdd. =F 0 Here Add and Add. are suitable relations between masses mk and state descriptions Yk functioning as premises additional to the conditions which define a Newtonian system. The .9-sets on the left side are the usual projections into o", The .9-set on the right side is the set of all state descriptions of.9 which are within an s-neighborhood of an element of the Kepler set .9Kep' Taking for granted for the moment the existence of additional premises as they appear in formula (9) we see that there exist approximative D-N-explanations between Kepler's and Newton's theories in both directions. There is, however, an essential asymmetry between them concerning their respective ranges of application. The explanation of Newton's theory

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by Kepler's theory leaves out the overwhelming majority of approximative Newton cases that can arise as possible explananda, the reason being that the overwhelming majority of s-neighborhoods of Newton cases will contain no Kepler cases and that, therefore, you cannot start your explanation using formula (8). On the other hand, the very same formula tells us that, given any Kepler case together with an s-neighborhood around it, there exists a Newton case within this neighborhood. We can therefore be pretty sure to find the desired additional premises such that (a) the Newton case can be subsumed under them and (b) the originally given Kepler case can be explained according to formula (9). In the case of two bodies it can even be proved that the additional premises can be chosen universally. At any rate the totality of all possible explanations of the form (9) will explain approximately every Kepler case whereas the explanation (8) running in the opposite direction will fail to give explanations of most of its virtual explananda. Therein lies the asymmetry between Newton's and Kepler's theories if seen in the light of their mutual explanatory possibilities. I now leave my case study to conclude with a general remark. Up to this point I hope to have shown in a particular, admittedly simple, case how the idea of an approximative explanation can be elaborated. What can be said in favor of that idea when we broaden our view and try to overlook the whole of physics systematically as well as historically? In answer to this question I would first of all point out the abundance of cases distributed all over physics to which the concept of an approximative explanation may be applied, or-to express it more carefully-from which we could learn by a thoroughgoing analysis to form such a concept in a precise and general manner for the first time. Let me pick out only one other type of case at random: the quantum-theoretical treatment of the hydrogen atom. Within quantum theory you can treat the hydrogen atom according to the composition of several alternatives: quantizing the Coulomb field or not, relativistically or not, taking into account the spin or not, viewing the thing as a two-particle system or as a one-particle system. Though these alternatives are not independent of one another, you get a good many cases of possible treatments out of them, all concerning one and the same object and all being quantum-theoretical in nature. What are the relations in which these several theories of the hydrogen atom stand to each other? Taking them pairwise and leaving aside those pairs in which both members contain improvements with respect to the other, the remaining pairs will presumably consist of members one of which can be approximately explained by the other. Consider, e.g., the most elementary treatment of the

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hydrogen atom using the Schrodinger equation for the electron in an external Coulomb field, and compare it with the two-particle treatment taking into account the motion of the proton. In the usual textbooks on quantum mechanics this comparison is performed up to the point where you are told that the energy spectrum resulting from the two-particle treatment approaches the energy spectrum resulting from the one-particle treatment if the mass of the proton approaches infinity. This result is surely part of an approximative explanation and even the most important part if viewed from an experimental standpoint. But regarding the claim to understand the entire one-particle theory as an approximation to the two-particle theory, this cannot be the whole story. To fulfill this claim one has to compare the two state spaces of the theories on lines similar to the given treatment of the Kepler-Newton case. Because of the rather formidable mathematical difficulties to be encountered here I have simulated the problem on the somewhat lower level of the harmonic oscillator. Here I tried to understand as my explanandum at least the probability distributions of position and momentum in the energy states of the one-particle system from the standpoint of the two-particle system. It turned out that these distributions are approximations from two-particle states provided you use a sufficiently weak topology allowing approximations already for finite times. The premises additional to the two-particle equation are then not only a sufficiently high value of the mass of the subsidiary particle, but also a sufficiently well-defined position of the center of gravity of the two-particle system, the momentum expectation value of the center being, of course, zero. From the calculations I got the impression that this result can be extended to yield an overall explanation of the desired kind (SCHEIBE, forthcoming). Thus we would have one more example for a positive answer to the problem of approximative explanations belonging to quite another field of present-day physics. But the matter is interesting also from the historical point of view. If we overlook the development of physics since the time of Kepler and Galileo, the most remarkable fact about this development is the progress physics has made. Believing in some continuity of this progress, I would say that at least one feature of it can be understood by applying the idea of an approximative explanation, viz., the feature that physical theories have relieved other theories about the same type of objects without, however, the older theories being given up definitely. One way to look at this process is to say that the old theory has been falsified empirically and that the falsifying phenomenon can be explained by the new theory. Another way to put it would be to say that the old theory has been recognized

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to have only a limited range of application, the range having been extended by the new theory. Now, in whatever way you want to have this, at any rate the old theory may have been corroborated very well, and this is reason enough to keep it up even after its shortcomings became known. The other day I was told that a star, a red dwarf of about 0.6 sun masses and rather near to the solar system, has been observed to have a trajectory deviating remarkably from a straight line. The data were put into a computer, and it turned out that the star must have either one dark satellite with about 1.8 Jupiter masses or two of them with 0.9 Jupiter masses. I am pretty sure that the computer had been informed about Newton's theory to do his job and not about Einstein's. But if we keep up our theories for further use, we must know the limits of their applicability. In the example just given it would probably have been worthless to tell the computer something about Kepler. How do we know this? We know it by an investigation of the limits of Kepler's theory in the light of Newton's theory. Were Kepler's theory applicable in the present case, the masses of the supposed satellites would have to be so small that they could not bring about the observed effect. In this way approximative explanations function to tell us to what extent our old theories remain applicable, and this seems to me to be an important function. Advocating approximative explanation, I neither mean to say that it works in every case nor that, if it works in some case or other, it thereby is the whole story in our endeavor to clear up the development of physics. At any rate I would maintain that it is an important complement to the purely deductive pattern of explanation. For the latter is scarcely applicable to any historically interesting case of two successive theories. Elaborating the idea of an approximative explanation, it may turn out that its application is confined to the progress within normal science as opposed to the scientific revolutions, to use the terminology of THOMAS KUHN (1962). But even here I am inclined to be optimistic. Of course, the role played by an approximative explanation in our understanding of a scientific revolution in physics will be relatively small. But I believe it can be part of such an understanding. Especially, I do not think the appearance of meaning variance to be an unsurmountable obstacle to approximative explanation. After all such an explanation amounts to a comparison of numbers. Thus, the essential requirement to be fulfilled to make it possible is to know what numbers are to be compared with each other. You have to correlate the quantities of the explanandum theory to suitable quantities of the explaining theory. The correlation may be looked at as being purely external in nature

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or as being a reinterpretation or whatever you want it to come out. As soon as the correlation is established it opens the way for quantitative comparisons and eventually for an approximative explanation. And as long as it is not established one simply does not know what the two theories have to do with each other. Thus by solving the problem of meaning variance in a concrete case you will be led at least to the possibility of making an approximative explanation part of your understanding of the relation between the two theories in all its semantical complexity. Concluding my lecture, let me say that, in view of my case study, the last remarks are to be taken as mere dreams of the future. References BORN, M., 1949, Natural philosophy ofcause and chance (Oxford University Press, Oxford) BUNGE, M., 1970, Problems concerning intertheory relations, in: Induction, Physics, and Ethics, Proceedings of the 1968 Salzburg Colloquium, eds. P. Weingartner and G. Zecha (Reidel, Dordrecht), pp, 285--315 DUHEM, P., 1954, The aim and structure of physical theory (Princeton University Press, Princeton, N.J.) (English translation from the second edition of the original, published in 1914 resp. 1906) F'EYERABEND, P., 1963, How to be a good empiricist-a plea for tolerance in matters epistemological, in: Philosophy of Science. The Delaware Seminar, vol. 2, 1962-3, ed. B. Baumrin (Interscience Publishers, New York), pp. 3-39 FEYERABEND, P., 1965, Reply to criticism, in: Boston Studies in the Philosophy of Science, vol. 2, eds. R. C. Cohen and M. W. Wartofsky (Humanities Press, New York), pp. 223-261 HEMPEL, C. G., 1962, Deductive-nomological vs. statistical explanation, in: Minnesota Studies in the Philosophy of Science, vol. 3, eds. H. Feigl and G. Maxwell (University of Minnesota Press, Minneapolis), pp. 98-169 HEMPEL, C. G., 1965, Aspects of scientific explanation (The Free Press, New York) KUHN, Th., 1962, The structure of scientific revolutions (The University of Chicago Press, Chicago) NAGEL, E., 1961, The structure of science (Harcourt, New York) POPPER, K. R., 1949, Naturgesetze und theoretische Systeme, in: Gesetz und Wirklichkeit, ed. S. Moser (Innsbruck) POPPER, K. R., 1957-58, Uber die Zielsetzung der Erfahrungswissenschaft; Ratio, vol. 1, pp. 21-31 SCHEIBE, E., 1971, Ein vernachldssigter Aspekt physikalischer Erklarung; I, Naturwissenschaften, vol. 60, pp. 1-6 SCHEIBE, E., Ein vemachliissigter Aspekt physikalischer Erkliirung, II, Naturwissenschaften, vol. 60 (forthcoming) SCHEIBE, E., 1973, Die Erkliirung der Keplerschen Gesetze durch Newtons Gravitationsgesetz, in: Einheit und Vielheit, Festschrift fiir Carl Friedrich von Weizsacker zum 60. Geburtstag, eds. E. Scheibe and G. Siissmann (Vandenhoeck & Ruprecht, Gottingen), pp. 98-118

HEGEL'S CONCEPTION OF 'BEGRIFFSBESTIMMUNG' AND IDS PIDLOSOPHY OF SCIENCE

G. BUCHDAHL University of Cambridge, England

One of the difficulties that confronts the modem philosopher of science when first turning to Hegel is the unusual order in which the familiar topics of his subject are discussed and that there is a considerable displacement by comparison with the conventional approach. When we consider, for instance, his treatment of induction and law, we do not find in Hegel's Logic any extensive discussion of induction and analogy, even though what he says on these topics is perceptive. But more strange, in the sections dealing with these matters there is no suggestion-unlike modem treatments-as to how such a topic might bear on the problem of scientific law. Not that Hegel has not a great deal to say about law, but what he says occurs in an altogether different compartment of his Logic, as for instance in the section which treats of the concept of the phenomenon [Erscheinung], headed 'The law of the phenomenon';' Let us consider this in more detail since it will lead us immediately into the center of Hegel's philosophy. Stripped of some of its technical jargon, Hegel's explication of 'the phenomenon' boils down to the following. In our analysis of this concept we must contrast the merely existing multitude of manifestations of sensory data, constantly changing, arising and disappearing, with what remains unchanging and subsisting-and this is 1 (References to Hegel's works are, first, to the German text, followed in brackets [ I by references to the corresponding translations. R = Remark; Z = Zusatz.) HEGEL (1934, ii, pp. 124-129; [HEGEL, 1969, pp. SOD-50S]). Cf. also HEGEL (1952, Ch. 3, pp. 102-129): "Force and Understanding, Appearance and Intellectual World". And a fuller treatment would also have to refer to the section in HEGEL (1934, ii, Ch. Ie): 'Reflection', especially C. 2: 'External Reflection', pp. 17-18; [HEGEL, 1969, pp. 403-4041, which gives an interesting use of 'reflection' in connection with Kant's 'reflective judgment' in the Critique of Judgment.

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'the law of the phenomenon'. 2 At first-this is of course only a logical first, a degree of abstraction-this law is contrasted by Hegel with the existential aspect of the phenomenon in its changing manifestations. The connection of factors asserted in the law is only posited, he says, warranted merely by experiment and observation, with which it connects inductively or hypothetically. As such, the connection still requires 'proof', and one expects Hegel to mean: proof in the sense of scientific explanation: though, as we shall see, this is not quite what he has in mind in the present instance. Thus, in the important example of Galileo's empirical law of free-fall (displacement, s, is proportional to the square of the time, t), the spatial and temporal variables are, as Hegel puts it, 'indifferent' to one another; there is no conceptual connection between them: "It is not contained [he means: so far] in the notion of the space traversed during fall, that it should correspond to the square of the time". And so the law does not possess any "objective necessity" (HEGEL, 1934, ii, p. 129; [HEGEL, 1969, p. 504]). Perhaps this much will not be strange to modern ears, though it may not be totally acceptable. But Hegel does not stop here, for he now imports into the discussion the vast complex of dialectical articulation of the structure of reality, of what he calls 'the Notion', or again, 'the logical idea'. First of all, whilst he recognizes and allows a place to scientific and mathematical approaches which supply a 'proof' of the above law, S = !gt 2 , we shall presently see that in his Philosophy of Nature he attempts also to provide an alternative set of notional considerations, which relate space and time in such a way that to s there must correspond, or that it is at least intelligibly plausible that there should correspond, the square of the time (HEGEL, 1847, Sec. 267 R, p. 88; [HIGEL, 1970a, p. 58] [HIGfL, 1970b, p. 255]). The quantitative aspect apart, we soon come to realize that the concepts of space and time are so articulated as to necessarily involve one another, each requiring another, whilst yet being nothing without the other. As is well known to students of Hegel, this unstable equilibrium results in an internal motion and the accompanying demand that it should be suspended [aufgehoben] within the next 'higher', usually more complex, categorial level, in order there to find a temporary rest. 2 Note that the association of properties of physical substances has frequently been stated to have the status of a law; cf. the 'Tubal-Cain's laws' in CAMPBELL (1957, p. 44). And of course, Kant had already analyzed the concept of phenomenon as containing an element of lawlikeness. On Kant's concept of phenomenon, as containing a reference to Iawlikeness, cf. BUCHDAHL (1969, especially pp. 625--626, 631--651; 1971, passim).

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Now the assignation of an element of reality, e.g., space, time, motion, law, force, within the dialectical flux of categorial placing, Hegel calls its 'notional determination' [Begriffsbestimmung]. It provides its justification or certification, and contrasts sharply with its position within, say, a scientific body of knowledge, a distinction anticipated in the Kantian division between the possibility and the probability of a hypothesis, or of a corresponding hypothetical concept." To this I shall return. But first we have yet to complete our account of the law, or rather, of lawlikeness in general-of the phenomenon-since we have found Hegel saying that as yet law stands on one side, its parts no more than posited, confronting the 'immediate' existential detail of the phenomenal chara-cters. What is it, we must ask, that supports this lawlikeness as such? Hegel's answer to this is unique and is drawn from the general approach of his dialectic. First, to take our example of the law of free-fall, the two sides of the equation of law, s = !gt 2 , are now asserted-in line with the general dialectical argument just sketched-to mutually imply and presuppose each other, where earlier they had been regarded as externally indifferent. For actually, "each of them [sc. each side] contains its other within it, and at the same time, as a self-subsistent side, repels this its otherness from itself, [so that] the identity of the law is therefore now also a posited and real identity (HEGEL, 1934, ii, p. 131; [HEGEL, 1969, p. 506])". Because of this, therefore, they no longer require theoretical foundation and mediation (Hegel speaks of 'proof and mediation'). And indeed, in the instance of Galileo's law, Hegel holds that no such scientific formal derivation is either required or possible. As for the philosophical mediation, here asserted to have been consummated, this anticipates of course the treatment in the Philosophy of Nature, the details of which are given at the end of this paper when we consider Hegel's 'derivation' of the law of free-fall 'from the Notion' (Hro-t, 1847, Sec. 267R, p. 88; [HrGlL, 1970a, p. 59]; [HEGEL, 1970b, p. 255-256]), with the familiar display of opposing tensions. Assuming this derivation, Hegel then goes on to argue that in consequence the kind of dialectical tension previously postulated as belonging only to the sides of the phenomenon (I have omitted the details of this 3 For the distinction between 'possibility' and 'probability' in Kant's doctrine of hypothesis, cf. BUCHDAHL (1969, pp. 512-514).

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here) obtains also between the sides of the law, and that therefore the two (the law and the immediacy of the phenomenon) are essentially one: "That which previously was law, is hence no longer merely one side of the whole whose other side was phenomenon as such, but it is itself the whole (HEGEL, 1934, ii, p. 131; [HEGEL, 1969, p. 506])". The upshot of all this is supposed to be that it is absurd to contrast sensory reality with its lawlike description, since upon closer inspection they collapse into one another-indeed, according to Hegel must do so, since their internal articulations are alike unstable, eventually leading to what he calls the 'dissolution of the phenomenon' (HEGEL, 1934, ii, p. 134; [HEGEL, 1969, p. 509]), and to the next stage, where the previous contrast between phenomenon and law is articulated as a new trio of contrasts, whole-and-parts, force and its manifestation, and the outer-inner relation; and so on from there. It will by now be clear that Hegel is not concerned here with an inductive foundation of actual laws qua laws, but with the attempt to articulate the concept of lawlikeness, by placing it within the dialectical flux of categorial tensions. Lawlikeness is not so much based on considerations relating to the inductive logician's stock-in-trade, but on the fundamental ideologies connected with the logic of the categories. Indeed, at this point Hegel follows up his argument with an additional and most interesting move. For, so he says, reflection shows that "the kingdom of laws" does not just involve the "simple, changeless but varied content of the existent world ", but it also "contains the moment" of the "immediately given [essenceless] manifoldness (HEGEL, 1934, ii, p. 131; [HEGEL, 1969, p. 507])". In other words, and to say the least, the intellectual [iibersinnliche] world of theoretical science is not in the end merely a theoretical complex, but must come eventually to be seen as being the world (HEGeL, 1934, ii, p. 132; [HEGEL, 1969, p. 507]); a conclusion not unlike that of some modern realist philosophers of science (SELLARS, 1963, p. 126). This completes our discussion of the concept of lawlikeness, leaving only Hegel's account of the particular form of Galileo's law, to which I shall return subsequently. For the moment, we need to consider something raised implicitly by the above discussion: Hegel's account of the theoretical concepts of science. To begin with, some general points. Whilst the concept of notional determination might have seemed strange twenty years ago, the general approach which it embodies can perhaps now be seen to have some interest for philosophy of science. Indeed, one way of

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removing the strangeness of Hegel's procedure is to link this to cognate viewpoints both on the part of Hegel's influential predecessors, as well as to later philosophies which, by having reached similar standpoints independently, can help us pick out what is still meaningful in the Hegelian framework. And if, though the approaches are parallel, the Hegelian realization seems strange and forced, its very strangeness-like a Wittgensteinean 'borderline case'-may help us grasp our own tenets the more clearly. Indeed, this way of proceeding is intended to illustrate a general thesis, implied in this paper, that the importance of this relativist-historicist approach, highlighting different interpretations of 'conceptual' or 'metaphysical foundations', can importantly clarify our whole notion of such 'foundations' in general. That 'metaphysical foundations of science', or basic underlying conceptual schemes, or again, global paradigms, articulating, creating or expressing intelligibility of specific scientific approaches, should be distinguished from normal science, has become a commonplace in recent years (cf. BURTT, 1932, passim; HARRE, 1964, Pt. i; HOLTON, 1964, passim; KUHN, 1962, passim; TOULMIN, 1961, Chs. 3-5; WHEWELL, 1967, vol 1, Bk. 1; vol 2, Bk 11). However, perhaps specific conceptual schemes and paradigms are too concrete; one may instead posit more basic and general intellectual standpoints, more abstract methodological principles. Thus one may show that scientific theory and theorizing have a complex methodological structure involving not only the formation of a system of laws, but also a considerable body of conceptual explication, some pretheoretical, some intratheoretical. Or again, one may distinguish a body of principlesI have elsewhere called them 'architectonic'-which embody a number of regulative ideas and maxims, preferred explanation types, and similar considerations (cf, BUCHDAHL, 1970, esp. table on pp. 226-227). Now it seems to me that conceptual explication-to pick out the most important of the strands just mentioned-is a genus of which Hegelian 'notional determination' is a species or special instance, so that if we understand it in this way, we may perhaps come to grasp more easily something of Hegel's intentions, just as the extreme form of Hegel's approach here may teach us something about the limits of this idea itself. However, these intentions can be made even clearer when seen in the light of certain earlier influences on Hegel, and here clearly the most important one was Kant, whom at some of the crucial junctures of his notional discussions of scientific concepts, in the Science of Logic, the Encyclopedia Logic and Philosophy of Nature, Hegel singles out as having been the first to have "awak-

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ened and inspired (aufgeweckt] the concept of a philosophy of nature (HEGrL, 1847, Sec. 262R, p. 68; [HEGEL, 1970a, p. 45]; [HEGEL, 1970b, p.241])". Let us give some of the context. Hegel, after having dealt with the concepts of space, time and motion, is here considering the concept of matter, and the problem he has inherited from his predecessors, the relation of matter and force: How can matter be conceived as attracting or repelling at a distance? Now interestingly, this discussion has its parallel in the chapter of the two Logics, entitled 'Being-for-self' [Fiirsichsein], discussed as what is involved under the concept of the one-and-the-many. That which is 'for itself' excludes, in the first instance, any 'other', since it is related only to itself. Hence it must be 'the one'. Yet it must also be regarded as having a relation to itself, so that the one at once becomes internally unstable; it so to speak repels itself, and posits itself as 'a many'. It thus becomes clear that repulsion is built into the very concept of the many (HEGEL, 1843, Sec. 97Z, p. 192; [HEGrL, 1892, p. 181]; HEGEL, 1934, i. pp. 156-170; [HEGEL, 1969, pp. 167-178]), although in the Encyclopedia Logic Hegel remarks that he there is using this term only as a 'metaphorical expression' [bildlichen Ausdruck] (HEGEL, 1843, Sec. 97Z, p. 192; [HEGEL, 1892, p. 181]), whereas in the parallel disquisition on matter in Philosophy of Nature, it is interpreted as straight physical repulsion, though still in an 'ideal' sense (HEGEL, 1847, Sec. 262, pp. 67-68; [HEGEL, 1970a, p. 44], [HEGEL, 197Gb, p. 241]). Finally, and per contra, in order to retain the conception of the one, as well as of the many ones, each repelling every other-where the 'each other' now denotes the opposite of repulsion-and thus in order to have the conception of a 'many ones', we also require the opposite of repulsion, i.e., attraction (HEGEL, 1843, Sec. 97Z, p. 192, [HEGEL, 1892, p. 181]; HEGEL, 1934, i, pp. 156-170; [HEGEL, 1969, 167178]). Applied to matter, as Hegel puts it in the Philosophy of Nature, the latter 'is inseparably both' attraction and repulsion; they are the 'ideal moments' of this real matter, for which reason, he adds, they must not be regarded as 'independent', or as 'forces on their own account (HEGEL, 1847, Sec. 262 R, p. 68; [HEGEL, 1970a, p. 45]; [HEGEL, 1970b, pp. 241-42]), as pseudoreified substances; an instance of which he gives in the Science of Logic, where he mentions the eighteenth-century habit of regarding electric, magnetic and gravitational forces as a sort of imponderable matter, as for instance various kinds of either (HEGEL, 1934, ii, p. 145; [HEGEL, 1969, pp. 519-520]). This is simply Hegel's general contention

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that 'force' must not be regarded as an ultimate and irreducible quasireified substance, for the purpose of scientific 'explanation', but that either (in science) it must be construed as a 'derivative' entity, or (in philosophy) it must be assigned its logical place in the context of categorial evolution, on the lines indicated just now, or (more general) in the section of the Science of Logic dealing with force, which will be considered presently in more detail. For the moment, let us consider further this problem in the special context of attraction and repulsion. We can, as I said, gather Hegel's general intentions if we consider at this point his debt to Kant. The main discussion occurs in the long Note appended to the section in Science of Logic entitled 'Repulsion and Attraction', where Hegel comments on Kant's method of providing 'metaphysical' accounts of the concepts of physical science, in particular, of the concepts of matter and force, as given in the Metaphysical Foundations of Science; an achievement "noteworthy, ... because as an experiment with the Notion it at least gave the impulse to the more recent philosophy of nature, to the philosophy which does not make nature as given in sense-perception the basis of science, but which goes to the abolute notion for its determinations" (HEGEL, 1934, i, p. 170; [HEGEL, 1969, p. 179])". In Philosophy of Nature we find, similarly, that Kant, through his attempt to provide a so-called "construction of matter" had "made a beginning with providing a notion of matter", and with this, of "the concept of a philosophy of nature (HEGEL, 1847, Sec. 262 R, p. 68; [HEGEL, 1970a, p. 45]; [HEGEL, 1970b, pp.241-242])". Therefore, if we can get clarity on what Kant meant by 'metaphysical foundation', and by 'metaphysical construction', we shall understand how Hegel conceived his own task. And this is indeed made very clear by his carefully worked critical discussion of Kant's procedure, even though it may not always be altogether fair, and indeed be contrived to persuade us of Hegel's own viewpoint. Considering here only some of the central points, Kant, as Hegel rightly notes, begins with an analysis of the concept of matter, as we have it 'empirically', as part of common consciousness. The impenetrability of matter, for instance, is analyzed as being explicable only in terms of repulsive and attractive forces. Here, the 'metaphysical' aspect, so far, clearly boils down to conceptual explication or analysis. In addition Kant claims however (though Hegel forgets to mention this point) that the analysis proceeds under the guidance of the categories, here, the category of quality, with

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its subdivisions, reality, negation, limitation. To the first, there corresponds repulsion, to the second, attraction, and to the third, the balance between these two 'forces'. So 'metaphysical' boils down to conceptual analysis under the guidance of the categories (cf. BUCHDAHL, 1972, pp. 159-164).

This gives us the connection with Hegel. The procedure is similar, though it is worked out in a far more complex, articulated and complete fashion. Hegel praises Kant for the basic idea, which was, not to add the concept of force to that of matter, as something 'foreign' to it, but to insist that we can "cognise [erkennen] matter as deriving from these two opposing determinations [HEGEL, 1934, i, p. 173; [HEGEL, 1969, p. 181])". On the other hand, he criticizes the insufficiently philosophical approach, which is still far too 'empirical', falling back on the sensory experience of the force of repulsion, and first assuming our sensory experience of the domain of 'matter', instead of offering us-as in Hegel-a completely logical analysis of 'being-for-self', into which repulsion-attraction is worked as analytical expression, serving ultimately as a means of illustrating the necessary language game here involved. True, he says, matter as it exists for sense perception is not a subject for logic, any more than are space and time. "But the forces of attracti onand repulsion, in so far as they are regarded as forces of empirical matter, are also based on the pure determinations here considered, of the one and the many and their inter-relationships, which, because these names suggest themselves most naturally, I have called repulsion and attraction (HEGEL, 1934, i, p. 171; [HEGEL, 1969, p. 179])". There is perhaps no better example than this through which to understand how Hegel conceived the relation between notional determination and theoretical science, and the relevance of the former for the latter. The Kantian method of 'establishing the possibility' of an entity or process or action, as exemplified in the analytico-transcendental approach of The Metaphysical Foundations of Science, and sharply distinguished from the question of 'probability', has developed into Hegel's dialectical approach which interprets the act of cognition [Erkenntnis] as the placement of a questionable concept within the transitional flux of the categorial advance of the logical idea. The notion of 'flux', of the transitional status of each of the intermediate elements in the dialectic advance, in which sense alone they are 'established',

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explains Hegel's attitude to some of the concepts of physical science, especially the concepts of matter, atom, and force. We can see this clearly in the case of force, as treated in the section of the Science of Logic to which I have already alluded (cf. HEGEL, 1934, ii, p. 146; [HEGEL, 1969, pp. 519-520)). We had previously noted Hegel's claim that in the advance of the logical idea the categorical detail of the phenomenon, or the merely sensory appearance of the manifold totality, and the hypothetical realm of lawlike determination of this world, must somehow come together. But this 'must' likewise needs some categorial expression, and Hegel finds this in what he calls 'the essential relation' (HEGEL, 1934, ii, p. 136; [HEGEL, 1969, p. 512)). The point is that the two sides of the phenomenon need an interpretation and exemplification which will exhibit them as essentially related; and the pictures which Hegel finds for this are the relation of whole-and-parts, which in turn advances to that of force and its manifestation." What Hegel says is best understood if we think of the example of central forces, e.g., mutual gravitational attraction, or of action-and-reaction. Here the action of one body on another manifests itself as the reaction of the other on the first, and vice versa. Neither, without the other, could manifest anything, and neither can stand on its own. Something like this Hegel gives as his explication of 'force and its manifestation': the purely logical conception of one thing positing itself in another, whilst this other can only posit itself at the same time in the one, in each case through observable mutual accelerations. Both this tension between the two, their mutual involvement and mutual repulsion-Hegel speaks here of the mutual 'solicitation' of forces (HEGEL, 1934, ii, pp. 147-149; [HEGEL, 1969, pp. 521-523))-and the manifestation of this as accelerated motion, is force-and-its-manifestation. There is no hidden center of reality, its nature ('force') unknown, known in quasifashion only through its effects, for the whole intellectual content of force, Hegel remarks expressly, is (1934, ii, pp, 144ff; [HEGEL, 1969, p. 518ff.]. Cf. also HEGEL (1843, Sec. 136, pp. 269-275; [HEGEL, 1892, pp. 246-251]). Hegel's architectonic here causes awkward divisions. The discussion of attraction and repulsion had occurred under the section on 'being', whilst the present section belongs to 'essence'. Each illuminates the other, although Hegel treated them apart, presumably because repulsion-attraction result only in the bare concept of continuous extended matter in general, whilst the later section on 'force' deals more generally with dynamic action between separated parcels of matter. In the Phenomenology ofSpirit, they-together with the concept of law, discussed above-still occur together in one chapter. 4 H!!GEL

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its manifestation; for which reason it is misleading to speak of the 'explanation' of these accelerations by means of a hidden force (HEGEL, 1843, Sec. 136R, p. 270; [HEGEL, 1892, p.248]). We should therefore say, he also tells us, that 'force is just that which manifest itself'; "and thus, in the totality of manifestation, conceived as a law, we at the same time understand and grasp [erkennen] this force itself (HEGEL 1843, Sec. 1367.2, p. 272; [HEGEL, 1892, p. 249])." Those who have followed the vicissitudes of the history of the concept of force during the interval between Newton and our modern period will recognize in this an instance in the long line of succession of attempts to provide an analysis of the phenomena of mechanics which reduces force to law, or to a secondary or derived notion (cf. JAMMER, 1957, Ch. 11). Hegel's point in introducing the picture of force is, of course, as a putatively more satisfactory way of articulating the 'many-one' relations involved in the concept of the 'phenomenon': the contrast 'sensory immediacy-law' being highly unstable, that between force-and-its-manifestations is hoped to yield a better model for the subsistence of the dialectical tension. However, we also see how the otherwise quite independent method of the dialectical logic is used to determine certain attitudes and conceptual approaches toward physical science; in the present case, Hegel's preference for Lagrangian as against Newtonian approaches," of phenomenological versus substantival theories. So let us in conclusion consider the question of the relation between notional determination (the conceptual explications of the dialectic) and the theoretical formulations as well as empirical results, e.g., empirical laws, of science. We have already mentioned atomic and gravitational theory, and lack of space forbids any discussion of these topics here. Instead, we will turn at once to the problem of free-fall, whose consideration I left incomplete in what has preceded. Earlier, we found Hegel saying that whilst from the point of view of science, the two sides of the law, sand (2, are merely juxtaposed, being based on experience, from another vista they must really mutually imply and presuppose each other; a connection only establishable through notional considerations. Hegel's argument for the example in question is unfortunately obscure, but we might keep in mind a specimen of considerations called 'dimensional', sometimes used in physical science. 5

Cf. the reference to Lagrange in HEGEL (1847, Sec. 267, p, 87n; 1970b, pp. 336--3371).

[HEGEL,

[HEGEL,

1970a, p. 58nl;

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The law s = tgt 2 relates space and time, concepts which-so he has argued (I will omit the details)-are internally related, mutually as well as through the concept of motion. Relative to time, taken as unity, space presents an 'asunderness' [Auseinandersein]: It is a magnitude which 'as coming outside itself, raises itself into a second dimension'; and the form or picture here presenting itself is that of the 'square'." Hence, in relation to space, its complement, time, will have to be raised to the second degree. This is an instance of a law "deduced from the notion", "derived from the notion of the subject [HEGEL, 1847, Sec. 267R, p. 88; [HEGEL, 1970a, p. 59]; [HEGEL, 1970b, pp. 255-56])", and it contrasts, Hegel insists, with the purely external approach of scientific mechanics. His main contention here, of course, is, really, to oppose a special case of scientific explanation, the attempt to 'explain' the law by reference to forces (of inertia and gravitational attraction) considered to exist independently. For, as also in the case of celestial dynamics, his position is that the empirical laws of mechanics and dynamics (Galileo's and Kepler's laws) are the ultimate court of appeal, and should be simply represented as a coherent system, of the type of which for him Lagrange's presentation in his Theory of Functions is the supreme example. This does not provide 'proofs' from ultimate forces, but is simply a representation of the empirical and lawlike facts. The only admissible sense of 'proof' that remains for Hegel here, then, is that of 'derivation from the notion'. But lest we jump to hasty conclusions concerning any misplaced apriorism in Hegel's philosophy of science, it should be noted that he was sanguine about the powers of such derivations. The ideal, of course, was to present the sides of the equation as a necessity of conceptual determination; but he expressed salutary and skeptical reservations about the extent of this method. Its only purpose was to make the subject intelligible, not to provide scientific explanation or prediction (HEGEL, 1847,Sec.270Z, p. 124; [HEGEL, 1970a, p. 82]; [HEGEL, 1970b, p. 281]). After all, science constantly stands in need of conceptual articulation. But in the battle of thought, the 'notion' may sometimes try to dictate, where theory ought to have had the prior say. But who, in our day, would wish to condemn Hegel for 6 For the general reference to the second power, i.e., the 'square', cf. also HEGEL (1843, Sec. 102R, p. 204; [HEGEL, 1892, p. 192]). Above all, it would be necessary to take into account the sections on space, time and motion in HEGEL (1847), to get a full appreciation of the argument.

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the general method of his procedure? Indeed, even in a 'hard' subject like mechanics, where the Newtonian paradigm in Hegel's time was so solidly entrenched, it was just this attempt to provide a totally different basic outlook that would seem to us nowadays to display one of the most exciting facets of Hegelian philosophy.

References BUCHDAHL, G., 1969, Metaphysics and the philosophy of science. The classical origins: Descartes to Kant (Blackwell, Oxford; M.I.T., Cambridge) BUCHDAHL, G., 1970, History of science and criteria of choice, in: Historical and Philosophical Perspectives, Minnesota Studies in the Philosophy of Science, vol. 5, ed. R. H. Stuewer (University of Minnesota Press, Minneapolis), pp. 204-230 BUCHDAHL, G., 1972, The conception of lawlikeness in Kant's philosophy of science, in: Proc. HIrd Int. Kant Congr., ed. L.W. Beck (Reidel, Dordrecht), pp. 149-171 BUCHDAHL, G., Conceptual analysis and scientific theory in Hegel's philosophy of nature (with special reference to Hegel's optics), in: Boston Studies in the Philosophy of Science (forthcoming) BURlT, E. A., 1932, The metaphysical foundations of modem physical science, second edition (Routledge, London) CAMPBELL, N. R., 1957, Foundations of science (Dover, New York) FINDLAY, 1. N., 1958, Hegel. A re-examination (Allen and Unwin, London; Humanities Press, New York) HARRE, R., 1964, Matter and method (MacMillan, London) HEGEL, G. W. F., 1943, Encyklopiidie der philosophischen Wissenschaften im Grundrisse. Erster Teil: Die Logik, ed. L. von Henning (Duncker und Humblot, Berlin) HEGEL, G. W. P., 1845, Encyklopddie der philosophischen Wissenschaften im Grundrisse, Dritter Teil: Die Philosophie des Geistes, ed. L. Boumann (Duncker und Humblot. Berlin) HEGEL, G. W. P., 1841, Vorlesungen iiber die Naturphilosophie als der Encyklopiidie der philosophischen Wissenschaften im Grundrisse Zweiter Teil, ed. C. L. Michelet (second edition, Duncker und Humblot, Berlin) HEGEL, G. W. P., 1892, The logic of Hegel, translated by W. Wallace (second edition, Clarendon, Oxford) HEGEL, G. W. P., 1934, Wissenschaft der Logik, 2 vols, ed. G. Lasson (Meiner, Hamburg) HEGEL, G. W. F., 1952, Phdnomenologie des Geistes, ed. J. Hoffmeister (Meiner, Hamburg) HEGEL, G. W. F., 1969, Hegel's science of logic, translated by A. V. Miller (Allen and Unwin, London; Humanities Press, New York) HEGEL, G. W. F., 1970a, Hegel's philosophy of nature, translated by A. V. Miller (Clarendon, Oxford) HEGEL, G. W. F., 1970b, Hegel's philosophy of nature, 3 vols, translated by M. J. Petry (Allen and Unwin, London) (translation with notes of Hegel, 1847) (Cf. also the review article of this work by D.M. Knight, The physical sciences and romantic movement, in: History of Science, vol, 10 (Heffer, Cambridge 1971), pp. 66-72

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HEGEL, G. W. F., 1971, Hegel's philosophy 0/ mind, translation by A. V. Miller (Clarendon, Oxford) HOI,TON, G., 1964, Presupposition in the construction 0/ theories, in: Science as Cultural Force, ed. H. Woolf (John Hopkins Press, Baltimore) JAMMER, M., Concepts 0/ force (Harvard University Press, Cambridge, Massachusetts) KUHN, T. S., 1967, The structure 0/ scientific revolutions (University of Chicago Press, Chicago) SELLARS, W., 1963, Science, perception and reality (Routledge, London; Humanities Press, New York) TOULMIN, S., 1961, Foresight and understanding (Hutchinson, London) WHEWELL, W., 1967, The philosophy 0/ the inductive sciences, 2 vols., eds. G. Buchdahl and L. L. Laudan (Cass, London)

LA CONTINUITE ET LA DISCONTINUITE EN CHIMIE ET EN PHYSIQUE AU XIX e SIECLE B. M. KEDROV Academic des Sciences, Moscou, U.R.S.S.

1. Introduction Le discontinu et Ie continu n'existent dans la realite qu'en unite inseparable. Mais ce n'est pas immediatement que cette unite peut etre decouverte et concue par la science qui en esprit analytique la dissocie d'abord en degageant ses contraires et en les etudiant non seulement separement mais parfois meme dans leurs contradictions reciproques. Ce n'est qu'ensuite, sur la base des details connus qu'on reconstruit synthetiquement l'unite detruite par l'analyse precedente. Done, de meme que dans toute histoire de la connaissance scientifique, l'analyse precede ici la synthese, la prepare. La synthese, de son cote, approfondit ce qui avait ete acquis au niveau de la recherche analytique, aide a surmonter ces aberrations dans la connaissance de l'objet qui avaient ete apportees au niveau analytique precedent. Ces reflexions sont legitimes aussi quand il s'agit de ces elements contraires de la realite qui se traduisent par les concepts de continu et de discontinu. Au cours de I'evolution historique de la science l'unique se dissociait en parties contradictoires qui pour quelque temps se revelaient en divorce et s'opposaient reciproquement. Le resultat en fut qu'une partie de la contradiction reelle se revele absolutisee, l'autre partie, contraire a la premiere, tantot se reduit a la premiere, tantot se voit niee, Dans l'autre cas, c'est une autre partie qui se voit absolutisee, la premiere ayant ete supprimee, Ce fut la cause de I'apparition des conceptions unilaterales, s'opposant l'une a I'autre, s'excluant reciproquement, dont les partisans discutaient, se tenant, aux points de vues extremes. En determinant les notions de discontinu et continu, Hegel ecrivait dans "La science de la logique": "pas une de ces determinations prises isole-

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ment n'est pas vraie, c'est par leur unite qu'elles constituent la verite. C'est ainsi, concluait-il, que precede leur consideration dialectique, ainsi que devient clair leur vrai resultat". Le developpement de la physique et de la chimie aux XVIIe et XVIIIe siecles creusait un abime entre ces determinations opposees, Au XIX" siecle ce developpement commence a eliminer l'opposition deja profondement etablie de la discontinuite et de la continuite, on tient maintenant a la reconstitution de leur unite primitive et a leur liaison reciproque. La reconstitution complete n'a eu lieu pourtant qu'au XX· siecle (la mecanique quanJique, la doctrine des particules elementaires). Tout le chemin parcouru par la connaissance peut etre illustre par l'histoire des theories de l'element, de la lumiere et d'energie. 2. Physique et chimie des XVll-XVme siecles Le probleme de la discontinuite et de la continuite en chimie et en physique a ete poses avant tout par rapport aux reflexions sur la structure de la matiere. Dans la philosophie de la Grece antique se sont formes deux differents points de vue: Leucippe et Demokrite reconnaissaient la structure atomistique discontinue de la matiere: selon leur avis Ie monde est compose d'atomes et des vides. Aristote au contraire niait Ie caractere discontinu de la structure de la matiere: il estimait que celle-ci remplissait tout l'espace. Au XVII-XVIII" siecles la discussion entre les partisans de la discontinuite et ceux de la continuite a pris une forme plus concrete. La doctrine de Newton s'appuyait sur la discontinuite de la matiere: selon Newton l'univers presente un systeme des corps discontinus qui sont en action reciproque dans l'espace vide. Descartes au contraire a developpe une doctrine selon laquelle c'est Ie milieu continu qui remplit tout l'espace de l'univers. Dans ce milieu il y a des flux et des tourbillons qui donnent naissance aux corps discontinus. C'est l'hydrodynamique et I'aeromecanique (les sciences etudiant les phenomenes mecaniques dans l'eau et l'air) qui auraient pu servir de base a telles reflexions. Dans la domaine de l'optique, Newton defendait une idee de la matiere lumineuse en forme des corpuscules emanees par les corps luminescents. Huygens, au contraire, avancait une doctrine de l'optique geornetrique postulant l'existence de l'ether, milieu continu dont la lumiere est un mouvement onduleux. La theorie corpusculaire de la lumiere est un cas particulier de la conception generale de la structure atomistique de la matiere predominante dans

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les sciences non organiques du XVII-XVlIIe siecles, Cette conception determinait I'evolution de la physique et de la chimie apparues en XVII siecle. L'atomisme qui a pris un caractere mecaniste correspondait parfaitement au materialisme mecaniste dont Ie representant fut Boyle qui, s'opposant it la monadologie idealiste de Leibniz, a avance sur la base materialiste sa "monadologie physique". S'appuyant sur cette doctrine, Lomonossov a su developper dans une forme hypothetique les postulats essentiels de l'atomistique future, elaboree par la physique et la chimie du Xl X" siecle, de meme qu'avancer certains pronostics remarquables. "Le principe de discontinuite", en tant que la loi generale pour la nature non-vivante, etait defendu par Leibniz bien qu'il soit en meme temps Ie fondateur de la doctrine proclamant I'existance dans la nature vivante de monades spirituelles (n'ayant pas de corps), en tant que formes discontinues. Plus tard, une idee de la continuite des forces et des formes materielles auxquelles ces forces servaient de base, fut affirmee au XVlII e siecle par l'idealisme (Kant) sous la forme de la theorie dynamique. Ainsi la discussion entre Ie materialisme et I'idealisme a la fin de XVIIle siecle a pris la forme de la discussion amorcee par les atomistes et dynamistes, les adeptes de deux conceptions differentes, dont chacune s'appuyait dans sa methodologie sur une categoric (concept) interpretee d'une maniere unilaterale. Les atomistes ont pris comme un point de depart la continuite abstraite qui leur servait de base. II est remarquable que la synthese de ces elements contraires de la matiere se soit averee acquise dans une forme pure de la philosophie de la nature par Boskovic. II a introduit la notion de "dynamide" (point de l'application des forces continues), en tant que celIe de la discontinuite, en liant ainsi les premisses essentielles de Ia doctrine de Leibniz (principe des monades et de la discontinuite) it la doctrine de Newton concernant les forces d'attraction et de repulsion, c'est-a-dire, it sa dynamique. L'idee principale de Boskovic, celIe de l'unite inseparable de la matiere et du mouvement, de meme que son atomistique dynamique qui s'en suivait, fut hautement estimee par Priestley ("Histoire de I'optique", 1772): par la suite, c'est lui qui influencera Thomson-Kelvin (XfX" siecle) et surtout J. J. Thomson, qui a decouvert l'electron (1897) et propose Ie premier modele statique de l'atome (1903). Selon Mendeleev, la doctrine concernant la substance it la fin du XfX" siecle peut etre consideree comme "une tentative de concilier et de concorder Ie dynamisme et I'atomisme", c'est pourquoi Boskovic est partout tenu en un certain sens pour Ie fondateur des idees modernes sur la substance. Au commencement du Xl X" siecle les debats eclaterent entre Proust Ie

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successeur de Lavoisier et Berthollet. II s'agissait de savoir si, etant donne les resultats des reactions chimiques, les substances forment des combinaisons selon des proportions constantes (discontinues) comme affirmait Proust, ou selon des proportions variables (continues), Ie point de vue sur lequel insistait son adversaire. La premiere doctrine confirmee par l'experience s'est enracinee pour longtemps dans Ie domaine de la chimie. En meme temps Dalton crea les bases de l'atomistique chimique, etablit son fondement empirique en decouvrant la loi des multiples simples (c'est-a-dire des proportions extremement discontinues, 1803). C'est a cette epoque aussi qu'il affirma la legitimite de la notion essentielle de toute la chimie du XIxe siecle, celle du point atomique. Ainsi la discussion non seulement entre Proust et Berthollet mais aussi entre atomistes et dynamistes fut reglee en faveur de l'atomistique, en tant que doctrine determinant la structure discontinue de la matiere. S'appuyant sur l'idee des forces continues, le dynamisme n'a pas trouve possible d'expliquer la continuite, de meme que la divisibilite, des rapports entre des elements chimiques faisant partie de compositions chimiques. Le chimiste Higgins le successeur de Dalton, s'etait approche de la decouverte de la loi des relations multiples simples, mais n'a pas fait cette decouverte, parce qu'il continuait de subir l'influence du dynamisme. Smith, le biographe de Dalton, devait ecrire plus tard que "le dynamisme ne trouvait pas de place dans la chimie. II foulait meme ses amis et Higgins a succombe etant devenu sa victime". Les tentatives de Hegel de rejeter l'atomisme et de galvaniser Ie dynamisme n'eurent pas de succes: la loi des relations multiples simples se revela inconsiliable avec le dynamisme, il ne s'accordait organiquement qu'avec l'atomisme. L'evolution suivante de la chimie au XIxe siecle, surtout de la chimie organique, non seulement affirmait l'atomistique de Dalton, en tant que base generale theorique de toute la chimie, mais encore allait reprendre l'idee de la discontinuite jusqu'a la liaison chimique des atomes. La notion de valence apparue a la suite de la theorie des types de Gerhardt s'appuyait sur l'hypothese que chaque atome d'un element ou d'autre ne pouvait s'unir qu'avec un nombre entier d'atomes d'hydrogene dont la valence fut pris pour unite. Ainsi I'idee initiale de Dalton sur la discontinuite et la divisibilite des relations s'est incarnee dans la notion de valence. Le principe de la discontinuite et de la divisibilite des relations a trouve son expression la plus profonde dans la loi periodique, decouverte par Mendeleev (1869). Mendeleev soulignait dans ses "lectures faradayennes":

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"En liant d'une nouvelle maniere les elements chimiques a la doctrine de Dalton sur la composition multiple ou atomique des corps, le principe periodique a trouve un champ tout neuf pour illustrer le pouvoir de la pensee." Ce principe a etabli que ce n'est pas sans cesse que croit la masse des atomes, la croissance s'effectue par sauts qui correspondent a ceux ouverts par Dalton. Done, au cours du XIxe siecle une idee de la discontinuite s'est affirmee dans la chimie. La doctrine atomistique soutenue par l'experience semble remporter une victoire definitive. Pourtant cette victoire fut ephemere, Des la fin du Xl X" siecle, en etudiant les solutions on s'est rendu compte de I'unite de la continuite et de la discontinuite qui representaient deux parties contraires de la structure chimique de la matiere. Pour suivre l'exemple avec Mendeleev, notons que, dans son ouvrage "Etudes sur les solutions aqueuses d'apres leur poids specifiques", Ie savant russe a montre que l'apparente continuite du changement de la composition des solutions aqueuses des differentes substances cache des combinaisons chimiques des molecules de la matiere dissoluble avec des molecules du dissolvant. Ces etudes ont ouvert Ie passage a la discontinuite dans Ie domaine de la chimie et de la physique ou la continuite regnait jusqu'alors. Deux contraires devenaient complernentaires, ils se penetraient reciproquement. A la fin du XIxe siecle et surtout au Xxe siecle ces idees ont ete developpees en Russie par Ie chimiste Kournakov qui a cree l'analyse physicochimiste. En etudiant des alliages, des verres et des solutions, il a decouvert qu'outre des combinaisons chimiques ayant une compostion discontinue (qu'il a appele "daltonides") il existe des combinaisons ayant une composition variable auxquelles il a donne Ie nom de "bertollides". De la sorte la position de Berthollet n'etait pas entierement fausse, elle contenait une partie de verite. Quant a Proust qui avait exclu l'element de la continuite sa position dans sa forme abstraite ne pouvait pas etre entierement correcte. C'est pourquoi la victoire de Proust n'etait que provisoire. Kournakov soulignait que la continuite et la discontinuite dans les systemes equilibres existaient en liaison reciproque, Les diagrammes se rapportant a l'etat continu et hornogene determinent en fin de compte les ruptures inattendues dans les performances de celles parmi des combinaisons chimiques qui peuvent-etre degagees sous forme solide par refroidissement. Si Mendeleev a trouve Ie discontinu dans Ie continu, Kournakov a decouvert le continu dans Ie discontinu. C'est dans cette direction que se developpait d'une maniere aussi evidente la theorie de la liaison chimique. La discontinuite abstraite initiale

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des "traits" de valence unissant les atomes donnait lieu a l'idee de la valence partielle (fractionnaire) des combinaisons organiques non saturees ce qui a fait penser aux forces de valence, creant autour de soi le champ continuo De telles reflexions furent confirmees apres le fondement de la theorie de la dissociation electrolitique et surtout apres la decouverte des electrons ce qui a permis de mettre en rapport les processus chimiques et physiques (electriques). Done, la rupture dans l'evolution de la chimie au XIX" siecle entre la discontinuite et la continuite allait d'etre eliminee par leur synthese, Un tout autre chemin pour parvenir a cette synthese fut parcouru par la physique au XIX" siecle, Si c'est la conception de la discontinuite (doctrine de Dalton) qui a triomphe en chimie au debut du XX" siecle en physique c'est la theorie ondulatoire qui s'est affirmee a cette epoque (grace aux travaux de Fresnel). Une idee de la continuite dans Ie milieu (ether de la lumiere) ainsi que dans les ondes de la lumiere survenant dans ce milieu, servait de fondement a cette theorie. La decouverte experimentale des phenomenes de la diffraction de la lumiere, expliques par Fresnel au moyen de la theorie ondulatoire, les influences de la polarisation sur l'interference, l'explication du phenomene de la rotation du plan de la polarisation - etc. - tous ces faits et d'autres semblaient refuter completement la theorie corpusculaire de la lumiere et affirmer definitivement la theorie ondulatoire. C'est en s'appuyant sur la conception de la discontinuite que Faraday approfondit l'idee du champ electro-magnetique; plus tard Maxwell crea une theorie electromagnetique de la lumiere. La ligne de la discontinuite s'est accusee d'une maniere aussi nette dans la domaine de la thermodynamique qui date du XIX" siecle ayant pris pour son point de depart les fonctions continues de l'etat des systemes thermodynamiques. En premier lieu il faut y mettre l'accent sur l'energie interieure du systeme. Un role extremement important a joue l'entropie, une fonction purement mathematique, introduite par Clausius et serappor tant au deuxieme principe. Toute la thermodynamique se developpait en science purement deductive, en s'affirmant comme un domaine mathematise de Ia physique ou le principe de la continuite mathematique semblait regner partout et definitivement. Ainsi dans la physique au XIX" siecle, c'est l'idee de la continuite des qualites et des proprietes physiques qui a triomphe provisoirement. Pourtant la discontinuite survivait dans la theorie cinetique-moleculaire des gaz et ensuite dans la statistique physique. Ces domaines de la physique

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faisaient evoquer un concept de la structure discontinue de la matiere, les proprietes thermodynamiques des systemes se deduisant par la sommation de la valeur du mouvement et des proprietes des particules (molecules), formant ces systemes, C'6taient Clausius, Maxwell, Boltzmann, Gibbs, plus tard Einstein, Smoluchowski notamment qui poursuivaient de telles idees renvoyant la discontinuite dans Ie domaine de la physique. Une decouverte de Boltzmann s'est revelee d'une importance particuliere, C'est celie qui montre Ie rapport entre l'entropie du systeme, composes des particules de la matiere et de la probabilite de son etat (1877). L'entropie, un des concepts thermodynamiques essentielles subissait done l'interpretation statistique et quantitative qui s'appuie sur Ie principe de la discontinuite. Neanmoins presque jusqu'a la fin du XIX e siecle la discontinuite restait eliminee par la continuite, C'etait l'idee dominante dans la methodologie de la physique. Max Planck ecrivait dans sa "Thermodynamique" en 1897, l'annee de la decouverte de I'electron: "La discontinuite dans Ie changement du poids equivalent caracterise la nature chimique de la matiere contrairement it ses proprietes physiques". En resume on peut dire que les changements physiques se poursuivent continuellement, les changements chimiques, eux, se poursuivent discontinuellement. C'est pourquoi dans la physique on retrouve par excellence les valeurs variables continuelles, tandis qu'en chimie c'est, dans la plupart des cas, a des nombres entiers qu'on a affaire. A partir de cette position Planck expliquait un celebre paradoxe de Gibbs concernant la mixture des gaz. II ecrivait: "La difference chimique de deux gaz ou de deux matieres ne peut pas etre presentee par une valeur continuellement variable, dans ce cas il s'agit des rapports, des relations dont les changements precedent par sauts, iI s'agit de l'egalite ou d'une inegalite, C'est ce fait qui de l'avis de Planck marquait la differenceessentielle entre les proprietes chimiques et physiques, ces dernieres devant changer continuellement. " Nous avons montre que la situation de la discontinuite dans la chimie avait cesse d'etre infaillible vers la fin du XfX" siecle et surtout au XX e siecle. II en va de meme pour la continuite dans Ie domaine de la physique. La decouverte de l'electron, qui marque Ie commencement d'une guerre declaree a l'idee de la continuite absolue dans la theorie de l'electricite, a temoigne de sa defaillance. Dans sa "Dialectique de la nature" Engels

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disait que dans le domaine de l'electricite etait encore a l'ordre du jour une decouverte qui pourrait etre comparee a celIe de Dalton, cette decouverte pouvant servir de centre et de base a la science et a la recherche". II s'agissait de comprendre, selon . Engels "ce qui constitue Ie substrat substantiel proprement dit du mouvement electrique, de savoir quelle matiere par son mouvement provoque les phenomenes electriques". La decouverte de l'electron par J. J. Thomson avait pour la theorie de l'electricite une importance analogue a celIe qu'avait pour la theorie de la substance la decouverte de Dalton. La notion de la substance a elargie celIe de l'electron qui maintenant faisait partie de la premiere. C'est Ramsay qui soulignait surtout ce fait. Une importance encore plus generale revetait pour la physique la decouverte faite par Planck (1900), celIe qui a decele Ie caractere discontinu de l'emission de la chaleur et aussi introduit Ie quant de l'action h, en tant qu'une constante nouvelle universelle. Peu apres, Einstein a avarice Ie concept du photon ou d'un quant de la lumiere, en tant qu'un corpuscule particulier de la Iumiere, C'est ici que prend son depart la physique quantique du xx· siecle, fondee sur la discontinuite. Done en physique, comme en chimie, Ie triomphe d'un contraire, notamment d'une continuite, ne fut que temporaire. De meme que la chimie du XX· siecle a preuve que Berthollet avait partiellement raison, la physique quantique dans Ie domaine de l'optique a rehabilite dans un certain sens l'idee corpusculaire de la lumiere, appartenant a Newton. Planck Iui-meme, qui peu de temps avant voyait dans la physique un citadelle de la continuite, a fait definitivement demolir par sa theorie des quanta ces doctrines unilaterales. C'est ainsi que s'effondraient en chimie comme en physique, les eclosions entre la continuite et la discontinuite, done entre ces sciences elles-memes. Mais c'est par la mecanique quantique que des Ie XX· siecle fut porte Ie coup definitif qui detruisit ces eclosions, en levant a la theorie de la lumiere, ainsi qu'a celle des particules microscopiques de la substance, l'ancienne opposition entre la discontinuite et la continuite, Ie corpuscule et l'onde. Louis de Broglie avance une idee fondamentale, selon laquelle a to ute particule microscopique (de matiere ou de lumiere), ayant une masse m et se deplacant a vitesse determinee correspond toujours une onde de la longueur relative a cette derniere, Plus tard, cette idee recoit une confirmation experimentale par la diffraction des electrons, des atomes et d'autres particules, ce qui conduisit a la creation du microscope electronique.

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3. Conclusion L'analyse de I'evolution chimique et physique, faite en termes des concepts de la discontinuite et de la continuite, nous permet d'aboutir aux conclusions suivantes: a) La connaissance de l'objet physique ou chimique de nature, de meme que d'un objet quelconque portait en soi sa contradiction interne, s'effectue selon un schema rappel ant la lettre "U": Ie mouvement part des deux bouts d'en haut et se poursuit vers Ie bas jusqu'a la rencontre de deux lignes. Tous les cas possibles se reduisent cependant a un schema unique: on passe de la dissociation primitive de l'objet - ne revelant que sa continuite ou sa discontinuite - a leur synthese, leur dependance mutuelle. b) Quant aux variantes de ce mouvement (selon Ie schema decrit cidessus) on rencontre Ie plus souvent les quatre cas suivants: 1) Ie mouvement commence des deux bouts simultanement, mais plus tard une des contraires commence a predominer, Ce n'est qu'au bout d'un certain temps qu'on s'avise de l'autre contraire qui se trouve etre incorpore dans Ie premier. 2) Les mouvements declenches des deux bouts vont toujours simultanement jusqu'a leur rencontre (I'unite des contraires dans ce cas .ne coincide pas avec l'unite des doctrines), c'est-a-dire jusqu'a leur unite interne. 3) Le mouvement commence a un bout, mais quelque temps apres une autre idee point. L'evolution ulterieure est analogue, selon les circonstances, a celie du cas 1 ou 2. 4) Le mouvement commence a bout et poursuit son chemin. Pourtant la connaissance approfondie de ce cas permet de deceler son contraire. C'est pourquoi, t6t ou tard, I'evolution de la pensee passant par Ie "Iinteau" d'en bas gagne l'autre ligne. c) Les particularites de l'objet etudie font comprendre pourquoi une conception unilaterale aurait pu remporter une victoire temporaire. L'objet chimique (la substance chimique) presente au premier chef des relations discontinuees des parties constituantes (par exemple les relations multiples simples) a cause d'une quantite peu nombreuse des atomes faisant parti des combinaisons chimiques simples. L'objet physique, a cause d'une quantite nombreuse de ses formes discontinuees ou du fait de leur caractere special, presente au premier chef une apparente continuite. Ce n'est qu'ensuite qu'on decele Ie caractere temporaire de la victoire remportee par une conception unilaterale. d) L'evolution de la connaissance scientifique va done a atteindre non seulement l'unite des contraires en termes de chaque science, prise isolement, mais aussi l'unite methodologique de ces sciences elles-memes. II

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est done impossible d'expliquer Ie paradoxe de Gibbs s'appuyant sur Ie divorce entre la chimie et la physique. II faut au contraire prendre comme point de depart l'idee de l'unite des contraires composant I'objet, de memes que les sciences etudiant les contraires.

THE CONCEPT OF PHYSICAL NECESSITY

M. MARKOVIC University of Belgrade, Belgrade, Yugoslavia

1. General concept of necessity

The concept of necessity, like all categories, is highly ambiguous and is being applied in several different ways: in logic, empirical sciences and ordinary language. The concept has three important philosophical dimensions: we use the term 'necessary' in order to characterize the way in which objective events may be connected ('ontological necessity'), or the way in which from some elements of our knowledge some other elements follow ('epistemological necessity'), or the way in which certain human actions are conditioned by some values, norms, goals ('axiological necessity'). In the history of physics until recently the concept of necessity was applied only to cases of strict determination on the basis of the so-called 'causal laws'. However, there is an increasing number of authors who argue that in so-called indeterministic physical structures we have good reasons to apply the concept of necessity, if not to individual events, then certainly to whole classes of alternative events exhibiting certain regular patterns ('statistical laws'). In spite of all these distinctions, there are some general features of the situation in which we speak about any of these specific types of necessity. First of all we must bear in mind a concrete totality, i.e., a definite system of objects S which are relevant for our problem. The system has, on the one hand, some general characteristics that, in any given moment t open a field of possibilities F, i.e., enable us to project a set of possible states of the system in various other moments (of the future or the past). The system, on the other hand, contains also some particular limiting conditions Le, i.e., some rules or laws that restrict the actualization of abstract possibilities. To say then, that an x is necessary means that X is a nonempty subset of the set of possibilities F and that all other possibilities of the

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field F except x are excluded by the action of the given limiting conditions LC. This definition holds for both logical and physical necessity. 2. Logical necessity

The concept of logical necessity is relative to a well-defined formal system. In the simplest and most general case the system is not interpreted. Rules of formation determine which of an indefinite number of combinations of (an indefinite number of possible) signs belong to the system; thus rules of formation play the role of the boundary conditions of the system. Initial conditions in this case are selected basic formulae (axioms); they determine the initial state of the system from which the system can be built up in many directions. The field of possibilities is enormous: it is constituted by all those combinations of signs that satisfy the rules of formation and can be derived by any modifications of basic formulae. Rules of transformation are the limiting conditions of the system S. They reduce all possible modifications to a subset of legitimate ones. To say that a formula IX is a necessary formula in S means (a) that IX belongs to a subset of formulae K within the field of all abstract possibilities F that can be derived from basic formulae by applying the rules of transformation of S, and (b) that all other subsets within F except K are excluded. The system S was an uninterpreted formal system. If we wish to build up an interpreted system that, in addition to syntax, comprises a semantics in which a set of semantical rules has been introduced determining meaning of individual signs and their combinations, we would get a stronger, interpreted system Sf, in which the concept of necessity would be considerably more narrow and applied only to those formulae which satisfy both syntactical and semantical rules (the sub-subset L within the subset K). If we wish to build up a logical system S" that would not have any arbitrary syntactical structure and any arbitrary interpretation but that would be applicable for demonstrative purposes in empirical science or ordinary thinking, we would have to introduce some additional limiting conditions. An example of these would be the requirement that each formula in S" must be applicable to a given empirical theory T as a condition of factual truth of its statements, e.g., as a scheme of inference by which from factually true statements of T other factually true statements can be derived.

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Consequently, the concept of logical necessity is relative to a given system and can be operatively defined only in a negative way: by a series of operations which exclude all those abstract possibilities that are compatible with the general definition of the system but are incompatible with at least one of the limiting conditions contained in the system. The qualification that logical necessity is apriori and analytical should be interpreted in such a way as to mean that logical entailment is independent of any particular type of experience and any particular field of reality to which it is applicable. If logical rule were independent of human experience in general and reality in toto, they would be arbitrary fictions of a totally alienated mind, or they would be linguistic conventions that might vary from one individual to the other. Logic has its permanent objective value insofar as its rules are general conditions of establishing truth, or in other terms, insofar as it has some kind of structural similarity with the world as a whole.

3. Empirical necessity All this holds for empirical necessity with the only difference being that it is relative to particular segments of reality and various concrete types of experience. While logical rules are applicable to all empirical fields so that it appears as if they are independent of experience, empirical laws have their definite, particular field of application. In that sense empirical necessity is synthetic and a posteriori. But there cannot be as sharp a demarcation line between logical and empirical necessity as envisaged by some empiricists three or four decades ago. Within empirical science one finds a few principles and laws, e.g., principle of conservation of energy, which are so well entrenched that they strongly resemble a priori and analytic statements of logic, whereas in concrete and nonformal empirical theories one meets laws of such a narrow scope that no one would have any doubt about their synthetic and a posteriori character. The main difference between logical and empirical necessity seems to be that in the former case relativity of the concept with respect to a specific system is obvious, whereas in the latter case we rarely deal with theories that are formalized to such a degree that they can satisfy customary logical requirements to be considered systems. That is why it was so difficult to conceive the relativity of the concept of empirical necessity. Outside the sphere of logic and mathematics the term 'system' should be taken in a more flexible way such as to mean any ordered set of objects that are relevant for the solution of a given problem. From the nature

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of the problem, the goal of inquiry and the level of previous theoretical knowledge it will depend which objects will be considered relevant for the solution of a given problem. Thus the system is neither simply given nor is it a purely subjective construct. Its objective dimension is constituted by real connections, relations of functional dependence, interactions among physical phenomena. Its subjective dimension is constituted by the character of the problem, our practical goal, the more or less incomplete amount of information at our disposal, our intuitive conjectures about the properties and relations among objects. In order to solve a problem in a practically satisfactory way we must collect a certain amount of information, and from the existing theoretical level it will depend how much and what kind of information is needed. An object 0 is therefore relevant to the given problem P when the information about 0 is one of the conditions for the solution of the problem P. To determine what information is needed for the solution of the problem P means to create the system S, i.e., to abstract from a huge universe of objects and relations those that seem to be in some meaningful connection with our problem. The boundaries of relevancy, i.e., the boundaries of a system are clearly established only at a considerable level of formalization, otherwise they are more or less blurred and vague. If we tend toward maximum precision and exactness the boundaries will be fixed and static; if concreteness, objectivity and practical applicability of our results are more important to us the boundaries will be more flexible and dynamic. The system is closed in the former case, open in the latter. Within the boundaries of the given system there is an enormous field of possibilities that may be projected. By projection we may understand any conceivable transformation of some actual object that satisfies some general limiting conditions. In empirical science projection is the set of all operations of interpolation, extrapolation, analysis, synthesis, idealization and establishment of limits that are compatible with the rules of logic. Projection is always liberation from some conditions and limitations of a given actual situation; it is always a relatively free and imaginative flight of thought from a momentary reality. However this freedom is always limited by some conditions of a more general character. As projection is possible at different levels of generality and abstractedness, different levels of possibility should be distinguished. Each level is characterized by a certain degree of freedom that depends on the number of limiting conditions which are included in the system.

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All systems in the field of science can be ordered into a continuum such that on the one pole we have the most general and most abstract system, with the largest field of possibilities and with the greatest degree of freedom, whereas on the opposite pole we should have the most concrete and specific system, closest to empirical reality, with strongest limiting conditions, therefore with the lowest degree of freedom. Thus the system of logical syntax will allow the greatest degree of freedom (say n) because in it all operations of projection will be legitimate insofar as they satisfy only one kind of rule: rules of formation. The next level of possibility with a slightly lower degree of freedom n -1 we will have in those systems whose operations of projection are limited not only by rules of formation but also by rules of transformation. The next degree of freedom n - 2 will characterize those fields of possibilities the projection operations of which are limited, in addition to beforementioned rules, also by semantical rules. The next degree of freedom n - 3 will characterize a system that among limiting conditions also comprises rules of application. This is the level of principles of a whole scientific field. Here we are still dealing with general logical possibilities, i.e., those which are determined by general logical conditions. Candidates for scientific postulates in a given field may be all those statements that clearly make sense, that are coherent and may be expected to function as conditions of factual truth (for example, in the sense of introducing a concept of great explanatory power, or a fertile scheme for generating factually true statements). To be sure, when we project we do not know whether our expectations will come true; that is why only some candidates for postulates will actually be accepted. At a critical moment of modern physics both Lorenz's contraction and Einstein's relativity principle were possible. The former was rejected because it involved empirically false consequences (fissures in swiftly movirig solid bodies). Einstein's principle did not have such consequences and it also offered a considerably simpler reorganization of the whole body of physical knowledge, by allowing all physical laws to remain invariant for all systems of reference. This is an example of how scientific thought progresses: by introducing limitations into a previously insufficiently articulated field of possibilities. These limitations are determined, on the one hand, by methodological considerations, and on the other hand, by new empirical evidence. The operational meaning of 'empirical necessity' is precisely this process of limitation, this series of eliminations of all those possibilities which are

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incompatible with special logical and methodological requirements on the one hand, and with empirical evidence from the given field on the other. At the next level of possibilities (with the degree of freedom n-4), which is the level of empirical theories of widest range and highest degree of generality, the freedom of projection will be limited not only by general logical rules but also by already adopted principles of the whole scientific field. This freedom is still very great and it leaves a very wide space open to imagination because it explores possibilities that lie far beyond established empirical data. Scientific thought at this level presupposes historical character of all actual knowledge and transcends its historically conditioned limits. What has been established so far holds also for all other levels of possibility, including empirical theories of medium and small range. For each level, for any given system S it is essential to distinguish between two types of possibilities: on the one hand a large class of preliminary (relatively) a priori possibilities, determined by higher-level theoretical considerations, and on the other hand, the subclass of a posteriori possibilities generated by empirical falsification of all other subclasses of preliminary possibilities. 'Empirical necessity' is an abbreviation for this process of reduction of the former to the latter. 4. Physical necessity

Physical necessity is a special case of empirical necessity. It is relative to systems of physical objects, i.e., objects that (a) are transcendent, i.e., exist independently of human consciousness, (b) can be located in space and time, (c) have properties of mass and/or energy. To be sure the concept of physical object has both logical and empirical basis and has, therefore, to be distinguished from the concept of a thingin-itself (Ding an sich). The belief that something is a physical object is grounded on a specific kind of sensory experience: immediate awareness of resistance to our bodily actions in a process of practical activity, immediate awareness that certain portions of space in some intervals of time are filled with something alien to us, exhibiting the properties of mass and/or energy. However a logical jump is needed to derive the concept of physical object from this specific type of experience. The ideas of space and time have a precise meaning only with respect to definite space-time systems. The structural properties of mass and energy are contingent upon some elaborated mathematical formalism.

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From the nature of a physical problem and from the purpose of inquiry it will, then, depend which physical objects, i.e., which objects satisfying the requirements of transcendence, space-time location, mass and/or energy possession, will be considered relevant and this will constitute the system with respect to which the problem of physical necessity will arise. These systems and the very notion of physical necessity are more restrictive than the notion of empirical necessity and the systems to which it is relative. They exclude from the field of possibilities all psychic events, and in case of events that have both a physical and psychic dimension they abstract from the latter and consider them as purely bodily events. A further restriction within a wide range of physical possibilities will be achieved through physical laws. To say, therefore, that a certain type of event x is physically necessary means: (a) that x is a subset of a field of apriori possibilities of the given field of physical objects; (b) that established physical laws are incompatible with any other possibility of the given field except x. It follows from the preceding discussion that the concept of physical necessity is relative in three following respects. It presupposes: (a) a real structure of objects in the external world independently of human consciousness ; (b) a body of knowledge and theoretical principles that precedes any specific inquiry at a given historical moment t; (c) a purpose of inquiry and a set of value assumptions from which it will depend what is to be considered essential, relevant, important, normal, significant, adequate, and what will be discarded as inessential, irrelevant, trivial, abnormal, inadequate. Without a minimum of regularity in the external world the concept of physical necessity would be pointless. It makes sense only if there is a certain order in reality such that whenever some specific conditions are given a specific event occurs. However, there is much less regularity and order in reality than in our theories and models, which inform us about reality in a simplified, idealized way. That is why the concept of physical necessity must be taken in a flexible and historical way. Our theories are incomplete and imperfect models of reality, therefore the occurrence of deviations from necessary patterns is a permanent possibility. These deviations are being called 'chance-events' and they are the consequences of conditions and factors that were neglected, underestimated or completely overlooked in our theory. When they happen and we observe them, the customary way to resolve this conflict between the theory and experience is to rationalize the chance-event by corresponding theoretical

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modification such as expansion of the system, generalization, reVISIOn of the law, building up a super system that will take into account the interaction between two subsystems. In such a wayan event that was accidental with respect to a system 8 1 will become necessary with respect to the system 8 2 , There is another kind of indeterminacy which may give rise to deviations from apparent physical necessity. In the former case it was objective vagueness, i.e., variability and irregularity in the structures of things themselves. In the latter case it is subjective vagueness, i.e., lack of a precise conceptual apparatus and lack of information about relatively simple structures of things. When we do not know the law or set of laws governing a set of phenomena, we have the impression of disorder and irregularity. A too abstract, narrowly specialized knowledge may be another ground for this impression. The existing division of scientific work has as a consequence the fact that an increasing number of scholars are able to offer explanation and prediction of a phenomenon only within the framework of a very narrow field. Knowledge from other fields that may be necessary and sufficient to show the necessity of a seemingly chance event exists for mankind, but does not exist for the given specialist. An economist who builds up an economic model but abstracts from the play of political forces, cultural processes, demographic tendencies, international relations, etnopsychic characteristics of the population, etc., will inevitably find many economic phenomena accidental, unexplainable and unpredictable. Therefore, in addition to ontological relativity of the concept of necessity, one should also take into account its epistemological relativity. A third kind of relativity concerns value assumptions and may be considered axiological. Whenever we are confronted with several seemingly equivalent theoretical systems we make a choice among them on the basis of a scale of values that expresses priorities of our intellectual and practical needs at a given historical moment. For example, the result of Michelson's experiment opened the dilemma: whether to relativize all laws while keeping space and time absolute, or to relativize space and time while keeping all laws invariant with respect to any system of reference. In well-known debates between Bohr and his followers versus Einstein and his followers about the validity of quantum mechanics, empirical data and mathematical formalism were not controversial. The dilemma was whether to preserve the principles of strict determinism and distributivity by allowing empirically untestable 'hidden variables' at the subelectron level, or to stick to empiricism with its demands for testability and operationalism while

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sacrificing determinism and logical principle of distributivity. Bohr's view prevailed because it expressed better the scale of values adopted by modern science: the substitution of operational concepts for speculative ones is considered more important than the conservation of any concept or principle in its classical form. It follows then, that the concepts of necessity and chance are relative to some fundamental value assumptions that are implicit in scientific method and have widely accepted, intersubjective character at a certain historical period. 5. Three types of physical necessity

We should distinguish among the following three types of physical necessity: (1) It is characteristic for the first one that existing physical laws in all cases exclude all possibilities except one. This is the case with necessity of death or necessity of many other physical processes that take place without deviations and exceptions, on the basis of very simple laws, e.g., eclipses of the sun and moon at regular intervals. This kind of necessity might be called strict necessity; it holds for simple systems of a high degree of uniformity and very small number of relevant factors. (2) It is characteristic for the second kind-tendential necessity that (a) the subclass of aposteriori possibilities that are compatible with the laws of the system contains several members; (b) the frequency of realizations of these possibilities has the form of statistical distribution with a central tendency and marginal cases that have relatively low probability of realization. This is no longer linear, strict necessity of individual events; here the concept of necessity makes sense only with respect to a whole class of different events: most logical possibilities are excluded. The basic differences between systems to which these two concepts of necessity are applicable are the following: (a) In the former case the system is fixed and closed. All relevent factors are included. In the latter case the system does not have fixed boundaries: some factors are at the very boundaries of the system or act in the system although we overlooked them, bringing about certain deviations from the normal course of events in the system. (b) To some extent tendential necessity is also the result of unpredictable and uncontrollable interactions within the system (such

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as interaction between the particles of the equipment for observation and examined microparticles). (3) To a third type of empirical necessity belong all those cases where there is no central tendency, where the laws determine a set of alternative possibilities with equal or irregularly distributed probability of realization. What happens when throwing a dice is a simple case of the necessity of this type. Physical laws exclude many logical possibilities: friction excludes permanent movement of the dice, gravitational action and inertia eliminate the possibility of the dice balancing on one edge, physicochemical homogeneity of the dice excludes the possibility of considerably greater frequency of one side over the others. In this situation it makes perfectly good sense to say that a set of six possibilities has been determined. The concept of determination here has negative meaning (as in famous Spinoza's dictum: Determinatio negatio est). The situation in quantum physics is much more complex but analogous. The movement of a microparticle could in principle be determined by means of the wave function '1jJ the arguments of which will be either spacetime parameters (x, y, z, t) or momentum parameters (px, py, pz, t). If we have the wave function in some initial moment to, the equation of Schrodinger will enable us to determine the wave function in any future moment tn' The state of microparticle in tn cannot be fully specified. However the equation allows us to specify the probability of the realization of any possible state of the particles. Therefore, it makes sense to speak about determination and necessity. The wave function excludes a very large number of logically possible states of a particle and reduces the number of (more or less) probable states to a limited set of real possibilities. Therefore when we have to deal with a whole system of particles at the moment to, the state of the system at t; can be determined with a high degree of reliability. This third type of physical necessity can obviously be derived' from the second type when one replaces the Gaussian form of the distribution of probabilities by any other form, preserving only the basic idea of determination: drawing a demarcation line between real and fictitious possibilities.

PROGRAM OF INVITED ADDRESSES AND SYMPOSIA FOURTH INTERNATIONAL CONGRESS FOR LOGIC, MEmOJ)OLOGY AND PHILOSOPHY OF SCIENCE BUCHAREST, RUMANIA, AUGUST 29-SEPTEMBER 4, 1971

Sunday morning, August 29

Opening Ceremonies Nicolae Ceausescu, President of the State Council of the Socialist Republic of Rumania Athanase Joja, Chairman of the Rumanian Organizing Committee Miron Nicolescu, President of the Academy of the Socialist Republic of Rumania Miron Constantinescu, President of the Academy of Social and Political Sciences of the Socialist Republic of Rumania Stephan Korner, President of the Division of Logic, Methodology and Philosophy of Science Mircea Malita, Minister of Education of the Socialist Republic of Rumania Sunday afternoon, August 29

Hour Address-Section IV. A. Tarski (USA), Reflections on the Present State of Set Theory. Symposium on Currents in 19th Century Philosophy of Physics-Section XII. M. Hesse (UK), Chairman M. Markovic (Yugoslavia) G. Buchdahl (UK) M. P. Asimov (USSR) B. Kedrov (USSR) Monday morning, August 30

Hour Address-Section I, A. Mostowski (poland), Partial Orderings of the Family of «r-Models. Symposium on Probability as an Objective Disposition-Section VI. (Dedicated to the memory of Rudolf Carnap.)

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P. Suppes (USA), Chairman R. N. Giere (USA)

I. Hacking (UK) W. Stegmiiller (BRD)

Hour Address-Section IX, F. Bresson (France), Possibilities of the Application of Mathematics for the Psychological Sciences. Monday afternoon, August 30

Hour Address-Section I, Gr. C. Moisil (Rumania), Elementary Logic. Symposium on Cellular Automata and Their Significance to the Foundations of Biology-Section VIII. M. Beckner (USA), Chairman A. Lindenmayer (The Netherlands) M. Arbib (USA) V. Varshavsky (USRR) G. Herman (USA) Hour Address-Section XII, S. Toulmin (USA), Rationality and the Changing Aims of Inquiry. Tuesday morning, August 31

Symposium, Can Psychology Bypass the Brain ?-Section IX. M. Constantinescu (Rumania), Chairman W. Rosenblith (USA) D. Davidson (USA) V. P. Zintchenko (USSR) Hour Address-Section XI, H. Schnelle (BRD), Problems of Theoretical Linguistics. Tuesday afternoon, August 31

Symposium on Theory of Hierarchies and Their Applications to LogicSection I. G. Sacks (USA), Chairman Y. Moschovakis (USA) Y. Ershov (USSR) Hour Address-Section II, Yu. V. Matijasevic (USSR), On Recursive Unsolvability of Hilbert's Tenth Problem. Hour Address-Section X, L. Mink (USA), The Divergence of History and Sociology in Recent Philosophy of History. Half-Hour Addresses-Section XII. A. Joja (Rumania), La Doctrine de l'Universal chez Aristote. E. McMullin (USA), Newton's Concept of Explanation.

PROGRAM

979

G. Patzig (BRD), Aristotle, Lukasiewicz and the Origins of Many-Valued Logic. E. Scheibe (BRD), The Approximative Explanation and the Development of Physics. Wednesday morning, September 1 Half-Hour Addresses-Section I. D. Gabbay (USA), A Survey of Decidability Results for Modal, Tense and Intermediate Logics. A. Lachlan (Canada), On the Number of Countable Models of a Countable Superstable Theory. M. Morley (USA), Countable Models with Standard Part. J. Robinson (USA), Solving Diophantine Equations.

Hour Address-Section IV, D. Prawitz (Sweden), Towards a Foundation of General Proof Theory. Half-Hour Addresses-Section VII. A. Fine (USA), The Two Problems of Quantum Measurement. N. Ovchinnikov & J. Akchurin (USSR), Concerning Unity of Knowledge in Physics. A. Shimony (USA), The Status of Hidden-Variable Theories. J. Stachel (UK), Metric and Affine Connection in Theories of Gravitation. M. Strauss (DDR), Two Concepts of Probability in Physics. Wednesday afternoon, September 1

Hour Address-Section V, G. H. von Wright (Finland), On the Logic and Epistemology of the Causal Relation. Symposium on Perspectives in the Philosophy of Mathematics-Section IV. H. Hermes (BRD), Chairman P. Martin-Lof (Sweden) A. Grzegorczyk (Poland) H. Putnam (USA) G. Kreisel (USA) J. Shoenfield (USA) Half-Hour Addresses-Section VIII. M. Beckner (USA), Behavior and Consciousness. D. Hull (USA), Reduction in Genetics-Doing the Impossible. V. Kremiansky (USSR)-read by A. S. Mamzin (USSR), Hyperstructures and 'Infa'-Systems of Organized and Organizing Information in Biology. L. Lofgren (Sweden), On Formalizability of Learning and Evolution.

980

PROGRAM

Thursday morning, September 2

Hour Address-Section III, D. Scott (USA), Models for Various Type-Free Calculi. Half-Hour Addresses-Section V.

J. Hintikka (Finland), On the Different Ingredients of an Empirical Theory. P. Kopnin (USSR) & V. Lektorsky (USSR), Gnosiological Aspects of Present-Day Science. L. Kruger (BRD), Falsification, Revolution, and Continuity in the Development of Science. F. Kutschera (BRD), Induction and the Empiricist Model of Knowledge. Hour Address-Section X, R. Moldovan (Rumania), New Trends in the Method of Social Sciences, and Especially of the Economic Sciences. Thursday afternoon, September 2

Half-Hour Addresses-Section III. D. Knuth (USA), The Dangers of Computer-Science Theory. M. Schiltzenberger (France), Sur un Langage Equivalent au Langage de Dyck. B. Trakhtenbrot (USSR), Formalization of Some Notions in Terms of Computational Complexity. G. Tseytin (USSR), Features of Natural Languages in Programming Languages. Friday morning, September 3

Half-Hour Addresses-Section II. A. Dragalin (USSR), Constructive Mathematics and Models of Intuitionistic Theories. Yu. T. Medvedev (USSR), An Interpretation of Intuitionistic Number Theory. A. Robinson (USA), Nonstandard Arithmetic and Generic Arithmetic. N. Shanin (USSR), Some Generalizations of Skolem-Goodstein Approaches to Developing of the Constructive Mathematics. Half-Hour Address-Section VI, C. Stael von Holstein (Sweden), The Concept of Probability in Psychological Experiments.

PROGRAM

981

Friday afternoon, September 3

Symposium onDeductive Models ofScience and Their Alternatives-Section V. W. Stegmliller (BRD), Chairman M. Hesse (UK) D. P. Gorski (USSR) A. Musgrave (New Zealand) A. L. Uyemov (USSR) C. G. Hempel (USA) Hour Address-Section VI, L. J. Savage (USA), Probability in Science: A Personalistic Account. Saturday morning, September 4

Half-Hour Addresses-Section VI. L. Laudan (USA), Induction and Probability in the Nineteenth Century. O. Onicescu (Rumania), Extensions of the Theory of Probability. Symposium on Causality in the Social Sciences-Section X. H. Wold (Sweden), Chairman A. Wellmer (Canada) J. Coleman (USA) D. Wiley (USA) S. Nowak (Poland) Saturday afternoon, September 4

Hour Address-Section VII, P. Achinstein (UK), Macrotheories and Microtheories. Half-Hour Addresses-Section XI. S. Dik (The Netherlands), Some Remarks on the Notion 'Universal Semantics' . J. F. Staal (USA), Uncontained Rules of Meaning. J. P. Thorne (UK), On the Grammar of Existential Sentences.