MATHEMATICAL LOGIC IN LATIN AMERICA Proceedings of the IV Latin American Symposium on Mathematical Logic held in Santiago, December 1978
Edited by
A. I. ARRUDA Universidade Estadual de Campinas Brazil R.CHUAQUI Universidad Cat6lica.de Chile Santiago, Chile N. C. A. DA COSTA Universidade de Sao Paulo Brazil
1980
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ISBN: 0444 85402 9
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Library of Congress CatalogIng in Publication Data
Latin-American Symposium on Mathematical. Logic, 4th, Santiago de Chile, 1978. Mathematical. logic in Latin America. (Studies in logic and the foundations of mathematics v. 99) Bibliography: p. Includes indexes. 1. Logic, Symbolic and mathematical.--Congresses. I. Arruda, Ayda I. II. Chuaqui, R. III. Costa, Newton C. A. da. IV. Title. V. Series. QA9.A1L37 1978 511' •3 79-20797 . ISBN 0-444-85402-9
PRINTED IN THE NETHERLANDS
to
ALFRED TARSKI teacher and friend
PREFACE
This volume constitutes the Proceedings of the Fourth Latin American Symposium on Mathematical Logic held at the Catholic University of Chile, Santiago from December 18 to December 22, 1978. The meeting was sponsored by the Pontifical Catholic University of Chile, the Academy of Sciences of the Institute of Chile, the Association for Symbolic Logic, and the Division of Logic, Methodology and Philosophy of Science of the International Union of History and Philosophy of Sci ence. The Organizing Committee consisted of Ayda I. Arruda, Rolando B. Chuaqui (chairman), Newton C.A. da Costa, Irene Mikenberg, and Angela Bau (Executive Secretary). Most of the sponsors were represented at the opening session. The Catho 1i c University was represented by its Rector Jorge Swett, its Vice-Rector for Academic Affairs Fernando Martinez, and its Dean of Exact Sciences Rafael Barriga who gave an address. The President of the Chilean Academy of Sciences, Jorge Mardones,a1so said a few words. Representing the Association for Symbolic Logic, Newton da Costa, Chairman of its Latin American Committee, opened the meeting. In preparation for the Symposium there was a logic year at the Catholic University. Advanced courses and seminars were given by Ayda I. Arruda (Universidade Estadua1 de Campinas, Brazil), Jorge E. Bosch (Centro de Altos Estudios en Ciencias Exactas, Buenos Aires, Argentina), Rolando Chuaqui (Universidad Cato1ica de Chile), Newton C.A. da Costa (Universidade de S~o Paulo, Brazil), and Irene Mikenberg (Universidad Catolica de Chile). Preceding the Symposium, there was a two-week Seminar consisting of short courses. Below are reproduced the Scientific Programs of the Seminar and the Symposium. A look at these programs shows the progress in research in Logic in Latin American in the last few years. (1) The papers which appear in this volume are the texts, at times considerably expanded and revised, of most of the adresses presented by invitees to the meeting. Also included are two papers by Australian logicians (Bunder and Routley) who could not come because of difficulties in booking space in airlines. Expanded versions of a few short communications are also included. This volume is dedicated to Professor Alfred Tarski. In his previous visit to Chile and Brazil, he stimulated the development of Logic and encouraged the Organization of the III and IV Symposia. His influence was decisive in getti ng the sponsorship of the Association for Symbolic Logic. Many of the participants from Latin America and the United States can claim him directly or indirectly as their
(1) For a history of the previous Latin American Logic Symposia see: A short history of the Latin American Logic Symposia in Non-Classical Logics, Model Theory and Computability, North-Holland Pub. Co. 1977, pp, lx xv i l , v
vii
viii
PREFACE
teacher. Although he could not be physically present at the Symposium, he followed the proceedings with great interest. The Organizing Committee would like to acknowledge the financial support giv_ en to the meeting and the publication of these proceedings by the following institutions: the Catholic Unjversity of Chile, the Academy of Sciences of the Institute of Chile, the Fundacion de Estudios Economicos del Banco Hipotecario de Chile the Comision Nacional de Investigaciones Cientificas y Tecnologicas, the Interna: tional Union of History and Philosophy of Science, and the Coca-Cola Export Co. The editors would like to thank Irene Mikenberg, who was instrumental in the preparing of the camera-ready copy. Most of the typing was done by M. Eliana Cabanas assisted by Rosario Henriquez. The editors wish to express their appreciation. The editors would also like to thank North-Holland Publishing Co. for the inclusion of this volume in the series Studies in Logic and the Foundations of Mathematics.
The Editors
Instituto de Matematica Pontificia Universidad Catolica de Chile June 1979.
PROGRAM OF THE SEMINAR
Ayda I. Arruda and Newton C.A. da Costa, (Brazil), TopiC6 on Modal LoMc. (Si x 1ectures). Jorge Bosch, (Argentina), TopiC6 in the
Phito~ophy
PanaCOn6~tent
06 $eience. (Six lectures).
Luis F. Cabrera, (Chile), Equivalence Retation6 and the Continuum (Three lectures). Ulrich Felgner, (West Germany), The Continuum
Hypoth~~.
Gen~zed
Hypoth~~.
(Two lectures).
Ulrich Felgner, (West Germany), Apptieation6 06 the Axiom 06 Algeb~ and Topology. (Ten lectures). Jerome Malitz, (U.S.A.),
and
Contnuetib~y
to
Quanti6ie46. (Four lectures).
PROGRAM OF THE SYMPOSIUM
DECEMBER 18. 9,30 - 12,00
Opening Session.
15,15 - 16,05
N.C.A. da Costa, (Brazil), A Model Theo~etieal Ap~oaeh to Vbto~. 'I R. Chuaqui, (Chile), Foundatiol~ 06 S~tieal Metho~ U~ing a Semantieal VeMnition 06 P~obab~y.
16,30 - 17,20
J.R. Lucas, (England),
14,00 - 14,50
T~h, P~obab~y
and Set
Theo~y.
DECEMBER 19. 9,00 - 9,20 9,20 -
9,40
9,40 - 10,00
~1.G.
ment
Schwarze, (Chile), AuomatizatiOn6 604
a -AdcU:Uve MeMMe-
Sy~t~.
M.S. de Gallego, (Brazil), The Lattice StAuetMe 06 4-Vafued LukMie.wi.cz Afgeb~. A. Figall 0 ,(Argent ina), The Vet~nant 'Sy~tem 604 the MO!Lgan Afgeb~M ov~ a Finite O~d~ed Set.
10,00 - 10,20
A.M.Sette, (Brazil), A Funetonial
10,30 - 11,,00
I. Mikenberg, (Chile), A Clo~Me 60~ P~al Algeb~.
ix
App~oach
to
F~ee
Ve
Int~p~etab~y.
x
PROGRAM OF THE SYMPOSIUM
11,15 - 12,05
U. Felgner, (West Germany), The Model and Undeeidab~lj.
ab~lj
Theo~lj
06
FC-G~oup~, Ve6~
DECEMBER 20. 9,00 -
9,50
E.G.K. Lopez-Escobar, (U.S.A.), Tnuth-value Semantico on.b.,Uc. Logic..
60~
Intuit-
10,15 - 10,45
A.I. Arruda, (Brazil), On
11,00 - 11,50
W. Reinhardt, (U.S.A.), S~6ac.Uon Ve6~on and Axio~ 06 InM-ni.tlj -in a Theo~lj 06 P~opMUu wUh Nec.u~Ulj OpeMtM.
14,00 - 14,50
J. Bosch, (Argentina), To~d a Conc.ept 06 Seienti6-ic. TMough Spec.-ltLe. RelaUvUlj.
15,15 - 16,05
O. Chateaubriand, (Brazil), An Exam-lnaUon 06 06 MathemaUco.
P~c.o~~tent
Set
Theo~lj.
Theo~y
GOdel'~ Phieo~ophlj
DECEMBER 21. 9,00 -
9,20
E.H. Alves, (Brazil), Some
9,20 -
9,40
M. Corrada, (Chile), A Fo~a1-lzaUon 06 the Imp~ed-ic.aUve Theo~lj 06 CW~U U~-ing Z~elo'~ AUMond~ng~auom W-lthout PMameteM.
9,45 - 10,35 11,00 - 11,50 14,20 - 14,40
R. Vaught, (U.S.A.), Model M. Benda, (U.S.A.), On
Theo~lj
on the Log-ic. 06
Vaguenu~.
and A~6-ibte Set6.
Pow~6ut Auo~
L. F. Cabrera, (Chile), Un-lVeMat Set6 ~e1.
15,00 - 15,30
Rem~k6
Set6.
06 Induc.Uon. 60~
Sel6duat
CW~U
06 80-
H.P. Sankappanavar, (Brazil), A C~ct~zaUon 06 P~nc.-lpat 06 Ve MM~an Atgeb~ and -lU AppUc.aUo~.
Cong~enc.u
15,45 - 16,30
C.C. Pinter, (U.S.A.), Topo.tog-ic.a.t Vua.tUlj TheMlj.in Log.ic..
A.e.geb~a-ic.
DECEMBER 22. 9,00 -
9,20
L. Flores, (Chile),
9,20 -
9,40
M. Manson, (Chile), Veontic., Manlj-valued and No~ve Log.ico.
9,45 - 10,45 11, 00 - 11, 50
Hempel'~
Nomo.tog-ical Veduc.Uve Model.
X. Caicedo, (Colombia), Bac.Iz-and-60IL.th Slj6tein6 6M A~b~lj Quan-
UMeM.
J. Malitz, (U.S.A.), Compact
F~gment6
06
H-igh~ O~d~
Log.ic..
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © North-Holland Publ ishing Company, 1980
A SURVEY OF PARACONSISTENT LOGIC
C*)
AlJda I. AMuda
ABSTRACT. This paper constitutes a first attempt to sistematizethe present stateof thedevelopment of paraconsistent logic, as well as the main topics and open questions related to it. As we want this paper to have mainly an expository character, we wil I not in general be rigorous, especially when an intuitive presentation is better for a first understanding of thequestions under consideration, as, for example, in Section 1. Section 6 is perhaps the only one where the reader wi 11 find some original results. The bibl iography, though large, is of course not intended to be complete. A general idea of the content of this paper is given by the Index.
INDEX,
1.
Informal
1
Introduction.
2.
Paradoxes, Antinomies, and Hegel's Thesis.
3
3.
Historical Development of Paraconsistent Logic.
6
4.
Objectives and Methods of Constructijn of Paraconsistent Logics.
11
5.
Da Cos ta 's Pa racons is ten t Log i c.
13
6.
Paraconsistent Set Theory.,
17
7.
Miscellaneous Topics.
22
8.
The Philosophical Significance of Paraconsistent Logic.
24
9.
Open Questions.
26
10.
Bibl iography.
27
L
INFORMAL INTRODUCTION,
Let £ be a language and IF the set of formulas of £; then any non empty set of F i s i sa'id to be a
ptWPO.6~UOl1at
.6lJ.6tem 06 £.
sub-
We say that a propositional
(*) This paper was partially written when the author was Visiting Professor at the Catho1 ic University of Chile, with a partial grant of the Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP), Brazi 1.
2
AYDA 1. ARRUDA
system S is ruv- thU-L.6 (or HeJUl.c..UtlL6-Hege£'J.> thu-LJ.» is the statement that there are true contradictions (cf. Petro v 1974). Sometimes, Hegel's thesis is also formulated as to imply that consistency is a sufficient but not necessary condition for the existence of abstract objects; concerning the existence of concrete objects, consistency is neither necessary nor sufficient. Clearly, Hegel's thesis can only be supported with the help of a paraconsistent logic. We shall show that at the abstract level Hegel's thesis is in fact true: there are paraconsistent theories (like the set theories described in Section 6) in which certain objects have inconsistent properties; for example, they belong and
A SURVEY OF PARACONSISTENT LOGIC
5
simultaneously do not belong to the same class. Therefore, one of the rna i n achievements of paraconsistent logic is to have proved that Hegel's thesis is true at the formal and abstract level. This means that an antinomy from the point of view of classical logic may be -surprising enough - a veridical paradox from the standpoint of paraconsistent logic. (Such result seems to signify that in the field of logic there is a certain kind of relativism, an issue which we will not discuss here.) Now, what can we say about the validity of Hegel's thesis in connection with concrete objects? It seems to us that paraconsistent logic is unable to settl e such problem. Only special sciences and epistemology can establish the truth or falsity of Hegel's thesis at the level of real, concrete objects; or, equivalently, if the real world is consistent or not. Anyway, we believe that Petrov is right in his interpretation of existing antinomies: they do not prove that Hegel's thesis is true for concrete objects, but at least they give us some hints on the plausibility of it. According to Petrov 1971, p. 388: "No elimination of fallacy in scientific knowledge has negative consequences for the adequacy or the completeness of knowledge. "Certain ... antinomies (as the classical ones originated in quantum physics by the wave and corpuscular aspects of elementary particles) , however, can in principle be eliminated only with the aid of theories and methods the acceptance of which encroaches too much upon the adequacy or the completeness of knowledge. "The conclusion is therefore plausible that certain ... antinomies are not fallacies as standard logic wants us to believe, but are peculiar obj ective truths." Therefore, we can conclude that the fundamenta 1 si gnifi cance of paraconsi stent logic in connection with the common antinomies is that we are now able to accept Therefore, most of them as veridical paradoxes, at least at the abstract level. from now on we should not try to exclude antinomies a priori, because contradictions are forbidden by logic. Only a posteriori elimination of antinomies is legitimate depending on logical, scientific, and epistemological reasons. Finally, it is worthwhile to remark that, among contemporary phi 1osophers, Wittgenstein maintained original views about contradiction and logic: "Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from contradiction." (Wittgenstein 1964, p. 332.) "If a contradiction were now a cctually found in arithmetic, that would only prove that arithmeti~ with such a contradiction in it could render a very good service. "
AYOA I. ARRUDA
6
(Wittgenstein 1956, p. 181e.)
3.
HISTORICAL DEVELOPMENT OF PARACONSISTENT LOGIC.
Several philosophers since Heraclitus, including Hegel, until Marx, Engel s, and the present day dialectical materialists, have proposed the thesis that contradictions are fundamental for the understanding of reality; in other words, they claim that real ity is contradictory, that is to say, that Hegel's thesis is' true in the real world. Clearly, if one accepts Hegel's thesis, one has to employ a new kind of logic (a paraconsistent logic), in order to study inconsistent but non-tri vi al theories. Strangely enough, philosophers who accept Hegel's thesis have not estab-· lished any formal system of paraconsistent logic. Instead of this, some of them have proposed the so-called dialectleal tagle, whose nature is rather obscure. Therefore, in our account of the historical development of paraconsi stent logic, we shall take into consideration only the work of logicians. According to ~ukasiewicz 1971, Aristotle had already an idea of the possibility of derrogation of the principle of contradiction, and consequently, of the possibility of paraconsistent logic. In effect, tukasiewicz writes: "Now according to Aristotle the principle of contradiction is not the highest law, at least not in the sense that it yields a neaesry pressuposition for all other l.oq-i c a.l. axioms. In particular the principle of the syllogism is independent of the principle of contradiction. This is gotten from a Long overlooked and misunderstood passage in the Second Analytics: An. Post. All, 77a 10-22 ... - 'The impossibility of joint affirmation and denial is pressuposed by no proof (syllogysm) unless the conclusion itself was also to have demonstrated such. Then it is demonstrated insofar as one accepts that it is true to predicate the major term of the middle term and not true to deny it. But as far as concerns the middle term and likewise the minor term, it makes no difference to hold that it is and is not. If, for instance, an object is given (e. g., Callias) of which one can truthfully predicate that it is man and insofar as man just a living creature and not also not a living creature; so will it be true to predicate that Callias is a living creature and not also not a living creature, even if man were not man and Callias not Callias. The reason for this lies in the fact that the major term holds not only of the middle term but also of other objects as well because it has a greater range (than the middle term); so that it makes no difference in the conclusion, if the middle term is the
A SURVEY OF PARACONSISTENT LOGIC
7
same and not the same.' "According to Aristotle this syllogism is valid (A = living creature, B =man, C = Callias): B is A (and not also not-A) C, which is not-C, is Band not-B C is A (and not also not-A) "However, if a syllo gism remains val id when the pri ncipl e of contradiction doesn't, then the principle of syllogism (and indeed the dictum de omni et nullo) is independent of the principle of contradiction. " (cf. tukasiewicz 1971, pp. 503-504.) Hence, perhaps Aristotle would deserve the title of founder of paraeonsis tent logic. Leaving out Aristotle, the two real forerunners of paraconsistent logic are J. tukas i ewi cz and N. A. Vas il 'ev. Both, i ndependently each other, argued between 1910 and 1911 that similarly to non-Euclidean geometry, a revision of the ba sic laws of Aristotelian logic would yield non-Aristotelian logics. And both suggested to eliminate the principle of contradiction (it is interesting to note that Vasil'ev also proposed the el imination of the te.ilium VtOVt da.tuJt, thus being sometimes considered as a precursor of many-valued logic, a field created by Post and lukasiewicz seve~al years later). tukasiewicz published in 1910, in Polish, a book and a Note (see tukasiewicz 19l0a and 19l0b), in which he tries to show that the principle of contradiction is not so fundamental as one usually thinks. He writes: "(A) The principle of contradiction cannot be proven by proclaiming it directly evident. For: (a') evidence does not appear to be a permissible criterion of truth; it turns out that false propositions as well are held to be evident (cf. the Cartesian proof of God). (b') the principle of contradiction does not appear to be evident to everyone: for the old eristic thinkers of Megara or for Hegel it was in all probability not evident. "(B) The principle of contradiction cannot be proven by setting it up as a natural law determined by the psychical organization of man. For: (a') it is possible td determine false proposltlons by our psychical organization (cf. e. g., many sensory hallucinations); (b') it is questionable whether the principle of contradiction can be validated as a iaw determined by the psychical organization of man. "(C) The principle of contradiction cannot be proven on the basis
8
AYDA 1. ARRUDA
of the definition of statements or negations. .•. For: (a') If one accepts that the negation "A is notE" means the falsity of the affirmation "A is B", then the principle of contradiction is not to be deduced therefrom. The notion of logical multiplication is not contained in the definition of negation, respectively falsity, and it is this notion which directly bestows on the principle of contradiction its characteristic imprint . . . . In terms of the definition of falsity or negation, however, it would still be possible to accept that the assertions "A is B" and "A is not B" hold at the same time in that they are both true and false at the same time. (b ") of course, if one prefers rather to avoid designating one and the same proposition as true and false, another definition of falsity can be set up which is of much greater account than the usual definition in terms of the basic thought in the concept, in that it is much more carefully formulated. The basic notion of falsity is, namely, that false propositions are no representation of the objective, or - in other words - that false propositions
correspond
to
If the principle of contradiction fails to hold now, there will be cases in which A is and is not Bat the same time. The principle of contradiction can in no way be derived from this definition of falsity." (eL tukasiewicz 1971, pp. 505-506.) N. A. Vasil 'eV was a medical doctor who becomes Professor of Philosophy atthe University of Kazan, Russia. His ideas abo~t the possibility of derrogation of the 1aws of contradi ction and exc1uded mi ddl e were pub1i shed ina seri es of papers between 1910 and 1913. In these' papers he proposed some views for the construction of logical systems in which these laws are not valid. "Vas il 'ev int ended to cons t r uc t a non-Aris tote l i an and universa l. logic, universal in the sense that it might cover an infinite number of logical systems. For him, a logical system is composed of two parts: that which he called Metalogic, i. e., 'an indispensable oore nothing objective.
of laws related to thought whioh are neoessary for any thinking and whioh oannot be eliminated from logio without its losing its logical charaoter' (Comey 1965); and a second part which we call here ontological basis of logie, i. e.,
'a varying range of laws which
are
funotions of the properties of the known objeots.' (Comey 1965.)
"Vasil' ev intended to see which postulates of logic could be changed or eliminated from logic without its losing itslogicalcharacter. Thus he was led to drop the (ontological) law of excluded middle and also the LAW OF CONTRADICTION which he took in the Kantian
A SURVEY OF PARACONSISTENT LOGIC form: ev
9
'no object can have a predicate which contradicts it'. Vasil'-
distinguished the law of contradiction from the LAW
CONTRADICTION: 'one and the same judgement cannot ly true and false'. 1965.)
OF NON-SELF
be s nul taneous-
Vasil' ev took these to be different
The latter helong to me t a l o g i c , and the first,
aws'. (Corney retained,
L
would belong to the ontological basis of logic. "Vasil' ev considered worlds in connection to which t h e re are only three sorts of different basic (predicative) judgements: tive, "S is P"; negative, "S is not P"; and indiferent
affirma-
(or contra-
dictory ), "S is and is not P", such that only one of these
ments can be true for a given ohject and predicate.
judge-
From these hy-
potheses he delineated an Imaginary Logic with an ontological law of excluded fourth substituting the ontological law of excluded middle.
Later he generalized these ideas to a logic with an ontological law of excluded (n+l) t.h , n
> 2.
He also tried to show
that
his
logic
with his law of excluded fourth has a classical interpretation,
as
it is the case with the Imaginary Geometry of Lobachewski." (Arruda 1977, pp. 4-5.) The first logician to construct a system of paraconsistent propositional calculus was S. Jaskowsk i (see Jaskowsk i 1948 and 1949), following a suggestion of tukasiewicz.
He called his system
diocU6~ive
(or diocounhive) logic.
hensive and developed account of discussive logic may be found in
A compre-
da Costa
and
Dubikajtis 1968 and 1977, and in Kotas and da Costa 197+a. Jaskowski motivated his discussive logic referring to several problems which originated the need of paraconsistent logic from Jaskowski 1948): 1)
\
(in the following, quotations are
'
The problem of organizing deductively theories which contain contradic-
tions, as it happens with dialectics:
" 'The principle that
dictory statements are not both true and false is the
two
contra-
most certain
This is how Aristotle ... formulates his opinion known as
of all'.
the logical principle of contradiction.
Examples of convincing rea-
sonings which nevertheless yield contradictory conclusions were the reason why others sometimes disagreed stand.
with the
Stagiri t e ' s
That was why Aristotle's opinion was not in the least
versally shared in antiquity.
His opponents included Heraclitus of
Ephesus, Antistenes the Cynic, and others century Heraclitus ideas were taken up by Hegel, classical
firm uni-
In
the early who
logic a new logic termed by him dialectics,
opposed
19 th to
in which co-
existence of two contradictory statements is possible." 2)
To study theories where there are contradictions caused
by vagueness:
10
AYDA I. ARRUDA
"The contemporary formal approach to logic increases the precision of research in many fields, but it would not be correct to formulate Aristotle's principle of contradiction as: 'Two contradictory sentences are not both true'. We have namely to add: 'in the same language' or 'if the words occurring in those sentences have the same meaning'. This restriction is not always observed in every day usage, and in science too we often use terms that are more or less vague (in the sense explained by Kotarbinski ... ), as was noticed by Chwistek ... Any vagueness of the term a can result in a contradiction of sentences, because with reference to the same object X we may say that 'X is a' and also 'X is not a', according to the meaning of the term a adopted for the moment." 3) In order to study directly some empirical theories whose postulates or basic assumptions are contradictory. " ... it is known that the evolution of empirical disciplines is marked by periods in which the theorists are unable to explain the results of experiments by a homogeneous and consistent theory, but use different hypotheses, which are not always consistent with one another, to explain the various groups of phenomena. This applies, for instance, to physics in its present-day stage. Some hypotheses are even termed working hypotheses when they result in certain correct predictions, but have no chance of being accepted for good, since they fail in some other cases. A hypothesis which is known to be false is sometimes termed a fiction. In the opinion of Vaihinger ... fictions are characteristic of contemporary science and are indispensable instruments of scientific research. Wheth~r we accept that extremist and doubtful opinion or not, we have to take into account the fact that in some cases we have to do with a system of hypotheses which, if subjected to a too precise analysis, would show a contradiction among them or with a certain accepted law, but which we use in a way that is restricted so as not yield a self-evident falsehood." Jaskowski had already constructed a paraconsistent propositional calculus. but N. C. A. da Costa is actually the founder of paraconsistent logic. Independently of the work of Jaskowski, he started in 1958 (cf. da Costa 1958), to develop some ideas which led him to the construction of several systems (see Section 5) 0 f paraconsistent logic. including not only the propositional level but also the predicate level (with and without equality). the corresponding calculi of descriptions, as well as some applications to set theory. Da Costa systems were extended and studied by several authors as. for example, J. E. de Almeida Moura. E. H. Alves, M. Fidel, M. Guillaume, A. Lopari c , D. Marconi. R. Raggio, etc.
A SURVEY OF PARACONSISTENT LOGIC
11
Da Costa and his collaborators investigated also several other systems of paraconsistent logic, some of them having intimate connections with relevant logic (see Arruda and da Costa 1965). In the last years many logicians contributed to the development of paraconsistent logic (some of them quite independently of the works of Jaskowski and da Costa), for example: F.G. Asenjo, L. Dubikajtis, W. Dziobiak, T. Furmanowski, J. Kotas, L. H. Lopes dos Santos, R. K. Meyer, A. Neubauer, J. Perzanowski, G. Priest, R. Routley, and J. Tamburino. It deserves to be mentioned that D. Nelson in a pioneering paper (Nelson 1959) studied a system of paraconsistent logic and applied it to arithmetic. The term pakaeon6i¢tent log~e was coined by the Peruvian philosopher F. Mira Quesada, and was employed by the first time in a lecture delivered by him at the Third Latin-American Symposium on Mathematical Logic, held at the University of Campinas, Brazil, in 1976 (cf. Mira Quesada 1976). Today, paraconsistent logic is a growing field of logic, and is being cultivated especially in Brazil (Campinas and Sao Paulo). Poland (Torun and Katowice), Australia (Camberra), and U. S. A. (Pittsburgh). We shall end this historical sketch by also observing that nowadays paraconsistent loqic is beinq employed in the formalization of dialectical logic (da Costa and Wolf), and to study aspects of the thought of some dialectical materialists (Apostel). Although dialecticians think that dialectics is in principle unformalizable, the fact is that formalization constitutes a powerful tool to make dialectical logic, and dialectics in general, more understandable. Here we have a situation similar to intuitionism: although intuitionists think that intuitionistic logic is in principle not. formalizable, they do not deny that formalization is important for a better understanding of their logic.
4. OBJECTIVES AND METHODS OF CONSTRUCTION OF PARACONSISTENT LOGICS. In order to better motivate the study of paraconsistent logic - its objectives and methods of construction - we refer to other problems besides those mentioned by Jaskowski. 1) To study directly the so-called logical and semantical paradoxes. For example, if we want to study directly the paradoxes of set theory. not trying to avoid them (as usual), we need to construct set theories ifl which such paradoxes are derivable, but are not formal antinomies. In this case we need a paraconsistent logic. 2) To have logical systems in which paraconsistent theories may be based. For example. logical systems to found certain paraconsistent reconstruction of the theory of Meinong, different and possibly stronger versions of the usual set the-
12
AYDA I. ARRUDA
ories, naive set theory, dialectics, and certain physical theories which perhaps are inconsistent (certain versions of quantum mechanics). 3) To study certain principles in their full strength as, for example, the principle of comprehension in set theory or in higher-order predicate logic. 4) To understand better the concept of negation. It seems that the notion of negation is precise only when we consider negations of simple propositions, as, for example, in the following sentences: "This sheet of paper is not white", or "These bottles are not cold". These simple and intuitive uses of negation can be extended in several ways, originating different categories of negati on, among which there will be the classical negation, and some paraconsistent negations. As we shall see, all negations that will be treated in this paper have some intuitive nucleus which is extended in several distinct ways. The creation of non-Euclidean geometries was one of the most important steps in the evolution of the human thought. Leaving aside their rel evance a s new mathematical disciplines, they have a philosophical significance. In effect, after the i r discovery, we understand better the meaning of some fundamental notions of science, and even the real meaning of scientific knowledge. We believe that the same happens in connection with paraconsistent logic: the construction of such logics, and of some heterodox logics, constitutes a fundamental experience of thought, whose byproducts are of basic importance for the understanding of the true meaning of logicity. There are two main methods of construction of paraconsistent logics: 1) Firstly, the syntactical method. By an appropriate modification of a given system of logic we can get an intuition of what would be a "good" paraconsistent logic. In this way, the systems C n , 1';;; n .;;; w , and the systems P and P* (see Arruda and da Costa 1965) were- obtained. Of course, after a system of paraconsistent logic is constructed, it is reasonable and convenient to look for a semantics for it. This was the case concerning the above mentioned systems, as well as many others, for instance, the intuitionistic system. 2) Secondly, the semantical method. Given a well developed semantics which satisfies the conditions for being a semantics for a paraconsistent logic, we try to obtain the axiomatic system determined by it. This method was used by da Costa and Dubikajtis to construct higher-order discussive logics (cf. da Costa and Dubikajtis 1977). Evidently, the classification of the methods of construction of paraconsistent logics just sketched is neither rigorous nor complete. The fact is that the construction of paraconsistent logics ;s done in the same way as that of mathematical and logical structures. There is no "royal road" for this endeavour, the most important to achieve this construction is to have a certain intuition of the fecundity of the structure to be obtained. The truth of this is showed by the fact
13
A SURVEY OF PARACONSISTENT LOGIC
that paraconsistent logics are intimately connected with other branches of logic and mathematics; for example, with intuitionistic and relevant logic, many-valued logic, algebra and topology.
We may say that one of the characteristics
0
f
the
importance of paraconsistent logic is its interconnection with various branches, already well established, of logic and mathematics.
In this way,
paraconsistent
logic has roots not only in the cultural tradition but also firm roots
in logic
and mathematics, from which it gets some of its inspiration, and for which it will contibute with new basic and interesting ideas.
5. DA COSTA'S PARACONSISTENT LOGICS. Loosely speaking, a paraconsistent logic is a logic in which a contradiction,
A & lA, is not in general an antinomy.
The propositional calcul i Cn
,
1 (B::> A)
3)
A::> (B::> A
6)
(A::>C) ::> ((B::>C) ::>(AVB::> C))
9)
A,A::>B/B
&B)
12)
B(n) ::> ((A::>B) ::> ((A::> IB) ::> IA))
13)
A(n)
&B(n)
::> (A::> B) (n)
The postulates of Cw DEF I NI TI ON.
THEOREM 1. -
2)
(A ::>B) ::> ((A::>(B::>C))::> (A::>C))
4)
A &B ::> A
5)
A &B ::> B
7)
A::> AVB
8)
B ::> AV B
10)
ilA::> A
11)
& (A &B) (n) & (A V
B) (n)
are 1-11.
In Cn , 1 .;; n < w, 1*A is an abbrevi ati on for
16
AV iA
I A & A(n) .
1- A -in the .tYltu-Ui.onv..uC'- po~ilive pftopo-l>ilional C'-a.f.cu.f.u-I>,
then, I- A -in Cw •
THEOREM 2. -
AU the we and val-id l.lC'-hemata 06 the uaM.tC'-al pOl.lilive pftOp-
ol.lilional C-alc.utu. one. a.f.!.lo val-id in Cn,
1';; n
C I 3x A(x) :J C
IV)
C:J A(xl / C:J 'tJx Af xl ,
V) In A and B Me. C'ongI1.Ue.11.t n0ltmulcu, (cf. Kleene 1952), 011. one. Ls obta-ine.d nl1.om -the. o-the.l1. by -the. eLUrvLnation 06 vaC'uou.6 quant-i6.{.elrJ.>, then A '" B iJ., an auom.
\Jx(A(x))(n):::> (\JxA(x))(n)
VI)
The postulates of CC,)* are those of Cw plus I-V above. The postulates of
c;; , 1';; n ';;·W,
are those of the corresponding
C:
pl us
C:
(C;),
the fo 11owing: I
)
x=
II"')
x
C:
I
\
Theorems 1-7 are easi ly extended to
x = y:J (A(x) :::> A(yl).
(C~ ).
The semantics for
n < w , is an extension of that of Cn • 1 ~ n < w (see Arruda and da Costa 1977), but a valuation semantics for C~ (C::O) is still an open problem.
1
~
THEOREM 10. -
The. C'atC'u£.{.
THEOREM 11.-
In r I-A .in Cn*, -them aU -the k-:t!l.a.n6n0ltm.o On A Me deduuble -in
C:
(C;;),
Cn , 1 .;; n ~ w, nl1.om -the. fz--tMn.o noJtm.o
THEOREM 12. Cn
-the n0ltmulcu, -in
r.
In -the. .cymbal = doe..o no-t OC'C'U!l. -in -the 60ltmula A, -them 1- A -in
-i.n, and only -in, 5.3.
on
1';; n .;; w , Me undeudable.
1- A
-in C: , 1';; n ~ w .
THE CALCULI OF DESCRIPTIONS D n , 1 .;; n
The calculi of descriptions Dn , 1
~
~
w.
n c w, are obtained from Cn,l';;;
n~
w,
17
A SURVEY OF PARACONSISTENT LOGIC
introducing the description symbol
t,
and the postulates DI-D5 below. The symbol-
ism and conventions are borrowed from Rosser 1953, with clear adaptations. If FIx) is a formula, them "the object x such that FIx)" is denoted by txF(x). If there is one, and only one, object which satisfies F{x), this object; otherwise,
rx
Ff x) will
txF{x) will denote an arbitrary object.
denote
The semantics
of Cn , 1";; n < w , can be extended to D n , 1";; n < w , as mentioned in and da Costa 1977; but D w still lacks a good semantics.
Arruda
The postulates of D n are those of C n ' 1";; n ..;; w, plus the following (where the restrictions are the usual ones):
Fxl
Dl.
IJx
D2.
IJx (PIx)
D3.
t x F[x) = t y F{y)
:J F[t Y Q{y))
== Q{x))
:J tx PIx) = t x Q[x)
D4. P[ty Q{y)):J 3x PIx) 55. 3jx Pix)
[lJx([tx P{x) =x)
:J
THEOREM 13. -
in
r
Dn ,
1";; n
Let A j ' A 2' . . . ,
Then
U {AL
Am be the pJUme componen:t6 06 :the
I-A in Do i6, and only i6,
A~n)
,".,
A~n)
r
r
in
I-A
w •
THEOREM 14.-
Let F be a 60tunuLa 06 Do, and
wbf.>:tduting ,* 601t "l ,
Then,
F* the 60tunuLa obtained
I-F in Do i6, and only i6,
6ltom F
1- F* in D n,
1";;
n<w. THEOREM 15. -
6,
Dn
if.>
a COnf.>eJtvative edenf.>ion 06 C n ' 1";; n ..;; w .
PARACONSISTENT SET THEORIES.
One of the aims of paraconsistent set theories is the study of the
conse-
quences of the schema of separation when we employ as subj acent log i c a pa raconsistent logic.
In the usual set theories one weakens the schema
and maintains classical logic as subjacent logic.
of separation
In paraconsistent
set theo-
ries, we try to weaken the subjacent logic and to strengthen the usual tions of the schema of separation, in order to obtain
"inconsistent
The basic problem of the construction of strong paraconsistent set which the schema of separation can
formul asets"
theories
. in
be formul a ted without restrictions to avoid
antinomies, but not formal paradoxes, has not been solved yet.
Particularly, be-
cause there has not been found adequate paraconsistent logics to attack the problem.
Nonetheless, while
not even trying to solve the basic problem of paracon-
18
AYDA 1. ARRUDA
sistent set theories, we can investigate the following two problems, whose solutions maygive some hints on the heterodox properties of paraconsistent set theories. PROBLEM 1. -
Admitti ng the exi stence of some sets wh i ch do not
usual set theories, to study their properties. ties of Russell's set. Ro =
x ixif x) .
ex i s tin the
For example, to study the proper-
PROBLEM 2.- To investigate the conjecture according to which when we weaken the subjacent logic we can obtain set theories ex.-Wtenc..i.aU-'1 .6bwngeJ1.
th a n the
usual ones. DEFINITION. guage. £ Fix).
Let T and T' be two non-trivial set theories having the same lan-
T is said to be ex.-WtenuaU'1 .6.tJz.ongeJ1. than T', if, t- 3£F(x) in T' implies that
for every abstract
t- 3£F(x) in T. and there
exists at
least one abstract £G(x) such that t- 3xG(x) in T but not in T'. 3£F(x) means that £F(x)
exists (cf. Rosser 1953, pp. 219-220).
Of course, (Informally
speaking, we say that T is existentially stronger than T' if all sets which exist in T' do also exist in T, and there exists at least one set in T that does not exist in T' .) For the study of these two problems, the systems Cn, i. c n c c , are adeOn the other hand, if we want to obtain paraconsistent set theories exis-
quate.
tentially stronger than the usual ones, it is easier to start with NF good development of NF, see Rosser 1953). first approach to the study of
Problem~
(for a
Since we are interested only
1 and 2, we consider here only a
version of NF. whose postulates are given in Rosser 1953, pp. 212-213. = as a primitive symbol.
ina weak taking
Since we have already mentioned that Do is essentially
equivalent to the calculus of descriptions given in Rosser 1953, then we axiomatize this weak version of NF. here denoted by NF o • in the following way. The postulates of NFo are those of Do plus the following: EXTENSIONALITY: \;/x\;/'1\;/z((XE'1=XEZ) :l'1=zl. SEPARATION: 3'1 \;/x(XE'1 == Fix)), in the case x and fJ are different variables. fJ does not occur free in F(x), and F(x) is stratified.
The conventions and set theoretical notations are like those of Rosser 1953, with clear adaptations. Starting with NF o we construct in this section a hierarchy of set theori es
19
A SURVEY OF PARACONSISTENT LOGIC
w, having the corresponding Dn , 1';;; n';;; w, as subjacent logics. Before sketching these theories it is worthwhile mentioning the problems re-
NF
, 1.;;; n';;;
n
lated to the formulation of the schema of separation in NFn , 1';;; n';;; w. Many forms of the schema of separati on for NFn , 1';;; n .;;; w, have been proposed, but most of them are proved to trivialize the corresponding NF n (see, for instance Arruda 1975b and 197+). tion for NF n
An apparently sure formulation of the schema of separa-
' 1.;;; n';;; w,
is proposed in Arruda 197+; nonetheless, this fonnula-
tion is not adequate for the study of Problem 2.
Taking into account this fact,
we present a weak form of NFn , 1';;; n < w , but a strong version of NFw • It is also convenient to clarify the meaning of the phrase "Russell set" that The Russell set for NF o is Ro =
will be used in this section.
xl x f/-
xl .
Th e
Russell set for NF n is Rn = x(x f/- x & (x E xl (n)). 1 .;; n < w. Since Rn trivializes the corresponding NFn • it cannot exist in NFn , but can exist in NFm , m> n.
In order to have a better understanding of the properties of Russell sets, it is convenient to strengthen the weak version of NFn , 1';;; n < w, by the introduction of Quine individuals (x is a QtL 1 < n < w .
von Neumann -
5) To see if the systems P and P* (Arruda and da Costa 1965a), plus the schema of separation without restrictions to avoid formal antinomies, but not formal paradoxes. are trivial or not. 6) To develop paraconsistent higher-order modal and tense logics. 7) To find a simplified formulation of the axiomatics of discussive presented in da Costa and Dubikajtis 1977.
logic
8) To try to adapt world semantics for the calculi Cn • 1< n<w. and to verify if the resulting semantics are more intuitive than those already known (as for example, those of da Costa and Alves 1976, and Loparic 1977).
9) To develop modal and tense paraconsistent logics (some hints are given in Alves 1976). 10) To complete the algebraic study of the systems Cn , 1 < n < co , in the various versions as, for example, those of Fidel 1977, da Costa 1966, and Sette 1971) . ~
11) To continue the study of higher-order paraconsistent logic began in Alves and Moura 1978. 12) To study systematically the relations among relevant and paraconsistent logic.
10.
BIBLIOGRAPHY.
In the following bibliography we list papers on paraconsistent logic and related topic~. Of c~urse, due to the vastness of the subject (especially the related topics), this bibliography is not intended to be complete. We no not mention, for example. abstracts of published papers, neither many papers on dialectical logic, relevant logics, and non-classical logics in general. The works with
28
AYDA I. ARRUDA
an * are not on paraconsistent logic but are mentioned in the text as references for notations, terminology, and axiomatizations of classical logical systems. G.
Achtelick, L. Dubikajtis, E. Dudek, and J. Konior.
197+.
On .the -independence 06 ax,[omb 06 JaifzOWO/U'6 d-i6cu66-ive
p'wp06ilional
ca.lculu!>, to appear. A. R. Anderson and N. D, Belnap. 1976. Entailment, The Logic of Relevance and Necessity, vol. 1,
Princeton University Press (vol. II, to appear). E. H. Alves.
1976.
1978.
Logica e Inconsistencia: Um Estudo dos Calculos Cn,
l';; n .;; w
(Master Thesis), preprint by the Institute of Mathematics, Universidade Estadual de Campinas, Campinas, Brazil. On .the decUdab-Uillj 06 a blj6.tem 06 d£a.lecUcal ptwpo6ilional £.og-ic, Bulletin of the Section of Logic, Polish Acad. of Sciences,7, pp. 179-184.
E. H. Alves and J. E. de Almeida Moura. 1978.
On !>Ome h£ghVt-OtLdVt pMaCOYl!.l-ib.te.nt calcuU, in
Mathematical Logic,
Proceedings of the First Brazilian Conference (Eds. A. I. Arru-
da, N. C. A. da Costa and R. Chuaqui), Marcel Dekker Inc., New York, 1-8.
pp.
L. Aposte1. 1967.
Log-ique. e.t: d£ale.cUque., in Logique et Connaissance Scientifique,
197+.
Log,[ca e d£a.le.t:Uca In. Hege..t, in La Formalizazzione della Dialet-
(Ed. J. Piaget), Ga11imard,
P~ris,
tica (Ed. D. Marconi), Rosemberg
pp. 357-374.
&Se11ier, Torino, to appear.
Aristotle. 1955. The Works of Aristotle, vol. 1 (logical works), Oxford University Press. A. I. Arruda. 1964. Considerac;oes sobre os Sistemas Formais NFn (Thesis), Universidade Federal do Parana, Curitiba, Brazil. 1967. SUIt WlC h£Vuvr.ch£c de cafcu..t6 ptwp06ilionne..t6, C. R. Acad. Sc. Paris 265, pp. 641-644. 1968a. SUIt Wle. h£Vr.aJr.ch£e. de cafcu..t6 p~op06£t:,[onne..t6, C. R. Acad. Sc. Paris 266, pp. 37-39. 1968b.
SUIt WlC
h£Vuvr.ch£e de ca.tcu..t6 ptwpo6iliorme..t6,
C. R. Acad.
Sc .
Paris
A SURVEY OF PARACONSISTENT LOGIC
29
266, pp. 897-900. 1969a. SWt une hi~c.hie de c.alc.tLt6 de p!CecUc.a.t6, C. R. Acad. Sc, Pari s 268, pp. 629-632. 1969b. SWt c.eA:tainu algeb!Cu de UiL6.6U vwn-uiL6.t>-lj.6teme NFw ' C. R. Acad. Sc. Paris. 270 A, pp , 1137-1139. 1971. La mathemat. on the log.-LC 06 vaguenco.6, to appear. 197+b. A.6emanUcal .6tudy 06 .6Qme .6Y.6te~ 06 vaguenco.6 log.-Lc, to appear. 197+c. On the log.-LC 06 vaguenu.t>, to appear. A. 1. Arruda and N. C. A. da Costa. SWt une hi~hie de .6y.t>temu 60!lJrlm, c. R. Acad. Sc. Paris 259, pp,
1964.
2943-2945.
o pateadoxo
de CU!C!Cy - Moh Shaw-Kwe..L, Boletim da Sociedade Matematica de Sao Paulo 18, fascs. 19 e 29, pp. 83-89. 1968a. On the po.t>tulate 06 .6epMation, Notices AMS 15, pp. 399-400. 1968b. FUl!.thete COn.6..Ldeteation.6 on the pO.6tulate 06 .6epMation, Notices AMS 15', p. 555. 1970. SWt le .6chema de la .6epMat.{on, Nagoya Mathematiaal Jurnal 38, pp. 7184. 1974. Le .6chema de la .6epMat.{on dan.6 lu .6Y.6teme.6 ]n' Mathematica Japonicae 19, pp, 183-186.
1965.
30 1977.
AYDA I. ARRUDA Une 4emantique po«4
te catQUt C~, C. R. Acad. Sc. Paris 284 A, pp. 279-
282. F. G. Asenjo. 1965. Viatect[c !og~c, Logique et Analyse VIII, pp. 321-326. 1966. A catcutu4 06 antinom~u, Notre Dame Journal of Formal Logic VII, pp. 103-105. 1972. On dialect[c tog~c, Teorema II, pp. 133-134. F. G. Asenjo and J. Tamburino. 1975. Log~c 06 ant{Y!om~e6, Notre Dame Journal of Formal Logic XVI, 278.
pp. 272-
N. D. Belnap Jr. 1977. A Me6ull 6oU/t-vatued !og~, in Modern Uses of Multiple-Valued Logic (Eds. J. M. Dunn and G. Epstein), Reidel, Dordrecht, pp. 5-37. J. Bl aszczuk. 1978. weakut Y!OJunat catcu.U wdh fLupea to MY! - counte.JtpCVtt6, Bulletin of the Section of Logic, Polish Acad. of Sciences, 7, pp. 102-106. J. Blaszczuk and W. Dziobiak. 1975a. RemafLk4 on Pe.fLzanoW4k~'.6 moM .0y.o tem6 , Bulletin on the Section of Logic, Polish Acad. of Sciences, 4, pp. 57-64. 1975b. Moda! .oy.otem.o fLe!ated to S4 n 06 Soboc.~i1.6Iz~, Bulletin of the Section of Logic, Polish Aca~. of Sciences, 4, pp. 103-108. 1976a. Moda! .oy.6tem.o ptac.ed ~n the "tfL~ang!e" S5-T 1 -T, Bulletin of the Section of Logic, Polish Acad. of Sciences, 5, pp. 138-142. 1976b. An a)(~omat~zat~on 06 Mn - c.ountefLpafLt.6 60Jt .oome moda! ca.ecu.e~, Reports on Mathematical Logic 6, pp. 3-6. 1977. Moda! !09~C.o connected w~th S4 n 06 Soboc~i1.6Iz~, Studia logica XXXVI, pp. 151-164. B. Bosanquet. 1906. ContfLad~ct~on and JteaLi..ty,
Mind 15, pp. 1-12.
D. D. Corney. 1965.
Review of V. A. Smirnov 1962, The Journal of Symbolic Logic 30, pp. 368-370.
A SURVEY OF PARACONSISTENT LOGIC
31
N. C. A. da Costa. Nata ~ob~e a ~on~eito de contnadi~o, Anuario da Sociedade Paranaense de Matematica 1, nova serie, pp. 6-8. 1959. Ob~~va~ou ~ob~e a ~on~Uto de ewt~nua em matemiiU.~a, Anuari 0 da Sociedade Paranaense de Matematica 2, pp. 16-19. 1963a. Sistemas Formais Inconsistentes (Thesis), Universidade Federal do Parana, Curitiba, Brazil. 1963b. ca£cuto p!!.Opo~ili.onnet6 pou): tu ~y~temu nOJlmet6 in~o~i6tanU, C. R. Acad. Sc. Paris 257, pp. 3790-3793. 1964a. ca£~uto de pJtedi~aU poun: tu ~yMemu 60~et6 in~o~~tanU, C. R. Acad. Sc. Paris 258, pp. 27-29. 1964b. ca£~uto de p~edi~ ave~ egaLUe poun. tu ~y~temu no~et6 in~~~ ~, C. R. Acad. Sc. Paris 258, pp. 1111-1113. 1964c. ca£c.uto de ducJL.i-ptio~ poWt tu ~ yMimu no~et6 inc.o~~tanU, C. R. Acad. Sc. Paris 258, pp. 1366-1368. 1964d. SWt un. ~y~teme in~o~i6tant de theoJL.ie du e~emblu, C. R. Acad. Sc. Paris 258, pp. 3144-3147. 1965. SWt lu ~y~te.mu no~e~ Cl ' C;, C~, Vi e.t NFi' C. R. Acad. Sc. Paris 260, pp. 5427-5430. 1966a. Algebras de Curry, Sao Paulo, Brazil. 1966b. Op~o~ non-monotonU da~ lu ;fAeJ.UM, C. R. Acad. Sc. Paris 263 A, pp. 429-432. 1967a .. Une nouvelle hiVLM~hie de theoJL.iu in~o~i6tanU, Publications du oepartement deMathematiques,Universite de Lyon, France, 4, pp. 2-8. 1967b. F~u e:t ideaux d'une a£gebJte Cn' C. R. Acad. Sc. Paris 264 A, pp. 549-552. ~ 1971. Rema~que~ ~Wt t« ~y~teme NFl' C. R. Acad. Sc. Paris 272 A, pp. 1149-1151. 1974a. RemaJtque~ ~Wt lu ~al~uto Cn' c~, C~ e:t Dn, C. R. Acad. Sc. Paris 278 A, pp. 819-821. 1974b. On the theoJty 06 in~o~~tent 6o~a£ ~y~t~, Notre Dame Journal of Formal Logic XI, pp. 497-510. 1975a. RemaJtk.~ on Jaik.ow~H di~cuMive togic, Reports on Mathematical Logic 4, pp. 7-16. 1975b. Review of Asenjo and Tamburino 1975, MatherRatica1 Reviews 50 # 9545. 1979. Ensaio Sobre as Fundamentos da L6gica, HUCITEC, Sao Paulo, Brazil . 1958.
32
N. C.
1976.
AYDA I. ARRUDA A. da Costa and E. H. Alves. Une -6eman..t6ive. plt.opo!.>\.tionaR. eateuR.u!.>, Bulletin of the Section of Logic, Polish Acad. of Sciences, 4, pp. 33-47. 1975b. Re.malt.11.6 on di!.>eu!.l6ive. plt.opo!.>I.tI.onaR. eateutu6, Studi a Logi ca 34, pp.39-43. Y. Gauthier. 1967.
LogI.que. he.ge.e.te.m 06 6uzzy 6e.t!.>,
to appear.
33
34
AYDA I. ARRUDA
R. Gotesky. 1968.
Philosophy and Phenomenological Research
The U6C¢ 06 ~nco~~tency,
XXVIII,
pp. 471-500.
J. Grant. 1974.
Incomplete model.o,
Notre Dame Journal of Formal Logic XV,
pp.
601-607. 1975.
Incon!>~!>tent and ~ncomplete logic!>,
Mathematics Magazine 48,
pp. 154-159. 1978.
Notre Dame Journal of Formal
CI'.lu.!>i~ca.aon 60Jt incon!>i!>tent theoJtic¢,
Logic XIX, pp. 435-444.
G. Gunther. 1958.
Me
V~e a.Jti.otot~che Log~11. dc¢ SUn!> und
dCJt Re6lexion.
~cht-a.Jti.otot~c.he Logil1.
Zeitschrift fur philosophische Forschung 12,
pp. 360-
407.
1964.
Va.!> PMb.e.em unCJt FoJtm~~CJtUJ1g dCJt .:tJta.lt6zcYldenta1.-Ma1.e/i..t.U,chen Log~l1.,
Hegel Studien I, pp , 65-130.
1971.
Vie ~to~che KategoJtie de.o NeueYl,
1972.
NatUnliche Za.h.e und V~el1.til1.,
Hegel-Jahrbuch,
pp. 32-61.
Hegel-Jahrbuch, pp. 15-32.
B. Hartmann.
1974.
Vie BegJti66I>loMI1. aU ~ek.:t;i!>che,
•
Hegel-Jahrbuch,
pp. 366-367 .
K. G. Havas. 1974.
V~e
Hege.f.!>che
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pp,
Hegel-Jahrbuch,
362-265. S. 1. Hessen. 1910.
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Logos 2,
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S. Jaskowski.
1948.
Rachunell. z:dan d.e.a !>y!>temow dedul1.cyjnyeh !>pueez:nych,
Societatis Scientiarun Torunensis, Sectio A, I, n95, 1949.
Studi a
pp.55-77.
0 Konjunkcj~ dy!>ku!>yjnej w Jta.chunl1.u z:dan dla !>y!>temow dedul1.cyjnuch !>pJtz:ecz:cych,
Sectio A, I, n9 8, pp. 171-172.
Studia Societatis Scientiarun Torunensis
35
A SURVEY OF PARACONSISTENT LOGIC 1969.
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A SURVEY OF PARACONSISTENT LOGIC
39
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A SURVEY OF PARACONSISTENT LOGIC
Universidade Estadual de Campinas Departamento de Matematica Campinas, SP., Brazil a~
Universidad Catolica de Chile Instituto de Matematica Santiago, Chile.
41
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui. N.C.A. da Costa (eds.) © North-Holland Publishing Company, 1980
ON STRONG AXIOMS OF INDUCTION IN SET THEORY AND ARITHMETIC MbtO.6.ta.v Benda.
ABSTRACT.
Althought Godel showed arithmetic incomplete, his work did not point out a natural extension of it which would make it more complete. This changed with Paris' discovery and Harrington's simpl ification of Paris' work. We continue in this direction by considering schemas which are more akin to general principles than the ad hoc sentences of Paris and Harrington. They can be shown to be a natural strengthening of the axiom of induction and their interesting feature is that they apply to set theory as well. The lecture shan illuminate the main points in proving their independence, shall outl ine their relative consistency strength and raise the issue of their appl icabil ity in arithmetical situations.
I NTRODUCTI ON. A lot of activity has occurred since J. Paris (see Paris 197+) discovered a mathematical sentence independent of the axioms Peano's arithmetic, PA.Harrington simplified Paris' sentence making it close to Ramsey's Theorem (see Harrington and Paris 1977). In Solovay 197+ , Solovay placed a function connected with the new Ramsey relation within a hierarchy of recursive functions which gave a new proof of independence for Harrington-Paris sentence. In Silver 197+, Silver gave an exposition of Harrington-Paris to which we owe some debt in that our combinatorial background resembles his, however, what we needed to prove flowed naturally to Silver-like partition result. More recently, alternative sentences appeared (see Ketonen 197+ and Pudlak 197+). The paper of McAloon, McAloon 1978, is a nice exposition. In another direction the combinatorial aspects of the strong Ramsey theorems were treated in Erdos and Mills 197+ and denda 1979a . In our view there are two issues of paramount philosophical importance which come out of Paris' result. One of them is the fact that we have a true sentence on the basis of which we can prove by purely finitistic means the consistency of arithmetic. In view of Godel's theorem, this is probably the ultimate we can achieve in proving the consistency of arithmetic. The other issue con ce r-n s axiomatization of arithmetic. Since Pea no nobody came with an axiom which was forgotten by him. Godel did find a sentence without derivation from Peano's axioms but it does not seem to be relevant for purely arithmetical questions. The sentences we have mentioned above are much closer to being applicable in arithmetic hence Paris' discovery may be regarded as a first step in finding a schema describing a universal method of proof much like the schema of induction does. We think that this is the most exciting open problem in this area. We would like to think that the present paper is a step in this direction. We 43
44
MIROSLAV BENDA
define certain axiom schemas AD, AI' ... which are stronger and stronger, the step from An to An+1 being an addition of the combination if 3 to the axioms in An" The schema AO is equivalent to the induction schema. The schemas are formulated in the language of set theory. It is well-known that PA is (up to relative interpretability) Zermelo's set theory with the axiom that everything is finite replacing the axiom of infinity. It turns out that the finite Ramsey theorem plays, in PA, the same role as the Erdos-Rado theorem in ZFC. In order to have the full benefit of this theorem in Zerme10's set theory, we take as our basis the following theory: T = Zermelo's set theory without the axiom of infinity and with the axiom "if K is a cardinal so is 2K " • Then by PA (an abuse of notation) we denote the and by ZP (p for power) we denote T + "there is much weaker than ZFC, since VK where K =]Lw which we prove essentially in T demonstrating with the schemas, are: 1.
PA + Al
~
Con (PA)
2.
ZP + Al
1-
Con (ZF)
theory T + "everything is finite" an infinite set". Note that ZP is is a model for it. Our results, that infinity has not much to do
Item 1. is not really new, we have shown the connection of Al to HarringtonParis elsewhere (see Harrington and Paris 1977). Item 2 is new, we were surprised that we could get by with ZP only. If we regard Al as true (see the last section for a discussion of this point) and think that the chances of consistency of ZP are better than those of ZF we see from 2, that, as in arithmetic, we have improved the chances of the consistency of ZF. As already mentioned, T denotes the Zermelo's set theory with the axiom "if K is a cardinal so is ZK (= Ip(K)I)" and without the axiom of infinity. In this theory, we have the usual notions as in ZF. We shall often refer to classes of T, these should be though of as their defining formulas. By C we denote the class of cardinals. If we say "for arbitrarily large x" we mean (ifKE C)(3x) [ixl "'K ... J. V is the class of all sets, N is the class (it is not a set in T) of natural numbers. Vex where ex E On (ordinal s) is the set of sets rank < ex. If R is a class then IIRII =
{Ixl:
xER}.
By Pw ( X) we denote the finite subsets of the set X. relations
~Je
use the Erdos partition
K ....
t
which mean that if 6: [K)t .... c (where [XJ is the set of subsets of X of size t) then there is If ~ K, Iff 1'" A which is 6- homogeneous that is 6 is constant on [ff)t. The finite Ramsey theorem is (vt)(lfc.) (ifm)(3n) [n .... (m)~) quantifiers limited to N) and the Erdos-Rado theorem is
(all
ON STRONG AXIOMS
OF INDUCTION
45
1. COMBINATORIAL PART. DEFINITION 1.1. - let R be a class of sets.
Then by R' we denote the class
{ftER I for arbitrarily large -6 ER,
ft~-6}.
We call R' the derivative of R. The derivative may be iterated: RCl+ 1 (R Cl), and RCl = n{R !iJO is limit. iJ The definition of the derivate is related to the definition of a derivative of a modeloid (see Benda 1979a). In one version of exposition a modeloid is a set of partial one-to-one functions on a set and its derivative consists of functions which can be extended within the modeloid to functions whose domain or range contain prescribed element, or, speaking loosely, which can be extended to a lot of functions in the modeloid. This analogy is not artificial for, as we shall see, there is a link between derivatives of classes of sets and derivatives ofmodeloids and the way we chose the definition of a derivative of a class gives the right analogue to the modeloid derivative. On the other hand, a slightly more cumbersome but weaker definition of a modeloid would yield most of the results in this paper; we shall state this precisely later. DEFINITION 1.2.- let mEN and R a class of sets. Ramsey if for some cardinal n a)
ft~-6ER
b)
litl>m and
c)
if 1-6
I;;;.
We say that R is m-
impli'es ftER (\1-6)[
6
C
It. ....
-6ERj
implies ftER
n then there is ftE R such that
lit I;;;. m and it C -6.
Definitions 1.1 and 1.2 are of paramount importance for this paper. definition enables us to state the axioms more neatly.
One more
DEFINITION 1.3.- If R isa class of sets then IIRII = {1ft
I: ftER}.
The axioms we are going to investigate have this form
First of all, let us get some examples of Ramsey sets: PROPOSITION 1.4..sezs homogeneoU-6 6O!t eVe!ttj
Let
6.£
:to
6.: [Vj..t .... C. for ..t .t
.£.;;;~EN.Then:thec..fM-606
Ls Ram;.,etj (6O!t -6ome m).
PROOF: let m=max{:t.I'£';;;Q}+1. We show that R is m-Ramsey. Condi.c tion (a) is clearly satisfied. For (b) let lit I > m and assume that any set -6~ft
MIROSLAV BENDA
46
t of smaller size is in R. We show that It is no-homogeneous. Let a, bE[t] 0 If au b has size smaller than It we are done, i.e. nOla) = nO(b). If not then lau b 1= It. Since lit I> to + 1 there are x E a - band yE b - a and tQe sets au{y} and bu{x} are in R so they are nO- homogeneous. Let cE[It] 0 be a set which does not contain x nor y. Then
because Cu (a - {xl ) U {y} is a proper subset of for the other equality. But for the same reason
n(a) showing n(a)
=
It,
n«a - {x}) u {y}) = n«b -
so is in Rand simil arly {y})U
Ixl ) = neb)
= neb).
For condition (c) the existence of n follows, in PA, from the finite Ramsey theorem, and in ZP, from the Erdos-Rado theorem. The sketch of proof in PA is, we take n suchtthat n til
nQ _ 1
~ (m)~Q
to o
~ (no)c
where nO is such that nO
Similarly, in ZP.
~ (n
t1 1 )c etc., un1
Alternatively, one can splice together results with-
the functions n{ into a single function and use the combinatorial out iteration. • We shall now state precisely the axiom schema AO: (A For: eVeJLy O):
mE
{n
N
R J.i, Rll.»U>ey then IIRII = N.
Clearly, this is a schema; note that even though m does not appear in the scope of the quantifier one should remember that R stands for a formul a which may have m as a variable besides, of course, some other variables which are quantified over universally. PROPOSI TI ON 1.5. - The naUaw.£ng (i) (t t )
The auom M.hema
(Aolt.
The {nduc.ti.an .t>chema {n
PROOF:
Me equi.valent:
PA.
For convenience we write the induction schema a follows: \;/x (\;/y (If
C
x
~ ¢(y))~ ¢(x))~ \;/x
¢(x)
This is easily seen to be equivalent to the usual induction schema axiom of foundation in this set-up). Now define ~(lf) by \;/x (Ixl.;;
Iyl ~
(which is the
¢ (x)).
We claim that ~ defines a Ramsey set with m = n = 0 and, indeed, it is easy to check the conditions (a) - (c). We now use (A O) to conclude that
ON STRONG AXIOMS OF INDUCTION
47
IJn IJx([xl < n .... ¢(x))
which is the same as IJ x e . To prove the converse, we use the finite Ramsey Theorem which is a consequence of the axiom of induction. Say ~(x, m ••• ) is a formula and m is given such that R=
Ix
11/1 (
x, m... )}
is Ramsey. From this, we get n satisfying condition (c). let be such that
n
Let q;;>max(m, n}, and
n .... (q)~.
Take a set t
of size
n
and for .6E[t]m
define
Let H~t be homogeneous for 6 of size IHI;;> n ; it follows from (c) that we have .6E[H]m such that 6(.6) = 0, therefore 6 is constantly 0 on [H] m. Now by (b) this implies that [H]m+l CR, which in turn implies that [H jm+2 C R etc , , finally getting HER. Therefore, qEIIRIi. • Having identified' (A O)' let us now discuss (AI)' It turns out that (AI) is independent of both arithmetic and set theory for the reason that it implies their consistency. We shall prove its consistency for arithmetic (in ZFC) in fact, we shall show that (An) is true. If we use the language Lw w in which the schema (A 0: ) may be formal ized for 0: < wI then even (Ao:) is true I for any 0: < wI'
Note that if 0: < "-' each of the sentences in (A 0: ) is first-order. Thei r complexity, as far as alternation of quantifiers is concerned, increases, each step adding a combination IJx 3y. (A) then may be formulated as countable conjunctions of first order sentences etc. In general, (A) may be formulated in L 10: [ + ,w (0: > w) . 0: PROPOSI TI ON 1. 6. (ZFC).- Fatz. Ra.m.6ey Theatz.em -i..6 eqtu:va1.ent to ,i6atz.
0: 0:
< wI' < wI'
V w 1= (A 0:)'
In 6a.c...t, the -in6btUe
VW 1= (Ao:)".
PROOF: Let R be Ramsey for some mEN; it need not be definable withinV w' Define 6: [V w]m .... 2 by if a.E R if a.'1c R
48
MIRaSLAV BENDA
By the infinite Ramsey Theorem, we find H £ Vw infinite homogeneous for 6. It we had 6 equal 1 on H this would contradict condition (c). Thus every set .6EVw' .6E [H jm i s in R. This implies that every .6 EVw' .6~ H, is in R. But then a IIR ll = N for any a, since any subset of H can be extended to arbitrarily large I(. ~ H, I(. E R. We have proved that every (A a) is true in Vw' Conversely, i f 6: [V,l m ~ C (E V ) then the class R of 6 - homogeneous sets is Ramsey by 1.4 (,~ W' +1 a so II R II = N for each a < wI' Since R is countable for some a <wI' Tf' =Tf'. Given I(.E Ra = Ra + 1 it can be extended to 1(.1 ERa = Ra + 1, I(. C ILl and so on; we find an infinite H such that every subset of H (from VI) is in is homogeneous for 6. This is the content of the equivalence. •
REMARK 1.7.-
R,i .e. H
In the first draft of this paper we posed the following ques-
tion: Is it true that for any a < wI there are c, mEN and 6: [Njm ~ c such that if R = {.6 E P W (N) 1.6 6-homogeneous} and Ril + 1 = Ril then il;;;. a. R.Laver has answered this question affirmatively. This shows that Proposition 1.6 cannot be improved. that is there is no il < co, such that the infinite Ramsey theorem would be equivalent to "60/l. ,,< il Vw t= (A,,)". is essentially the axiom of induction. Then (AI) In 1.5, we showed that (A a) and its successors may, in this light, be called strong axioms of induction, another (a better) reason 'for this being the fact that their form does resemble the induction schema. However they are called. the important thing is to find a use for them. Below we prove a combinatorial result which will be heavily used in the next section. THEOREM 1.8. t = (a , ... , al ••• l . Let fL be an ordered sequence of 1ength q from H such that for some 13 1,13 2 , [33 E H we have "4 < 13 1 < [32 < 13 3 < fL. Assuming (IJ yEa) (3 zEV )~.6 and realizing that all parameters in this formula are in V an ap"4 "1 plication of 2.2 (i) yields ( II yEa) (3 z The comprehension axiom then gives Formally, we get
E
V
"4
(II yEa)
) q, fL
•
bE V such that 13 1
(II yEa)(3 z
Using 2.2 (i) and the fact that the parameters are limited to
V
"1
E
b) q,fL •
we obtain
MIROSLAV BENDA
54
The comprehension axiom has a simple proof. Let ~(z, v) be a formula of rank from H (-6 of length q) and let .tE Vaoand aE Val be given. We should show that
q,
"o < £\'1 < "z < "s < -6
but this is true since any subset of a is in Va
2
The infinity axiom: If we work in ZFC then because H can be extended to arbitrarily large H such that (a) 'V ({3) where a E H, {3 E Fr, we see that all ~E H must be > w (if k ~ 3). The axiom then will be in Til.' On the other hand, if everything is finite, it will be so in all
V~ -6.
~Ie are now ready to prove tha t PA + (AI) 1- Con (PA) and ZP + (AI) 1- Con (ZFC) We take the latter case the former being entirely similar. Let P =(~ 0' ~1"" ~n) be a proof of ZFC. Let kE N be such that the universal closure of any ~,[ in P has quantifier rank';; k. Using (AI), we get H satisfying 2.2 (i), IHI ~ 11.+ 3. By Claims 1, 2 and 3 and induction, we see that every ~,[ET(k., H) and by 2.4 we find that ~n cannot be a contradiction. •
DI SCU SSI ON.
Let us summarize what we have accomplished so far in order to put the results into a broader perspective. We have defined the axiom schemas AO' AI' ... for the theory T, which is essentially Peano's arithmetic when the axiom of finiteness is added to it while it becomes the extended Zermelo's set theory, ZP as we call it, when the axiom of infinity is added to it. The axiom schema AO is provable in T because in PA, it is the finite Ramsey theorem while in ZP it is the Erdos-Rado theorem. The axiom schema Al is provable in neither of these theories. For ZP, this can be easily seen if we work in ZFC (and assume that it is consistent). A reason for this is that VK where K =.J-w is a model for ZP so if Al were provable in ZP we would obtain VK F Con (ZFC) which is impossible. Now, as far as PA is concerned, the axiom schema Al is true. We can say this because we have a standard of reference for PA, namely the natural numbers. Concerning Al an ZP we see that Al is false in a natural model of ZP and so the question arises whether ZP + Al is even consistent. One way out of thi s difficulty is to point out that Al is a much weaker axiom than, say, the existence of a Ramsey cardinals. In fact, as for arithmetic we just need a cardinal K> w for which the Ramsey theorem is true (i .e. homogeneous sets of power K exist for any 6: [K] n -> c}, Although this gives some credence to the consistency of ZP + Al it is not completely satisfying. Let us present a justification of the augmentation of ZP with Al in a way which is based on the uniformity of the proofs of independence we have given. This approach lacks, so far, preciseness however this is, hopefully, compensated by its possible applicability to other question of this sort. The axiom of infinity
ON STRONG AXIOMS OF INDUCTION
55
played such a minor role in the above proof because the sentences Al are formulated using mainly the notion "for arbitrarily large" and because the properties of this quantifier are very similar in PA and in ZP (or ZFC). We wish to make a sweeping generalization of this which consist in: (1) Isolating that which is common to PA and ZP other arguments.
in the above proof and
(2) Defining a class C of sentences which do satisfy the requirements of (1), that is sentences which speak of properties for which the infinity axiom irrelevant. The sentences of AI' A2 ••• should fall in~o this class. (3) Adopting the following General Principle: the hereditary finite sets is true of all sets.
Any sentence from C true of
On the basis of the principle we answer the question whether ZP + Ai is consistent by saying, of course, Al is in fact true of all sets. Some may say that this is the same escape as the one taken when justifying large cardinals (since there is one measurable cardinal (w), why not have two). The principle does differ from this line in that it refers to certain similarities of the respective universes in their totality rather than in what they contain. The goal, and the principle is nothing but a goal, is to bring these similarities out of the shadows.
REFERENCES. M. Benda 1979
a
On H~ngton'~ Pantit£on Relat£on,
1979 b : Modeto-f.d6 1,
to appear in Combinatorial Theory.
to appear in Trans. of the A.M.S.
P. ErdBs and G. Mills 197+
Some Bound6 nO~ the R~ey - p~ - H~ngton Numb~,
to appear.
L. Harrington and J. Paris A Ma.:thcma:ticcU. Incompletene-6~ -f.n Peano A~hme.:Uc, Handbook of Mathematical Logic, J. Barwise (ed.), North-Holland Pub. Co. Amsterdam,
1977
pp. 1133 - 1142.
J. Ketonen 197+
,
Set Theo~y nO~ a SmaLl Un-f.v~e,
to appear.
K. McAloon 1978
FO~e-6
comb-f.na.:to-ULe-6
du theMcme d'-f.ncomple.tude. Seminaire Bourbaki N° 521.
J. Paris 197+
Independence Re-6uL'U
nM
Peano A~hme.t-f.c U~-f.ng Inn~ Modw,
to appear.
MIROSLAV BENDA
56
P. Pudlak 197+
AnotheJt CombinatoJUal
pear.
Sen;tence I ndependen;t
06 Peano AJU:thme:Ue,
J. Silver 197+
HaNling:ton',5 veMion 06
.the
PaJr.{A lLUuU:.
R. Solovay 197+
Rapidtlj GlLowing
Ram-6elj
Function-6,
to appear.
Department of Mathematics University of Washington Seattle, Washington 98117
to
a p-
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © North-Holland Publishing Company, 1980
TOWARD A CONCEPT OF SCIENTIFIC THEORY THROUGH SPECIAL RELATIVITY Jo!tge E.
Bo
Otherwise the relativistic LA K W is called
0 ... v(lt, K}
>
O.
11011-E{.n6;(:Un-LaI1.
It can be shown that axiom (v-i) is equivalent to the existence of an invariant velocity, in the following sense: a number c is called an -il1vaJr.-ia11:t vetoe-i:tyfor a LA K W = (E, K) iff there exists a schema of uniform motion It in E such that V (It, K) = c nOll evetty K E K. Axiom (v-i) is equivalent to the following: (v-i')
W hal>
al1
-il1vaJr.-ia11:t vetoe-i:ty.
And each of these conditions (v-i) (v-i")
The eha.ttac:tett-i.6tic
k
On
and (v-i') is in turn equivalent to W -ih "Willy pMU-ive
(I 0).
It can be shown that in an Einsteinian LA K of (positive) characteristic k, there are exactly two invariant velocities, given by the formula
±. c = 1 Iv'k . In a non-Einsteinian relativistic LA K there is no invariant velocity, as in are the classical case. Naturally, equivalences between (v.i.), (v.i.') and (v.i.") referred to the framework of definition 3, i.e., to a ltetaUv-ihtic LA K W. We pass now to the problem of units. It turns out that it is possible to establish an adequate concept of equivalence for units without introducing any metrical concept; this will be done, then, in the framework of pure affine geometry. DEFINITION 4.- In any LAK we say that the couple (K, K') of allowable frames has eqa-ivalel1:t time un-i.t.6 (respectively "pace un-i.t.6), if the time coordinate (respectively space coordinate) in K of the time unit (respectively space unit) of K', equals the time coordinate (respectively space coordinate) in K' of the time unit (respectively space unit) of K. It can be shown that the couple (K, K') has equivalent time units if and only if it has equivalent space units; and each of this conditions is equivalent to the following: the determinant of the matrix K- K' is equal to 1. DEFINITION 5.- A LAK is 110JrmaUzed iff every couple (K, K') of allowable frames has equivalent time units. For every normal i zed LA K, transforma ti on rna tri ces K- K' are of the form 1
Ipeuctl E.£n6:te.£n1ctn I1..£nema.tic..6 is a couple (2, W), where 2 is a PF and W is an EM. The schema (2, W) is val1d if and only if there exists an embedding n: 2 -+ W. Thus, according to this view, a physical schema may be valid and may be non valid. Observe that if the schema (2, W) is valid, all real-like theoretical physical frames F (belonging to F) are such that the time axis of their corresponding frames in Ware well defined, and consequently we may speak of the time axis (and the space axis) of a real-like theoretical physical frame F. Finally, observe that the notion of schema of uniform motion can be introduced in 2 via the embedding 6, and it results in an invariant concept with respect to the class F in 2.
§3.
STEP III:
EMPIRICAL INTERPRETATION.
From an empirical point of view, a physical schema (2, W) as stated in § 2 is still a rather theoretical entity. What the experimental physicist does is not to perform a global test concerning the whole universe, but to establish the actual empirical conditions in which a test should be relevant. The specification of such conditions constitutes what I call an empirical interpretation. This is formally given in definitions 12, 13 and 14 below. DEFINITION 12.-
An ex.peJL£men:t.a..e 6Jtamwonk for kinematics is an
ordered
JORGE E. BOSCH
64
(RE, 1, M), where RE is a well defined class of ~etevant eteme~y eT is a class of measuring instruments, and M is an empirical method (ordered set of rules) such that the following conditions hold: triple
ve~,
(i)
e wUh
06 I and :to any membM e 06 numbeM [:t, x l, c.aU.ed :the c.oMdinateJ.> 06
16 muhod M JA appUed t» any membM i
RE, it pIlOduc.eJ.> an
~eJ.>pec.:t:to
o~dMed p~
L
06
~ea.l
(U) 1 has at .teM:t two di66MeJU: membeM, and 6M eac.h i in 1 :thMe intu.Ui.ve notioVL6 06 .6imu1.:tanUty and c.o-.6paUa.U:ty wah lleJ.>pec.:t :to L The.6e UOVL6 genMate ~etmoVL6 SF and CF M in Ve6.[nition 8.
Me no-
The relevant elementary events are those which are actually interesting for the experimental- physicist: then, the class RE may be finite. All the problem of "equivalent time units" lies in the requirement that the method M must be :the ¢ame for every member of T (system of measuring instruments). In fact, if method ~r involves an explicit use of units, the requirement is such that method M itself must establish the correct way for producing such units. For example, if the unit "meter" is used, method M may contain a rule 1ike thi s: "Take a ugid nod whic.h, at ~e.6:t wah ~e.6pec.:t :to :the 1>:tandMd me:tM 06 PMJA, coinc.ide.6 experY1>:tem 06 meMu.4ng iVL6Uume~ unde!l c.oVL6ide!lation". For time the situation
is similar: if comparison with a standard clock is not judged a satisfactory operation, we can choose a rule like that proposed by Bondi 1965: "A and B can u.6e
M :thm una 06 time :the ha.t6 U6e 06 a nuc1.eu.1>
made up 06 a 1>pec.i6.[c.
numbM 06
It is tacitly intended that this phenomenon of decay must be produced in a box at rest with respect to the corresponding observer. If method M does not involve an explicit use of units, then the units implicitly used are equivalent by definition, due to the uniqueness of method M itself. The idea is that method M should be c.anonic.a! with respect to the elements of 1. and ne.utnoYL.6".
P~OtoVL6
DEFINITION 13.-
An experimental framework (RE, T, M) will be called ade-
quate if it satisfies the following conditions:
... , in' with n;;' 2, be eteme~ 06 I, and.tu m ... , m bema1, n, 1, 1>uch:that m E i 601L k = 1, •.. , n, Then 601L each k it ill ~e k k q~ed ihas: at .teM:t :tMee due!lminmoVL6 06 m p~oduc.ed by :the 6.[xed iVL6~ument k i mu.1>:t be eve~ betonging:to RE; 1 (U) FOIL each k = 1, ... , n, :the meMMeme~ Ilegill:te!led by i co nce~ning :the de:te!lminmoVL6 06 m Ilevea! :that :the vetoc.i:ty (in :the OILdinM~ .6eVL6 e) 06 m k k JA C.OJt6:tant 601l eVMy meMMed intMVa!; (i)
:te!lia.t
Lu i
poi~
Two remarks are in order: (1) the point mk is supposed to be a representative point of the 1>:ta:te.6 of i k; it may also be said that mk is ~gid.ty belonging to i k ; (2) A due!lminmon of a material point is a primitive notion whose intuitive meaning is the instantaneous state (or apparition) of thi s rna teri a1 point; in Reichenbach's terminology, the determinations of a material point are the genidentical events which constitute this material point. In this sense, the determinations do not depend on any measuring instrument; but, for the sake of briefness, I speak of :the de:te!lminmoVL6 pMduc.ed by an in.6Uument i 1 as signifying :the cOOlLdinateJ.> 06 a due!lminmon e.6:tabffihed by i 1.
TOWARD A CONCEPT OF SCIENTIFIC THEORY DEFINITION 14.-
65
The physical framework (in the sense of Definition 9) framework (RE, 1, M) is the quadrupl e of Definition 13) by:
genvwA:ed by the adequate experimental (U, F, R, A) defined (with notations
U = RE
(.i)
at:
U
V, wheJte
V JA the -6e;t 06 aU :the even:t-6
pok, p/, p/,
-6ueh
Po k
-if.. :the event eOl1-6-if..:Ung .in :the (poM.ibly .ideal) de;teJln1.ina.:Uon 06 m k .il1-6mnt 0 w.i:th ll.e-6pee:t:to .[k; PI k JA de6.[ned .in :the Mme way 60Jr. .il1-6mnt 1
:that :
06 .i p 2k k; mneouf., wUh 6JtOm
Po k, (ti)
JA :the event cOl1-6JA:Ung .in :the de;teJtm.ina.:Uon (poM.ibly .ideal) , -6.imul-
Pok w.i:th ll.e-6pee:t:to .i 06 a mctteJUa1 po.int -6Uuctted ctt dJAmnce k, .in :the J.,el1-6e 06 p0-6.[:Uve meMWtement 06 .il1-6:tJtument .i k; k, F JA :the clM-6 06 ll.eal-Uke phy-6.ical 6Jtctme-6 (P p/' p/);
1
o
(.u.i) FOJr. each F .in F, R(F) JA :the MdeJted pct.iJt (SF' CF), wheJte SFand CF Me th« ll.uct:UOl1-6 geneJtctted (in the sense of Definition 12 (ti)) by :the meMwUng
.in-6:tJtument M-60c.ictted:to
F;
(.iv) Fon: each F .in F, A(F) JA :the .6y.6:tem 06 cOOJr.d.inctte-6 uemen:t-6 06 U by :the .in-6:tJtument M-60c.ictted:to F.
M.6.igned:to
:the
It is evident that the physical framework generated by an adequate experimental framework is a quite ideal (or abstract) entity. It depends, of course, on some broad spectrum terms and on some primitive ideas, such as "simultaneity with respect to an instrument (or to an object)", etc. DEFINITION 15.- Let Y = (RE, 1, M) be an adequate experimental framework, 2 the physical framework generated by it, and W an Einsteinian model. Then we say that the experimental framework Y con6.iJtm.6 the physical schema (2, W), if this schema is valid in the sense of Definition 11. Otherwise we say that Y ll.e6u:te-6 the schema (2, W). Until now, no concept of :the-OJr.fj has been introduced. In fact the precedi ng machinery suffices to give a formal account of scientific practice (at least concerning relativistic kinematics). But from an epistemological standpoint the following questions are relevant: what kind of object is (if any) the Special Theoll.y of Einsteinian Kinematics? And if such a :theoll.fj exists, does it give some information about reality? Which is its cognitive status? A possible answer is: there is no thing such as the Theory of Special Einsteinian Kinematics; all there is in this connection is a mathematical tool, the Einsteinian model of Definition 6, and the possibility of making up some experimental devices in order to obtain confirmations or refutations according to Definition 15. In a case of refutation, physicists would look for a convenient change in the mathematical tool or in the concepts of physical and experimental frameworks. This is a pragmatic point of view. But there is an alternative, which consists in defining a theory: several ways are possible; it seems to me that one of the most convincing is the following:
tem
DEFINITION 16.- The TheOJr.y 06 Spec.ictl E.in-6:tein-ian K.[nemct:UCf.. is the sysin the following entities:
consist~ng
(.i)
The clM.6 '06 aU E.il1-6:tein-ian modei..6 .in :the -6en-6e 06 Ve6.[n.i:Uon 6.
(ti) The .6:tJtctt.i6.[ed clM.6 a6 aU uemen:tMy evenU: .6:tJtctt.i6.[ect:Uon JA g.iven by :the d.i66eJtent levei..6 06 mean.[ng 06 :the bll.oad .6pec:tJtum :teJtm "uemen:tMy event" . (.u.i)
The clM-6 06 aU (a.e:tual 0Jr. .ideal) adequctte expeJt.imen:ta.e. 6JtctmewOJtk.6
.in
66
JDRGE E. BOSCH
;the .6l!JUe 06 VeMnil.iol'l.6 12 and 13: ;thM.u, ah..o a .6.tJLa.:t.i.Med c1.a.6.6, ac.c.olld.ing w.£:th ;the .6.tJLa.:t.i.Mc.at:ion 06 elemen;taJr.1j evenU; (.iv) A .6.tJLa.:t.i.6.ied c1.a..6.6 06 c.on6~a.tion 6unc.;tJ.oYl.6: 601l eac.h un.iVeJr..6e U 06 elemen;taJr.y evenU 06 a g,(.ven level, deno.te by Y ;the c.oMe.6pOncUng c.lau 06 adequa..te expeJUmen.ta.e. 6!tC1.YY1e.w0llk.6; then the c.oMe.6pOncUng c.on6~a.tion 6unc.;tJ.on Ls the 6unc.;tJ.on 6u :Y ..,. {D, l l , wh.ic.h to the adequate expeJUmen.ta.e. 6!tC1.YY1e.wOllk Y E Y M.6.igYl.6 the value 1 (c.on6~a.tion) .i6 thelle ewu an Ee.ln eVeJr.q Jr.ea!.>onable pJr.e!.>ci'l..:ta.tion 06 Cla!.>!.>.lcal K£nematic!'>. More precisely, there is a definition of the TheMy 06 CwMcal K£nema;t[c!.> which is obtained from Definition 16 by changing only part (.l) of this definition: this is accomplished by taking the class of all c£aJ.>!.>.lcal model!.> (easily defined) instead of the class of all Ein!.>.:te.ln-lan model!.>. As the experimental framework is the same for both theories, a campa ri son between them is straightforward. Commensurability is thus obtained as a result of formalization, and this may be considered as an advantage (last but not least) of the axiomatic method. §6.
TOWARD A CONCEPT OF SCIENTIFIC THEORY.
TOWARD A CONCEPT OF SCIENTIFIC THEORY
71
The above treatment of Special Einsteinian Kinematics suggests a generalization to other theories. A more complete account of this program will be developed in another paper: here only a brief sketch of this idea will be proposed. I believe that much of the advanced scientific theories may be presented according to a schema 1 ike tha t of Definition 16 (§ 3). This schema woul d comprise four parts: (~)
({)
A class of mathematical models, defined in a set theoretical framework. A (perhaps stratified) class of entities designated by a broad spectrum
(ill)
A class of adequate experimental frameworks, closely related to enti-
term.
ties in ory.
(~).
({v)
A confirmation function depending on the internal structure of the the-
It is highly probable that a mathematical model of class ({) will contain more elements than those designated by the broad spectrum term of part (~) : then the definition of a sort of embedding will be in order, and this embedding will supply an interpretation of the theoretical terms of part ({). An experimental framework will be a structure concerning one precise meaning of the broad spectrum terms appearing in (~); and its definition performed in a sufficiently canonical and universal way so as to exclude sibility of ad-hoe devices. Moreover it would be of a nature appropriate account of aecepted 6a~.
level of would be the posto give
In this manner it would be possible to test theories and even to establ ish comparisons between them. But the experimental meaning will always depend on the choice of one level of'meaning of a broad spectrum term, plus a number of auxiliary hypotheses and empirical statements concerning accepted facts. This is just a program, which has been illustrated by Einsteinian and Classical Kinematics. Its success in a larger domain depends on further research.
REFERENCES. H. Bondi 1965 Some -6peuo.£.
06 the UlUtebuan eqUll-'UoM. In Lectures on GenBrandeis Summer Ins titute in Theoreti ca 1 Phys i cs , Prentice-Hall, New Jersey. -60,[tI.UOIU
eral Rela tivi ty .
M. Born 1962
Einstein's
theory of Relativity.
Dover Publications, New York.
J. Bosch 1971 On the ax{omat{c 60undat{01U 06 -6pe~ ~e.r.at{v{ty. cal Physics, Vol. 45, N° 5, p.p. 1673 -1688.
Progress
of Theoreti-
A. GrUnbaum 1961 Law and convention {n phy-6{co.£. theo~y. In Current issues in the philosophy of sc I ence , H. Feigl and G. Maxwell (edS.). Holt, Rinehart and Winston, New York.
72 1963
JORGE E. BOSCH Philosophical
problems of space and time.
Alfred Knopf, New Yorl :;:HI this lemma means that no term and its negation are provable in the system. SET THEORY. The set theory of Bunder 197+ c based on the higher order predicate calculus outlined earlier, the equality axioms (E1), (E2), (E3), (E4)., (E8) and (E9) and the rule W QX, W QY I- H (QXY)
(replacing (E5» was shown to be relatively consistent with the higher order predicate calculus alone. From this we could derive comprehension, pairing, replacement and extensionality. To have the sum set property as well we need: (E6)
I- F AH x:>
x
Ax.
We can extend Lemma 6 as follows: LEMMA 7. The .6y.6tem 06 BundeJr. 197+ c. -f.nc.tucUng (E1), (E2), (E3), (E4), (E6), (E8), (E9) and WQX, WQ Y I- H( QXY)
6.
PROOF.
We add (E6) to the system(s) we considered in Lemmas 2, 3, 4, 5 and
Lemma 2 goes through as before. In the proof of Lemma 3 we need to cons i der the extra case where Z = :;:(FAH)A. T is then Z or AW for some W where F AHW is in a previous step. As any previous step must have a normal form so must T. Thus Z mus t be a hypothesis and the res t of the proof goes through as before. Lemmas 4 and 5 hold as before and in Lemma 6 the extra case again is Z but as :;:HI* :;:(FAH)A or AY for any Y, this case does notarise and so Lemma 6 holds.
:;:(FAH)A,
(E1), (E2), (E3), (E4), (E6), (E8), (E9) and the rule replacing (E5) are relatively consistent with this as in Bunder 197+ b so Lemma 7 holds •• For the set theory of Bunder 197+ b which includes the comprehension, pairing, replacement, sum set and power set properties, we need to add to the higher order predicate calculus: ( A)
I- Ax:>
x
FAHx.
If we also want the axiom of infinity to hold in this system we also require: (0)
I-AO
A HIGHER ORDER PREDICATE CALCULUS
where O=oBf(WQ1)
and Q
1
=0
81
AXAy(FAHxt\FAHyt\ ::A.AU(XYVlyU)).
We can again extend Lemma 6. LEMMA 8. J.>.{.J.>:te.n:t.
PROOF.
2 to 6.
The. J.>yJ.>:tem
on
BundeA 797+b .{.nc.£u.di.ng (AJ, (0) and (Q1 J
.{.J.>
c.on-
We add (A), (0) and (Q1) to the system(s) we considered in Lemmas
Lemma 2 goes through as before. I n Lemma 3 we need to cons i der the extra cases where Z = ::A(FAH) or ::A( BAQ1)' T is then FAHW H(W ( ) or A(QIWI) I 2 I, where WI' (and in the second case W2)' has a normal form as it appears in a previous step AWl (and AW 2)' so only in the second case may T be without norma 1 form. We now consider the proof of AWl (AW The last step in this cannot be by 2). DT:: or DTP'. If t t comes by (0) WI W = I' (Q, W ( which has a normal form, z 2 2) so AWl must come from ZI by a sequence of P,:: and Eq steps, where Z' does not r
come by DTP, DT::, or clearly, by (A) or (0).
If
Z' comes by (QI) i.e., is
FAAQ then WI is Q W for some W in normal form, so WI W2 has normal form. I 3 3 I Thus Z' is a hypothesis. Now Z' can be cancelled only by a DT:: step, so Z' = UV for U E U and V indeterminate which can only lead to a WI = VW 3 W4 ••• Wn for some W3 W4'···'W n in normal form. Then WI W2 is also in normal form, which is impossible. Thus the hypothesis Z' cannot be cancelled and Lemma 3 holds. Lemmas 4 and 5 holds as before and in Lemma 6 we need the extra cases Z ::A(FAH) and Z = FAAQ1' but as ::HI *FAHY, H(YW) or A(Q1Y) for any Y or W, thi s case does not ari se and Lemma 6 holds. Therefore, Lemma 8 holds .• Note that, as is to be expected from the inconsistency proof given in Bunder 197+b, the above proof fails when we have both (A) and (E6) in the system. In the section dealing with Lemma 3, AWl and AW could ~ave been derived by (E6). 2
REFERENCES. M. W. Bunder
on
197+a
Plte.di.c.a:te. c.a1.c.u1.u.J.>
197+ b
A one.-auom J.>e.:t :the.OILy bMe.d on Mghe.1t OILdeA plte.di.c.a:te. c.a1.c.u1.u.J.>. published.
aJtb~y
Mgh OILdeA.
Not yet publ i shed. Not yet
82 197+ c
M. W. BUN DER Set theOlLtj in plLecUc.a;te ealeu£.u.6 w.Uh eqWLUty.
Not yet publ ished.
Department of Mathematics The University of Wollongong P.O. Box 1144 Wollongong, N.S.W. 2500 Australia.
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © North-Holland Publishing Company, 1980
BACK-AND-FORTH SYSTEMS FOR ARBITRARY QUANTIFIERS Xav.leJL Ca..leedo
ABSTRACT.
Loow(K) is the logic obtained by adding a
KX1 ••• Xk (¢I(X 1)'" ¢k (X k ) ) to the logical operations of L oow' The corresponding
lindstrom's quantifier
finitary logic is L w w (K), and L oo w (K-i.) -i.E I is obtained by adjoining a family of quantifiers. In this paper, we give back-and-forth systems characterizing elementary equivalence in those logics and their fragments of bounded quantifier rank. This general izes work of Fra'l s se and Ehrenfeucht for L w w ' Karp for L oow ' Brown, Lipner, and Vinner for cardinal quantifiers, Badger for Magidor-Mal itz quantifiers, and others. Our systems apply to higher order quantifiers also.
INTRODUCTION. Ehrenfeucht 1961 and Fraisse 1955 gave back-and-forth or game theoretical characterizations of elementary equivalence in first order logic, L ww ' later generalized by Karp 1965 to infinitary logic,
L=w' These characterizations were used
to obtain results about definability of ordinals and preservatio~ of elementary equivalence by operations on structures. Lindstrlim 1969 used Fraisse-Ehrenfeucht games to characterize L ww ' Back-and-forth systems for logics with cardinal quan-
tifiers are due to Vinner 1972 and others. Badger 1977 gives systems for 1ogi cs with Magidor-Malitz quantifiers (Magidor and Malitz 1977), and shows the fail ure of interpolation and preservation of elementary equivalence by products in these logics. Krawczyk and Krynicki 1976 give systems for certain monotonic quantifiers, without any appl ication. Makowsky 1977 a has similar systems and he studies monotonic quantifiers in detail. Back-and-forth systems for Stationary Logic, L{aa) (Barwise, Kaufmann and Makkai 1977), were given independently by Kaufmann 1978, Makowsky 1977 b and the author (Caicedo 1977 b). In our doctoral dissertation, we presented back-and-forth systems characterizing elementary equivalence in logics obtained by adding to first order logic
quantifiers of the form Q; ¢ (x), this means binding one or several variables in and a single formula, and gave various applications, particularly to L w w (Q1 ) L(aa). In this paper, we introduce back-and-forth systems appropriate for quantifiers binding several formulas: Qx·1· .. ~n(¢1(x·1)' ... , ¢n(Xn)) 83
XAVIER CAICEDO
84
Although the methods may be applied successfully to second and higher order quantifiers, as they were applied to L(aa) in Caicedo 1978, the corresponding results will be published elsewhere. We assume as known the notions of an abstract logic, as well as the extension relation between logics, and the notion of a generalized quantifier (lindstrom 1966, Barwise 1974, Makowski, Shelah, and Stavi 1976). Ol,;;&- , ... denote classical structures, and A, B, .•. denote their universes. In Section 1, we introduce quantifier symbols and their interpretations. Instead of considering quantifiers in the sense of Lindstrom 1966 and Mostowski 1957 only, we deal with the more general case of so called "weak models" where the quantifier interpretation forms part of the structure. Lindstrom-Mostowski quantifiers are recovered as families of weak models where the interpretations of the quantifiers are determined, up to isomorphism, by the domain of the structure. In Sections 2 and 3, we define the back-and-forth systems and prove the characterization of elementary equivalence. In Section 4, we consider monadic quantifierS, those where the quantifier binds a single variable in each formula, and extend a result of Friedman 1973 about the failure of Beth's definability theorem in cardinality logics to logics with these quantifiers. Also we show that any extension of Lww(QO) by monadic quantifiers satisfying interpolation must satisfy the downward Lowenheim-Skolem theorem. In Sections 5 and 6, we give a simpler version of back-and-forth for cOn~~ quantifiers, which becomes PC definable. The main applications deal with (infinitary) extensions by monadic quantifiers of L,,)w(Ql)' logic with the quantifie r "there are uncountably many". These include an analogue of Lindstrom's theorem for L ww(Q1)' a relative interpolation theorem in L ww(Q1) with respect to such extensions, and the existence of models satisfying few types in those extensions. Makowsky and Stavi discovered independently the relative interpolation theorem in L ww(Ql) and L(aa), with respect to their infinitary extensions. Finally, in Section 7 we show that elementary equivalence is preserved by cartesian products in a natural extension of logic with Magidor-Malitz quantifiers.
§1 GENERALIZED QUANTIFIERS. A qua~n~~ ~ymbot is a symbol Q together with a sequence of positive integers (111"'" l1 il > caned the type of the quantifier symbol. Given asetof relation, function and constant symbols, the language Lww(Qj) jEJ is obtained by addi ng to the usua1 forma ti on rul es of Lcos» for atomi c formul as, "l, A, and 3, the new rule: If Q. is a quantifier symbol of type (111' . . . , ttk), ¢1""'¢il are j ~ ~ formulas, and Xl' ••• , x k are lists of ttl' ..• , 11 k variables, respectively, then Q. j xl"'" X. k( ¢ 1"'" ¢ k) is a formula. It is understood that only those free variables of ¢ ~ which appear in the 1ist are bound by the quantifi er.
x~
type
If Ol
is a structure in the ordinary sense and Q. is a quantifier symbol of 11 > , thenan~n;(;eltplte;ta..t[ol1nOJt Q ~11 a is a family k
(111"'"
BACK-
Q~
9
"i nR (A ) x .•• x 9(A ).
Qj
where
AND-FORTH SYSTEMS
85
An L ww(Q.j)jE]-.6t!u.Lc.tuJLe has the form (Ol;Qj)jE]
is an interpretation of
a.
Q. j in
The semantics of Lww(Q.j)jE] is
defined in the usual way, except for the additional clause: (Ol; Qj)jE] 1=
Q. j
x1,···, xn (.
iff
interpreting the symbol
Qj
the language writing
1>
for
Loow(lJ../)jEJ
L+.
Qj is understood, we will abuse
The existential quantifier is the Lindstrom-Mostowski quantifier
3(A)
defined by the function well know.
=
{S
C
-
A
Is*-
0}.
of type (1 )
The following quantifiers are
CMd£na.t qua.ntiMe.M, type (1), for each ordinal a: Q.a(A) = {S:=.Allsl ;;;. wa}. for each ordinal a and finite 11., the quantifier of Q.~ (A) = {S :=.An 13 I ~A such that In ~S and III;;;. w } . a
Mag~do4-Maiitz qua.nti6~e.M,
type
(11.):
Chang qua.ntiMeJt, type (1) : Q.(A) = {S ~ A I HaJz.t.i.g qua.ntiMeJt, type (1,1): H(A) HenlUn qua.ntiMeJt, type (4) : Hen(A)
lsi = IA\}. {(S, T) I S ~A, T ~A, lsi = ITI}. 4 {S ~ A I 3 6, 9 : A -+ A such that 6Xg ~S}
Note that the logics obtained from these quantifiers do not incl ude those where the meaning of quantifier is not determined by the domain of the structure, like Sgro's topological logic (Sgro 1977). However, if we consider only logics for classical structures with a finite number of relations, functions, and constants, then any logic is a sublogic of some Lww(Q.j)jEJ' Even more, if the logic L is closed under substitution of relation symbols for formulas then L is equivalent to some Lww (Q.j ) j E ] '
§
2 BACK-AND-FORTH SYSTEMS,
Through this section (a, q) and (~, 4) will be quantifier structure where q 4 are interpretations for a quantifier symbol of type (11. " " , n ). A and k 1 B are assumed to be disjoint. Sequences in An u s'' (11. E w) wi 11 be denoted
and
a,
a, a', a', T, T ' , h, h'; the value of catenation is denoted by juxtaposition.
11.
will be clear from the context.
Con-
DEFINITION 2.1.- A baek-and-6olLth be;tween (a; q) and (~; 4) consists of a linearly ordered set p = (P, b,i isapartial isomorphism from 01- to&. As (-U:), interchanging the roles of
(.i..u:)
DEFINITION 2.2.(01-; Cij)jEJ
each
jEJ.
(p, {E~I pEP, nEW})
isa
:to ($-;Jr.j}jEJ ifitisonefrom (OI-;Cij) The existence of such a relation is denoted by
baek-and-6oJvth
to
(:6-';"j)
(a;Cij)jEJ~
61tOm
for
()&..;\)jEJ
REMAR K. We assume that any back-and-forth sa ti sfi es the extensi on property for the existential quantifier. It is enough to postulate in (,i,i) , for each p' p a 'V T and p < p' the existence of a function 6: A-+B such that o a. 'V T 6 (a). Property (B) holds automatically. DEFINITION 2.3.- {E I nE w} is a bael 61tOm n ioi, Cij)jEJ to ($-; "j )jE] if En is an equivalence relation in An U Bn for each n , and properties (,i) to (,iv) of Def. 2.1 hold, dropping the parameter conditions.
§
Such relation is denoted by (or; Cij) jE]
'V
(~; "j) jE]
.
3 CHARACTERIZATION OF ELEMENTARY EQUIVALENCE.
Let or and!tJ., be classical structures. C = {Cij I jE J} and D = {\ I jE J} are interpretations in 01- and~, respectively, of the quantifier symbols
XAVIER CAICEDO
88
{Q.[ jE]}. j
THEOREM 3.1. -
(ot; C) ~ (~; D) then (ot; C)
I6
Suppose (ot;C) ~ (~; D). Bn define:
E
a
{J
(ot, a; C)
iff
"val
iff
(:e-,1';D)
(ot, a; C)
iff
and a, a'EA Yl
For every {J
( ;g,; D) (Xr; D) (-Lil).
wheJte P 1.1> non-well. ondened, (bac.k-and-6oJtth wUhout Pa.JLamet:eM).
The same as the proof of Theorem 3.1, but
simplifying
the definition of back-and-forth to: a 'V 0' iff (Ol, a; C) ~ (Ol, 0'; D), etc. Instead of t~ one defines t .. = {4> (y) I (Ol. 0; C ) F 4> (a)}, By Lemma 1. 2, a. a. fI t .. is a sentence of L (Qj') j' E ] ' The rest of the. proof is simpler because a. oow we do not need parameter conditions. Choose any non-well ordered set P and define 0 ~ t iff a » t , Let PI> P2 > ,., be an infinite descending sequence of P , One shows, as in the proof of Theorem 3.3, by induction on the complexity of 4> (y) , that for all n: => (U).
(-Lil)
(U)
=>
(i),
o
Pn 'V
implies:
r
(Ol; C ) F 4>(a)
iff
(~; D ) F 4>(r),
To use the extension property in the inductive step for one chooses
§
4
P n + .:\
< P n +.:\ _ I < .. , < P n ' •
Qj
xl.· ... xm(4) l' ,, . , 4>m )
INTERPOLATION AND MONADIC QUANTIFIERS,
A quantifier symbol of type n(x ,This includes the cardinal quantifiers Qa: "there are at least wa eln ments", as well as Chang's and Hartig's quantifier, but notthe Magidor - Malitz quantifiers. Throughout this section L M will denote the logic L oow (Q.). E ] , j j where Qj runs through all the possible Lindstrom-Mostowski monadic quantifiers.
»·
LEMMA 4. I . -
a =L
oow
16 Ol and :t; Me .:\:tJtuc..tu!l.u 06 poweJt at
(Q):tr impUu
I
a
=L;,t-
M
mO.6:t
wI' then
•
PROOF. Let 2n = {o I 0 : n ... 2 }, for seA 1et sa = Sand sl = A - S • For each nEw and F: 2n ... {K 'K cardinal, K" wI} define the quantifier:
~(A)={(SI,.. "S)',;/OE2nl n
l~~)I=F(O)}, Since a monadic structure n L« n ,(. (A, SI".'. Sn) is completely determined by the above set of cardinals, there corresponds to it a unique F such that (Tl'".,T n) E Q F (8) i.ff (8, T1,.", Tn) "" (A, SI" ." S). Let 3!K x 4>(x) mean that the truth set of, .4>(x) has exaclty K n elements (K';; wI)' Clearly, for structures of power at most wI' Ol=L (Q);6implies
a
=L
(Q
3,K)
Ir.
and so
oow l' , K definable from the former quantifiers.
oow 1 (Q),(",' because the QF' s are oow F F If ~O is 4> and 4>1 is I 4> :
a
=L
92
XAVIER CAICEDO
() () ) QF"l"'" ( '1>1"1"'1>" rt
rt
rt
=>
f\ I>E 2 rt
1 (,,)1> (i I 1 . [3.' F(I> ) x i Sup {] 1= q,} = x , 16 not have Lowenheim numbeJt such. !>entenc.e haJ> modetJ> 06 MbWuvt.Uy iMge -i..ty.
PROOF.
L+ doe!> c..etJLd,[nat-
q, with infinite models of size X, X < K
Suppose that there is
,
but not of any size between x+ and K (included). By definition of U5wenheim number, there is a sentence l/J with all its models of power greater than X. Let e be a sentence whose models are the equivalence relations. The classes K 1 (respectively K ) of models of e having as many equivalence classes as a model of 2 q, (respectively, a model of l/J) are PC classes of L+, defined by the projection of the sentences: 8 /\
"6
is a function onto V" /\ 'ifx 'r1 Y (x E Y +-+ 6(x) = 6(Y)) 1\
q, V (resp.
1/1 V).
Since these sentences do not have common model s of power 1ess or equal than", K 1 are disjoint PC classes of L+. However, they are inseparable in L+.If 2 has X has equivalence classes of power X', and (A', E ') E K has (A,E) E K 2 1 X' equivalence classes of power X', an easy back-and-forth argument shows tha t and K
(A, E) =L (A', E'), and so (A, E) =L (A', E'). C M terpolation fails in L+. • COROLLARY 4.5. -
Le;t L+ be a tog-le between
tiv-lzaUon and -baU66y-i.ng -i.nteJtpotaUan, then Sk.atem theOllem.
From this we conclude that in-
L ww (12. ) and
L hav-lng IletaM L+ -6aU6Mu the dOwnwalld Lowenheim-
0
PROOF. There is a sentence in Lww(Q.O) ' and therefore in L+, which has models of power w but not larger. By Theorem 4.4 it must have Lowenheim number w .•
§
5
COFILTER QUANTIFIERS, A SIMPLER BACK-AND-FORTH.
n Let 12. be a quantifier symbol of type (n), q ~ 9(A ) is a eoMUeJr. -lnteJr.pllUaUon 06 Q. if it satisfies Monoton-i.U.ty: SEq and S ~ s' imply S' E q, and V-f.,f,Wbu.tiv-i..ty: SuS' E q implies SEq or S' E q. Obviously, q is a
q
cofilter interpretation iff the "dual" interpretation = {S I A - S €I- q } is a filter over A. In terms of the language, (m; q) must satisfy the schemata:
\!X(¢4l/J) 4(Q.xq,4Q.xl/J),and
q
tifier is w-eompte;te if u {Sn I nE w} E q implies
QX(¢V1/I)4(Qx¢VQxl/J).
is an
w- complete filter. for some n. •
Sn E q
Acofilter
quan-
Equivalently, if
Qa is w-co'mplete if its cofinality Ql' Magidor-Mal itz quantifiers are not
The cardinal quantifiers are cofilter, is greater than
w,
as is the case of
cofilter, however the quantifier
n H
o'
where
H~ xl'"
xn ¢(x 1"'"
eto x
is equivalent to the cofilter
quantifier
means "there is an infinite set n)
1
such
XAVIER CAICEDO
94
tha~
a 1, ..•• an
for all distinct
06 c.G6ille.!L quan..ti..Me.!L btte.!Lp/te:tatiOft6 and a.Mume the numbe.!L 06 /te!atiaft6 in Mc.h -6tJr.uc.tu!Le. is MMte, then: (a)
1'1
(aL; C) == (i:-; D)
r 6 in
ad~a1'1
the quan.ti Me.M
1'1
(b)
(al; C ) ==
(e.)
t
6
(z, ;
(oz; C)
(oZ; c)
in6
~
(
.t
D)
S(w)
i66
Me
(1'1, and !J;; if they do not have an interpolant in L w w ( Q1 ) ' then the PC classes K = {OZ r- T I or 1= 1>} and 1 K = {tr t T \;e, F 1/1} are inseparable. By Lemma 5.3, there are structures OZ I=(jl 2 and Z, F 1/1 such that (J1.. t r ~ L :t, ~ T. Therefore, 1> and !J; do not have interpolants in
L W
•
M
L ww (12 1 ) , In T ha.6 a (uncountab.tel model., it ha..6 a (u,ncouYl.ta.b.£.e) model. .6aU~nying at mo.6.t coun.tab.£.y n0lt each nEw. many n-.typv.. in L W THEOREM 6.5. -
PROOF.
Take
Le..t
K
1
T be. a couYl.ta.b.te. the.oJty in
= K 2 = Mod(T)
in Lemma 5.3.
Theyare
obviously insepa-
97
BACK-AND-FORTH SYSTEMS
rable; then we have of power at most
a-6-;'.
(a, Q1) If
wI'
t
i~wJ,
(JtJ-; Q1) with P non-well ordered, in
AIt , find
bEBIt such that
ia , a)=\
a
and f(;..
;-6;
then
(Q )(i6, b)=L (Q)(Ol, a'); see Theorem oow 1 oow 1 3.6. By Lemma 4.1, ta , =L (m, Therefore, it and satisfy the same M type in L Since the number of equivalence classes of - is countable, the same
Hence,
a)
a').
M.
is true of the number of types.
a'
•
In L w w we have that a theory with infinite models has an uncountable model satisfying at mostcountably many types OVe!L ea.eh eoun;ta.b£.e ~u.b~e.t. Thi sis no't true here as shown by the counterexample:
Q x P(x), --, Q x R(x), 1 1 If x If y(P(x) /\
p(y) /\
"
I 1/!)=P (1l)=jl' (Mod ModK(1/!,(1l), ModK(1/!) where the measure u ' defined on subsets of Mod K (1/-.) is the restriction of u to Mod K (~ ). A few remarks about this concept are in order. 1) We could be tempted to write P(p IK) instead of PK( (T), then 1);
"'F F
6(0)) ;
otl
x 1/.{ 6] iff X: (T)E R¢; 6 (iii) for negation, disjunction, and conjunction, same clauses as usual; (i v) if l/J = 3 v e, then
"n.
~ F x l/J[ 6J if there is an I'l E W such that Ot. -r:F x l/J! n( t1 (~) is the assignment which coincides with 6 in every variable except rly) v where it assigns n .
1\
For each one-one Mod
x E wT , and sentence 1); of ot F F x ~} •
(1j;) = { F :
where (possi-
s'
L
r K,x: As for simple probability structures, we define
T
(p ). Mo~ K K, x The a-field 11)- of subsets of TK where the measure should generated by these sets ModT (1/) • K. x 1j.
r'V ¢ if K, x
A measure on this a-field
Mod
r K, x
11-
(1jJ)
invariant under
T
K
is
be defined is that
the product measure ~T
of
the measure u defined on the a - field M generated by the sets ModK (¢ ) for 4> a sentence of L,and invariant under "'K' We then defi ne PK, x ll/JJ=
108
ROLANDO CHUAQUI
llT (ModK, x (~.))
for every sentence
\j.
of Ls
.
Bernoulli's law of large numbers can be expressed and proved in this representation. We have, for each real number E > 0, a sentence of LS that expresses that the re 1ati ve frequency of the truth of the sentence jl of Lin the n fi rst terms of the sequence x differs from PK (~) by more than E. This sentence can be written by (*) V (3:mV(v1) " O. K, x These laws can be proved in the usual way, since the measure . ~is the product measure wjl. All other laws of large numbers and central limit theorems also can be obtained. In this type of models, the classical methods of estimationand hypothesis testing can be justified. I shall illustrate this with an example. My discussion follows closely Lucas (1970), Ch. V. Let us suppose we have a coin that looks unbiased and that we have no other relevant information about it. The natural simple probability structure which describes the possible outcomes of tossing the coin is that which assumes the coin's being symmetrical with weight distributed evenly between the faces. Thus, we assume a simple probability structure K defined as KoofExample 5 in Chuaqui 1977, Section 6. The reasons for assuming this model may be varied, just as the reasons for accepting any scientific hypothesis are varied. Among these reasons we may count, for instance, past experiences with similar coins. We now proceed to test this hypothesis. In order to do so, we toss the coin The probability that the frequency of, say,heads in these tossings is computed by building the compound probability structure wK (it is really enough with nK) and proceeding as in the Frequency Model constructed above. If this probability of the actual outcome is very low we reject the orign times for some large number n,
FOUNDATIONS OF STATISTICAL METHODS ina1 hypothesis that K is the adequate simple probability structure, The ability level at which we reject the hypothesis is arbitrary. However, the is justified (cf. Lucas, 1970, Ch. V) because given any significance level obtain an n large enough where this level can be attained. Thus, there is quence of events with probability approaching zero,
109
probmethod we can a se-
If we reject the original hypothesis, using methods of estimation, we may assume a new simple probability structure K' as a model for the coin tossing, This K' has to assign a reasonable probabil ity (using again WJ iff T
~S.
R ~ p. and
we define:
113
FOUNDATIONS OF STATISTICAL METHODS (v)
Let t E T, then Tt={~: s ETand~ '" ( T, RI'S > ,then S E F.
T
E
It follows immediately that if T E F and x E T, then T and
x
Ix E F.
DEFINITION 4.6.- K =( I, H) is a compound pltOba.b.u..u:y hbtu.cxWte (i)
I =(
(i i )
H is a set of functions such that if 6 E H, then
if:
R> is a causal structure.
F,~
Va
6E
F.
(iii) For each 6 E Hand;t EVa 6=T the set H(6,x) = {g(x) : 9 E Hand 9 I'T " 6 I'T } is a simple probability structure of a fixed simix x larity type v. (f v)
For each T
E
F , there is an 6
E
H, such that Va 6 " T,
is called the causal structure for K and ,I the set
K.
of compound outcomes of
DEFINITION 4.7.-
Let K= (I, H> be a compound probability structure, Va 6. Then, 6 : 6 E A and VA 6 z: T J,
A CH, T E F, 6 E if and x E
(i)
AT = {
(i i)
A(T) "
{6 I'T : 6 E A }
I
(iii) let A ::. H T ' then, A(6,;t) " {g(x) : 9 E A and 9 I' T;t " 6 I' T } • x We now assume that for each simple probability structure H (6,x)there is given a symmetry relation between its subsets ~ 6,t. We extend this definition to an equivalence relation -v between subsets of H, A, B,such that A ::. ~ and B £ H T" for
T , T'
E
F.
First we introduce a definition of isomorphic simple probability structures. DEFINITION 4.8.- Let K and K' be simpl e probabi 1 i ty structures with universes A and A'. Then, (i)
K~
9
K'
iff K'
9
E
AA',
={ g* q{,
:
9-1
at.
E
A' A and
E K}.
Let ~ K ' ~K' be equi va1ence re 1at ions between subsets of K and K', B £ K and C £ K'. Then B '" C iff K ~ K' and g**B ~ K' C. 9 9 If B"'g C, then Band C are symmetric with respect to '1 O. Let S be a component of T and
e.
E Sn T
with
t~
E '!' with'!' C 1> and y'1- fv 4> : z " {x :
4>} .1.-6, and only .1.-6,
z = Uy I 'v' X (xE Y
InS we can introduce the fo 11 owi ng objects of (i) (f t )
+->
3 u(x E u ] A 4» ]
S:
a .opec. a, if a" T wi th fv T a coUection 06 cfu6!.>u B, defined by
"
0
B{z) if, and only if, 4>(z) with 4> E '!' and fv 4> (iii) a unaJt.y op~on F defined by x = F(z) if, and only if, x = T with fv T ~ {z} •
~ {z} •
Now, we restrict ourattention to systems S including the impredicative schema of class specification and some assumptions such as extensional ity and others that will be clear from the context. We define the ordered pair of two classes a and b, I a,b] , by z
"[a,b]
if,
and only if, z = a
x
{O}
U
b x {l}
We have the following definition schemata, for every TEA and 4> E 1> : z =
{T :
xl'" .xn
z=u
x , •• • ,x
4>} if, and only if,
z
>
{y :
3 xl'" 3 x~ ..
{T:4>}if,andonlyif, z"{Y:3x •••
n
3x~
,c.
(y " TIl, <j>)}
(yETA4>)}
Collection of classes F defined by F(z)
if, and only if, h(x E f A z = F(x) I,
where F is a unary operation, are called .6upeJt.cfu6!.>u. Superclasses can be represented using the relation It = UX{TX{X} : q,} which represents an operation defi ned by the term T, in the sense that for every 4> E 1> , I- 'v'x (q,'" T " It*{x}). S Hence if we define z = x In q,] if, and only if, z =
U
{T
x {x}: q,} with TEA and q, E 1> ,
we can represent the superclasses F by the relation F =i F(x): x E f] in the sense that any class z of this collection is the image under F of {x} for some x E f. F is called a !.>upeJt.c1M.6 ltelat.i.-on. F is called the de6.1.-ni.-ng opeJt.a..Uon of the superclass relation F and is denoted by Oe6 F j f is the set of
124
MANUEL CORRADA
c.odu of F , in symbols
Cd F. Bold face capital letters F, G, H , ... will be used to denote arbitrary superclass relations and the corresponding bold face italic letters the respective superclass. The corresponding light face capital letter will be used to represent its defining operation. If a superclass relation is represented by an arbitrary bold face capital letter, say F, it will be assumed that the correspondi ng sma 11 bo 1d face letter, f, denotes the set of codes of F.
In order to give full proofs of various result stated in the subsequent discussion we would have to use some properties of superclasses. As we have pointed out, superclasses can be represented as a special class of relations, that we have called superclass relations. As a consecuence of this,various results in the theory of superclasses can be obtained from analogous results concerning superclass relations. First at all we introduce the b~nany pte~eate eta between classes and superclass relations, as an abreviation: y 1/ F is an abreviation of 3 z(zEf 1\ F(z)=y). 3y 1/ F will be an abreviation of 3y(y 1/ F). By means of the eta predicate the fact that superclasses can be represented by their corresponding superclass relations can be stated precisely as follows in the equivalence of the following two schematic propositions: (a) F is the superclass relation that corresponds to the superclass F (b)
F(z)
~6,
and only
.(6,
z 1/ F
Due to this fact we will use interchangely the terms superclass and superclass relation. In most of the cases we will say superclass instead of superclass relation. Let ~ be a binary relation. class f~( F) as follows:
For an arbitrary superclass F we define
f~(F) = {<x,y): xEf II yEf 1I - z ={{x}, {x,y}} ,
from (vi). from (vi), (vii) and P.
z e u x v - zEV !\3x3y(X Eu!\y EV!\z = } n A = {x : xEA A <j>} • • An alternative, weaker, formulation of Metatheorem 2 would be {TI,T2}=P+ECEO' Let xE = CEO U {T 3, T4, T5, T6}. By the second part of Metatheorem 2 and Lemma 2
PARAMETERS IN THEORIES OF CLASSES 1, (xii) we have XE2 I- P+E 1.
129
Using Metatheorem 1 and 2 we get:
We believe that the above metatheorems, in particular Metatheorem 3, are specially convenient, in the axiomatic foundations of theories of classes that include the impredicati ve comprehensi on axi om schema, such as the Genera 1 Class Theory or oneof its extensions such as the Impredicative Theory of C1asses. This leads to considerable simplifications in the metamathematics of class theories with the schema CEo As examples of this kind, 1et us quote the construction of £ , the constructib1e universe, [cf. Chuaqui 1980 l , and the techniques of replacing classes by numbers in Corrada & Chuaqui 1978. It is known by resu1ts in Gi1more 1974 that the extensionality axiom, that we have called E, is inconsistent with some set theories. Thus, for foundational purposes it will be convenient to eliminate Axiom E. In all the metological relations that implicitely involve the derivability relation I- , that appear in the above metatheorems, this re1ations are relativized to sentence E. In order to avoid this relativization there is a device that we sketch here and wi11 prove to be particulary useful in the next section. In xEl +P, the sentences that refer to the existence of some particular" classes and operations between them insure the existence of this classes and operations but not it uniqueness. Consequently locutions such as "y is one of the classes of elements in a such that ~", "y is one of the unions of", "y is one of the universal classes" applies properly in this case. We shall show how to avoid E by an example of the step-by-step procedure involved. By a usual theorem of first order logic [cf.Monk 1976, Theorem 11.32 pp. 2101 even in the absence of extensionality we can introduce denotation symbol s for classes and operations between them. In order to illustrate this step - by - step procedure we take the last part of Metatheorem 2. Here it is used the fact: CEO + I ~ A () V = A. We have: (i) (i i) (i i i) (iv) (v) (vi)
{CEO' {CEO' {CEO' {CEO' {CEO' {CEO'
I}
I-
I}
I-
I}
I-
I}
I-
I}
I-
3yVx(xEy 3yVx(x Ey \fx(xEA-+ Vx(xEAf\ Vx(xEAf\ \fx(xEAnV
...... 3 u Ix e ul f\ x = x) , by CEO' ...... 3u!x EuJ), by (i). 3u(xEu)), logical inference. 3u(x Eu)-+x EA), logical inference. 3u(xEu) ...... x EA), by (iii), (iv).
by I, (i i ), (v) and the corresponding denotation of the symb01s. Arguing in this way we obtain the ana1ogous of the previous metatheorems, not invo1vi ng the sentence E: I}
I-
METATHEOREM
1'.
METATHEOREM
2'.
METATHEOREM
3'.
CE=pXE2.
A set of sentences
W is ca11ed
...... xEA),
CE=pxEl' HI' T2}1- CEO' CEO I- T2 and, {CEO ,I} I- T1.
n~~e a~omat£zab!e
OVe4 a
~et
On
~entenc~
MANUEL CORRADA
130
r
if there exists a sentence
such that
~
{r,~}
=W.
It is easily shown that for every n;' 1 in the metatheory, CE +1 is a finite n extension of CEn' This can be proved adding to CE n all the possible classes and operation between them definable in CE n +1 but not in CEn' This large number clearly is finite. The restriction n;' 1 is not essencial; in fact as a consequence of Metatheorem 3 or Metatheorem 3' we obtain:
CE and because CE
,fA
MrU:te1-y ax..ioma.t.i.zab!e oveJt CEO
is not finitely axiomatizable we have in consequence: CEO u not 6-{.rU:te1-y
aJUoma..tizab!e.
To conclude we may mention that in spite of the simplicity of the schema CEO' a very weak theory of classes based upon this schema can be proved to be undecidable. §
2.
THE SETS OF SENTENCES
pr ed
XE1
AND
XE 2pr ed
We denote by Tired Zermelo's predicative Aussonderungsaxiom without rameters, t , e. : Tpred 1
Let
Ifx3yltz[ZEY+->ZEX
XEpred 1
=
{Tpred T l'
2'
A~) where ~ES, fv ~ ~
{x}
pa-
and x « y.
T} . 3
It is easily seen that in XEired + p + E one can prove the existence of singletons' the empty class, pairs, ordered pairs of sets and the union of two classes. With this, using Theorem 1 in Levy 1974 with minor modifications we obtain the analogous of Metatheorem 1 for the predicative comprehension axiom: METATHEOREM 4. The analogue of Metatheorem 2 is as follows: METATHEOREM
5.
{Tired, T2} ~ CE6. CEo ~ T2 and {CEO'
I}
~ Tired.
If we set XE~red = {CE6' T3 , T6 } the analogous of Metatheorem 3 can be lished using Metatheorems 4 and 5:
e s ta b-
Due to the remarks at the end of the above section it turns out that the sentence E can be eliminated from these metatheorems, and we have: METATHEOREM 4'.
CE S=-pX E1pred
PARAMETERS IN THEORIES OF CLASSES
METATHEOREM 6'.
131
CE S;: PX E~red
It is easily verified that for every '1re F),
characteristic of e. We designate by t: the smooth operator that associates to any nonempty the set of all functions 6 of PIA) into A, fulfilling the following
set A
condition:
6(B) f/- B if B *- A, and 6(A) is any element of A. t xF [x]
is defined as
'an x such that iF [x] "
and t and r are associated;
in several cases, d(r) may be restricted to functions which satisfy the require-
¢ V if V is a definable set different from the universe, d[r) (V) is a fixed element of the universe otherwise. The axioms of rare:
ment that: d(r)[V)
and
rxFz[xl = rYFz[Y]' \lx(P
+-7
Q)
-> r
xP = r xQ
and the scheme 3x(x = r x F A F) .... \lxF.
THEOREM 6.-
vbto.6 b., e 06
Let L e [LI.' L r) be the tanguage L .ill wlUeh one 06 ,U.6 [I., r l , with the eotl.Jl.e.6ponding axiom.6, and T u {F} be a set.
t.,
eto.6ed 60tLmuta.6 06 1)
e (I., r ] b., notLmat;
3)
16
(LL,L
4)
r)
(LI. ,L r
l.
We have: 2)
inn
In L
rl-F
1= F
i66
I' 1- F;
PROOF.- Obviously,
6O!Lmuta 06 the 60tLm i3xF .... exF=ex(x*-x)
L,
e and r are normal.
[rxF *-
2 and 3 are consequences of the corWith respect to 4, it suffices to
that if i3xF, then we have \Ix "l F and \lx(LF
THEOREM 7. -
e
b., a theOlLem.
responding results for vbtos in general. Therefore,
I' 1- F in L
in L e (LL,Lr ) without e (L,r);
e [L r ) any
r x (x= x) .... i\lxF 1
axiom.
r
e [I., r ] doe.6 not oeeM in the 60tLmutM r U{F}, then:
\lx(F
Let LL e
+-7
x*-x), and
x = x), since
u
16
x = x
note
is an
exF= ex(x*-x) . •
be the tanguage L in wlUeh two
and e, wah the tLe.6pective axlom.6. theotLy Twho.6e wnguage
+-7
amo ng i l l vbto.6 Me I.
I. X (x *- x) = e x (x *- x) b.,
vaHd -in
then LxF=ex«3: xFI\F) V (i3:xFl\x=ex(x*- x))) Ll.e,
cs a theOlLem 06 T 601L any 60tunu.ta F. PROOF.- Suppose that 3:xF. = eX(3:xF/\F)=exF.
Now. suppose that
Then, ex«(]:xFI\F) V (i 3:xF 1\ x= ex(x *-x»)
But exF
and I.xF satisfyF; hence, I.xF = exF.
i 3: xF. Therefore.
ex ( ( 3:xF 1\ F) V ( i 3: xF /\ x =
ex(x*-x»)=ex("l3:xF/\ x =ex(x*-x)) =ex(x=ex(x*-x»=ex(x*-x);
a
NEWTON C. A. DA COSTA
140
since LX(X*X)
= ex(x*x),
it follows that
t
xF
=
€xF . •
From now on, when we say that a is a structure for L we suppose characteristic postulates of the vbtos of L are valid in Ol.
tha t
th e
MODEL THEORY AND NORMAL YBTOS.
Now we shall indicate how most results of elementary model theory can be extended to first-order languages with normal vbtos. First of all, it is immediate that we have the following propositions: THEOREM 8 (Compactness, first form). A 6oltmu£.a A M, a .:theoltem 06 a .:theolty T 06 L -i66 A M, a .:theO!Lem 06 bOme 6bu.:te.£.y ax-i.omatized pM.:t 06 T . THEOREM 9 (Compacteness, second form) • -
Let T be a .:theMy -in L;
T hall
a mode.£. i66 evelty MvU.:te.£.y axiomatized pM.:t anT hall a mode.£..
Let a = is a weak isomorphism of a and J} iff the following conditions hold: 1) rf> is an isomorphism in the usual sense of A and B; 2) rf>(rJl(vxF» = J}(vxF¢) for any formula F of L{Ol) having no free variables other than x, and any v E V. The bijective mapping ¢ from the universe of Ol to the universe of J} is called a strong isomorphism of ol and J}, which are said to be strongly isomorphic, iff we have: 1) ¢ is an isomorphism of A and B; 2) dry) (K) = k if and only if g[v)(i(K)) = ¢[k), for any v E V, K c loll and kE 1001,where if, is the canoni ca1 extension of ¢ to the power sets of lOll and I;;' I . An isomorphism is a weak or strong isomorphism. Ol and J} are isomorphic if they are weakly or strongly isomorphic. Let rf> be an i6omO!Lp~m 06 Ol and;;'. c1.o-6ed .:te.Jtm a 06 L (Ol), and OleA) = J}(A rJ»
THEOREM 10. -
nOlL
eVVLIj
Then
nOlL
¢ (Ol
evVLy
(a»
= J} (a rJ»
c1.O-6ed noJtmufu A
06 L (Ol).
PROOF.- By simultaneous induction on the lengths of a and A. If ¢ is a weak isomorphism, the proof is as that of Lemma 1 of Shoenfield 1967, p. 172, taking into account that ¢ (Ol (v xF) = .G-(vxF¢) by definition. If ¢ is a strong isomorphism, the proof is Similar, but we have to show that ¢(Ol(vxFl) = J}(vxF rJ». By induction hypothesis, we may assume that a (F[a]) = T iff J} (FrJ>[arJ>])= T. Therefore, ¢(d(v)([al rJl(F[a] )=T and a is the name of a l ) = g(v) i( [al
141
VARIABLE BINDING TERM OPERATORS
OZ(F[a]) =T and a is the name of a]»
= g(v)([¢(a)l.fr (F¢[a¢]) =T and a'"
is
the mane of ¢(a)]) = .G- (vxF"'), by the definition of strong isomorphism, hence ¢ (07, (vxF)) = fr (vxF'" ) •• COROLLARY. -
I.>bwng -LoomOltphfAm -Lo a weak -LoomofLphfAm
EVVtIj
and
I but not c.on-
vVt.Qetlj) .
r is a set of formulas of L, Ol and .fr are structures for L, and lOll subset of I £.1.
is
a
In this case, if a is an element of 107,1, we employ the same name
for a in L(07,) and in L(.fr), as in Shoenfie1d 1967.
We say that Ol is a r -sub-
structure of.G- (and that.f,. is a r-extension of Ol) if, for every closed formula A
a. (A) = T
in r (Ol),
imp1 ies that .fr (A)
= T.
It is clear that if r is
of all formulas of L which do not contain quantifiers and vbtos, and and £. = < B, 9
>,
the set
07,= in
If I' is the set of all formulas of L without vbtos, we say
elementary substructure for r-substructure, and if I' is the set of all formulas of L, then the r-substructures are called complete substructures. THEOREM 11.-
16
e -Lo a c.on.atant 06 L and x -Lo a valUable, x =e betongl.> to I",
and Ol -Lo a p; l.>ubl.>tJtuc.tuJte 06 .fr, thcn 07, [e ) =
PROOF.If
We have:
j} (e J •
See Shoenfie1d 1967, p. 74 . •
a
and .fr are structures for L, and lOll C l.fr I, then the structure
is defined as in Shoenfield 1967, p. 74.
The notion of r-diagram of
.f.r 0l OZ is also
the same of that book, and denoted by Dr ( Ol). THEOREM 12 Me I.>bllic.tWLe.6
.(.oo
(Diagram Lemma) • -
oOlt L
I.>uc.h that
16 r -Lo
a
se:
06 60fLmula.a .{.n Land
107, I C liT I, then 07,
iTOl-Loamodel06 0r(Ol).
PROOF. -
If
a.
07, and .fr
-Lo a r - l.>ubl.>tJtuc.tWLe 06 .fr
is a p; substructure of .fr and A is an axiom of
°r (OZ),
then A is,
by definition, valid in iTOl ' and froz. is a model of 0r(Ol). Conversely, if.f.rOZ is a model of Dr(Ol), then every formula of rrOZ) which is valid in a , is also val id in .fr, and
o:
is a I' - substructure of iT . •
A set of formulas r is said to be regular if x = y and x"* y belong to and for evei-y formula A in r, the formulas of the form A[ xl' X2 ' ••• , xu] belong to I", THEOREM 13
(Keisler I s model extension theorem) • -
r, also
Suppol.>e that 07, .u, a
142
NEWTON C. A. DA COSTA
-6vwc.twr.e 601t L, T Ls a .theO!llj wah laYlguage L, aYld I' .u, a lLegu£M -6et 06 601tmulM 06 L. TheYl, a has a r-Qx-teYl-6ioYl wlUch.u, a model 06 T i66 eveJtY -theOltem 06 T wlUch if.> a d-WjUYlCUOYl 06 negatioYl-6 06 60ltmu£M in I' .u, va1.id iYl (Jl • PROOF.-
The proof of Shoenfield 1967, p. 75, of Keisler'S theorem for standard
model theory remains valid, since all results on which it is based (the reduction theorem for consistency, the completeness theorem, the diagram lemma, etc. ) are true for model theory with vbtos.
•
Let rand t. be sets of formulas. t. is said to be associated to I' i f formula of the form
I;fx 1 I;fx 2
•••
I;f x n A
every
is in t., where A is a dinjunction of
negations of members of r. Bya reasoning similar to that of Shoenfield 1967, pp. 75-76,
we establ ish
the following proposition: Let I' and t. be -6W 06 60ltmu£M,
THEOREM 14. -
c.ia-ted to
r',
wdh I' lLegu£M and t. aMO-
OZ a -6-tltuc.twr.e 60lL L aYld JJ a t.- ex-ten-6-i-oYl 06 OZ; -theYl, -theJte exJA:t6
a I' -ex-teYl-6ion t 06 JJ wh-i-ch if.> a complete ef.>-teYl-6ioYl 06 OZ (and a 60!ttioM an elemen-tMlj ex-teYl-6ioYl 06 OZ). Suppose that T and T'
al-6o
are theories of L; it is clear that T' is an exten-
sion of T iff every model of T' is also a model of T. If I' is a set of formulas in L, r' is a subset of I' composed of all formulas of I' which are theorems of T, and every structure in which all formulas of
r'
are val id is a model of T, then T is equivalent to a theory having all nonlogica 1 ax i oms in r. THEOREM 15 (Los-Tarski). Let ~ deno-te the. se: 06 aU 60ltmu£M 06 L which do Ylo-t con-tMn bound vM-. A theOltlj T .u, eQu- do no-t COn-tMYl bOUYld vAAiablef.> -i-66 eveJtlj ~ - -6ubJ.>-tlLuc-tlllLe 06 a model 06 T if.> al-60 a model 06 T.
PROOF.-
If T is equivalent to a theory T' satisfying the conditions of the the-
orem, then every definition of
~
~
- substructure OZ of a model .r;. of T' is a model of T', by the
- substructure; thus, every
~
-substructure of a model of T is
model of T.
Conversely, suppose that every
mode1 of T;
we have to show that if every theorem of T whi ch does
~
bound variables is val id in the structure C. , then ( Keisler's theorem, also a model of T.
C has a •
Analogously, we have:
~
a
- substructure of a model of T is a is a model
- extension which is a model of T,
no t con t a i n of
T.
and so
r:
By is
VARIABLE BINDING TERM OPERATORS
143
LiU be the. ~iU 06 aU. 6011mulal> in L in wlUch :thel1.e Me no oc06 qua.n:t.-LMeM, and T a :theMy wUh £.anguage L. T ~ e.qiUvatelVt t» a :the.oJr.y who~e non1.ogicat axiom~ Me in i66 eve.Jr.y -Mb~bwc:tUJte 06 a model 06 T ~ a model 06 T. THEOREM 16. -
CU!rJLenc~
Le:t e be. the: set: 06 aU. 6011mulal> in L wlUch do no:t con;ta.{.n vb:tM.
THEOREM 17. -
FM anlj :theMIj T with .tanguage L, T ~ eqiUvate.1Vt to a :theMIj who~e nonC.ogicat ax.iom~
e i66
belong to
eve.11.1j elemelVtMlj ~ub~:tI1.uc:tUl1.e 06 a mode£. 06 T~ amode.t 06 T.
The sequence of structures O1- j = , 01- 2 = < A 2,d 2>, ... is a chain if for every n , An+ 1 is an extension of An in the usual sense of model theory. A union of a chain O1- j.01- 2 •... is any structure 01-= such that A is the usual union of Aj• A2.... Consequently. a given chain Olj. 01-2 •... has in general several unions. An elementary (complete) chain is a chain O1-j. (Jl2 . . . . in which. for any n , a ; is an elementary (complete) substructure of (Jln+l Clearly. if , ,... is an elementary (complete) chain, then Aj• A2 constitutes also an elementary chain in the usual model - theoretic sense. A strict union of a chain of structures (Jlj, 01- 2 " " is any union ot of (Jll' Ol2"" satisfying the extra requirement that if A is a formula of L ((Jln) without free variables, then 01- (A) =Oln(A). for every n; in other words, a strict unionof a chain is any union which is a complete extension of every structure of the chain. Obviously, the next two propositions hold: LiU r devw:te :the ~iU 06 aU. 6011mulal> 06 L without bound van.i.a.nlj union 06 a chain 06 ~:tI1.uc:tUl1.~ 6011. L ~ a r - ex.;teMion 06 eVel1.lj elemen:t a 6 :the chain. THEOREM 18.-
ab.t~;
THEOREM 19
(Tarski). -
Any union
06 an
elemen;ta.Jr.y chain Ls an e..teme.1VtM1j ex-
.te.n6ion 06 e.Vel1.lj eleme.1Vt 06 :the. chMn. THEOREM 20. -
Anlj comp.tiUe chMn hal> a
~:tJr.,{c:t
union.
PROOF.- Let01- j= , Ol2= , ... be a complete chain of structures for L ; L' will denote the language obtained from L by the adjunction of the names of all elements of U= IAjl u IA 2I u ... (the names are supposed to satisfy the conventions of Shoenf te Id 1967). We define the denotation .vu(t) of a closed term t of L' and the value V"U (A) of a closed formula A of L'. relatively to U, as follows: If t (A) is a closed term (formula) of L'. there exists n such that t(A) is a term (formula) of L(Oln); then we pose .vu(t) =
otn ( t)
(V"u (A)
= OZ n ( A)
).
144
NEWTON C. A. DA COSTA
Since Al ,
A2
, •••
A,
constitutes an elementary chain, its usual union
having U as universe, is a structure for L without vbtos and also an el ementary extension of any structure of AI' A2
, ••.
Let d' be a function defined on V and
whose values are functions from the power set of U into U, such that: and there exists a formula A of L', having no free variables having also the property that name of a I, then d'(v)(VI
d' Iv)
V = lal a
=JJu
E
U and
(vxA); if V CU
IV) is an arbitrary member of U.
V-ulAxla 1)
if
VC U
other than x, and
= T,
where a is the
but there is no formula as A,
Clearly, d' is well defined because
OLI,
ct 2 , ... is a complete chain.
Reasoning by induction on the length of terms and formulas of L',we show that
a' =
( A, d') constitutes a pseudo-structure for L.
Fi rstly, one has to prove
that d'(v), for vEV, is such that it makes Iv and II v valid. offers no difficulty.
In connection with I I v
""u ( A x Ia I) = T and a is the name of a
I
'
The proof of Iv
we have to show that i f
I b I ""u( B x Ib I ) = T and b
la I
the OLI , ct2 , ... is a complete chain. Secondly, since for any formula A of L(OL n ) we have ct(A)=T iff OZn(A)=T, the characteristic logical axioms of of v a 1so ho1din OZ' . name of b],
then
vxA = vxB,
=
but this clearly holds by the fact that
Therefore, since the vbtos of L are normal (and axiomatizable), that there exists a structure
the chain Oll
'I
OL z , ...
is
it follows
OZ = (A, d ) for L which is a strict union of
•
An existential formula is a prenex formula which does not contain vbtos,
and
such that all quantifiers in its prefix are existential. As in Shoenfield 1967, p. 77, we can prove the following proposition: THEOREM 21
(Chanq - Los - Suszko) • -
A theMy T .u., equivalent to a theMy
havIng oVJ1.y noVJ1.ogIcal auom/.) whIch Me. e.wte.~ I6, and only ,[6, 06 any c.hMn 06 mode.lJ., 06 T .u., a.t!.lo a mode..[ 06 T . PROOF.tions.
any un,[on
The same as in the case of usual model theory, with obvious modi fica•
The cardinal of a structure OZ is the cardinal of the universe of OZ; the structure
ct is ca 11ed fi nite or i nfi nite accord i ng to whether its car din ali s
finite or infinite; similarly, we define a countable or uncountable structure. Suppose that m is an infinite cardinal.
A first-order language L with vbtos
is said to be an m-language if the set of its symbols has cardinal
.;;;; m.
ory T is an m-theory if L (T), the language of T, is an m-language. language (l{o - theory) is called a countable language (theory). ory
without vbtos, we have:
A theAn l{o-
As in model the-
VARIABLE BINDING TERM OPERATORS
THEOREM 22
16 m v., an -e
that T and T' Me srandarui theoJUe!.>. T U T' v., -v.,teYlt - a UOI.> ed 60Junuia A, wdhout vbtO!.>, !.>llcch that A v., a theoftem 06 T and "l A ~ a theoftem 06 T'. PROOF.- Evidently, if there is such formula as A, then
T U T' is inconsistent.
Now, let us admit that there exists no formula in the conditions of A; under this hypothesis we have to show that Shoenfield 1967, p. 80.
T U T' is consistent.
vJe proceed an in
One has to construct an elementary chain AI, Az, ...
models of T without vbtos and another elementary chain
A;, A; , '"
T' without vbtos , satisfying the condition that Al
A~!L'
/
u
of
of models of
Az!L'
A~!L
is an elementary chain, where L is the language whose nonlogical symbols are those common to the 1anguages of T and of T', both wi thout vbtos.
The c ha ins
AI, Az, ... and A~, A~, ... are easily constructed. If A and A' are the usual unions of AI' Az, ... and of A'l, A~, ... , then A is a model of T without vbtos, and A' is a model of T' without vbtos. In consequence, we can construct a model Shoenfield 1967, p. 80). have to expand
B to a model .G-
culty at all, since the vbtos of model and is consistent.
B of
T U T' without vbtos
In order to obtain a model for
•
= (
B, d ) for T
U
T'.
T U T' are all normal.
(see
T U T' (with vbtos), we But th i s offers no diffiTherefore, T U T'
has a
146
NEWTON C. A. DA COSTA
The preceding theorem could also be proved as a consequence of Theorem 4. Precisely as in usual model theory, we deduce from the preceding proposition Craig's interpolation lemma: THEORE~1
25
(Craig interpolation lemma) . -
Let T and T' be two MandaJu:i
theouV->, A a 601Unu1a 06 L(T) wUhouX vbt0-6 and B a 601Unu1a 06 LIT') without vbt0-6. 16 A-->B J.A a theOltem 06 T U T', then then« J.A a 601Unu1a C, a.L60 withOuX vbtol.> , Mch that 'i A --> C and 'if C --> B •
PROOF.- Shoenfield's proof remains valid, with minor modifications. • Q will denote a set of nonlogical symbols of the language L of a theory T. We say that an n-ary predicate symbol p not belonging to Q is defi nab1 e in terms of Q in T if there exists a formula A,without vbtos , whose sole free variables are n distinct variables xl' X 2, ••• , x n ' and whose nonlogical symbols belong to Q, such that IT px lx 2" .. x n +-+ A. An n-ary function symbol f wh i c h does not belong to Q is definable in terms of Q in T if there exists a formula A, free from vbtos and satisfying other obvious conditions, such that 'T Y= fXIX2,,,xn +-+ A (y ,x] ,x2'''' ,x n are distinct variables). The notion of u-weak isomorphism, where u is a predicate symbol or a function symbol, is defi ned as in Shoenfie1d 1967, p, 81. Then, it is immediate that we have: THEOREM 26 (Definability theorem of Padoa-Beth) • Let Q be a I.>et 06 noVl-togica.t I.>ymboto 06 a I.>tandaJtd theolLY T, and let u be a pftedicate OIL 6unction I.>ymbo£. not in Q. Then u J.A de6inab£.e in teJtml.> 06 Q in T i6 and oVl-ty i6 60ft eVeJlY two model!.> o; and ;;. 06 T and eVeJlY bijection 1> 6ftom I all to I;;' I which J.A a kweak if.>omoftphiJ.>m 60ft eveJty k in Q, if.> a.L60 a u -weak J.AomolLphiJ.>m.
Though we shall not go into detail s , we note that the Theorem of Padoa - Beth has a more interesting formulation (whose proof is based on the main resu1 ts of Corcoran, Hatcher and Herring 1972), as follows: THEOREM 26'. AI.>I.>u.me that Q J.A a se: 06 noVl-togica.t I.>ymboll.> 06 the language 06 the theOfty T and that u J.A a rtonlogica.t I.>ymbol 06 T whic.h does not be£.ong to Q. u Is de.6inab£.e. in teJtml.> 06 Q in T i66 60ft any two model!.> al and ;;. 06 T and any bijemve. 6unmon 6ft om Iall to 1;;'1 whic.h J.A a k-weak if.>omOftphiJ.>m 60l( eveJly k in Q, 1> J.A alI.lo a u-we.ak J.AomOftpftJ.Am.
Another result which constitutes a complement to Theorem 26', and can be established by the methods of Corcoran, Hatcher and Herring 1972, is the definabi1ity theorem for vbtos below.
VARIABLE BINDING TERM OPERATORS
147
Let at and JJ. be structures for L, v a vbto of L, and is any model of T whose cardinal is m, we can obtain another model if = (B r h > of T of cardinal m, which is not strongly isomorphic to a, by changing the function g(E); this can be done as follows: let V be a subset of the universe of OZ such tha t it is not defi nab1e in the 1aguage L ( a), i , e., there is no formula F of L(a), having at most one free variable, satis-
VARIABLE BINDING TERM OPERATORS
fying the condition:
V = [a I Ol(Fx[a] ) =T l .
=
the sole difference that if g(e)(V)
149
Then.& is exactly 1 ike
k, we make
= l,
h(e)(V)
with
l
a,
with
=1= 11. • •
It is a little tedious to check, but Morley's theorem remains true when normal vbtos are added to first-order languages.
That is to say, if a countable theory
T (with vbtos) is m-categorical for a given uncountable cardinal m,
then it is
also m - categorical for any uncountable cardinal m. The symbol Sn (T) and the notions of type of an n-tuple of individuals of the structure Ol for L, n - type of a theory T, principal subset of Sn (T), etc., are defined as in Shoenfield 1967, pp. 91-92.
By the same proof as in that book,
we
establish the following proposition: THEOREM 31. T Ls a theoJty and r a non void !>ub!>et 06 5 n (T), !>a;(:i/.,6ying the condiJ:ion!>: 1) T i/., countable; 2) 16 A i/., a cU6junction 06 negation!> 06 60Jtmula!> 06 r, then A it> not a theOltem 06 T. UndeJt thue condiJ:ion!>, theJle ewtt> an n - type K in a countable model 06 T !>uch that r C K.
Any n-type in a countable theOlty
COROLLARY. -
model
i/.,
an n- type in a co untable
06 thi!> theMy.
An n-ul tra filter in T, JF, is a subset of Sn (T) but JF =1= Sn (T);
A, A -> B E JF, then
If
THEOREr~
in
T
IT
2) If
A, then A E JF;
B E JF;
PROOF.- Let T' be the theory , ••• ,
4)
T
it> an n - type in
T
(and any n -type
T to which we have added as new axioms all the for-
fnl, where A belongs to JF and fl, ... ,f n are n new constants.
T' is cons i stent, because otherwi se "1 Al V
to IF, would be a theorem of T;
Sn (T), which is absurd. an=
JF C Sn (T),
it> an n - uUJta 6iUeJt in T).
mulas Al r j long
1)
A 1\ B E JF;
5) For any formul a A in Sn (T), A E JF or "1AE JF .
An n - uUJta 6iUeJt JF in
32. -
such that:
3) If A, B E JF, then
Ol(t n),
A I ' ... , A k
be-
but in this case IF would be identical
to
...
V -lA k '
Therefore, T' has a model
then IF is the n-type of
THEOREM 33 (Ehrenfeucht). -
Let
a ,
where
and if al = Ol(rl)"'"
(al.···' an)' •
T and
r be Jtupec:Uvely a countable con-
t>i/.,tent theoJty and a !>Ub!>et 06 Sn (T) which i/., not punupal. UndeJt thue condiJ:io Yl!>, theJte it> a co untable mo del Ol 06 T !>Uch that r i/., not included ina n y n - type
06
oi .
PROOF.- The proof of Ehrenfeucht's theorem of Shoenfie1d's book, pp. 90-91, mains valid when vbtos are added to first-order languages.
•
re-
150
NEWTON C. A. DA COSTA With exactly the same proof as in Shoenfie1d 1967. pp. 91-92. we have:
THEOREM 34 (Ryll-Nardzewski). FoJt a eomple..te and eountable .the-oILy T hav-tng only -tnMnLte model-6 , :the f,oliowinq pMpe!LUe-6 aILe equivatent: I} T.iA K 0 - ea:tegofL,{,catj 2) Fon. any na:twe.a1 nwnbe!L n, :the -6et: 06 n - :type-6 in T .iA Mnliej 3) FOIL any natWtat nwnbe!L n, ali n - :type-6 -tn T aILe plLinupat.
Among the
immed i a te consequences of the precedi ng results. we menti on th e
following: THEOREM 35. -
AMwne :that e .iA :the onty vb:to 06 :the funguage 06 a :theMy T.
16 ali aUOm-6 06 T aILe logieat, .then e
e
06
-w
no.t deMnable in T.
~:t - OILdeIL COMe-6pOnmng aUom-6). A .the My T .iA equivatent :to a .theMY WhOM auom-6 do not eontain e -t66 eVe!Ly e - -6ub6UudWte 06 a mode£. 06 T -w a mode£. 06 T .
THEOREM 36. -
Let:
be .the MU
ali 60lLmulM w.Uhou.t e in .the
jJfLemea:te ea£.euJ:U6 with equatUy and Hilblli'-6 .oymbot e (w.Uh .the
There exist infinitely many vbtos. most of them having interesting properties. For instance, the following ones. mediately
whose characteristic postulates are listed im-
after their introductions (they are also governed by the basic
princi-
ples Iv and II v; and the smooth operators associated to them will not be exp1icit): en:
If F [xl does not define the universe. en xF [ x I
made
is an individual x
such that x satisfies F iff there are at least n individuals satisfying F. Characteristic postulate of
en:
*"
3xiF->((3x (x = en xF II F) /\ xn-l
*" Xu /\
where the variables ew :
+-> 3xI 3x2'" 3x n( xl x2 1\ xl F [xI! 1\ F(x2l /\ .,. 1\ F (xu]»,
Xl. x2 ..... xn
and
*"
x3
.,.
x are distinct.
If F[ x ] does not define the universe. then e(,)xF [xl denotes an i n-
dividual x such that x satisfies F iff there are infinitely
many individuals
sat; sfy i ng F. Characteristic postulates of e w : (3xiF II 3x(x =€wX F II F)) -> 3xI'"
for n = 1. 2 •...
3x n(xI
*" x2
II Xl
*" X3 II ... 1\
151
VARIABLE BINDING TERM OPERATORS
(3 x , FII 3Xl ... 3xn\ly(F[Y)~(Y=Xl VY=X2 V", V y=Xn»))~ I 3x (x = e'J x F /\ F)
for
,
2, '"
n = I,
n Let us admit that F[x] does not define the universe; then, L n F[x) denotes an individual x such that x satisfies F iff there are exactly n i ndinxF ual s satisfying F. If 3n xF and 3 abbreviate respectively the formula stating that there exist at least n individuals satisfying F and the formula stating that there are at most n such individuals, then the characteristic postu1a te of
L
n
is the fo 11owing: 3xIF ~(3x(x= LnxF/\ F)-- (3nxF /\ 3
Similarly,
enx F [ x ] could be introduced:
nxF».
for F [x J not defining the uni-
verse, F[enxF) asserts that at most n individuals satisfy F. We call Frege's symbol the vbto f syntactically defined by Iv' II
v
and
f xF = fxG ~ iix (F -- G) .
Consequently, we have: iix (F +-+ G) -- f xF = f xG THEOREM 37. -
WhateVeJL smooth. openaron.
6 we a.!l.6ouate to
f, f, associated to
6, .u., not nOILmM.. PROOF.-
Let us suppose that L is a usual first-order language having at least n
constants cI , c z , ... , c n , and that
a
is a structure whose universe has n el e-
ments.
We shall denote the elements of lOll by al,aZ' ... , an' and admit that
Ol(ci)
=
ai' i
= 1,
2, ... ,n.
able by a fonnula of L.
It is clear that every subset of lOll is defin-
Therefore,
a
cannot be expanded to a structure for
L
with f, since the cardinal of loZI is less than the cardinal of the power set of
lOll.
•
When axiomatizable but non nonnal vbtos are included in the set of
vbtos of
L, the 'corresponding' predicate calculus with vbtos is not in general a conserv· ative extension of the predicate calculus without vbtos. The proof of the next theorem, which is a little long,' will be omitted here. We only remark that it is based on an idea of Corcoran, Hatcher and Herring Corcoran, Hatcher and Herring 1972):
(see
in order to prove the completeness theorem
for a particular language with vbtos, they replace it by a convenient usual first-
152
NEWTON C. A. DA COSTA
order language, with new fur-ct i on symbols, subjected to appropriate axioms. THEOREM 38. -
r 6 any 1L6u.a.l
6Q!1. a 6illt-Q!1.dVL !a.vrguage. £ (waho ut = (A, d ) 60!1. I: ptlL6 the. cou.oma.Uzabte.
f..Vtue.tWte. A Ol:
vbtof..) cart be. e.xpa.vrde.d to a f..Vtue.tWte.
vbto v, in which d ( v) if.> any e.!ement 06 v (IAI) a.vrd the auom-6 0 6 v Me v a!i d, the.n v if.> noftma! IwnVLe. v if.> the !.lmooth opeJLatOl1. aMouate.d to v l .
The notion of ultraproduct (Shoenfield 1967, p. 104) can be adapted to structures for L, and its main properties remain true. Of course, generalizations of several other results of standard model theory offer no difficulties; this is what happens, for example, with Lyndon's theorem (cf. Shoenfield 1967, p. 94, exercise 6) .
In Corcoran, Hatcher and Herring 1972 there is a method of converting structures for L in structures for L without vbtos but with extra function symbol s . Witr. the aid of such nethod we can systematically reduce most problems of model theory with vbtos to standard model theory; for instance, Morley's theorem already mentioned, for first-order languages with vbtos , can be proved by thi s method. It may also be employed to establ ish the soundness and completeness theorems for vbtos in an illuminating way (see Corcoran, Hatcher and Herring 1972). Some of the above results are adaptable to the case of first-order predicate calculus with vbtos but without equality. Postulates Iv and II v have to be changed as follows: I~.
G[vxP z [xl]
F).
Originally, Hilbert envisaged the symbol e only as a formal device to facilitate metamathematical research; so, the sole scheme which e had to conform
with
was the following: or, better, 3xF --> lix(x = e x F --> F). Clearly, an interpretation of the e -symbol so axiomatized has to be nonstandard. A second nonstandard axiomatization of the e -symbo l is obtained if we impose that it has to be governed only by two specific postulates, to wit: exFz[x1= eyFz[yl, and 3xF --> lix(x = exF --> F) . The third treatment of e so far considered is the standard one , studied above. A variant of it results by requiring that the function d(e) in the structure Ol = (A,d)
be such that d(e)(IOlI)
=d(e){~);
we get a complete system of axioms
for the e -symbol so interpreted by the procedure of addi ng to the standard ax ioms for e the fo 11owing new one: ex{x=
x)
= ex{x ,p
xl ,
Nonstandard presentations have also been proposed for the description symbol L •
One of the best known nonstandard handlings of t is that of Hilbert and
Ber-
nays (see Hil bert and Bernays 1934, vol. 1, p. 384, and Lei seri ng 1969, p. 101 ) . It is introduced through the
-rule:
t
If 3:xF is a theorem, then txFis a term
and one may infer that Ii x (x =
t
defined and the semantics for
so treated cannot be standard.
t
xF --> F) .
In consequence,
t
xF is not always
Other trea tments of the descri pt i on symbol are the fo·ll owing, a11 of them in-
155
VARIABLE BINDING TERM OPERATORS
vestigated by Hailperin (cf. Hailperin 1954), and some of them by other authors (Liu 1974,
Montague and Kalish 1957,
and Scott 1967):
according to the first
treatment (Russell's), any atomic statement containing an improper l
xF, i. e., such that
description
3:xF is false, is also false; according to the
second
(a modification of Russell's), any atomic statement containing an improper description is true;
finally, according to the third, which is Frege's,
proper descriptions denote a given (fixed) object. second treatments are not standard.
all
im-
Evidently, the first and the
In Rosser 1953 there is a calculus of
scriptions to which only a nonstandard semantics can be supplied,
de-
i f soundness
and completeness are required. Notwithstanding, for most nonstandard presentations of vbtos nonfunctional semantics can be provided.
In order to give an idea of the method one has to em-
ploy to construct such semantics, let us sketch the semantical analysis of a particular version of the e -symbol.
We suppose that e has to satisfy only one spe-
cific postulate: e)
3xF~\fx(X=EXF~F).
The language to which E is adjoined is a first-order language with equality, with the notions of term and of formula conveniently modified.
The axioms
and
primitive rules of inference are the usual ones, plus the scheme e. Let us denote the language so obtained by A structure
LE
•
for L E is an ordered pair
waht6uYlkA:ioYleYl,
Grund. d. Math. 3,
Zeitschr. f. math. Logik u.
pp. 30-68.
N. Bourbaki.
1966.- Theorie des Ensembles, Chaps. 1 and 2,
Hermann, Paris.
J. Corcoran and J. Herring. 1971.- Notv., OYl ct !.>eman-Ucat anaty!.>.i.6 06 vcttUabfe biYlcUYlg tV1m opeJtcttO!L6, Logique et Analyse 55, pp. 644-657.
J. Corcoran, W. S. Hatcher and J. Herring. 1972.-
VatUctbfe bbrcUYlg tV1m opeJtcttO!L6,
Math. 18,
Zeitschr. f. math. Logik u. Grund. d.
pp. 177-182.
N. C. A. da Costa.
1973.- Review of Corcoran, Hatcher and Herring 1972, Zentralblatt f. Math. 257, pp. 8-9. 1975.- Review of Liu 1974, Math. Rev. 50, pp. 7-8. I. F. Druck and N. C. A. da Costa. SuJt fv., " vbt o!.> " !.>efoi'! M. HCltchVL, C. R. Acad. Sc. Paris 281 A, pp. 741-743.
1975.-
VARIABLE BINDING TERM OPERATORS
161
M. Guillaume. 1964. -
Recherches sur Le symbole de Hilbert,
Clermont-Ferrand, France.
T. Hailperin. 1954. -
RemaJLquu on -,
Harvard Univ. Press, Cambridge.
Fund. Math. 43,
J. B. Rosser. 1953. - Logic for Mathematicians,
pp.156-165.
McGraw-Hi 11, New York.
D. Scott. 1967.- EUf.>tence and duCJl..i.ption in 60nmaf logic, in pher of the Century, Little, Brown & Co., J. Shoenfield. 1967.- Mathematical Logic,
Addison-Wesley,
B. Russel;L,
Philoso-
Boston.
Reading.
A. N. Whitehead and B. Russell. 1925.- Principia Mathematica, Cambridge Univ. Press, I (1925), II (1927), and III (1927).
Cambridge, 2nd; vols.
162
NEWTON C. A. DA COSTA
Universidade de Sao Paulo Instituto de Matematica Sao Paulo, SP., Brazil and Universidad Catolica de Chile Instituto de Matematica Santiago, Chile.
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Co~ta (eds.) © North-Holland Publishing Company, 1980
THE MODEL THEORY OF FC-GROUPS UWc.h FelgneJt
Dem 1.FC-KOln gewidmet ABSTRACT. Following R. Baer and B. H, Neumann a group G is called and F-C Group if for each 9 E G the conjugacy class {x- 1gx;xEG}of 9 is finite. In her celebrated paper W. Szmielew solved the problems of decidability and internal characterization of elementary equivalence for the class of abelian groups. The model theory of abelian groups is by now well established. For the next slightly larger class. i.e. the class of nilpotent groups (even nil-2) only a few sporadic results are known. We found it more promising to look at another generalization of commutativity. namely the class of FC-groups. We have the following results: (1)
(2) (3) (4) (5) (6)
For all nEw: the theory of {G; [G:Z(GJ] :::. 11} is· decidable. The theory of {Gj [G: Z(G)] < ~oHs undecidable. The theory of all FC-groups and the theory of all BFC-groups are both undecidable. We classified all stable, w-stable and ~o - cat~ gorical groups G for which [G:Z(G)] is finite. We determined the ~2 - theory of all periodic FCgroups. We have the surprising result. that the Hypercenter of every FC-groups of finite exponent is first-order definable without parameters,
It seems to us that it is possible in the near future to get elementary invariants for periodic FCgroups.
§
1.
INTRODUCTION.
The model theory of abelian groups is more or less thoroughly accomplished. Starting from W. Szmielew's fundamental classification of all abelian groups up to elementary. equivalence (1955) a large number of the most important problems in this area has been solved in the last two decades. We may mention here the numerous investigations' concerning the elementary theories of sundry classes of abelian groups, their decision problems, the work on categorical abelian groups, Horn theories of abelian groups and finally results concerning infinitary logic and 1st-order logics with generalized quantifiers. In the case of non-abelian groups there are only sporadic model theoretic re163
164
ULRICH FELGNER
sults known. Most of these results, however, are very impressive and substantial. They usually combine deep results and difficult methods from group theory, recursion theory and model theory. We may mention here the work on (1) algebraically closed groups, (2) X -categorical stable groups and groups of small Morley rank, (3) the elementary t~eory of free groups, (4) the elementary theory of symmetric groups, (5) the undecidability of the theories of various classes of groups, and finally (6) the work on varieties of groups and verbal subgroups. One may hope for further modeltheoretic results if the groups investigated are not too far removed from abelian groups. Besides the class of nil-2 groups another prominent candidate is the class of FC-groups. Following R. Baer (1948) a group G is called an FC-group if each element has only finitely many conjugates in G. In this paper we start to investigate FC-groups from a model-theoretic point of view. We also investigate the related class of BFC-groups and the class of groups with finite central factor group. In § 2 we collect some algebraic results on FC-9roups from the literature and add a few results which have hitherto been unnoticed. In § 3 we generalize R. Baer's characterization of the hypercentre of a finite group to the case of FC-groups. This has the rather surprising modeltheoretic consequence that the hypercentre of an FC-group of finite exponent is first-order definable: In § 5 we show how the algebraic machinery of factor sets (§ 4) can be used to deal with model-theoretic questions about centre-by-fi nite groups (e.g. saturatedness and stability). We also solve the problem which centre -by-finite groups are models of the first-order theory of fjnite groups. The notion of 'FC-group' intends to generalize the notions of 'abelian group and 'finite group'. In § 6 we therefore compare the 1st-order theories of finite groups Th(Fin), of BFC-groups Th(BFC) and of FC-groups Th(FC). These theories are all different. We are mainly interested in questions of the following sort: up to which quantifier complexity are these theories identical. In several cases we shall obtain the exact answer. These are not purely technical questions because the answers tell us how 'close' two theories are. In § 7 we calculate the FC-centre of a restricted wreath product and generalize some earlier results of G. Baumslag. We use the results of § 7 in § 8 where we solve Baldwin's problem whether locally-nilpotent stable groups are nilpotent. The answer is negative. Among other things we classify the stable FC-groups and the Xl-categorical FC-groups. We use the following notation. N = {O,l, ... } is the set of all non-negative integers, Z is the infinite cyclic group of all integers. For n EN, Is. n, 2(n) denotes the cyclic group of order n. (X) denotes the subgroup generated by X wh{le {a,b, •• } only denotes the set of the elements a,b •• ; xY = y- 7xy and [x,y] = x- 1y-l xy. Further [x,y,z] = [(x,y],z] and = {x9 ; 9 E G} is the conjugacy class of x in G. If A and B are subgroups then [A,m denotes the subgroup gener and ated by all commutators [a,b) for a E A and bE B. Thus G' = [G,G) = G(I) G(n+l) = [G(n) G(n)j Put G(W) = n G(n) Let Z (G) denote the terms of the
i
"
,1
EN
ex
transfinitely extended upper central series of G. Thus Zo(G) = 1, ZI(G) = ZIG) is the centre of G and Za+ IIG) = {x E G; 'tJ'J E G: [x,y] E Zex(GJ} and for limit ordinals '" Z"rG) is the union of the previous terms. :IC(G) = if ZorG) is the hypercentre of G (ex runs through all ordinals). The centralizer X in G is denoted by CG(X).
§
w,u., the 6iMt .(.n6.(.nUe otr.d.(.nal and IXI
2. ALGEBRAIC PROPERTIES OF FC-GROUPS
.(.0
the cMd.(.naUty 06 X.
165
THE MODEL THEORY OF FC-GROUPS In this section we recall some known facts on Fe-groups for future We also add a few results which have hitherto been unnoticed.
reference.
Recall that a group, each element of which has only finitely many conjugates, is called an FC-gnoup. This terminology is due to R. Baer (1948). Following B.H. Neumann a group G for which there is a positive integer n such that each element of G has at most n conjugates, is called a BFC-g~oup (i .e. boundedly FC). A qroup is called toc.illy nMmat if each finite subset is contained in a finite normal subgroup. THEOREM 2. 1.
(J.)Sllbg~Ollp,;
and homomMpYU-c- ,[mage,; On
FC-g~Ollp,;
Me FC-
g~oup,;.
(UI J.,;
The weak d~ec.t p~odllet On any numb~ On FC-g~oup,; J.,; an FC-gnollp.
(ill) 1 n G J.,; a nVii.tety MnUety 9 en~a..ted.
gen~a..ted FC-g~oup,
(J.v]
(R. Baer) In G J.,; a.n
(v)
(B.H. Neumann) The
(vJ.)
Pwod,[c FC-gnoup,; Me toC-illy
toC-illy- nJ.nUe) •
FC-g~oup,
then G/Z (G)
(J.')
Mnile and Z(G)
then G/Z(GI J.,; toC-illy no~at (and henc.e
c-ommu.ta..to~ ';llbg~Ollp
G' On an
FC-g~oup
G J.,;
pe~odJ.c..
no~at.
(vU] (S.H. Neumann) In G J.,; an FC-g~oup, then the On Mnile Md~ no~ a ehMa.d~tic ,;ubgftoup. THEOREM 2.2.
J.,;
(1. Schur-R. Baer)
~e.t
t(G) on i l l eteme~
In nM Mme pMJ.tive
J.n.teg~
1->- - - - - i »
1
G - - -... >:> H
1
------,»
1
---....;> M ---~> G* ---~» H - - - - - - i > 1 PROOF A = {( oJ;(a), a-I); aEN} is a normal sub\Jroup of M x G. Put G* " MxG/A. Then m 1-+ [m, l)A is an embedding of Minto Z(G*J. If B = {(oJ;{a)b,a- I); a EN AbE M} and if A : G* -+ IMxGJ/B is the canonical projection, then rMxG}/B~ H and the kernel of A is {1m, l)A; mE M} ~ M. Thus G* is a central extension of M by H. The mapping 0: 9 ~ (1, g) A is a homomorphism from G into G* such that the above diagram commutes. It remains to prove that G* can be described as EIM,H,6*J, where ~*" v c «, By definition, G = E(N,H,6J = {(u,al; uEHA aE N} with the multiplication given by 1~,a)o{v,bJ = Iuv, 6(u,v}abJ. Put N = {(I, aJ ; a EN} so that H ~ GIN = {{u,a)N; uEHA aEN}, The natural isomorphism(u,a)H....u from GIN onto H will be called a. Consider the following transversal for the cosets of N in G, Tru,a)NJ = (u,lJ E G. We finally define a factor set F:G/NxG/N-+N in the usual way:
F(lu,ct)N, (v,bJN) ",cru,ct)o(v,bINI-1o,((u.,a)NiO,((v,bJNJ (uv, 1) -1 0 (u,1) 0 (v,l) (v-Iu.- I, 6(v- Iu- 1,uv)-I) 0 (u, 1) 0 (v,l) -I-I -I -I-I -I-I -I (v u. uv, 6rv. ,vJ6rv u ,u)6rv u ,uvl )
II, 6(u,vl} EN (the last equation follows from (0) wi t.. a v· 1u- l , 0 = u, tv = v) .Thus F(a- 1 (u ) , a-I (v)} " II, 6Iu,v)). Thus, if Nand N are identified and Hand G/N are identified (via a-I), then F and 6 are identical. Using this identification we may, hence, say that 6(u,v) = T(uv)-lT{u)T{v) V u,v E H, where T is a transversal for the cosets of N in G. But then oJ; 0 T : H -> G* is a trans versa 1 for the cosets of M in G*. Hence, by Schreier's extension theory, G* and E(M,H,1/:o 6J are isomorphic in such a way that the above diagram (with exact rows) commutes, Q.E.D.Lemma 4.1 has a number of consequences.
LEMMA 4.2. Mn..i.te.
16
Let N be a ,c.ent!tal -6ubg!toup 06
oJ; denotu the endomo!tphMm
momo!tphMm o:G
-+
a
1-+
G -6uc.h
N ffi(G/NJ -6u.c.h that the 6ollowing
d..i.ag!tam commutu:
N > - - > G -l»G/N
oJ;
J
1
that n" [G:~
an (60!t a E NJ, then thelLe
II
0 N - - Nffi(GIN)-;,>G/N
16 in add..i.tion N..i.-6 tOMion-6!tee, then 0 ..i.-6 an embedd..i.ng.
..i.-6
..i.-6
a ho-
172
ULRICH FELGNER
PROOF. Put H=GIN and choose a factor set tI: HxH"'N such that G= We have \&(tI(u,v))=(tI(u,v) )n=tln(u,v) by the definition of multiplication in the group Cext. Since G/N=H has cardinality n, the exponent of Cext divides n (cf'. M.Hall, "The theory of groups" (1964)p.223, Theorem 15.2.1). The extension G*=E(N,H, W 0 til is hence equivalent to the direct sum N Ell H and the claim follows from Lemma 4.1, Q.E.D. • E(N,H,tI).
LEMMA
4.3.
Let N be a Llbg!l.oup otl G I.>uuA be a factor set. Let e1 be the neutral element of A and e2 the neutral element of H. Intro duce the following ternary relations: ])1 (a,b,c.) x· Y=z holds in H, Ro(x,y,al => o(x,y)=a. In correspondence with the group extension EIA,H,o) we introduce the following first-order structure: &(A,H,ol = (AuH, ])1' ])2' R e1, e2), o' In the above definition we tacitly assumed that AnH=~. Notice that A is definable in &(A,H,ol by 3V3W])I(Ll,v,w) (similarly for H). If 01 and 02 are equivalent factor sets from H into A then &IA,H,01 1 and &(A,H,02) are not necessarily elementarily equivalent although E(A,H,011 and EIA,H,021 are even isomorphic over A. LEMMA 5.1. let
61:HlxHl->Al
G aYLd Gz be gl[ou.p.6. Put A.=Z(G.I, H.=G./A. 001[ ,iE{7,Z} and ~ ~ ~ ~ ~ 1 be a oa.c.tOJt.6et .6Llch that G1=E(AI'HI'0II. To Gz .t.6 wea.k1.y.6at!!:.
Let
Jta.ted and HI Mnde, then the ooUow,ir;.g Me equ.iva..tent:
W
G] and Gz Me eie.mentMay equ.iva.ient,
(Ul theJte.t.6 a oa.c.tOI[.6et oZ:HZxHZ->AZ .6Llch that
GZ 2!; E(Az,HZ'oZ) and &(A 1,HI' 0I) =&IAZ,Hz'oz)' PROOF. (i) =>(ii): Assume that G1=G2, G(E(A1,Hl'011 and that HI= GI/Z(G I) is finite (where AI=ZIG1)). If A2=Z{G21 and H2=G/A2, then AI=A2 and HI~2' Let T .... G a choice function (i .e , a transversal for the cosets of Al in G1) such I:H1 1 that TI(AI)=l and 01(x,yl=T (y) for all X,yEH 1 , Put H1={h1,h2, •• ,hm } I(xyl-1 T1(xlTl and consider the following m-type p={(v 1 , .. ,vml;Gl~ 1>[TI (hI)'" ,T I (hmlJ}'It follows from G that p is consistent with the theory of G2, Since G2 is wea~ly I=G2 saturated (cf. Sacks 1972, p.119) there are dl, .. ,dmEG2 .uch that G2f[d l", ,dm I for each Ep. Let 0 be an isomorphism from HI onto H2 ~nd put g,i=o (h,i) , If T2:H2 ->G 2 is defined by T2(g,i)=d,i (for l~,i~), then T2 is a transversal for the cosets of A in G (a 1st-order description of this fact is in pl. Define finally 2 2 02Ix,yl=T2(xy)-lTQ(xIT2(y) ,
174
ULRICH FELGNER
then n2 is a factor set and it is clear by Schreier's theory that G2~E(A2,H2,n2 I. We have (Gl,·,1"1(hl), •• ,1"1(hm»:(G2,·,dl, •• drJ by the choice of dl, •• ,dm• Also (HI" ,hI''' ,hrJ" (H 2,' ,gl'" .grJ and it follows, by the Feferman-Vaught theorem, that S(Al,Hl,nll and S(A2,H2,n2) are elementarily equivalent. (ii) ~(i): Let ul be the ternary relation describing the multiplication in Al and let u2 describe the multiplication in HI and Gl~E(Al'Hl'nl]={(u,a) iUEHl II a EA 1}. Define in S(A1,H1'6 1),P1 (xl'x2'x3'yl'y2.y3) .... 3Z13Z2[ u2(x 1,x 2,x S ) II ul (zl' z2' 1f31llul (lfl'lf2,z21IlRn (x l , X2, L:l JJ · Thus PI (xl" .,lf3Icodes the relation (x l'Yl)o(x2'Y2)=(x3'Y3)' where lY3=61Ixl,x2)YIY2 and x3=x1x2. Using PI we can transform each statement about E(A l , H1,nl] into a statement about ~{Al,Hl,nll. If we define P2 similarly for ~(A2,H2,62J. then the elementary equtva Ience of I> lAy, Hl,nll and ~(A2,H2,n2] implies the elementary equivalence of E(Al'Hl'nl) and E(A ,6 i.e. of G1 and G2 ' Q.E.D.• 2,H2 2), COROLLARY 5.2. Let GI and G2 be gftOUp.6, A-£=Z(G-£l and H-£=G/Z(G-L) 60ft -LE{I,2}. 16 HI M n-Lnile and G2 -L.6 weaktY.6atuttated, then the 6oUow-Lng Me equx"J=H !1-
7"
2
(A 2,·, n2 (cr(x),a(y))>
x.1f
6 .:H .xH ....A. - is ~Cl -saturated. It follows from Chang and Keisler 1973, Proposition 5.1.1. (iii) (page 21S) that S(A,H,nJ is ~Ci-saturated. To prove the converse introduce the relation PI as in the proof of Lemma S.l.Using PI each I-type over some XcG can be transformed into a certain 2-type over some y~
E(A,H,61, Q.E.D. • COROLLARY 5.4.
16 G/Z(G) -L.6 6-Ln-Lte, then nOft aii
o~~ Ci~O,
G
175
THE MODEL THEORY OF FC-GROUPS
THEOREM 5.5. 16 G/Z(Gl .,£f, 6.,in.£te artd Z(G)e1.emen:taJu:.ly equ.i.valertt:to a.rt
u1.:tJr.a.pJr.oduct 06 u1.:tJr.a.pJr.o duct 0 6
6.i.rtUe 6.i.rtUe
thert G"£!'
a.be..Ua.rt gJr.OUpf" 9 Jr.O up6•
e..temerttaJtily
equ.i.valertt
to
a.rt
PROOF. Assume the hypothesis of the theorem. Bya result of Basarab 1975, ZIG) is either finite (and in this case there is nothina to prove), or there are pairwise non-isomorphic finite abelian groups B. (for .,£EN) and an w-incomplete ultrafil ter F on N such that Z(Gl =g"T B./ F. -t -tc."
-t
Put H=G/Z{G), A=Z{G) and let 6:HxH-+A be a factor set such that G!::!!EIA,H,6J. Consider the following n-type p={(v 1, .. ,v J ; Z(GJI' 1J[6(h1,h11, 6{hl'h2 J, .. ,6(h n m,hm)] } , where H={hl'~"" h } and rt=m 2. Since D", B./ F is w-saturated and elementarily equivalent to Z(G) m -tc." -t there are d . . in B=D"T B./F (1 generated by g. (ii) Consider the followins !1-sentence: 3a]b3e.3d (a;lI/lb-lab=a2/1e.-lbe.=b2/1d-le.d=e.2/1a-lda=d2).
Higman 1~5l has shown that there is an infinite simple group in which this sentence holds. He also proved that a,b,e. and d have infinite order. Since a,a2,a4, are all conjugate this sentence is false in each FC-group, Q.E.D. • LEMMA 6.2. gJtoup~
Ul Any TI J-untel1.e.e blue ..L11. ail. Mnile gJtoup~ (even (n ail. gJtOUp.6 wh..te.h Me toc..all.y FC-gJtoup~).
(li)
.u blue ..Ln
ail. Fe-
Any ~l-~entene.e blue ..Ln aU Mnile e.yme. gJtOUp.6 06 pJt.£me-poweJt OJtdeJt
blue ..L11. ail.
gJtOu~
I
.u
{1}.
PROOF. (i) Assume, if possible, that there is a TIl-sentence el? true in all finite groups which is false in the FC-group G. By the Lowenheim-Skolem theorem we may assume that G is countable. G is the union of an incrnasing sequence of finitely generated subgroups An (~N), G=~N An' Since ¢ is TIl(hence also TIZ} it follows from the Chang-~os-Suszko Theorem (cf , Chang and Keisler 1973, Theorem 3.2.3) that Anf i¢ for some ~N. The group An is a finitely generated FC-group. A IZ(A) is therefore finite (Theorem 2.1, (iii)) and Z(A) is finitely generated. n n 11. Hence Z (AI1.~F $ B with B fi nite and F free-abel i an of fi nite rank. Put H=A/F. By Lemma 4.2 An can be embedded into F$H, hence FffiH~i¢. Let m be the rank of F, hence F=Zm. Notice that , where e=[a,bl. Put F=( a;;$( b2) Then F is free-abelian of rank 2, H/F is the dihedral group of order 8 and H/Z(H) is elementary abelian of order 4. Moreover HF¢, Q.E.D .• REMARK. The metaphysic of the proof of Lemma 6.4 is the following. If G is a group such that G/Z(GlFI and Z(G)Fl then certain elements of ZIG) arp . possibly first-order definable (just by the way how G/Z(GJ sits on top of Z(G))which are not first-order definable in the substructure Z[G) alone. In our case we could speak in G about the generators of the three direct summands of Z(G). Notice that this fact is hinted by our Corollary 4.5 (there we could only produce an almost divisible group 0). We finally emphasize that the finitely generated nilpotent FCgroup H produced in the proof of Lemma 6.4 is in a way the smallest possible example: the number of generators of H is the smallest possible number, and the order of the central factor group is the smallest possible number and H' is as small as possible. A formula ¢ is said to be poIt, On the other hand >It is obviously false in the weak direct product of w copies of the symmetric group S5 of degree 5, for example. Q,E,D, •
EXAMPLES. We gi ve a few examples of sentences true. i'll all FC-groups, but false in some infinite groups. . 4>1'
"16
no pa~Jt 06 d~~t~nct conjugate eiement~
commute, then
gJtoup ~~ abei~a.n".
2'
Thi s sentence is false in al1 non-abelian free groups; it is 2=1 "16 the gJtoup hM oniy two conjugacy ciM~~, x 60Jt aU. x",
!o2'
the
182
ULRICH FELGNER This sentence is k 2 and false in some HNN-extension. m "In x =1 nM all x then nOll all e.temel'Lt6 u ,v On p-powVt onde»: thVte excsxs w
-+6 x-b I . The fol l owi nq Lemma 1,b generalizes a result of Baumslag 195Q, Corollary 3.2.
LEMMA 7.1. 16 Afl
and B. (60Jt the 1>ame pJWne p) 6u~_h that B ..u, 61n1.te and A 16 an FC-fl'LOUP. I6m..u, the n.Upotency cfM6 06 B, then A"'B 16 hYPQflcen:Ota.t and 1.'1. 5= Zw + m(A",~) • PROOF. By Theorem 7,2 the base group AS coincides with the FC-centre of ;'\,8. Since A is locally finite by Theorem 2.1. (vi), A"'B is a locally finite pgroup.Hence FCZ(A"'B) ~Zfl!(A"'B) by a theorem of McLain (cf. Robinson 1972, Theorem 4.38). Our claim follows fromA"vB/A 6==f3 , Q,E,D.·
We shall apply these results on wreath products in the next section to solve a problem of Baldwin on locally nilpotent stable groups. We shall briefly outline some other uses of wreath products. LEMMA 7.5. Folt an!:! g1ven gltoup fI .theJc.e ew.t6 a gJtOup G and 60me gEG .that CG(g)~G and G/CG(g)~ H.
1>1.lc.h
PROOF. Let At1 be any abelian group and let G be the restricted wreath prodduct, G=f"v'1. If aEA then let 0a be the function 0a(x)=a iff x=l and 0a.(x) =1 otherwise. It follows readily that for any lFaEA, CG(f 0a, 1» ={( 6, 1} ; 6EA(H)} 9! (H' A ~ G and hence G/C G( lOa' 1),'" H, Q. Eo D.• Notice that in contrast to Lemma 7.5 not every group H ap~ears as the central factor group of some group (cf. Beyl, Felgner and Schmid 197+). Similarly not every group H appears as the commutator subgroup of some group G, Lemma 7.5 can be used to prove the undecidability of the 1st-order theory of FC-groups and some other related classes of groups.
FOJt each p06.£Uve 1ntegeJr. m: .the 16t _o!tdeJc. theo!tlf 06 aU g!tOUp1> G .t>uch .that I G/Z(G) I:.m ..u, dec.1dable. (11) The lJ.>t_ OJtdeJc. the-OJty 06 aU g!tOUp)., wUh 61nUe c.entJtal 6ac..t0Jt g!toup..u, THEOREM 7.6. W
undec.1dable. (111)
(1v)
..u, undec.1dable. The 61Jt1>t-OJtdeJr. theo!tlf 06 aU FC-gftOupl> ..u, undeUdable. The 61Jt1>t-o!tdeJr. theOJty 06 aU SFC-g/tOup1>
PROOF. (i) is immediate from Szmielew's result that the 1st.order theory of abelian groups is decidable and the fact that there are only finitely many groups of order < m. (ii), (iii) and (iv) are immediate consequences of a rather deep theorem or A. Cobham which states that the 1st-order theory of all finite groups, Th(Fin), is hereditarily undecidable (ct , Vaught 1960). A proof of this result is, however, not yet published. Therefore we shall give here a short proof of (ii), (iii) and (tv) based on a (published) result of A.I.Mal 'cev , s t.order theory of all nil-2 groups of (tt ): If p is any odd prime, then the 1 exponent p with finite central factor group coincides by Corollary 5.7 with the 1st-order theory of all finite nil-2 groups of exponent p. The latter was shown to be undecidable in Mal 'cev 1961. Since the class of all nil-2 groups of expo-
185
THE MODEL THEORY OF FC-GROUPS
nent p with finite central factor group is 1st_order definable in the class of all groups with finite central factor group our claim readily follows. (iii) and (tv}: Let u be a variable which is not in the language.c of group theory. u is fixed throughout in the sequel. If ~1' ~2 are terms of .c, then let (~1=~2)# be ]w!wu.=uw A~lw=~). Put (A 'JI)#= K. Let T be a transversal for the cosets of Z(G) in G and 6(x,Y)=T(xyj-1T(x)T(Y) (for x,y EH=G/Z(G). Then ~E(Z(G), H,6) by an isomorphism which leaves Z(G) pointwise fixed and induces the identity
ULRICH FELGNER
186
mapping from H onto H. We can thus translate each I-type pES! (X) into a 2-type of &(Z(Gl,H,6) and conclude that &(Z(G),H,&) is not K-stable. But H is finite, and hence ZrG) would not be K-stable,a contradiction. (iii) This is immediate from (ii) and Berthier's theorem according to which all abelian groups are stable. Q.E.D .• If G is an ~o-categorical FC-group, then G/Z(G) is not necessarily finite. In fact, let G be any infinite extra-special p-group. Then Z(G)=G' is cycl i c of order p,G/Z(G) is an infinite elementary abelian p-group and G is ~o-categorical {the case of extra-special p-groups of exponent p was considered in Felgner 197~. THEOREM 8.2. (' EH are elementary equivax,y x,y lent and G*=E(Z(G*),H*,&*). By the Engeler, Ryll-Nardzewski, Svenonius theorem there exists an isomorphism ~ from ZrG) onto Z{G*) such that ~(&(x,x))=n*{cr(x), a (y)) for all x,yEH. It is clear then that (u,a)l-> (a(u),~(a)) (where uEH,aEZ(G)). is an isomorphism from E(Z(G),H,6) onto E{Z(G*~ H*,6*). Hence G and G* are isomorphic, Q.E.D .• We may reformulate Theorem 8.2 as follows (using J. Rosenstein's classification of ~o-categorical abelian groups): 16 G/zrG) .fA 6.table gILOup 06 exponent p iJ.> nilpotent. In fact the stability implies that the lattice of all centralizers has finite dimension and the group is hence soluble by Baldwin and Saxl, The claim follows from a theorem of N.D. Gupta (cf , Robi nson 1972, Volume II, Theorem 7.18). If we strengthen the hypothesis in another direction we obtain that ~7"eategolL{ea! hypeILeentlLaf gILOupl.> 06 6{nUe exponent aILe YJil.potent. (Mall cev proved that hypercentral groups are locally nt lpotent l ) To see this, work with the connected subgroup GO and the fact that GO has finite Morley-rank. It follows that GO is ni lpotent. The nilpotency of G then follows from Baumslag 1959, Lemma 3.8, Q.E.D ••
REFERENCES. A.D. Asar 1973
A eonjugaey theoILern 60IL loeally 6{nUe gILoupJ.>, Journal (2), vol 6, pp 358-360.
London Math. Soc.
R. Baer 1953
GILOUp element!.> 06 plL{me POWelL {ndex, Transactions Amer Math. Soc. vol.75, pp 20-47.
J .T. Ba 1dwi n
189
THE MODEL THEORY OF FC·GROUPS stability
197+
theo~
and aegeb~, to appear.
J.T. Baldwin and J. Saxl 1976
Log~eae ~tability ~n g~oup theo~y,
A), pp 267-276.
J. Australian Math. Soc.vol.21 (series
S. Basarab Modetete teo~~ etement~e a g~oup~o~ cetari Matematice vol. 4 (Bucarest), pp
1975
abe~ene 6~~e,
381.~86.
Studii Si
Cer-
G. Baumslag Wneath pnoducto and p-g~oup~, Proc. Cambridge Philos. Soc. vol.55, pp224231.
1959
F.R. Beyl, U. Felgner and P. Schmid On
197+
gnoup~
whieh
oee~ ~
eentne
6aeto~ g~oup~,
J. of Algebra, to appear.
C.C. Chang and H.J. Keisler Modet
1973
theo~y,
Studies in Logic, vol.73, North-Holland Publ.Comp.Amstermm.
U. Felgner 1977a
Stab., in
logic,
~th
odology and Philosophy of SCi •• Stanford 1960 Congress, Nagel, Suppes and
Tarski (Eds).
Mathematiscnes Institut Universitat TUbingen German Federal Republic
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © North-Holland Publishing Company, 1980
SEMANTICAL MODELS FOR INTUITIONISTIC LOGICS E.G.K.
L6pez-f¢cob~
ABSTRACT.
It is ironic that intuitionism, whose origins are rooted in the concept of "proofs", shou I d produce so many (apparently) different kinds of models: Kripke models, Beth models, topological models, realizability, Swart models, and so on. Furthermore there appears to be a general view that most of the model! ings are equivalent, al though occasionally it is observed that they are not! In this talk we consider the concept of an abstract semantics for a logic L which we believe satisfies the minimum requirements in order to be called a "truth-value semantics" for L. We then discuss possible notions of equivalence between different semantics for L and in particular we catalogue just about all the truth-value semantics for intuitionistic logic and some of its extensions. We conclude with a Beth-l ike model I ing for the extension CO (constant domains) of intuitionistic logic.
§
O.
INTRODUCTI ON.
One of the distinguishing features between Kripke models and Beth models for the Intuitionistic Predicate Calculus (I PC) is that the latter have constant domains while the former need not. Furthermore it was known, almost from the time of their conception, that the Kripke models with constant domains correspond to the extension of the 1 PC obtained by adding the schema: IJx (P x V Q)
~
IJx P x V Q
where x is not free in Q. Not surprisingly such an extension of the I PC is often called the "logic of constant domains", or simply" CD" (1). Si nce it is often claimed that Kripke models and Beth models are equivalent, it was natural 'to try to determine which Beth models correspond to the loqf e CD. In order to be able to give a meaningful answer to the above problem, it
is
(1) The logic CD has turned up in other contexts; for example in the extension of CD obtained by adding the w- rules is equivalent to the extension obtained by adding the restrictively restricted w- rule.
191
192
E. G. K. LOPEZ-ESCOBAR
first necessary to be more precise about "correspond". In trying to do the latter one is quickly made aware that there are many different types of models for the I pe; even amongst the Kripke models one finds many different definitions, usually accompanied by a remark to the effect that the class of models so defined is equivalent to the one originally defined by Kripke (although no one seems to bother to define what is meant by two classes of models being equivalent). It may be worth observing that a similar situation occurred with classical logic in that a certain moment a large number of extensions of the Classical Predicate Calculus (cr c ) were being investigated. It wasn't until the appearance of lindstrom's 1969 paper, giving a characterization of the epe amongst the co11 ection of ab.6:t'ute:t logic.!>, that some order was imposed on the various extensions of the
cr c .
There was however an essential difference. In the classical case the languages varied but the kinds of models did not, while in the intuitionistic case what changed were the models, the language did not. What we propose to do in this paper is to introduce the concept of an "a.b.6:t'ute:t .6emawc.!>" for (extensions of) the IPe and discuss some of its ramifications. In passing we shall obtain a (non - standard) Beth semantics for the logic cn . §
1.
THE LOG I C I P
e
AND ITS EXTENS IONS.
1.1. The language. We shall assume that we are given a fixed first-order 1anguage La which includes amongst its symbols:
propositional constant: 1 propositional parameters p, q, ••• individual parameters a, b, individual variables x , lj, R, S, relational symbols \f x, 3 x, /\ , V , logical and other symbols
::l , (,) •
If the language L is obtained from the language La by the addition of a set C of individual constants, then we may write "LO(C)" instead of "L". For the most part we shall only consider languages L which have a finite number of relational symbols and propositional parameters. SeVl-t L is the set of sentences in the 1anguage L (i .e. those formulae of L with no individual parameters). 1.2. The c.alc.ullL6. One of the purposes for introducing model theoreti ca 1 methods is to avoid formal proofs. Consequently we shall avoid as much as possible reference to any parti cu1 ar proof-theory of the IP e. We do the 1atter byassuming that we have a consequence relation Cn such that
A E Cn (f) iff the formula A is an intuitionistic consequence of the setf of sentences. The principal requirement we place on Cn is that Cn(0), where 0 is the empty set, be preci se1y the set of theorems of the I p e (say, accordi ng to Kl eene 1952) . 1.3. TheoJUu. Given a set T of sentences in a language L the.theoltem.6 of T, ThmL(T), are those formulae A in the language L such that A E Cn(T). A set T of sentences in the language L is an L-TheOltlj just in case that
193
MODELS FOR INTUITIONISTIC LOGIC
Log~eh. A set H of formulae in a language L is a log~~ iff for all A, if A E Thm (FJ) and B is a formula of L obtained from A by the
1.4.
formulae
L
replacement of relational symbols or propositional parameters by other relational symbols or propositional parameters respectively, then B is also a theorem of H. A theory T(logicH) isB-~on.oLltentiffB'1'ThmL(T)(B'1'Thm L(H». iff it is 1- consistent.
It
is
~on.o..utent
A theory T is L--6a.tuJuLted iff for sentences A,B, hCx of L: (1) if AVBET then either A E T or BET, (2) if h C X E T then for some individual constant ~ of L, C~ E T.
§
2.
2.1.
logic
MODELL! NGS FOR THE LOG I CS IJ. By a semantical interpretation S
Semafttic.a..t ~nteJLpJteta.tion.o.
H we understand a pair:
S
such that AI Msx Sent
s
=
(M S'
for an L-
F S)
is a non-empty set and FS is a subset of the cartesian product
subject to the condition that: L
60Jt a.U
m: EMS'
~6
and
m
(2)
A, B E sentL and a.U B E Cn( IJ U {A}) then
F s B.
m F SA
(we omit the subscripts "SO when there is no chance of confusion; furthermore write "m FA" instead of "(m, A) E F "). An H-semantical interpretation S and a11 sentences A E ThmL( H ):
is
we
-6ound just in casethatforallmEM s
An H- semantical interpretation S is ~omplete iff for all L-sentences A, if for all EMs' F SA, then A is a theorem of H.
m
m
An
H- semantical interpretation
s i stent H - theory if
'ilrrtEM
S
T
S
is -6tJtongly c.omplete iff for every
con-
and all sentences A:
('ilBET(mFSB)
... mFSA)
then
AECn(HUT).
(2) This is what we bel ieve to be the minimal requirements for a "semantical" interpretation: Observe that it excludes mcst real izabil ity interpretations. Another, advantage of the condition is that the set of sentences "true" in a given model Ms will be closed under H-consequence and thus we can talk about the "theory of a model".
mE
E.G.K. LOPEZ-ESCOBAR
194 2.2.
IP C.
Examplu 06 flOund and -6tMngly complete -6emant..L
both S
and
Me p.1>.L
both S
and
I I
Me !.J emafttica! intvr.pJte;taUo I1!.J
then
S==* I,
then S == I,
Me a.p.1>.i.
and S
noJr. H,
iJ.,
1>ound and comple:te
noJr. H
1.
then 1>0 -l.!.J
§ 4. SUMMARY. The following are some of the well-known results concerning Kripke and Beth models expressed in terms of the relations between semantical interpreta ti ons.
4.1.
(Kripke 1965)
K == K == K == X 3 O 1
4.2.
(Kripke 1965)
Xl:::; B O
4.3.
4.4.
B O :::; B,
IU
(Gabbay 1977)
i KO B:::; B
1
4.5. B:::; B 2 ' B 2 i B 4.6. (de Swart 1977) B 3 '" B 2 §
5.
NON-STANDARD BETH MODELS.
198
E.G.K. LOPEZ-ESCOBAR
5.1. I NTRODUCT ION. A1 though some authors appear to object to the various attempts to make Kripke (and Beth) models a plausible interpretation of intuitionistic reasoning, it nevertheless is a fact that people are more interested in models which have some semantical interpretation (perhaps that is a reason why Kripke models have been studied much more than algebraic models). The attempts to give a plausible interpretation for the Beth and Kripke models center around the view that intuitionistic logic is a logic of "positivistic research". In such a positivistic research we assume that we have various states of knowledge which form themselves into a partial order. In both Kripke and Beth models if a sentence is asserted to be true on the basis of a given stateofknowledge it will also be asserted to be true on later states of knowledge. The difference appears on how it is asserted to be true: in Krinke models, it depends only on the later states of knowledge, in Beth models it also depends On the paths through the states of knowledge. In an intuitionistically correct Beth model the partial order of the" states of knowledge" forms a OM (classically: a finitely branching tree) and the paths considered are all the paths through the fan. In fact, it is customary to restrict oneself to fans which have no finite (terminating) paths and so the paths in question are the so called .£pfJ ('£'Il6.{VI-Ue£y pft(!.eecUllg fJequelleefJ) admitted by the fan (law) . The Beth models thus described form a complete and sound semantics for the fPC and even an intuitionistically acceptable proof of the completeness is possible (see de Swart 1977). We will call the Beth models in which all the admissible ips are used in the definition of validity the .£./ltuitiOn.{.fJt. F ~) and A and L - f'orrsul a, then A.u., tnue. -eCU/led -
on
Beth L>:tJw.c..:tuJ1.eL>.
that we are dealing with a single Beth structure. 5.4.1.
LEMMA.
Tn
s.
n FA
In this subsection we assume
n~mEo~
and
then~,
m FA.
PROOF. By induction on the logical complexity of the formula A. If A is either a conjuction, conditional or universal formula then the result follows immediately from the induction hypothesis. For atomic, disjunctions andexistentia11~ quantified formulae one needs to use the fact that one is dealing witha cOll'prehensive set of paths. More specifically, one should first verify that no matter what property P( ) : 'V
t< nEt< L
Fz. 3 x p( a (Q)) f\ 'Vex mEt< E F;t;.
5.4.2.
COROLLARY.
5.4.3.
THEOREM.
empty L>equence. qu..{vaLent•
:t
F A -
•
~,O F A, wheJLe 0 Ls .the code 0011. the
"z:
(1)
.t-,I'lFA,
(2)
'VexI'lEt<EF~ 3 Q(~, a(Q) FA) •
the nou.ow-
aJz.e
e-
PROOF. The condition that Fx. is a comprehensive set of paths gives us immediately that (1) => (2). That (2) ~ (1) is proven by induction on the complexity of the formula A. If A is an.a!omic formula then it follows from the definition of ~, n FA. Of the rema'mt nq cases the only one which might not be immediate is the one corresponding to the existential quantifier. Thus assume that
(3) See, for exanp l e de Swart 1977. For another context in whi,ch plays an important rBle, see Gabbay 1976.
1 E V
(n)
201
MODELS FOR INTUITIONISTIC LOGIC
3k(~,a(k)1=3XCx). Then, from the definition of
lfanEaEF;&
secu-
rability we obtain that:
Using the properties of a comprehensive set of paths we then obtain: If a
nEa E Fg.
3:t 3 a( ~ , a:t 1= C a ) .
From which it follows that:
t-, 5.4.4
DEFINITION.
5.4.5. THEOREM.
Beth
n
1=
3xCx.
•
= {~: ~ is a Beth structure}. Ls a .6ormd and eomptete .6emanlie.6 60IL
Beth,
IPC.
PROOF.
Since Beth includes the original Beth structures it follows that B, the or iq ina l Beth semantics. Since the original Beth semantics is known to be a complete semantics for IrC, it follows from Lemma 3.3. that Beth is also a complete semantics for the IPC. It only remains to show that Beth is a sound semantics for the IPC. That is, we have to show that for all 1tYE Beth and all
Beth
~
theorems A of IPC, :c.- 1= A. One way to do the latter is to choose some formalization for the IPC and then (1) verify it for the axioms and (2) check that the rules of inference preserve that property. If one chooses the formalization given in Kleene 1952 then (1) and (2) are straightfoward enough. •
§
6.
BETH STRUCTURES AND CLASSICAL LOGIC.
6.1. In:t!Loduc.:t.i.on. We now turn to consider Beth structures in relation to extensions of the IPC. In this section we consider the Classical Predicate Calculus, cr C. The crc can be obtained from the rrc by simply adding all the Lformulae of the form A V"lA (where"l A is an abbreviation for A:J 1).
Suppose that }& = (0 ~, VI'-' FI'-) is a Beth structure such that all the ips in F~ are sharp arrows (see Definition 5.2.), then such a Beth structure will be called a Sha!Lp Beth stnuc.:twr.e. Then we let SB be the semantics of a.e..e. (standard and non-standard) sharp Beth structures and SB be s the semantics of tre .6tandaILd Beth structures. The following lemma is immediate. 6.2. ShMp Beth .6:tJwc.:twr.e.
6.2.1. LEMMA.
SB
s
s SB.
Since the CPC is obtained from the Irc by the addition of the schema AV,A, the following lemma is preparatory to showing that SB is a sound and compl ete semantics for CPC : 6.2.2. LEMMA.
16 Z=
(ax.' V,j:.' Fx,.)
E
SB, a
E
F~ and A .u,
an L-6O!L-
E.G.K. LOPEZ-ESCOBAR
202 thVte. .w a k a (k ) a a(k)
1=
k 6ueh that
Ca
(4) The argument is simply as fol lows: Since the semantics cannot be SB there must be a structure in the semantics with at least one non-sharp Ips. Choosing then an appropriate V one can then fail to satisfy a sentence of the form 'if x (Px V Q.) ::> 'if x P x V Q.
MODELS FOR INTUITIONISTIC LOGIC then:
thVte.u, a
Iz !>uch that 60lt aU piVl.ameteM ~.u(r.:}
F
203
a
Ca
If we use Beth's idea of using the natural numbers as the parameters of language then the condition (*) can be expressed as follows: 'tj P E :IN 3
Iz
E :IN
(it, u ( II)
7.2.2. DEFINITION. a sharp arrow of a.
A fan
7 .2.3. DEF I NI TI ON.
~ 3 II. E IN 'tj P E :IN (~,
F Cp}
u (1t-!>hMY.' iff every node of
o
the
F Cp} .
a
lies
on
A Beth structure ~ = (al'-' v~, FJI-) is almo!>:t-.6hMp
if and only if the underlying fan
a
J¢-
is almost sharp.
The use of almost-sharp Beth structure is that they can be used to obtain definite structures. For given an almost sharp Beth structure ~= (ax..' V~, F~) , then the Beth structure .to = (ax,.' Vx,.' Slf,) where S~ is the collection of sharp arrows of "s » will be a sharp Beth structure and hence a defi ni te Beth structure (5). Note however that not all definite structures need be sharp, e.g.
-,
THEOREM.
7.2.4. :tllel1.
J?
PROOF. then
16.t-
'tjx (Ax V B) ~ if x A x V B
'tjQ
11.
E
=
(ax.'
(wheJte
v~, F)C.) .u, x
dOe!>
It suffices to prove that for all EF H(J&.,u(lz} F 'tjx A x or Q ' it-
that ~,11. F ~x(AxVB) have that
11. E
and that
1?,
11.
a de6-{.nUe Beth
11.0t OCCUlt
6ltee -i11.
AX. • if
~,u (Iz)
!>:tltuctUite
B) ~ va..Ud
1?,11.
F if x(Ax VB}
F B}.
Thus assume
Then for all parameters F
.£11.
a
Aa V B.
(5) The almost-sharp Beth structures are sound and complete for the extension of the IPe obtained by adding the schema ifx'l 'lAx ~'l'l'tj xAx.
we
204
E.G.K. LOPEZ-ESCOBAR
In particular we obtain that for all parameters a there corresponds a ~ such that (either Z , a (~) 1= Aa or Z-, a (~) 1= B). If there is a I< such that it- , a (I k (fl (t) =
i3 (Iz)). According to Lemma
206
E.G.K. LOPEZ-ESCOBAR
8.3.3.
we then have that Ii n 3.t(~.i3 (t)
where:? is the standard Beth structure Aplying Lemma 8.3.2.
t= P(n)) ,
(0, V, Path o ) '
we then obtain that Ii n 3
But then using the fact that
aa , L( 73 (:t)) 11E F~
~
P(n)).
we obtain
Ii n(OL , ~ (1 60ft ~n:tuit{onJ.1,:t£e loMc, J.S.L. vol. 42 , pp. 306 - 308.
S. Gornemann 1971
A LoMe .6:tMngeJt. :than
~n:tLU.tionJ.1,m,
J.S.L. vol. 36, pp, 249 - 261.
MODELS FOR INTUITIONISTIC LOGIC
207
S. C. Kleene 1952
Introduction
to Metamathematics, Van Nostrand Publ ishing Co.
S. A. Kripke 1965
Sema.ntLcat ana1.y.s.u,
06 .£n:tJ.U.tion.i..1.:Ue .tog.£e, in
Formal systems and re-
J.N. Crossley, M.A.E. Dummett, editors. North-Holland Publishing Co. Amsterdam. pp. 92 - 130. cursive functions.
P. LindstrBm 1969
Ort
ex.:teYl<S,[OYl<S
06
E.temen.t:aJty Log,[c.,
Theoria vol , 35, pp. 1 - 11.
C. A. Smorynski 1973
06 KJUpke mode£.6, Chapter V in: Metamathernatical investigations of intuitionistic arithmetic and analysis. A. S.
AppUc.a:UoYl<S
Troelstra editor.
Springer Verlag Publishing Co.
H. C. M. de Swart 1977
Art .£n:tJ.U.tion.i..1.:Uc.aUy p.taUA.£b.te '[Yl:teJtpJte:ta:UOI1 06 .£n:tJ.U.tion.i..1.:Uc. .tog,[c., J. S. L. vol. 42, pp. 564 - 578.
A. S. Troelstra 1977
A-6pew 06 c.On6bw.c.:Uve ma:thema:Uc.-6 in: Handbook of Mathematical Logic. J. Barwise, editor. North-Holland Publishing Co.
1978
Some JtemMM on :the c.omptexuy 06 Henk.£n - KJUpke mode£.6, Indagationes Mathematicae, vol. 40, pp, 296 - 302.
Department of Mathematics University of Maryland College Park, Maryland 20742 U. S. A.
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © North-Holland Publishing Company, 1980
TRUTH, PROBABILITY AND SET THEORY J. R. Lu.c.<MI
ABSTRACT.
The concept of probability has largely, but not entirely, been shaped by the formal requirements of the probabil ity calculus. These requirements are satisfied by several different accounts of what probabil ity really is. Probabil ity could be interpreted as a general ization of truth and falsehood or as a propensity or some other sort of property, or in terms of set theory. The first account has the advantage that the concept of truth manifests much the same complexity and many of the same difficulties as do the various accounts of probabil ity. Most logicians, however, favour an account in set-theoretical terms, but this runs into a number of technical difficul ties. We cannot give a satisfactory explanation of probabil ity in terms of set theory, although set theory can elucidate particular issues. One example from frequency theory is considered, and certain aspects of recent work by Chuaqui are discussed.
One of the paradoxes probability gives rise to is the fact that there is a large measure of agreement about the formalism coupled with wide disagreement on how it should be interpreted. Apart from certain qualms about the legitimacy of infinite operations, which will surface later, we all, objectivists, subjectivists, frequency merchants and the rest, accept the axioms of the probability calculus, while maintaining stoutly that probability can only be understood as an objective property of things, as a measure of one's own belief, as a limiting frequency in an infinite collective, or what have you. This not only presents philosophers with a problem but has led mathematicians and logicians to turn away their attention from the question of what probability is, and to take an excessively formal viewof their subject. It is not just due to wickedness or the intellectual myopia which philosophers are fond of finding in their colleagues in other disciplines. For the formal approach has been to a large extent constitutive of the subject. As Ian Hacking has pointed out (Hacking 1975), the word "probable" has changed its meaning in the last four centuries, and has only gradually come to mean what we understand by it. In the Eighteenth Century Joseph Butler used the word to characterize arguments which were not demonstrative but none the less carried some weight and were worthy of our credence. That usage survives into the present age, but only in attenuated form. Although there have been valiant attempts to develop a formal calculus of a~gu. me~, they have not been very successful, and we have begun to sense that even if they were successful the result would be something radically different from the calculus of probabilities, if only because an argument not only can support or fail 209
210
J. R. LUCAS
to support a conclusion but can also count against it. In the typical case which Bishop Butler was attempting to characterize there are arguments tending some one way, some the other, and we have to strike a balance. If we were to represent these in any mathematical form, we should need to assign them positive and negative measures respectively, assigning zero to arguments which hold neither in favour nor against the suggested conclusion, arguments, that is, which were entirely irrelevant. Such a theory would be very different from the calculus of probabilities , and would need another name, perhaps confirmation theory, perhaps something else. The words "probable" and "probability", and, to a lesser extent so far, "probably'; are felt to be less andless appropriate to cases which will not fit into the formalism of probability theory, and thus our usage and correspondingly our understanding of the concept is being moulded by, or, perhaps better, is growing into, the formal structure of the calculus. From this the moral to be drawn might appear to be that the formalist who operate with great subtlety and sophistication within the axiomatic system, but do not consider how their results are to be interpreted or how considerations arising outside the formalism should bear on the calculus itself, are entirely right. Nevertheless, I believe their approach is wrong both practically and philosophically. It has led to many mistakes and fallacies in the application of probability, and in particular to the disrepute in which statistics has come to be held. Scientists who have occasion to use probabilistic concepts, notably biologists in one direction and physicists in another, have been handicapped not by lack of formal manipulative competence but by an inability to handle probabilities conceptually. And anyone concerned to grasp the truth must echo the protest made by Plato in the seventh book of The RepubL{c (1).
Where one does not know the starting point, and the intervening steps and the conclusion are woven out of what one does not know,
how
on
earth can their mere consistency ever become genuine knowledge?
If we taKe both these points we must set ourselves to explain what probabilities are, granted that their logical shape is such as to fit the probability calculus. The probability calculus requires that probabilities be numbers - usually real numbers in the closed interval [0, 1] - which are assigned to entities themselves subject to Boolean operations: that is, if p has a probability so does its complement, and if p and q have probabilities, so do their meet and their join. We therefore have to ask what Boolean entities, as I shall call them, are candidates for having probabilities. Among the front-runners are propositions or propositional functions, beliefs, properties, and various sorts of classes - finite classes, linearly ordered denumerable classes, non-denumerable classes. For my own part, I would give the palm to the first, and say that the philosophically basic sense of the concept is that in which probabilities are generalised truth values and are ascribed to the same sort of things as can be said to be true or falsepropositions and propositional functions in my terminology, but also sentences, statements, well-formed formulae, theories or whatever else can be given the truth values True or False. Whereas an assignment of classical truth values can be viewed mathematically as a mapping from the set of propositions, or whatever, into the
(1)
533c 3 - 5.
TRUTH, PROBABILITY AND SET THEORY
211
2-membered range {O, l} - or {F, T}, or {L, T} - , an assignment of probabilities is a mapping from the set of propositions into the closed real interval [0,1] . The main virtue of speaking of probabilities in the same logical breath as truth and falsehood is that it bridges the gap between subjective and objective theories of probability, and integrates de. dido and de. fte. understandings of the concept. Sometimes our usage suggests that probabilities pertain to bel iefs or relations between beliefs; at other times that they are the properties of things or classes of things. But truth is similarly ambiguous. Sometimes we use it of be1iefs, and it seems to be an operator warranting our assertions: at other times the truth is something objective, independent of our subjective opinions or assertions, that in virtue of which our statements are true, if indeed they are. Basically, we have to distinguish between making a claim and what is claimed. To make a claim is something which I, or you, or the other chap, can do, and for which the words 'true' and 'probable' are the correct counters to use. In this way I can say that the Special Theory of Relativity is true or that the General Theory of Relativity is probable, and then I am asserting them, guardedly in the latter case, unguardedly in the former case, and claiming them as worthy of credence by any reasonable person. But of course I may be wrong. My say-so is not self-authenticating. And whether I am right or wrong depends not on what I say but on the facts of the case. Philosophers have found it extremely difficult to oive any satisfactory account of truth: but their embarrassment is our advantage. For although philosophers do not know how the word is used, we do know how to use it. Although we cannot formulate a precise definition which does justice both to the operational use of the word 'true' and to the criteria for its justified use, we are able to use it with a fair degree of confidence. To say that probability is like truth would be helpful and illuminating. And if the word 'true' is used to express belief, but is correctly used only when the thing referred to by the subject term has the property described by the predicate, we shall not be surprised at the word 'probable' b~ ing used to express a degree of partial belief, or at its being correctly used only when a thing under certain conditions - e..g. a coin being thrown - has a certain sort of property - a propensity. With probability, as with truth, its dual aspect stems from its gerundive force. To say that a proposition is probable, or true, is to say that it ought guardedly, or unguardedly to be believed (2). But to say 'ought' is always to stick one's neck out and invite a demand for justification, which may ultimately prove unsuccessful. With 'ought', both the rational 'ought's of academic discourse and the moral 'ought's of practical life, one always may be honest and sincere, but mistaken, and may, on challenge, have to withdraw claims put forward in perfect good faith. The divergence between what is meant by the word in actual use and the grounds on which its use can be justified is not peculiar to probability but recurs throughout moral philosophy, and there are systematic parallels between the philosophical problems of probability and those of morals (see Mackie 1973, ch, 5, p. 227). The suggestion that probability might be a branch of moral philosophy will be distasteful, not to say alarming, to most mathematicians and logicians, and will reinforce them in their conviction that probability should be studied aseptically, and that if the question "~:hat is probability?" is to be answered at all, it should be answered in terms of set theory, and we should say that probabilities are some sort of measures - non-negative, normalised and additive, or more often, countably additive - defined on the subsets of any given set, which is usually calleda prob-
(2) Compare St. Paul '5 formulation as translated by the Engl ish Prayer Book from I Timothy, 15 "This is a true ~aying and worthy of all men to be received". The Greek word, however, is 1ft aTO~ not dA'T/l1i", and Cranmer may have been in fluenced by the German word tJte.u.
212
J. R. LUCAS
ability space. But this cannot be so. Although, as I freely concede and will make use of later, we can give set-theoretical interpretations of probability, we cannot simply define probabil ities as bung measures defined on the subsets of a given set. For although we might naturally suppose it unobjectionable to assign to every subset of a given set a non-negative normalised measure subject to some fairly natural conditions, it turns out not to be possible to assign a measure, anda 60fLUOni not a probability measure, to every subset of a given set (see Royden 1968, pp. 52-3, 63-4). \ole have to conclude, as a restriction on measure theory generally that not every set is measurable. The proof of this surprising result turns on a re-creation of Zeno's paradox. A set is effectively divided by Aleph nought, KO' by forming a certain quotient set, and then multiplied by it again, using countable additivity and some criterion of equivalence: if we assign any positive measure to the set after division, we shall have to assign an infinite measure after remultiplication, while if we assign zero, then, thanks to countable additivity, we must assign zero also to it after remultiplication, either of which conclusions leads to inconsistency. This argument can be parallel led in the special case of probability, where, again, Zeno's paradox reappears if we allow that every set has a probability, together with some natural assumptions about equality of measures to be assigned to sets. Consider, for instance, the following problem (3). \oIhat is the probability of picking a natural number at random and finding it even? It seems natural to answer 'one half'. Similarly there is a one half probability of finding it odd. There is, we are inclined to say, a one third probability of finding it divisible by three, a one quarter probability of its being divisible by four, a three quarters probability of its not being divisible by four, and a one fifth probability of its final digit being 0 or 5. But what is the probability of our picking the number two from among the infinite set of natural numbers? If we are accustomed to construe probability as measures of sets, we feel obliged to say that it is zero. It is a singleton set, and all singleton sets must have the same measure, and unless it were zero, we should have the total probability infinite, contrary to the postulate that it should be unity. But if it is zero, and if the probabil ity of our picking four is zero, and likewise for six, eight, ten, etc., then the total probability of our picking some even number must be the infinite sum of the probabilities for each particular even number that we should pick that one, and that will be zero too. So whatever probabil ity we give to our picking the number two, or any other particular number, we shall be led to an inconsistency, so long as we have among our postulates of probability that the total probability should add up to unity and that the probability of a countable infinity of disjoint sets should be the infinite sum of their probabilities. It is clear that something has gone wrong. What it is can best be seen if we adopt the approach of the non-numerate philosopher and question what we mean by picking a natural number at random". In the f tnf tecase it could be explained perfectly well, for instance by supposing we were to choose one out of a large pack of cards, each card being inscribed with a different numeral. But we could not be presented with an actual infinity of cards to choose from. No physical process can produce an actual infinity. But perhaps mental processes can. Perhaps we should just think of a number. But most people think of rather low numbers. Perhaps we should first pick a mathematician at random out of those having tea in the Maths Institute one day, and ask him to choose a number. But even then we should not get very high: he might, if we had explained that it need not be a low number, oblige 10 10 10 by picking 1010 + 1, but would be unlikely to pick 1010 + 2 or 1010 + 257 ; and there are a lot of numbers larger than that. Although we might get some larger number, the probability of doing so is low. So we should not say that the probability of picking two was the same as that of picking any other number. At a very rough approximation, we might suppose that the probability of picking any particuII
(3) Which lowe, together with much valuable discussion, to my colleage, J.A.D. Welsh, Fellow of Merton College, Oxford.
TRUTH, PROBABILITY AND SET THEORY
213
lar It - digit number was 1/ (io" - 1 . 9· 21t ) . If that was so, we could calculate the infinite sum of all the probabilities, and find that they added up to unity. Another way of picking a number at random would be to have something 1 ike a roulette wheel, with the pointer coming to rest at just one of the ten digits, 0,1, 2,3,4,5,6,7,8,9, or an extra space representing HALT. A first spin would not be counted unless the pointer came to rest at a non-zero digit: after a valid first spin, spins are repeated until HALT is pointed at. If the probability of HALT is that of each of the other digits on second and subsequent spins would be·-/o 1, the series will converge to a limit, and by multiplying by a suitable constant factor, can be equal to unity. These resolutions of the problem resemble those of the problems posed by Bertrand (4), and depend on insisting that the random procedure be specified sufficiently exactly for us to be able to specify what outcomes are equiprobable. With a set-theoretical approach, however, it is more difficult, because considerations of measure theory incline us to an assignment of equiprobability which gives trouble. Unless therefore we abandon either the axiom of countable additivity or all definitions of equiprobability which would allow us to construe some random event as an infinite sum of equiprobable events, we must allow that not every set can be assigned a measure, and that therefore probabilities cannot be defined as being simply the measures that sets have. This carries the further unwelcome corollary that we cannot unrestrictedly postulate a sample space of etementafLy events.In the standard set-theoretical interpretation, ultimately due to Kolmogorov (Kolmogorov 1956), we consider an ordered triple (n, F, p ) consisting of a sample space, n., which is the set of all possible elementary events, a set, F, of subsets of n called "random events" - and a measure, p, on F. It is reasonable then to argue that F is not only a Boolean algebra but a eomp£.ue Boolean algebra: that is, given any (possibly infinite) set of members of F they have a least upper bound and a greatest lower bound; for if we consider any set of random events, we can consider also whether any of them will happen and whether they all will, the former constituting a least upper bound and the latter a greatest lower bound. If, then, F is a complete Boolean algebra, it would follow that, were it also atomic, it would be isomorphic to the field of all subsets of n (Halmos 1963, Theorem 5, pp. 70). But not all subsets have a measure. So F cannot be atomic. Not all Boolean algebras are atomic, and it could be that the field F of random events was not an atomic Boolean algebra. In that case we should not define n as the set 06 anything, but rather as the maximal element of F, or, more economically, but in terms of complementation as well as union, we could say that for any A E F, n = AUA, where A is the complement of A. If once we grant that the set of random events is closed under complementation and finite disjunction - that is, that if it 'is possibl e to ask what the probabil ity of the outcome of sene trial being A, then it is reasonable to ask what the probabil ity of its being not A, and of its being either A or not A, is, - then we have already specified what the sample space is, and further specification of n is redundant; further witness to which is the fact
(4) Bertrand 1889, pp. 4-5; or see Kneale 1949, pp. 184-5, or Lucas 1970 a. pp. 117-8.
214
J. R. LUCAS
that in most treatments of the subject the sample space, once introduced, all but disappears from view (Scott and Kraus 1966, pp. 219). For all these reasons we should drop n. We have simply a set, F, of random events, which can be conjoined, disjoined and complemented in the usual Boolean way. We may, with some hesitation about admitting infinite operations, idealise random events as a complete Boolean algebra 6("),
cP(It~,,».
Then for each a> 6(0)
M 1= (llelt(a,(3»
6(a, (3) such that for all
et(a,(3,'Y» II
choose 6(a) such that for
~(It~,,), 1t(~,(3». Finally for each such 'Y
> 6(a, (3) we have M
1=
~ (It~a), It(~,
",(3 (3) ,
•
•
DEFINITION 3.14. Xl' x2 ' x3 ' x4 will be the natural order of these variables. Given a permutation p of these variables lll' ll2' ll3' ll4 say that llj is m.wplttc.ed if some ll.{, .{ < j follows "! in the natural order. If V = (tj,z, d)
where
!f., z -
E
Kl
WI
(1M1)11
I
!f. n z = 0 -
and d = (d. : 1';;.{';; -t
-6)
and 0.;;
-6 .;; 4,
i:« 2 and d. elz for .{ > 2 then V is ttUowa6te. Given a cont -t'l B (u 1 e VI) ••• (u 4 e v 4 ) cP of '" and an allowable V, the cO!Vl.e6pond. ... . V l l d 1 .•• dt A.ng -6ll6-6titu.t£on -tlUtttnc.e 15 I/J p = (llt+l e ht+ 1) ",(u 4 e h4) ~ (u ••• u.t) where 1 l llt+ 1 is the first misplaced variable .;; -6 and h.{ =!J if u.{ e !J and ": = z d. el!f. A.
junct
for
"'p =
if u . e lz. -t
We 1et
I/J V
= 1\
P
I/J V •
P
DEFINITION 3.15. Let F be a countable family of prunings. By -6UP F we mean the pruning 'I with Dom 9 = Dom 6 and 9(a 1 , ••• , a ) = sup 6(a 1 , · · · , am)' m 6E F If 9 and 6 are prunings write g .. 6 if Dam g:2. Dam 6 and 9("1"'" "m)"
6(al' ..• ,am) for each for all
6E
F.
(a l, ... ,a m)EDom6.
Clearly if g=
sup
6E F
6 then
g"6
FRAGMENTS OF HIGHER ORDER LOGIC
DEFINITION 3.16.
229
Let
eEKk+i(IMI) and let J beasubsetofu1uie. wI if for every allowable V = (y, z, d) where y,z e ke and
Say that J qu.tLU6-{.e-6 Rngd ~ J we have M r= "" V • Let
LEMMA 3. 17.
eou.n.:tabie. d > Ci and
c
E
K k +i
(I M I)
Then n0ft eVeAy Ci and eVeAy J u {d} qu.tLUMe-6,
and flu.ppofle
J
qu.tLUMe-6 and
we ke theJte J..fl flame
d eiw
J
flueh
Ls that
PROOF. Suppose J qualifies. Choose Ci > sup J and !':!.ek~. For each conjunct r, of ",,8 and each allowable V= (!J..,~d) with !Lek~ and f.=[Q. or y = wand z e k~ we get a correspondi ng subs tituti on ins tance "" V whi ch has the - C P form (u P e a) cP V. When a = w be P _ we apply Lemma 3.13 and obtain nV' Let 0I the sup of these nv I s and choose for q any element > Ci of the form q = n(~)(Cil'·"'Ci.e) where Ci > n(Ci , .. ·, Ci _ ) for each m
(For the general case as described in the statement of the theorem we pa rt it ion where
[ ,,)11.
11. =
and given 1" j " P(
I 0'
g,
d 6(0).
m = {a ..... all.} 1
11..)
-(.,t,j)
.. :.(';;.e., m is the set of all sequences (6-(.(,j) ... m.)
to be a member of P
h)
.e.
such that for each
m
= () {p(
m. Since such that where d{3
I 0'
K
g,
h):
(6, g, h)
E
K l . m
ca11 them xa •
M a
let
Then PEP m
and is the color we assign to
is weakly compact. there is an m and a K - powered subset He" c P. Given a E P and {3 E H let V {3 = {d(3 (a) : "1 E H} m m a ."1 (a) is the projection of d{3 into M Clearly. if {3 =I={3' then
[H]7
,'Y
M",
'"
,r
x, y Hence for all a
V",.{3 n V",,{3' = 0 and for all distinct
we have
the projections of
g(O)' d6(O), g(l)' d 6(1). h(O)' d 6(1). h(l)
Ya' za' wa respectively. are distinct and p
a E P
1=
E
Pm
M",
1=
1/1
z , WE V",.{3' a", as we needed to
show. • As in the first order case. the ultraproduct theorem yields the Compactness Theorem.
FRAGMENTS OF HIGHER ORDER LOGIC
233
THEOREM 4.2. Le..:t K be weak.£.y compact and.te..:t V be:the se: deCfleal".{.ng quan:UMe.M. Then L [V) .{.I" A - compact nO!(. eve.!ly A < K, A < K and L a modu.
K
.{.I"
a MlU:tuy M..t.LIMab.te -6ub-6e..:t
on
L (V)
06 pOWe.!l
K
on
.{..
pflOpeJU'.y e. r .{. n
A then L hal"
L:
REMARK. The proof of compactness for W for K weakly compact gi ven in Magidor and Malitz 1977a is much more elaborate than the ultraproduct argument just presented. However, the combinatorial proof yields a completeness theorem whereas the present proof does not. We strongly suspect that the method used in Magidor and Malitz 1977 b can be extended to the new quantifiers so as to gi ve a completeness theorem but we have not done so.
§
5.1
SECOND ORDER DESCENDING QUANTIFIERS.
In this section we consider the compactness problem for quantifiers of the form U1 U2 ... Un + 1 where U1 is either 'tlX or 3X for some X, and for.{. > 1 U.{. is either 'tlY::, 3Y::, 'tlY::E X, or 3Y::E X for some sequence Y:: of first order -t -t -t -t -t variables. Such quantifiers we call -6econd oflde.!l de.-6cend-
Y = z]]
which asserts that every large set X contains a subset x in the universe. Let be a first order sentence which says that any two elements of B are order i3 somorphic under 6ab and that the common order type is infinite and has no limit
0
elements.
a.
to
Now let a
Clearly, if X;;;.
=
K .
"i !\
"t " a 3
and Xw = X then
(X ,E)
can be expanded to a model
On the other hand if (A, e, B , 6) 1= a we can take so that each b E B must have order type w.
X to be order isomorphic
K
PROOF OF i v) . It is enough to show that the quantifier of case pressed by this quantifier. We express by
of
iii) be ex-
'if X 3x 'ify 3zEX ¢xyz 'if X 3x1, X2EX'ify3zEX ¢6(x
1,
x
2)
qz ,
If K = X+ = 2X then the first is equivalent up to satisfiability to the by the lemma. • In contrast to .cv) above, the quantifier 'if X3 x fact 3X'ifxEX3y'ifzEX¢XYZ is equivalent to
E
X'if Y 3z
E
X
second
is compact. In
FRAGMENTS OF HIGHER ORDER LOGIC
We say that the quantifiers of L(QJ there is a formula exchangi ng Q' and Q.
cp
LEMMA 5. 3. .6 ub!.leq uenc.e 06
16 Q E V ;
235
() and ()' are eqlLi.valent if for every formul a ~ of· L(QIJ such that J= 1/1 ...... cp and similarly
and L K (QJ is c.ompac;t .then Q
.w
eqlLi.valent:to a
PROOF. Let Q be a compact member of V ~ and wri te Q as 'if X [0]. If nothing of the form 3 u E X occurs in [0] then Q is logically equivalent to the first order quantifier string obtained from Q by deleting 'if X and replacing 'if u. E X by 'if u.•
Now assume that 3 u. E X occurs in [0]. By Lemma 5.1 i), no compact Q has a subsequence of the form 'if X 3 u. E X 'if VEX. $0 if 'if v E X occurs then Q has the form (1)
'if X [lJ 'if v E X [2] 3 x E X [3] ,
where 'if v E X is the last occurrence of a bounded universal quantifier and 3 XE X is the first occurrence of a bounded existential quantifier. By Lemma 5.1 ii) , nothi ng of the form 3 z occurs in [I] and so the terms of the subsequence [I] are either of the form 'if.6 or 'if.6 EX. Hence Qcp is equivalent to (2)
{t.
'if X 'if""i E X [ 2] 3 x
E
X [ 3] cp
Let .6 =.61' .6 2' ••• ,.6 n and let Itl , 1t 2, ... , Itm be a list of those variables It occurring in cp such that 'if It E X or 3 It E X occurs in Q. let F be the set of all I - I functions 6 such that Vom 6 ~ {ltl' ..• , It m} and Rng 6 ~ {.61"'" .6 } . For each 6 E F let CP6 be the result of replacing each It.f.. E Vom 6 by n 6(1t.f..)' Let r be the set of all such CP6' Clearly, 2) is equivalent to (3)
Here [2 J
[3 J •
'if1'if"i 'if X [2] 3xEX[3J V r
•
is a fi rst order sequence and nothi ng of the form 'if u. E X occurs
in
$i nce [2 J can not cor.tai n 3 v 'if u. as a subsequence we can suppose that [2] has the form 'if v 3u for which we have the equivalent quantifier (4)
'ift'if"i 'ifv 'if X hEX 3u[3J
by permuting quantifiers. sider (5)
If [3] is first order we are finished.
'ifX 3x EX 3U[3].
If not
con-
J. MALITZ AND M. RUBIN
236
If 1n II > 0 then by 5.1 iii) we have no occurrences of the form If w [4] 3 z EX in [3]. Permuting existential quantifiers we get (5) equivalent to If X 3\1 EX (5) where [5] is first order. If ln U = 0 then (5) has the form
which is equivalent to IfX3x EXlfp3z EX3q [6].
The argument presented for [3] is now applied to leads to the desired form. DEFINITION.
C" = {Q.E
Let
THEOREM 5.4. -il In" to the 60Jun
u
v~:
[6]
and so on.
Q is compact in the"
weakly c.ompac;t then Q E
c" -i66
Clearly, this
interpretation}. Q.
u
equ-Lva.len:t
IfX3ltE Xlf6"3zE Xlfu3vE X •••
U) 16 Q.
E
C"
tc
>
-in6 Q
wI and
u
o
wI
, on: -i6
K
= X+ > wI and o
X+
and X<X = X
then
equ-Lva.len:t to If X 3 it E X If 6"3 Z E X
where ln It ,;;; 1. PROOF OF i).
Immediate from Theorem 4.2 and Lemma 5.3.
PROOF OF i i). By Lemma 5.3 we can assume that Q is a subsequence of If X 3it E XIf:63 z E Xlfu3v EX
By Lemma 5.1 iv)
Q. is a subsequence of If X 3 It E X If 6"3
ZE
X
where ln It ,;;; 1. Our remark following Lemma 5.1 shows that such quantifiers are . <w reductb l e to L . The result now follows from the compactness of L"<w . ( For " = wI' see Magi dor and Malitz 1977a , for " = X+ with A < A = X and " see Shel ah 197+.)
§
6.
PROBLEMS.
237
FRAGMENTS OF HIGHER ORDER LOGIC
Here we mention only two open problems, ones that we consider to be of particular interest. It was shown in Magidor and Malitz 1977 a that Ow is not necessary for the <w 1 countable compactness of L • In fact a model of ZFC + I CH is descri bed wI there in which L< w is countably compact. Is there a model of ZFC in wh i ch wI L <w is not compact? If the answer is yes, is there a simple combinatorial statewI ment equivalent to the statement "L < w is countably compact" or a simple combiwI natorial statement weaker than which imp 1i es the countable compactness 0 f
°wI
L < w? wI
Such statements, if elegant enough, mi ght be i nteresti ng new axi oms for set
theory. In Morgenstern 1977 it was shown that if K < X < M and if M is weakly com<w <w . <w pact and LX is K - compact then L{X,M} 1S K - compact. (The language LL; , where L; is a set of cardinals, is the smallest set of formulas containing the first order formulas which is closed under conjunction and negation andquantification by any quantifier occurring in u U:w:K E L;}. Hence quantifiers havi ng different interpretations may occur in the same formula of L~w.) Burgess [unpub1i shed ] has shown that the 1anguage havi ng the quantifi ers of L <win the 1 wI interpretation along with the quantifier Q. of L in the w 2 interpretation is countably compact in L. The second problem we want to mention is this : suppos<ew L; is a set of cardinals such that Is LL; X compact?
L: w is
X
compact for every
K
E
L; •
REFERENCES.
J. Barwise, M. Kaufmann and M. Makkai 1977
S;ta;tLOn.Mlj log- .....
k, .e. E I wUh the OpeJla.t. /R)
IRENE MIKENBERG
246
b) h .u:, one-one. from bellow, that is;
~ Fi(h/a O)"'" h/a n_1)) • To prove thi s fact, we have to use a constructi on
0
f
R
we define 6·,9·EB)/\y.ESV 3'Y1' ••• ,3'Y(-r1""''Y = G1(h", ••.•• h '1 q E ~}
a has the property that q '1-
n { - a
Furthermore, one can show quite straightforwardly that if (4)
It follows that if lowing properties:
n: A... B
for each
pa.ce of the cylindric algebra A. We will see that it is a very useful tool, not only in algebraic logic, but also in model theory. Algebraic properties of logical properties.
1"
Thus,
and 3"
dKA
translate routinely into
and
topo-
3 A commute, so do theirduals.
1 x is also an open and closed equivalence relation on A, as is, for
1 w ' For each permutation is a homeomorphism 6. .... t.. •
example, ~T
0
3", S~
For example, since
The equivalence relation
T
of
W,
ST
is an automorphism of A, hence
lw is especially interesting: if A is the cylindric
algebra of a theory T, then T is complete iff for lw' If T is an arbitrary theory, each
A has only one equivalence class class in t.. is
I w- equivalence
the dual space of one of the complete extens ions of T.
4.
TOPOLOGICAL DUALITY AND MODELS OF THEORIES.
One of the most impoYcant aspects of topological duality theory is
the
fact
TOPOLOGICAL DUALITY IN ALGEBRAIC LOGIC
263
that all the countable models of a theory T are determined up to isomorphism by certain points in the dual space of T. To understand this, we should recall that in Henkin's method for constructing a modelof T, we prove the existence of a com-
plete extension T * of T,* in a language having an infinite set C of new constant symbols, such that T has the following so-called Hen~n PkOPe4tlf :foreach ~(v) with only v free, (7) T*
if
then
(3v)~(v)E T*,
canonically determines a model of
Now consider an ultrafilter ing the following property: (8)
q
T whose universe is a quotient set of C. F I=r of T, hav-
in the algebra of formulas
For any formula
~(v1)' if
« E w,
q.
~(v/{) E
for some c.EC.
~(c.)E T*
(3 v 1) ~(v 1)
E
q, then for some
It is easy to see that q determines a model exactly as T* of the model determined by q is a quotient set of Iv : /{ sentially a quotient set of w. «
does; the uni verse that is, es-
E w}
To transcribe this notion in the language of cylindric algebras is easy: if A is a cylindric algebra, we are interested in an ultrafilter q of A wi th the property: (9)
For any then
a E A,
Sltc-a
E
there is a
/{ E w such that if
3 1a E q ,
q.
!:.;
Now, the topological dual of an ultrafilter of A is a point of are interested in a point q E!:. having the property: (10)
a
For any then
q
(Reca 11 that by (4),
E
E A, there is a -1 S 1 (a) . -/{ -
S~ a
-1
= ~~ (~) •)
tc E
w such that if
q
E
thus, we
i 1 (~),
Such a poi nt wi11 be called a.modetpo-i.n:t.
The completeness theorem for the predicate calculus is therefore equivalent to showing that every non-trivial cylindric algebra has a model-point. In fact, this is very easy to show topologically. (The argument is due to Rasiowa and Sikorski 1950.) It is simplest to deal first with those points which are fto.:tmodel points; they are the points p which satisfy the condition: (11)
there exists a E A such that for every -1 1 and pf/. ~/((~), that is,
(12)
there exists
a E A such that
NoW the set (13)
-1 3 (a) _ u S 1 (a) -1«e.» -/{-
/( E
p E~ l(a) -
W,
u
/{Ew
PE
~1~)
-1 S 1 (a). -/{ -
CHARLES C. PINTER
264
is obviously closed, and by familiar methods it can be shown to be a boundary set. The set of all points which are not model points is the union of all the sets (13) over all a EA. By the Baire category theorem, this is an FG set with an empty interior. Thus its complement, which is the set of all the model points, dense Go set. Thus, not only is it true that !:- includes at least one point, but in fact, the model points are a dense Stone space, any two dense dense
Go subset of
~
Go subset of
A.
is a model
Since in a
Go sets intersect non-trivially, it follows that any
includes a model point.
This statement is known to be the
topological version of the omitting-types theorem. If q is a model point, it determines canonically a model a whose universe is a quotient set of co, If T is any permutation of co, then ~T (q) is clearly also a model point, and determines a model which is isomorphic to the set
Q[. In fact,
{~T (q) : T E permut (w)}
ex.
is called the otrb-U of q, and corresponds to the isomorphism class of It is an interesting fact that, if
A has only one
!w class (that is,
A
is the space of a c.omplete. theory), then e.veJr.fj oJtb-U if:, de.YUle..in A. Thi s may be shown as follows: let a ~ 0 be any basic open (clopen) and, to simplify the argument, suppose A a = {l}, that is, a is ~/{-saturated for every /{ ~ 1. ~l (£0
Then
~.,(£0 ~
=~
because
A has only one
3 _l-w
class.
If
there
were no model point in ~, there would also be none in ~ ~ (~) for any /{ E W • -1 (Indeed, if pE ~~(£0,then ~TPEf!,.,where T is the permutation ofw which interchanges 1 and /{.) Thus, all the model, points of A waul d be -1 ~ 1 (£0 - u S 1 (£0, and this is impossible, as we have seen above. /{ Ew - /{
in
If A is the dual space of the algebra of formul as of a theory T, and rz represents the set of all the model points of A, then as we have seen,.n is a Go subset of A. But it is known that any Go subspace of a Stone space is again a Stone space. Thus, .n is a Stone space and is, in fact, homeomorphic with the model .6pac.e. of T, studied, for example, in Suzuki 1970.
5. If
DUAL SPACE OF SUBALGEBRAS OF THE ALGEBRA OF FORMULAS. A is a cylindric algebra and
nEw, let
B
n
designate the
subalgebra
of A consisting of all the elements of A whose dimension set is inc 1 uded in {O, ... , n}. Thus, we may think of B as the algebra of all the formulas with at most the variables
v o' "
"
n
vn free.
These subalgebras of
A, and their dual
spaces, play an important role in many arguments of model theory, and it is ofimportance to establish exactly the relationship between A and the B , nEw.
n
First, it is worth noting that for each
nEw, Bn
is the range of the quan-
TOPOLOGICAL DUALITY IN ALGEBRAIC LOGIC tifier of A.
3w _ {a, ... , n}' and therefore
Let 0
be the mapping from w to
e(n)
=
Bn
265
is a relatively complete subal gebra
w defined by 0 for every
nEw.
We may think of 50 as the substitution which replaces every variable by vO.Clearly 50 is a homomorphi sm of A onto B0' and it is easy to show that the kernel of 50 is the ideal of A generated by {-d OK: K -:F-O}. The dual of this ideal is the open set U {-!fOK: " -:F- O} , so by (5), the range of 50 is n {~K: K -:F- O} = n d , . It follows (using the fact that B is the range of a quantifier) O K-:F-A -K" that the dual ~e of 50 is an open and closed continuous injection with domain ~O and range n ~"A' In particular, ~O is homeomorphic with n ~,. K -:F-A K-:F-A K/\ Furthermore, if e : B -+ A is the inc1usi on rna ppi ng, its range is a1so B0 ' O hence ~: 6. ~ §.O is an open and closed continuous surjection. By (6), eq(~) = lw_{O}'
In general, Bn is homeomorphic with n {£i. OK: K > n}, and there is an open and closed continuous surjection ~: A~]n such that eq(~) = ~ _ {O } . W , .... ,n Next, consider Bm and B where m < n , Let 0 : n-+ m be the n defined by
function
for for
K>m
Then So is a surjective homomorphism Bn ~ Bm with kernel (- d O(m+ l )" " ,-dan)' Thus, ~o is an open and closed continuous injection.§m ~.§n with range ~O(m+ 1) n ••. n ~On' In particular, §m is homeomorphic with the subspace ~O(m+ 1) n ... n ~On of ~n' Furthermore, if e : B .... 8 is the inclusion function, n m then ~: §n ~ §m is an open and closed continuous surjection whose equivalence relation is the restriction to § n of ~m+l 0 ••• 0 ~n
REFERENCES. N. Bourbaki 1966
General
P. Halmos
Topology, Addison-Wesley.
266 1956
CHARLES C. PINTER AlgebJuU:c .f.og-
Px)& - O( 3 y(y =1= x) & Py»
i.e. 03x(0(3y(y= x)-> Px)& O'l:fy(Py-> Y = x»,
or, perhaps better, essences under the equivalence 03x(Px & Qx), which is the same as 0 I:fx(Px +-> Q.x) for essences. (Of course for Montague's system it follows that P = Q , which it does not for Plantinga.) Now Plantinga avoids discomforts a) and b) by supposing that an essence of x always exists, even if x does not. Thus one avoids asserting that x exists, while retaininq some way of referring to x (namely reference to its essence which does exist). There is sti 11 a third discomfort, however: c) If x does not exist, it is hard to see why the property of being x should exist either, or any essence of x , More generally, if the individuals which exist depend on the world, why shouldn't the properties which exist also depend on the world, especially since many are defined in terms of individuals (being to the left of a, etc.). The treatment given are here will avoid these discomforts. (We will consi der 0 (Px V i Px) to be tautologous, which may give rise to a fourth di scomfort, however: for we are inclined to say that if x does not exist in a world, neither Px nor i Px makes sense.) I:fx
We are taking the case of set theory as a sort of model of a s,,'SfcC' t 0 \"y framework. As a philosophical framework, however, it is not adequate, or at least it is incomplete, in certain respects. It seems appropriate to remark on these at this point. First, the expressive power of any classical language does not include its own semantics; second, whatever formal theory we adopt in the language will be incomplete; third, intensional notions are conspicuously absent. Regardi ng the
A THEORY OF PROPERTIES
271
third, in particular we have no adequate interpretation of intuitionisticmathematics, which concerns itself from the outset with intensional entities such as proofs constructions, etc. (Although we are used to thinking of formal proofs as strings of symbols, it would seem that what carries conviction when one reads a proof is not the formal string but some interpretation which we attach to it.) While it is not obvious that no adequate account of these can be given in set theory, it is at least highly implausible. The frame is thus incomplete to the extent that intuitionistic mathematics is part of extant mathematics. Fourth, there is no account whatever of epistemology; we have only plausible axioms adequate to most purposes, but not theory about how we know they are plausible, etc. A fifth remark is analogous to the third. To the extent that extant mathematics uses self-applicable concepts (as some have argued is the case inca tegory theory), set theory is inadequate for an account of the mathematical world we live in.
§1.
LANGUAGE AND LOGIC.
1.1. Our language has as basic truth functional connectives -> and "'; /\, V , etc. are defined as usual. We also have equality, =. In addition we have the modal operator 0 for necessity; 0 is defined as ",0",. We have one binary relation symbol E. This is used in the usual way to indicate membership (x E tj ) when tj is a set, but also to indicate that tj holds of x , or x has tj , when tj is a property. There are three unary relation symbols: U(x) for 'x is an individual (or combination of individuals, i.e. set of individuals)', Set(x) for' x is a set', and A(x) for 'x is the actual world'. We shall use Greek letters for possible worlds, reserving Ct for the actual world. The formation rules for formulas are the usual; if 0 is a formula, so is DO. A sentence is a formul a with no free variables.
The basic ontology for the intended interpretation of this language is: individuals (and combinations thereof ), properties of these, sets and properties of properties of individuals, etc. Propositions and possible worlds will be defined in terms of these. In fact, --', -v , 0, U, Set, A, will also find their place in this ontology; =, E, and 3 will only have "truncated" forms (as we expect from the set theoretic analogy, where E does not exist, but EA ={(x,Y)lxEy&x,tjEA} does). It is interesting that these are the logical symbol closest in meaning to the verb 'to be'. 1.2.
The logical axioms are given by Du
Ll
-> ( I ,
where
(I
is a sentence.
By the closure of a formula 0 we understand the sentence
where the free variables of 0 are among ioms are the closures of: L2
Tautologies.
ts
Vx(O
L4 L5
0
->
cP)
-+
'tJxO ,
'tJu(
Vx 0
-+
(Vx 0
-+
O' . , . , xI! -1'
X
VxcP).
provided x is a variable not free in ->
e[x/ul}, provided x is free for
u
0 •
in
O.
The remaining logical ax-
272
WILLIAM N. REINHARDT
The notation e(xlu) x in e by u
is used for the result of replacing all free occurrences of
L6
x = x.
L7
x = Y -> [e(ulx) -> IJ(uly) I , u free for x , yin IJ •
L8
DIJ -> IJ.
L9
D(IJ-> (DIJ-> D D\ixD[3y(y= x) -> IJ I. The only rule of inference is modus ponens: from 01 and 01 -> 02 to infer a2 • It is easy to prove the deduction theorem: If "i- ... , on 1- o then f- 01 /\ ... /\ on -> 0 , and the necessitation rule, if f- a then f- Do (but of course not from 01 f- a to infer 01 f- Do. Also if f- e(x) means f- \ixIJ(x), we cannot infer that if f- IJ(x) then f- DIJ(x).) 1.3. TERMS. So far we have avoided terms, even individual constants. extensional logics we have the theorem:
In
x
THEOREM 1.2.1. 16 T JJ., a theoILy and T 1- \i 3 ~yIJ (x,y), and T' JJ., ab.tIUned 6!LOm T by adcUng a new 6unc.tion "'umbo! 6 wah auom 'if x IJ (x, 6x ), then T ' JJ., a c.onl>e!Lvllive extenl>ion 06 T. In intensional logic the same theorem is true, yet the introduction of terms requires some caution. There are two natural approaches which conflict (or give rise to two distinct kinds of terms). We illustrate with the case of individual constants. The first is to treat individual constants like variables, so that e. g. L5 yields the closure of \i x
e -> ()
(xl c)
for an individual constant c.. The second is to think of an individual constant c. as corresponding to the phrase 'the C' in English, so that "by definition" C(c.), and hence D C(c.). Now it may happen that in every possible world there is a unique C , but different ones work in different worlds. Then 3 ~ xC x but 'V3 x D C(x). The latter is \ix'V DC(c.), contrary to the "definition" of c . We shall adopt the former course, retaining L5, which we now state in the proper form for terms. Note that from Lll, L5 yields D \i u D (& u
where
/I,
u
is an abbreviati on for
-> [ \i x e -> () (xlu) I ) ,
3 y( Y =u},
Thus the natural generalization to allow terms is the closure of L5'
&(T)-> [\ixIJ->IJ(xl r ) ]
273
A THEORY OF PROPERTIES where
x
is free for
T
in
O.
We may now state various analogues of the above theorem on adding terms. The simplest is 1.2.1 (of course this only says T' I- 0 c , not T' I- OOc). Also: THEOREM 1.2.2.
I6
T .u, a theMy J.>uch that
T I- 0 \:Id y(O 0 (x,y) A 0 \:Iy' (0 0 (x,y') .... y = y'))
and T'
o \:I x
[8,
.u, obnuned 6ltom T by adriLng a new nunction J.>ymbo! 6 wah auom 6x A 00 (x, 6x) 1, then T' .u, a conJ.>eltvative edeYl!.>.{.on 06 To
1.4.
INTENSIONAL EQUALITY.
In set theory we have axioms which say that certain entities are coextens i ve in membership with certain concepts. For example, if 0 (t) is a formula expressing 'union of A', e.g. 3 u(tEuEA) then we say that a set z has membership coextensive with 0 by \:I t(t
E
z
+->
0 (t» ;
this is a special case of the statement that \:I t(O (z )
+->
O(t),
¢(t)
are coextensive,
¢ (t» .
Now for an intensional theory we need to say that z is cointensional with 0 ,or, slightly stronger, O(t), ¢(t) express the same concept (or mean the same thing). Clearly this requires a 2-place, variable binding operator. We shall use 8 "" ¢
t
for this. In the case where we identify concepts which necessarily have the same extension, we can use
x
(where lists the free variables of 8 other than z , and lj those of ¢ other than z ) and ""t thus becomes definable from O. In the interests of simpl ifying the language and logic we shall adopt this treatment of
:t'
This course appears
to be fairly satisfactory for the semantic and set theoretic objectives stated above. From the broader perspective of a philosophical frame, however, it may be better to take ""t as primitive. The reasons for this have to do with 'knows' and 'believes' and will be elaborated on below.
Although we will not officially treat
""t as primitive, we will try to write axioms in a form which would also be suited to such a treatment. Also, we indicate now the new rule of formation and main axioms which would be required. The formation rule is that if 8 ,¢ are formulas, then (8 =t ¢) is also a formula, whose free variables are those of 8 and ¢ except for z :
Fv(O
t¢) - (Fv 8 U Fv ¢)
"v
{z} •
WILLIAM N. REINHARDT
274
The main axioms are simply analogues of the usual equality axioms. the 0 liD 11 •.. 0 closures of: El
e '" q,
E2
e
0=
E3
e
= q, A t
E4
t
E P
t
e
if
t
q, ... q,
0=
t
0=
t
= q,
E
is a bound variable variant of
q,.
0 •
¢(p/e) ... t
e
or
Specifically.
¢(P/q,)
where P/O indicates proper substitution for Pin e ,
of
e
Q... P = Q •
We also allow simplification of conjunctions
e A q,
E5
0=
t
q, II
e,
eAq,o=ei\q,i\q,
t
We define
0
~
ep
to be
o lit 0 (0 ...... 0), and De
0
0=
0 A ep; note that
=0, t
t
e
0=
t
q,'" 0 (0
0 A ep
Step.
=q,) A O\l:t. 0 t:
Also since
(0 ...... q,). It is philo-
sophical problems involving belief that suggest it may be better to treat
=t
as
primitive. For if intensional equality is defined as above, for necessarily existing propositions p , q we would have p'" q iff 0 lit 0 (p ...... q) iff o (p ....q). which in 55 would give rise to the following difficulties with belief. Let B(p) be a predicate meaning (a certain person) 'believes that p ", If this means 'believes or can be persuaded by rational means', it does not seem completely unreasonabl e to say p = q A B(p) ... B(q), p/tov-i.ded that the set of p , q such t ha t p'" q is recursively enumerable (r .e. )for then we can conceivably do something to persuade: namely start listing such q's and a rational being will modify beliefsaccordingly. This however is not compatible with 55 for 0, because for a from say the 1anguage of arithmeti c (with D), a)
{a IDa}
is not r.e ••
b)
if
10 (a ... a 2)} is r.e., then by taking 2) l 0 p ...... 0 (P'" T)).
{(aI' a
is also r.e. (use
a
2
= T,
N
{aiD a}
Toseea),notethatinS5. Oa or O'Va •• ·.OaVOO'Va. if N= is r.e •• it is decidable (a If. N is shown by OO'Va. and O'Va is in the modal language of arithmetic. hence is enumerated). But of course ~ = {a I o (Q'" aj) extends Q. which is an essentially hereditary undecidable theory: thus k is undecidable. Evidently a decision procedure for N would yield one for ~ ,so N is undecidable and hence not r .e.: (The non-r ,e , character of the necessities of 55 was pointed out to me by M. Lob.) {a IDa}
§2.
AXIOMS FOR SETS AND PROPERTIES.
2.1. ON INDIVIDUALS, SETS. AND PROPERTIES.
A THEORY OF PROPERTIES
275
We regard sets as being constituted by their members. Thus, although they involve the abstract element of combining or collecting, they otherwise have the same sort of being as their constituents. For this reason we include sets of individuals among the individuals. The pure sets (e.g. the empty set) have no constituents other than pure sets, so that their being lies entirely in the element of combining; these we regard as the mathematical individuals. Properties on the other hand are not constituted by the individuals to which they apply; we have the property of being red even F there happen to be nored individuals. Also, the property of being a or b - :t(:t=aV :t - b) - has for constituents a, b , = , V , whereas the set {a, b} has only a and b. The class C of things satisfying a property P (the extension of the property) evi dently is arrived at via the property; we regard P as a constituent of C, or at least consider that in general C depends on P for its existence. One advantage of including sets of individuals among the individuals is that ordered pairs of individuals are again individuals, and we may treat relations-inintension between individuals as properties of individuals (rather than properties of higher type). In the case of contingent individuals a, b one might attempt to explain the relation of constituency to {a,b} as follows. If {a,b} exists, so must a, b; but a, b could exist without being combined to form {a, b}, It is not clear how good this is as an analysis of 'constituent', since it would make any necessarily existing individual a constituent of everything. Moreover, it seems to requi re giving up the necessity of Ijailb 3 z(Set z A Ij :t(:t E z ....... :t = aV :t = b). Nevertheless, we shall be guided by the idea in the following way. i)
We regard the pure sets as being mathematically necessary, but (in general) not logically necessary (and similarly sets in general as mathematically necessary relative to their ultimate constituents).
ii)
We keep the convenience of necessarily existing ordered pairs, and other usual set theoretic operations (thus mixing a bit of mathematical necessity into our logical necessity).
iii)
We keep the distinction urged above between sets and properties. According to iii), we define Px, "x is a property", to be 'VSet(x) A 'V U(x). 2.2.
SET AXIOMS.
Closures (i .e.
0 't} x 0 ...
closures) of
Sl
Ux .... 0 Ux ,
S2
Set x ....
S3
Set X/\:tEC
S4
Set x /\ :t E x/\8.x .... 8.:t • (Since sets are constituted by their elements.)
S5
Set z/\'t}t (:t E z .... U:t) .... Uz . (U is individuals and sets thereof.)
S6
tE
x/\
o Set
x•
" .... 0 (J.: i::
Ux .... Set x ,
x) •
276
WILLIAM N. REINHARDT
S7
Extensionality: V x V tj [Set x /\ Set tj /\ V t(t E x - t E tj) ... x = y] •
S8
3zVt(tf/cz). Set z
rp 3 z , (3 z,¢), 3 z ¢ 3z(¢ /\ .•.. ).
We shall use the notations to the formula
¢:
to indicate relativization
Sg
Pairing:
'Ix Vlj 3z Vt(.tE z - t = xVt= lj). Set z
SID
Union:
ilx 3z ilt(tE Set x Set z
Sl1
Power set:
S12
Zermelo Scheme:
3u (t E u/\ uE x)). Set u
Z -
ilx 3z ilt(tEz-Sett/\ ilu(UEt"'uEX)). Set x Set z D 'Ix 3z Vt(t E z +-+ 0/\ t E x}, where Set x Set z Notice that this gives,
z is not in O.
D ilQ D ilx 3z illj(tE z ....... tEQ/\t Ex) , PQ Set x Set z so that we allow all possible subsets of x , S13
Regularity:
'Ix [V t(t EX'" 0) ... o(t/x) ] ... ilx O(t/x). Set x Set x
S14
Replacement:
D[ilx3~te ... Va
o. S15
3z ilt(tEz ....... 3xEaO)), a,znotin Set a Set z
Dependent Choice (any number of choices): D(ilx 3ljO ... ila f 3F(F)[(O,a) E F/\ ilv ilv' 3x3tj((v,x) E F /\ cn ordtv) v' = v+l (v',lj)EF/\O)]) , fcn(F)
is
VxV£jVlj'«x,lj) E F /\ (X,tj') E F'" lj = lj') ,
ord(v)
is
trans (v) /\ iI x (x E
trans(v)
is
where:
Vi
= v +1 is
(x,lj)
2.3.
E
F
is
ilx iI Y (x E lj /\ Y E iI t(t E v I
+-+
t
trans (v)) ,
V ... V ...
E V /\
X E v)
t = v) ,
also as usual.
AXIOMS FOR PROPERTIES.
We define P(x) ties (see §1.5).
to be
'V$et(x) /\ 'V U(x); intuitively, these are the proper-
Dilx D iI£j (t E x == t E lj'" X = lj) Px Ptf t (This is a repetition of E4 if ==t is regarded as primitive.)
PI
A THEORY OF PROPERTIES P2
277
(Zermelo-Montague)
o
'd"a 'd 0. 3 S (t ESE; e ("a, t ) /\ tEO.) , P(S) t
e
where the free vari ab1es of We shall use the notation.
for the property given by P3 where
teo. is
P(o.)
u E
-
and
e -u ~
t: =u r u E 0. (reca 11 that -+
u. EO.)
P4
0 3S (t E S t U{t)). P(S)
PS
Possibility:
P6
Union:
P7
Classes:
cjJ
do not Occur free in
means
1\ 0 'dt D(&t
Thus P3 is an analogue of the power set axiom. erty given by P3.
where
S, Q.
3 S (t ESt t ~ 0.) , p(S)
0 Ii u 0 (u E t
is equivalent to
it, t:
(J.
P2.
o 'd 0.
Higher types:
are among
-+
e /\
&0.).
We shall use
cjJ
=u
(J,
so t ~ 0.
P 0. for the prop-
0 'd Q 3S (t E S =t 0 t Eo.). P(S)
D'do. 3S (tES
P(S)
0 Ii a. 3s 'd t(t P(S)
t3U(tE ul\u E Q)). E
S ...... tEa. 1\ (J) ,
S does not occur free in
(J.
See
§
§3.
AXIOMS FOR POSSIBLE WORLDS.
3.1.
PROPOSITIONS AND POSSIBLE WORLDS.
7.3
for a discussion of P7.
In order to state the remaining axioms, we must introduce propositions and possible worlds.
p
~
DEFINITION 3.1.1. 1; that is,
where ~n
1
= {OJ
and
0
We say that p is a pltOpo.6ilioYl
and write
Ps(p) in case
is the empty set.
Evidently every property o.(x) determines a proposition p by taking x = O. the other hand, if p is a proposition, we may take 0. to be the property
t(p /\ tEl).
Clearly then
o
(p ...... 0
E
Q) •
WILLIAM N. REINHARDT
278
It may be that intuitively p , pAD = 0, p ADE 1 are all distinct propositions, but they are also clearly extremely close in content or meaning. Thus since Q is intensionally the same as p A·:t E 1 (that is, :t E Q :;:t p A :t E 1) it seems reasonable to identify the proposition p with the property Q. This is whatwe have done in the above definition. In case p is a proposition according to Def. 3.1.1, we shall write simply p for 0 E p . DEFINITION 3.1.2. case
We say that
Ps (.6)
/I,
0 {3
/I,
0
is a possible world and write
{j
W(/3) in
if P [0 ({3 .... p) V 0 ({3 .... -v p ) ) •
Ps(p)
Thus possible worlds are propositions which determine or decide all possible propositions. We then have PROPOSITION 3.1.3. a)
W({3)
0 'ri x [0 ({3
&x) V 0 ({3
b)
W({3)
D'riX[D({3
"'£/) V
wheJte £/
.u anq 60Jtmufu wUh
o ({3
-v 8. x)
)•
£/))
,
6!Lee valUa.ble;., among
x
o' ... ,
x _1 . n
PROOF. a) is a. special case of b) (8. X is the formula 3y(y = x)). To prove b), given e and take p to be
x
according to P3. Then :t E P :;:t e(x)
/I,
:t E
1 ,
so D(p -. £/) •
It now follows from the definition of possible world that b) holds•• We remark that one could ask for stronger or weaker notions of possible world. For example, one could require
o 'ri x 0 or only that
o ({3....
the latter requires /3
if Y [0 ({3 ....
'rip (Op-+ 3/3
Ps(p)
W(/3)
y) V 0 ({3 .... x ~ y)) ,
if P [0 ({3 -+ p) V 0 ({3 -+ -v P)]) ;
Ps(p)
to decide only those p that exist for it.
3. 2. AX I OMS.
WI
X E
D({3-+p)).
A THEORY OF PROPERTIES
279
This just says that possible means true in some possible world. The next axioms introduce the notion of the actual world. W2 says that there is a unique actual world (necessarily), W3 that if ~ is a possible world, then in ~ the actual wor1 dis ~,and W4 tha t truth is truth in the actual wor1 d. W2
0 3~ [W~ A A~ A IJ'Y(A'Y ~'Y = ~)]
W3
IJ ~ W~
W4
0
(~ ~
A~ A
8, ~ ) •
Closure of 0[0 ...... IJ {3 (A~ ~ 0 (~ ~ 0))
1•
By W2, 3 ~ ~ A~ ; thus by Theorem 1.2.1 we may add an individual constant c< such that A(ca..tu..Med by .:the 0A x = 0 IJx 601L .:the u~veJU>a..e. quan.-tiMeJt..
PROPOSITION 4.2.3. ab.60lute 60JrJnul.a..6, .:talUng
e,
PROOF. We observe tha t 0 IJ 0 IJ••• 0 closures of the fo 11owing schemes (for ¢ o-absolute) are theorems (writting A x for 0 IJ x ) : a)
A x(e -. <jl) -. (A xe -.A x<jl) ;
b) c)
(J -. /\ x(J,
x
not free in
/\ y [/\xe-. (J(xly) 1,
d)
x
(J;
free for
if
in e;
tautologies.
Now d) is already an axiom of our system; a) follows immediately from 0 [IJ x(e-.¢) -. (IJ x.B -. IJ x c ) 1 and the distributive property of 0/-.. b)
Since
(J
is
o-absolute,
o(e
-e-
IJx e)
e .... «(J -. e s o v ;»
0'"
c)
0 IJ u (lJxe -. e (xlu)
o IJ u o IJ u.
0/-'
IJ x e)
-.
(&u -.
distribution, tautol~gical
from above ,
axiom, olJx(J) tautological from above, tautological from above axiom
L5
0 [ g u -. ( IJx (J -. (J (xl 1/) ) J 0 [ IJ x e
Also,
axiom,
o e -. 0 IJx (J o (J -. (e -. 0 o",(J-.",e
O(J V 0'" (J.
(J (xl u))
l
use
Lll ,
tautologous,
A THEORY OF PROPERTIES
o o
281
0/ ... distribution, Op"'p,
IJ u. [0 IJx IJ ... 0 (8.u. ... II(X/u.)) ]
IJ u. [0 v c e » (&u. "'1I(x/u.)) 1 OIJu.[OlJxIJ ... IJ(x/u)] qed.-
4.3. TRANSLATION INTO PSEUDOCLASSICAL FRAGMENT. DEFINITION 4.3.1. By the p.6eu.doc1.a.o.6.f.c.a1. n!La.gmen-t PC we shall understand the (O-absolute) formulas built up by 'V,"', 0 IJ x from the following formulas: 1)
o [~ ...
2)
o [e> ... 8.(~)
3)
O[~
4)
o [~ ... Ux I ;
U~ (x) •
5)
o [~ ...
Set ~ (x)
6}
D[~
7}
Ox = u
THEOREM
8.~ (x)
3 lj(q = x ) 1
...
for short.
EWe> (~).
& W(~) ]
X E lj]
X
Set(x) 1
E~
lj .
A~ (x).
... A x ]
x =
;
4.3.2.
FO!t ea.eh
one new nJtee vevUa.bR.e
~
(Fv II
lj
II .theJte.f..6 a p.6eUdoc..e.aM.f.c.a.l noltmtLla Fv II
=
I
U
{~},
~
IJ I (~) wUh
"f. Fv IJ) .6uc.h .that
IJ~ IJx [IJ~ --II'(~) 1
W(~)
(wheJte
X
exhaU.6t the nJtee vaJt.f.a.b.eu
0n
IJ
J. In paJt.t.f.c.u..ta!t,
PROOF. If IJ is atomic, we just take II' to be O(~ "'(1). i.e. IJ~. If IJisassigned II'(~}, 'VIJ is assigned 'V1I'(~};simi1arly lI ... cp is assigned II'(~} cp'(~}. These work because of Proposition 3.1.3, which says 0 (~~ II) V o (~ 'V IJ). Thus, we have 'V 0 (~ ~ IJ) -- 0 (~ .. 'VII): i}
it)
'V 0
(~
... IJ) .. 0
'V IJ)
(~
0 (~ ~ 'VII) ... (0 (~
Prop. 3.1.3.
,
IJ) ... 0
(~
... F)) ,
• ... 'VO(~-+II)},
"'(O('V~"'IJ}"'O'V~) -e-
('V O'V
... (0
~
... 'V 0 the last step since
W(~)
~
... 'V 0 (~
(~
.. II)} ,
... IJ) ,
... 0 ~.
But then
('V IJ)~ ....... 0 (~ "''VIJ) • ....... 'V 0
(~
... IJ) •
WILLIAM N. REINHARDT
282
'"
0 {3
'"
0 I ({3) ,
as desired.
Similarly,
,
("'O)'({3) , (0 (3 -+ <jJfJ)
(0 -+ <jJ)fJ
(0 -+ <jJ) fJ -+ 0 (fJ -+ (0 -+ <jJ)) ,
-+O[(fJ-+O)-+(fJ-+<jJ))
,
-+ 0 (fJ -+ 0) -+ 0 (fJ -+ <jJ) , -+(OfJ-+<jJfJ), "'(0 -+ <jJ)fJ-+ "'0 ({3 -+ (0 .... 4J)) ,
o ({3
-+ 0 1\ '" 4J) ,
-+ C({3 .... 0) 1\ 0 ({3 .... '" 4J) , ... 0 fJ 1\
4J {3 ,
-v
... '" (0 (3 ... 4J (3) • (0 ... 4J) {3
Thus
For
Ijx 0
o
(0 (3 -+ 4J (3) ,
0 ' ({3) -+ 4J' ({3) ,
(/1 -+ 4J)'({3) ,
as desired.
we take 0 Ij x [ &(3 x -+ 0' (13) ) :
(fJ -+ Ij xO) -+ 0 Vx (fJ -+ 0) , -+ 0 Ij x 0 (& x
(fJ -+ 0))
,
... 0 IjxO [fJ -+(&x -+0))
,
-e-
-+0 VxO [(fJ... &x)-+ (fJ -+8)), -+0 h[O(fJ -+&x) -+O(fJ -+0))
Now by the i nducti on,
0 (fJ ... 0)
0' (fJ),
... Oh [&(3(x)-+O'(fJ)]
Thus
(ljxO){3 -+ (ljxO)'(fJ).
1)
o
2)
o
3) 4)
5)
o o o o
Ij x Ij x 'Ii x 'Ii x 'Ii x Ij x
[ 0 ({3 -+
&
&
x)'" 0 (fJ ... 0))
x ]
[ fJ 0 (fJ [ fJ -+ 0 (fJ [{3 -+
x) ]
0) ]
(fJ -+ 0 ) ]
6) 7)
D'lixU-+O)
8)
o
[fJ -+
&
'lixO).
•
For the other direction,
[ 0 (fJ -+ & x) V 0 (fJ -+ '" [ fJ
so thi sis
,
ax ) ]
since W({3) , axiom, from 2,3, from 1,4,
o
p-+p,
tautologous from 6 , qed.
This proof essentially shows also the following lemma, which will be
useful
in
A THEORY OF PROPERTIES
283
the remainder of the proof of 4.3.2. O-ab-6o!u..te, then we have the
16 0 .iA
LEMMA 4.3.3.
0 'tI 'tI••• c!O-6u.Jr.e 06
0(/3 -- 'rIxO) - 0 'rIx[8./3x--8] In pa.Jr.t.lcu..ta.Jr., we have .the (McUna.Jr.lj
PROOF. 0(/3--0)-0 above proof gives
since 0
O!I.
'rI
is
J
c!Mu.Jr.e 06
O-absolute.
Thus the first part of the
0(/3-- 'tIxO) ->O'rlx[O(/3->8,x)--O]
,
and the second the reverse. In particular, 'rIxO- ('rIxO)O/ - 0 'tIx[8,O/x -> 0]. Finally, Fix
DO
goes to
0 'rI/3 [EWO//3 --0'(/3)]:
x, and using
P2 let
p be the proposition such that
O(p Thus
0 p -
08 ; ,
-v
0 -v 0
so
O(x» .
0 -v 0 (x) .
0", P -
o so
Now by WI ,
p -- 3 /3 [W 13 1\ 0 (J3 -- '" p) 1 ,
x--
3 /3 [W /3 1\ 0 (/3 ->
-v
0
x)] ,
x)] -- -v 0 '" 0 x (/3 -- -v '" 0 x) - 0 (/3 -- 0 x) ,
'rI /3 [W /3 -- '" 0 (/3 -- '" 0
But '" 0 (/3 -- -v 0 so this gives
x) -
0
'rI /3 [W /3.... 0 /3
Since
This proves thelemma.-
xl --
0 0 'X
.
0 p -> 0 (/3 -- p}, the reverse holds as well, and
DO;-'tI/3[W/3--0/3X] By the induction,
0/3_ 0'(/3), so we get
DO! Since both
'rI/3 [W/3 --0'(/3)]
W/3 and 0' (/3) are D-absolute, the lemma applies and we get 08 -
0 'tI /3 [8. 0/ /3 0 'tI 13 [8.
0/
-> (W 13 -- 0 1.(13 ) )] ,
/3 1\ W/3
->
8 ' (/3) ] •
WILLIAM N. REINHARDT
284
Since EW a (I)) = O{a .... 1'.(3 A W(3 } is equivalent to O{a for D-absolute ¢, 0 (a .... ¢) is equivalent to ¢,
and thus since De is equivalent to
as desired.
-e-
1'.(3 ) A O{o: .... W(3), and
{OIJ)(3,
This completes the proof. •
We remark that we could have strengthened the conclusion of 4.3.2 to read \/(3 0 V! [IJ(3..-. IJI {(3) 1 W(3 if we had adopted an axiom slightly stronger than WI, namely WI'
0
\/ p {O p .... 0[0: ....
Ps(p}
3(3{W(3 AD {(3 .... p»] } •
The proof is the same except for the case of DO, where WI' is used.
§
5.
5.1.
RELATIONS IN INTENSION. I NTRODUCTION.
A reasonable theory of intensional entities must take care of relations as well as properties. In the extensional case we are used to the reduction of everything to sets; extensional relations become sets of ordered pairs. We wish in a similar way to reduce intensional relations to properties. It is likely that the method used here is not entirely satisfactory, since it depends on treating ==:t as definable. Also it depends on the following special axiom W5
o \/ x
D \/ Y o 3 u 3 v(u. = x A v = y} ,
which allows for joint realization of arbitrary entities. This could perhaps be avoided by taking (x,y) as a primitive term, with axioms the closures of \/ xy {{x,y} = {{ x L, {X,If}}) {x,y}
=
(X',if')"-'
x = x'A Y = If'
(the first merely to have (X,If) be the usual thing when x,y exist). There is a possible philosophical objection to W5 , namely that some individuals (or other entities) may be incompatible. It is not clear how strong or well directed this objection is, for the following reasons. First, if it only means that certai n properties of individuals are incompatible, so that e.g. 0 3 xPx, o 3 xQx, but 103 x(Px A Qx}, then it is no objection at all. Second, if metaphisical incompatibility is intended, it appears misdirected since 0 is very broad logical possibility. Third, even if there is such a thing as logical incompatibility, there may be a viable distinction between logically possible worlds (3, (3' and mere numerically possible worlds r , which would be constituted e.g. as some kind of combination of f3, (3' for which it would be understood that existence in either (3 or (3' entails existence in r, but r would have no other structure except set for-
A THEORY OF PROPERTIES
285
mation. We now turn to a discussion of ordered pairs. 5.2. ORDERED PAIRS. Db(t,x,lf) = Ir..t /\ Set(t) /\ xEt /\ IfEt /\ 'ifw(wEt ... t= x V t:» If) OC(t,x,lf) = 8> t /\ Set t/\ 3u3 v [Db tu v /\ Db u x x /\ Setu Set v Db v Xlf].
DEFINITION 5.2.1.
PROPOSITION 5.2.2.
c£'O.6UJte
Db tXIf'" 0 (8)t ... Db tXIf).
06:
Db txy'" 0 (8)t ... &t /\ Set t /\ x E t: /\ If E t)
PROOF.
is clear from Set Axiom 3.
Also, clearly,
o 'if w(Db t x If /\ wEt ... w = x V v = If) , o 'if w 0 (II, w /\ Db t x Y /\ wEt ... w = x V w = If) , but
Db t x lj
...
Clearly, from and we get But
8> t /\ Set t , so Db t x If /\ wEt ... 8> w . ' . OllwO(Dbtxlf/\WEt ... W=xVw=y).
o (Db t x If /\ wEt ... w = x V o (w E t: /\ w * x /\ w * If) 0'" Db t x If •
0 II w(w E t /\ w
* x /\ w * If
0 (w E t: /\ w
o II w(w E t: /\ w * x /\ w * If o (3 w(w E t /\ w * x /\ w * If) 0 0 'dx 'd If 08.
e.to!.>Wte 06:
0(1], x A 11 If) /\ 0 'd x 'dIf 0 8 -+ 0 (11 x A 11 If A 'd x'd If 0 8) , -+008, -+ 08
But by W5,
0 (11 x A 11 If ) 0'dx'dlf 0 8-+ 0 8
i.e.
0'dxO'dIf[O'dx'dlf08-+08] 0'dX'dlf08 -+0 'dxO'dlf08.
-
Clearly the same proof shows: PROPOSITION
5.2.14.
0'dx00'dx1 ... 0'dxn08
C.c0!.>Wte06:
o PROPOSITION PROOF.
5.2.15.
'd
X
o 'd xl
+->
.•• 'd x nO 8 •
0 h03lfOc(t,x,lf) +-> Oh 3lfOc (t,x,If).
C.co!.>Wte 06:
From 1eft to ri ght is tri vi a 1.
For the other di recti on,
OC(tXIf)-+I1X,
o 3x 0 31f Oc (txlf) -+ 0 h 0 31f(Oc tXIf A I1x)
PROPOSITION
5.2.16.
-+ 0 h
030' 3x OC(txlf) ,
-+ 0 h
30' Oc(:t x If) •
C.co!.>Wte 06:
-
0 3:t(OC(:tX0')/\ 08)+-> 0 'd:t(OC(:tX0')-+ 08) .
PROOF.
In general, 03:t 1J A 0 'd:t(1J -+ 8) -+ 0
3:t(1J A 8) .
Thus since 03 :tOC(:t,X,0'), this gives right to left. left hand side but not right; then
o
3 t(Oc(U 0') A 08) ,
For left to right, assume
WILLIAM N. REINHARDT
288
o 3 ;(;(Oc(;(; x If) f\
De)
"v
I n genera1, if
c o
and
o ¢ f\ o ¢;(;' ... ;(; = t:", then o 3;(; ( ¢ f\ D () f\ 0 3;(;' (
E
v«
A{3
0 ({3 -> t
R) ,
E
-> 'tf{3 0 (Oc t x Y /\ t A{3
E (3
-> 'tf{3 0 3t (Oc t: x Y /\ A{3
t
E (3
R) ,
o Oc t x Y /\ R x Y .... 'tf{3 o 3t(Oc t x y /\ t A{3
E (3
R) ,
R) ,
->if{3 O(Oc txy/\ tE{3R), by Prop. 5.2.8 , A{3 .... if {3 ;(; E{3 R , A{3
PROPOSITION
C.tO.6Wte 06: Re1 R /\Re1 Q/\ 0 'tfxtj 0 (Rxtj QXtj) /\
5.3.5.
o (a R PROOF.
Re1 (R) /\
t
E R -> 0 3 x tj Oc
t
x tj
0 Oc ;(; x y -> 0 (t E R -> R x y)
and
0 Oc ;(; x y .... 0 (Q x if -> ;(; E Q) ,
Q x y) •
and by assumption,
0 if x y 0 (R x if
So from
0 3 x Y Oc t: x y we get o 3 x Y 0 (;(; E R .... ;(; E Q) o 0 (;(; E R ... ;(; E Q)
o
(t
E
R -> ;(;
E
Q) -> R = Q •
j
but
-e-
If,
Q)
Re 1 R /\ ;(; E R -> ;(; E Q • Thus Re1 R -> (t E R -> t E Q). if ;(; 0 ( 0 p ... 0 8) , we have
o
Since
Re1 R is
Re1 R -> 0 'tf t 0 (;(;
E
0 -absolute, and (Op ->0 ;(;08)->
R -> ;(;
E
Q)
The reverse is symmetrical, so Re1 R /\ Re1 Q -> 0 itt 0 (-t E R
t E Q) ,
.... R - Q • • PROPOSITION 5.3.6.0 if Ii ifQ 35 (5 xy Rel Q Rel 5
wheJl.e 8
=0
x,y
q, me.ano
0 if x ify 0 (8
8(Ii,x,y) /\ Qxy) ,
=0
XLj
q,) /\ 0 (a Yo /\ ••• /\ ay
wheJte y wu the 6Jtee vaJUab.te.6 06 8 otheJt ;(;han vaJUab.te.6 06 q, o;(;heJt than x, y • PROOF.
Suppose
l£, Q
given, and let
5 =
x, y, a.nd
t te
E
n
z
zO/\ .../\8,z) n wu the 6Jtee
8,
Q /\ 'tf{3 0 'tfxy(Oc ;(; x Y -> A{3
A THEORY OF PROPERTIES
II~
(;;:xy)) l .
Evidently, (Rel Q.... Rel Oc t x Lf ; we have
assume
o o
We must see that
S).
(t
E
Q on 0
aILe among
x O"'"
O"'"
x
... ,
so
xn _ 1
n_ 1)],
xn _ I'
The idea is to proceed as in the usual definition of satisfaction except that
WILLIAM N. REINHARDT
294
we assign to each formula 0 with free variables among x o' ... , x/'l_l an i ntensional /'I-relation rather than an extensional /'I-relation. (A similar treatment of satisfaction is suggested in Leeds 1979.) Perhaps the most natural way to proceed is to generalize slightly our notion of an n-relation, to allow F-relations, where F is a finite set of natural numbers. Then to 0 with free variables Iv.(..1- ((3) as asserting that the actual world is so compatible. Then we have PROPOSITION 7.1.3.
me
a)
0
b)
ome
+->
06 :
CiOf>W1.e6
if C< if{3 [c<
'tic
B =Of 'V( A & 'V B); A B =Of (A .... B) & (B .... A);
(x)A =Df (Ux) A; (Px)A =Df'V(x)'VA.
Where xl"'" xn are all the free variables of the formula A, (xl)'" (x n) A . is a (uI'liVeMal)cto-6uJz.e of A. A -6eVl.tence of LQ is a formula in which no variables occur free. The systems that are central in what follows - they are by no means the only important quantified relevant logics - are the affixing systems, the constant domain quantificational extensions of the affixing systems of Routley and Meyer 1978. The basic affixing system SQ has as postulates these schemes: AI.
A .... A
A2.
•
A3. A & B .... B A7.
(A .... C) & (B
A8. A & (B V C)
A9 . 'V'V A.... A.' QAl. (X) A .... A (t Ix),
A & B .... A •
M. (A .... B) & ( A.... C) ..... A .... (B & C) .
C) ..... (A VB) .... C (A & B) V C.
where t
is a term.
SEMANTICS OF QUANTIFIED RELEVANT LOGICS. I. QA2. (x)( A .... B) A .... (x) B, QA3. (x)(A V B) A V (x) B, QA5. (x)(A B) (Px) A .... B, R1. A, A B .. B R2. A, B .. A & B R3. A B, C .... D .. B .... C ..... A.... R4. A "'B .. B .... "'A QR1. A.. (x) A
307
where x is not free in A. where x is not free in A. where x is not free in B . (Modus Ponens). (Adjunction). D (Affixing). (Rule Contraposition). (Generalization).
Here AI' ... , AI'! .. B abbreviates: where AI"'" AI'! are theorems so is B. BQ may be reaxiomatised, along the lines of the axiomatisation of RQ in Meyer, Dunn and Leblanc 1974, to eliminate QR1 as a primitive rule: the procedure is to take the closure of all axioms as axioms and to compensate for theorems lost thereby adding new axioms, such as (x) A & (x) B .... (x)( A & B). But as a primi ti ve rule of BQ, QR1 is no worse than R3, for example, which likewise is "abnormal" in having no direct theorem scheme analogue. Nor need it have: it is essentially a device for generating theorems from theorems, not itself a theorem analogue. Extensions of BQ which have full contraposition 04. A .... "'B ..... B .... "'A as a theorem scheme can eschew axiom-schemes A7 and QA5 which then follow respectively from A4 and QA2. Additional axiom schemes and rules drawn from the list in Routley and Meyer 1978 may be added to BQ singly or in combination to yield a wealth of stronger systems with the same (constant domain) quantificational structure. Here a fairly short list of optional extras will keep us quite busy enough. The list will however be at least large enough to include such systems as RQ, EQ, TQ and S4Q. B1. A & (A
B) .... B •
B4. A B6. A
C .... A ..... C .... B
B
(A
B ) .... B •
01. A V -v A •
04. A .... -v B
.B
-v
A
BR1. A .. (A
B)
B .
B3. A B5. A BID. A 03. (A 05. A &
B
B
(A
B)
B
B.
"'A) ....
C
A .... C
A 'V
B
A.
('" A VB) .... B
Some of the more or less familiar systems which result from BQ in this way are these: -. GQ: BQ + 01. TQ: BQ + {B3, B4, B5, 03, D4}. The addition of B3, B4 and 04 means of course that rules R3 and R4 can be derived. EQ: TQ + BR1. RQ: TQ + B6. S4Q: EQ + BID. More compact axiomatisations, of RQ in particular, are presented subsequently. An LQ model ~.tw.c:twr.e (LQ m.~. J is a structure S = (K; 0 > where K is an L m.s. and 0 is a non-null set of objects (cf. Routley and Meyer 1973, p. 238). In particular, a BQ n.s , S,from which other LQ m.s. are derived, is a structure S = (T, 0, K, R, *,0> where 0 is a subset of K, TEO, R is a 3-place relation on K, * is a I-place operation on K, and V is a non-null set, constrained for every a, b, c, dE K, by the following conditions, in which a" b Of (Px)(Ox & Rxab):-
RICHARO ROUTLEY
308
pI. p2. p3. p4.
a
• B ~ 'V A .
Ala. A::> B ::> .
'V
(B & C ) ::> .
All. A12.
(x) A ~A (t Ix), for any term t. (x) (A~ B) :J. A ~ (x)B, where x
is not free in
A.
A13.
(x) (A VB)
is not free in
A.
~.
A V (x) B , where
x
'V
(c & A) .
Rules: Rl. A, A :J B ~ B R2. A~ (x)A
(Material Detachment). (Generalisation).
The axiomatisation of FDQ given is a simple extension, to include basic entailment principles, of classical quantificational logic Q. The semantical analysis of FDQ, the adequacy of which is established in Routley 1978c, is likewise a straightforward worlds elaboration of classical quantificational semantics. An FDQ-mode£. M is a structure M= (T, K, *, V, 1) where K is a set of worlds, T E K, * is an operation on K such that T* T and a** = a for every a E K, V is a non-null set of items or objects, and 1 is a valuation function (in the
SEMANTICS OF QUANTIFIED RELEVANT LOGICS. I.
313
FOQ modU1>:tJtuc..tWLe (T, K, *, V) ) which assigns to each term of FOQ an element of V, to each n-place predicate at each world a of K an n-place relation on VI"', and to each sentential parameter at each a E K exactly one of the holding values {I, a}. That is, an interpretation is an unconstrained LQ interpretation. I is extended to all wff of ply for every a E K: I(6
n(t
1,
••• , t
n},
a)
FOQ by the following evaluation rules, which ap1 iff
I(6
(On an alternative, truth-valued, semantics ed 1 or a by the model without ~nalysis of cate components.) I(A&B,a) I('VA,a)
1 iff
= 1 iff
I(A,a)=1
n,
a) (I(t
1),
••• , 1(t
n)).
I(6n(t ... , tn)' a) is again assign1, n 6 (t 1 , ... , t n ) into subject and predi-
I( B, a).
I(A,a*)=F1.
iff I'(A,a) = 1 for every x-variant I' of I . 1 iff for every b E K, if I( A, b ) = 1 then I (B , b )
I«x)A,a)- 1
I(A .... B,T)
1.
Derived evaluation rules include the following: I(A V B, a)
I (A :J B, a)
I«Px)A, a)
iff I(A, a) = 1 or I(B,a) - I'; iff I ( A, a*) =F 1 or I ( B , a) = 1; - 1 iff I'(A, a) = 1, for some x-variant I' of I. 1
1
Semantical notions such as truth and validity are defined just as for LQ. As shown in Routley 1978 c, the theorems of FOQ are precisely the FOQ valid wff, and corollaries such as the Skolem-L6wenheim theorem for FOQ follow from the completeness argument. The first degree restriction of FOQ is incorporated in the semantics through the interpretation rule for .... , which assigns values to implication formulae only at T. To move to the higher-degree this limitation to T has to be removed , something that may be done in a variety of ways, e.g. replacing T by a in the rule would yield a kind of rigid S5, higher degree irrelevant, logic. ~ore interesting is the minimal adjustment - yielding at the sentential level thenonreplacement systerrs of Routley and Lopar lc 1978 and Routley and Meyer 1978 - which assigns values from {I, a} arbitrarily to implicational wff at worlds other than T .Th is simple step furnishes a semantical analysis for the system FOQt, which differs from FOQ only in the removal of the formation restriction of FOQ to fi rst degree wff (2). THEOREM 2. WheJLe A .w a w66 06 1>lf1>:tem FOQt [.-t.e. 06 :the IUglteJL degil.ee J.>lj1>:tem auomat-Wed exac;Ulj Uize FDQ excep:t:that w66 Me no:t c.onMned :to :the 6fu:t
degil.ee), A
.w
a :theMem 06 FOQ4- -£66 A -if.>
FOQt-vaLid.
(2) Observe however that FOQt is not simply a substitutional extension of FOQ. The effect of Material Detachment has also to be considered. Thus, for example, «p .... p) .... q) :J q is a theorem of FOQt (and valid) but does not result by substitution from a theorem of FOQ (this example is due to J. Slaney).
•
314
RICHARD ROUTLEY
PROOF
is almost the same as that given in Routley 1978 c for FDQ.
•
In the canonical model used for completeness r( A B, a) is assigned value 1, for each world a distinct from T, iff A -> B Ea. There is good reason for dissatisfaction with FDO t - apart from the weakness of its higher degree logic, and the failure in particular of the rule of Replacement of Equivalence (i.e. coimplications) - namely the excessively classical character of FDQt, most conspicuously its inclusion of the rule 1 of Material Detachment as a primitive rule, and its consequent inadequacy for dialectical purposes. The first stage in removing classical assumptions involves reaxiomatising FDQ t without use of 1 (i.e. R1) as a primitive rule, and correspondingly adjusting the semantics so that the classical assumption T = T* is removed. The semantical adjustment is the simpler. A partial ordering (or inclusion) relation";; is added to FDOt models, and the requirement T = T* replaced by the requirement T* ..;; T, which while ensuring completeness of T of one sort (that symbolised in the law of excluded middle A V "v A) does not preclude inconsistency of T. The semantical adjustment also turns out to have the advantage that it enables various sublogics of FDO t of interest to be semantically encompassed. Specifically, then, to proceed downwards in the direction of weakness and generality, one moves first to a model ~:twctwte. lt1 = (T, K, ..;;, *, V) with T E K, * an (so far unrestricted) operation on K, ..;; an order relation on K such that if a";; b then b*";; a*, and V a non null set. The way down leads all the way down to a system P+Q , a quantified version of the minimal positive system p+ (of Arruda-da-Costa: see Routley and Loparic 1978), P+Q, which is a minimal relevant logic with respect to the modelling and the style of completeness proof, has the following postulates:-lo
PI. A -> A . P2. A , A B => B. P4. A & B -> A . P3. A -> B, B -+ C => A C . P5. A & B -> B . P6. A, B => A & B. P8. A -> A VB. P7. A -> B, A -> C=> A -> (B & C) . P9. B -> A VB. P10. A -> C, B -> C => A V B -> C P1l. A & (B V C) (A & B) V C. QPl. (x) A -> A (t Ix) . OP2. A->B=>A-+ (x)B, x not free in A. OP3 .. (x)( AV B)->. A V (x) B, x not free in A. QP4. A .. (x) A . QP5. A (t Ix) -> (Px) A • OP6. B -> A=> (Px)B -> A, x not free in A. QP7. (Px)(A & B) ->. A & (Px) B, x not free in A. -lo
-lo
-lo
•
Furnishing good semantics for P+Q, unlike the stronger affixing systems, is not a difficult feat (and is carried out in Routley and Loparic 197+). However the present investigation concerns not the way down from way up, the way to relevant affixing systems.
FDQt, but the
§4. STRENGTHENING THE HIGHER DEGREE; AND TRIVIAL AND LESS TRIVIAL SEMANTICS FOR SUCH SYSTEMS.
SEMANTICS OF QUANTIFIED RELEVANT LOGICS. I.
315
The trivial semantics (3) for extensions of such systems as FOQt simply stipulate semantical postulates for each scheme of the extension beyond schemes of FOQ. e.g. if extension includes the axiom A .... B ..... B .... e ..... A .... e. the semantics has the postulate that. for every ct. if I( A.... B. ct) = 1 then I( B .... e ..... A .... C • ct) = 1. The method, which includes I in the modelling, foregoes recursive specification of I from atomic beginnings, and instead specifies I as a function on wff and worlds which takes values in {I.O} , and which is subject to a set of conditions. namely all those characterising I in the case of FOQt and also further conditions for schemes of the extension. The method may be usefully illustrated by the trivial semantics for RQ. RQ gets selected throughout for illustrative purposes not because there is any special interpretational virtue about it - it fails badly for all the important notions relevant logics aim to explicate such as conditionality. implication. lawlike connection. entailment. propositional inclusion - but because in lands of deviant logics it's moderately well known as logics go. and also because it's technically exasperating. It's so close to classical to be good for practically nought but enthymematic purposes. yet though only one (albeit large and irrelevant) step removed as it were from classical. nothing much classical appears to work either at all or at all well. A tJu:v;.a..e modet M for RQ is an FOQt model, i.e. M= .£oVLl> on FOQ eOVLl>.£deJled, .£nRQ (ctI1d noft ma.ny othe»: h-tgheJl deg!tee exteVLl>.£oVLl> LQ 06 FOQl, A .fA a theO!lem 06 LQ .£6n A .fA tJu:v.£a.U.y LQ va.t.£d.
c1.u.d.£l1g
PROOF enlarges on the proof of Theorem 2. Consider an axiom scheme of the form C .... D where C or D is an implicational wff. The axiom is reflected semantically by an interpretational requirement if I(e. a) = 1 then I(D. a) = 1. So it is immediate by the evaluation rule for .... at T that the axiom is valid. For completeness set I( A. b) = 1 iff A E b. and use closure and the fact that T is regular. Observe that if e E ct. then DE a. in virtue of theory closure under provable implication; hence I( e, a) = 1 implies I( D. a) = 1. • In the case of relevant logics such as
RQ
there is one considerable further
(3) There are more trivial semantics. (4) This scheme, U distribution, is the only quantificational postulate of higher degree character that occurs in the main axiomatisations considered.
RICHARD ROUTLEY
316
difficulty, that of showing that the logic does extend FDQt . The main difficulty here is that of showing that the rule (~) of Material Detachment, i.e. R1, is an admissible rule. But this can be proved, for all the common relevant systems which include A V 'VA as a theorem, by the method of metavaluations (for main deta i 1s see Meyer 197+). The "trivial" semantics are more trivial than the relational semantics introduced because, at least in the case of higher degree implicational wff, the trivial semantics place no great distance between the axiom schemes and the corresponding semantical conditions on the valuation function. Still the "trivial" semantics (Which have been presented as giving a satisfactory semantica1 analysis of certain relevant logics by Hunter 197+) have merit in that they enable various results to be proved, including one important result needed subsequently. Let the classical negation enlargement LQ' of LQ be formed by adding to LQ a negation symbol, - , subject to the following schemes:C1.
A .... A .
A (trivial) rule: 1(A, a) = 1
CR1.
A & B .... C=>. A & C .... B.
LQI model is an LQ model which conforms to the classical iff
1(A, a) '4= 1,
for every
a
negati on
in K.
THEOREM 4.WhC/te LQI .L6 ;the c-f.a.6-6.f.c.a.lnegation enR.aJtgemen;t 06 ctI1lf 06 ;the ertelU-LoIU 06 FDQ c.olU-LdC/ted, A .L6 a ;theO!lem 06 LQ' -L66 A .L6 :tJUv. A: , using eRl , by R+ principles and Rule Contraposition .• A .... A: A
A .... B -> . A VB. PROOF
•
that T15 is interderivable with T16 is as in Routley and Meyer 1978,
(i.e.C2).
A&(A->B)~.B
PROOF. Apply T16 to T14. •
T18.
(i.e.MR3). PROOF.
(A .... B) & (C ....
D)~.B
.... C ..... A .... D.
Apply T16 to (A.... B) &(C .... D) ..... B .... C ..... A .... D • a theorem of R+ .•
SEMANTICS OF QUANTIFIED RELEVANT LOGICS. I.
321
To cope semantically with CBQ (= CGQ), it simplifies matter to extend the system (conservativelY as it turns out) by the sentential constant t: , subject as always to the two-way rule Rt.
(t rule).
A -- t ... A
The resulting system is called CBQ s:
.
§6. CONSTANT DOMAIN MODELLINGS FOR THE CLQ SYSTEMS. Finding a modelling for CRQ is simply a matter of building on the semantics for CR of Meyer and Routl ey 1973 (showing the adequacy of the mode 11 ing is another matter). A CRQ mode£. 1.>:tItuc:tWtc (CRQ m.s.) adds to a CR r.m.s. (as previously explained in Meyer and Routley 1973) a non-null domain V of objects. Precisely, a CRQ m.s, S is a structure S = (T, K, R, V) ,where K and V are non-null sets, T E K, R is a three-place relation on K, subject generally to these requi re ments: q1. q2. q3.
iff a = b ;
RTab
Raaa;
R2aebd, where Rabed =Df (Px) (Rabx & Rxed)
if R2abed then
A CRQ model adds to a CRQ m.s. an interpretation, or valuation function, I, which is defined as for FDQ systems. Specifically a CRQ mode£. ftf is a structure }If = (T, K, R, V, I), where the substructure (T, K, R, V) is a CRQ m.s., and I is a function which assigns to each subject term t (i.e. subject variable or constant) an element, I( t) , of V, to each ft-place predicate parameter at each world a of K, and n-p l ace relation on K (extensionally, a subset of Kft) , and to each sententi a1 parameter at each a inK exactly one of the values {l, 0 } . Interpretation I is extended generally to all wff as follows:
IC6(t 1,.··, tft)' a)
=
1 iff
pretations of terms t 1"'" icate 6ft at world a;
O(t I ) , ... , I(t
ft)}
t
nt, a),
i.e.
iff the inter-
t ft instantiate the relation assigned to a-pl ace pred-
I ( A & B, a) = I
iff I ( A I a) = I = I ( B I a) ; iff r ( A , a ) oF 1 I{A ... B, a) = 1 iff, for every band c in K, if then materflllly I( B, c) = 1 ; I ( A, a) = 1
iff I X ( A I a) is an x-variant of
I ((Ux) A, a) = 1
before IX ments to x ,
I
Rabe and
I(A,b)
for every x-variant IX of I I where as iff IX differs from I at most in assign-
To model other CLQ systems, i.e. classical "relevant" systems wfth the I.>a.me. 1.>:tItuc:tWte., it is largely a matter of varying the modelling conditions. For systems which admit reduced modellings (in the sense of Routley and Meyer I978)a basic system is CAQ for which only the one semantical postulate qI is required. That is, a CAQ model ftf is a structure subject just to the requirement qI, or equivalently to: qua.~6~e~onal
q1a.
if
RTab
then
a
=b
and qlb.
RTa.a..
RICHARD ROUTLEY
322
In each case the interpretation rules are extended precisely as for CRQ. Where, however, the CLQ system (strictly CLQt system) includes t both the model structure and the interpretation rules have to be enlarged. A CLQt model. M is a structure M= (T, 0, K, R, V) which differs from a CLQ model primarily in containing a set 0 such that TEO C K. The ru 1e for evaluating t is again simply iff
1(t,a)=1
Oa.
For many systems the semantical postulates have then to be adjusted. CBQt, on which other systems are built, are as follows: qla'. qlb'.
For x E For some
if Rxab then a x EO, Rxaa.
(1
Those
for
= b .
Awff A is.twein M just incase I(A,T)=I,and6aUein M otherwise. A is CLQ-valid iff A is true in all CLQ-models, and invalid otherwise. A set S of wff is CLQ ~~uttaneouoty ~~6iabte iff for some CLQ-model M, every wff A in S is true in M. Truth-valued semantics for CLQ systems are again simpler. A CLQ TV m.~. is simply a CL m.s.; and a TV valuation in such m.s. is a function which assigns to each atomic wff at each a of K an element of 11. The extension of 1 for wff compounded by connectives is as before, but the extension to quantified wff becomes: 1((x) A,a) = 1 iff
I(A(t/x),a) = 1 for every term r .
TV truth, validity, and so on, are defined, in terms of TV valuations, as above for truth, etc.
§7. ADEQUACY OF THE SEMANTICS FOR 'WEAKER CLQ SYSTEMS. Soundness is straightforward and succeeds not only for every system considered but for a great range of additional systems.
aUo
THEOREM 5 (CLQ and CLQ t: Soundnu~). CLQ-TV-valid. S~CV11.y 601t CLQ t .
EveJty theOltem 06 CLQ
J.J.,
CLQ-valid, and
PROOF is, for the most part, straightforward case by case verfication, showing that the axioms are valid and that the rules preserve validity. Some strategic examples serve to illustrate the method. ad A2. Suppose 1((A & B) ... A, T) 1. Then for some a, b in K, RTab and 1 (A & B, a) = 1 1 ( A, b). By ql, a = b, so 1 ( A , a ) 1, but 1 ( A, b) = 1 which is impossible.
"*
"*
More generally whenever 1(A... B, T) "* 1, for some a, RTaa and 1(A, a) = 1"* Thereafter the procedure can follow the details of Routley and Meyer 1973. Thus, for example, the R+ axioms schemes can be verified as in Routley and Meyer 1973 or 1978. I(B, a).
adR1.
Suppose
I(A,T)=I=I(A ... B,T).
Since
RTTT,byqlb,
I(B,T)=1.
SEMANTICS OF QUANTIFIED RELEVANT LOGICS. I.
323
ad R3. Suppose I(R3, T) =1= 1. Then I(A ... B, T) = 1 = I(C'" D, T) =1= I(B .... C ..... A .... D, T). Hence for some a, RTaa, r (B .... C, a) = 1 =1= r (A .... D, a). Thus for some b , c , Rabe and I (A, b ) = 1 =1= I ( D , e). As Rabe and I (B .... C, a) = 1, either r(B, b) =1= 1 or I(C, e) = 1. Since I(A ... B, T) = 1, rCA, b) = I and, by qlb, RTbb, I(B, b) = 1. Hence I(C,e) = 1. Similarly then, as rrc ... D, T) = 1, I( D, e) = 1, which is impossible. ad CRl. Suppose that in some model, I (A & C .... Jr, T) =1= 1. Then for some a, RTaa and I ( A , a) = I = I ( C , a) =1= I ( Jr, a ) • Thus I ( C , a) =1= 1 = I ( B , a) • Hence as RTaa, riA & B .... C, T) =1= 1. In sum, if A & c .... 1f is not CRQ valid neither is
A & B .... C.
Verification of the quantificational postulates is like that given in Routl ey and Meyer 1973 and Routl ey 1978a ). For the weaker systems containing t
§
2 (and in
there are some complications.
ad A=>t"'A. Suppose t ...A is not valid: then in some model, r(t .... A,T)=I= 1. Then for some a, RTaa and n.e, a) = I*" rCA, a). Then for some a, 0 a and T(.A, a) *" 1. Form a new model with base a. in place of T. This is permissible since Oa, given that all semantical postulates are stated, not in terms of T, but generally for x in O. Then A is not valid. ad t .... A => A.
The rule is tantamount to Axiom z •
For it yields t since t .... t; and t yields the rule by Modus Ponens. And t is val id in virtue of OT. Also verification of the rules is a little more complex. ad R1. Suppose r(A I T) = I but r( B, T) = I for some model. Then, since by q1b' for some x in 0, RxIT, T(A ... B, x) =1= 1. Form a new model M I on base x , Since it is indeed a model A .... B is not val id. ad R3. Suppose B .... C ..... A ... D is not valid, i.e. for some model I(B ... C ..... A .... D, T) *" 1. Then as in the previous case for R3, r( A, b) = 1=1= I( D, e) and either r (B , b) =1= I or !( C , e) = 1. Si nce for some x in 0, Rxbb and A B is valid, !( A, b) I materially implies I( B, b) 1. Similarly, as C D is valid, T(C, e) = I materially implies I(D, e) = I; and contradiction results. • Proofs of completeness are somewhat more arduous, and require many prel iminaries. (For indications of the origins of these preliminaries see Routley 1978a.) Since the same notions will recur in completeness proofs for a range of quantified implication systems LQ, the preliminaries are, as in Routley 1978a, stated more generally than required simply for CLQ systems. The definitions are intended to apply both to LQ - a representative relevant system without perhaps a classical negation - and to linguistic extensions of LQ - also designated on occasion by LQ, though maintaining distinctions here is of critical importance in avoiding fallacious argument - obtained by adding further (at most denumerably many more) variables or constants to LQ (and accordingly inflating the supply of wff and logical axioms). An LQ-thcony T is any set of wff of LQ which is closed under adjunction and provable LQ-implication, i.e. foranywff A,B if AET and BET then A & BET, and if A E T and I- LQ A .... B then BET. More generally, a the.ony (linguistically construed) is a set of wff closed under certain operations. An LQtheory T is neg ulan iff all theorems of LQ are in T; a theory is pn£me (Vcomplete) iff whenever AVBET either AET or BET; 4ieh (U-complete) iff, whenever
RICHARD ROUTLEY
324
A(t/x) E T
for every subject term t of LQ, (x)A ET;-6atwr.a.:t.:ed (P-complete) iff whenever (Px) A E T, A (t/x) E T for some term t of LQ. A theory T (for LQ) is quant{6~~-eomplete iff both rich and saturated; -6~ght iff prime and quantifier-complete; and adequate iff straight and regular. T is non-degen~e (n. d.) iff T is neither null nor universal (i .e. contains every wff). For systems with a classical negation - , such as CRQ, the canonical model is, in one way or another, built out of n.d. straight theories. The way depends on whether the modelling is reduced or not. Consider the unreduced case first. Let
KLQ_be the class of
LQ-theories, and K LQ
the class of n.d.
s tra i ght
theories; 0LQ the class of regula~ LQ-~heories, and 0 LQ the class of n.d. adequate LQ-theori es. For a, b, c E :LQ' RLQ abe iff whenever A ..... B E a and AE b, BEe. RLQ is the restriction of RLQ to n.d. straight LQ-theories. V is the LQ class of terms of LQ; thus VLQ is denumerable. Where T
CLQ
is any adequate
on TCLQ is the structure eal ~ntenpnetat{on I in I(p, a) I (t, a)
n6, a)
for each n.
Sc Sc
=
CLQ-theory, the (unnedueed)
°
eano~eal
. It is:SUBLEMMA. 16 I-
CLQ A::> B and A E T then BET.
leads to Suffixing C -+. A -+ using R3
RICHARD ROUTLEY
330
PROOF. A
T
AE T
Suppose
and
Then as T
I- A J B.
is regular A
BET, so
B) E T. Hence, as T is closed under ~ , by T3, BET. Otherwise, since is CLQ regular, all results for CLQ and CLQ theories extend to T and T theorie~
ad U.
For every
1«Ux)A, a) = 1
a in
c'
K
iff
IX(A, a) = 1
iff
r (A (t /x), a) = 1
iff A(t/x) E a
for every
iff (Ux) A
cation.
E
I (t) E V ,
for every t
c
of CLQ', by applying
=1
iff
a, since a is rich and closed under CLQ' impli-
iff
t
=1
I(A(t/x),a)
iff A(t Ix) E a ness,
the
I(t) = t ,
For the truth-valued semantics the matter is still simpler. 1«x)A,a)
r,
IX of
x-variant
for every term
induction hypothesis and the equation
ad
J
& (XV
(x)AE a •
a E Kc '
For
for every subject term t;
for every t , For if
A(t Ix) E a
for every
t then, by rich-
(x)AEa; and the converse follows by instantiation and CLQ'-closure.
l(t, a)
(where applicable).
Suppose, fi rs t , suppose
0CLQ a.
Then
a
=1
iff
0CLQ a
iff
tEa
is regul ar, but I- CLQ-t, so tEa.
tEa. Then since for every theorem A, I- CLQ :t
~
ad (~). It is at this stage that results thus far achieved become more system dependent. ad ql.
There are two cases:-
ad qlb.
Reduced case only.
theories in the model to case. ad qla. whence
Suppose
R Tab. c
a C b follows.
Then
RcTaa
T-theories. If further But then
a
conspicuously
holds in virtue of the restriction
Such a condition fails in the A E a,
=b
Conversely,
A , A E a, so 0CLQ a .
then, since
of
unreduced
A ~ A E T,
A E b
r
by the following:
SUBLEMMA. WheJLe a and b Me n.d. l.>:tJuUght (CLQ-) theolUe!.> (in 6act c.onl.>iJ.>tenc.y and - - c.omple:tene!.>,f, Me enough),.i..6 a c b then a = b, Le. :the theo1Ue!.> Me maximal .i..n anotheJL .i..mpoJr-ta.n;t -(A&B). For the present l e t V be defined AVB=Df -(- A &-B): In fact A VB+---> A 'it B. Now A & A -e- B and A & B -+ B , so A & AVA&B-+B, whence distributing A& (AVB)-+B, i.e. A&-(-A&-B)-+B. Relettering ==-A& -(A & B) -+ ,- B, so by CRI, ~ &::B-+- (A & B). ad 02, i.e.
-(A V B) -+. -A V =-E.
Similar.
ad D3. By Double Negation for each negation. ad RDl. Suppose A -+ B; to show -A -+
-:::-n.
Apply Rule Contraposition twice ••
Proof that ,t may be eliminated, i.e. that it conservatively extends systems to which it has been added, may take either syntactical or semantical form. Here a semantical version using the trivial semantics is given, subsequently in the case of stronger systems a syntactical version is outlined. THEOREM 9.
(,t removal).
LQ,t.fA a c.oft6etLvative eueft6'£oll 06 LQ and CLQ,t
SEMANTICS OF QUANTIFIED RELEVANT LOGICS. I.
335
06 CLQ.
PROOF. Since LQ ~ LQt one half is tnmedfate , For the converse suppose A is a wff of LQ but not a theorem of LQ: to show A is not a theorem of LQt, it suffices to furnish a countermodel to A in an adequate semantics for LQt. This can be done in a straightfo~lard reworking of the trivial semantics inwhich world T is replaced by set of worlds 0 and truth is determined at an arbitrary element of 0 (or alternatively validity in a model is characterised in terms of holding at all elements of 0). Theorem 3 goes through as before, and it has a corol1aryan exactly analogous result for LQt. where t . subj ect to the t rul e, is evaluated by the semantical rule: I (t. a) = 1 iff Oa. Now, as A is not a theorem of LQ, there is by the reworking of Theorem 3 and LQ countermodel, built on 0, to A. But this countermodel is in fact an LQt countermodel to A; hence by the corollary to Theorem 3, A is not a theorem of LQt. The argument for classical systems is similar .• COROLLARY (Completeness for CBQ and weaker extens ions). WheJtc CLQ .u., a wcakeJt cx.tcl'l4-Lon 06 CBQ, -L6 A .if, CLQ - valid thcn A .if, a thcOJteJn 06 CLQ. PROOF. Suppose A, which is a wff of CLQ, is not a theorem of CLQ. Then, by Theorem 9 it is not a theorem of CLQt • Hence, by Theorem 6, it is not CLQtvalid. But A is a wff of CLQ, so, as its semantical assessment does not involve t: , A is not CLQ-valid. (Strictly the last move involves a step like that used in Theorem 9, namely:- Reformulate CLQ semantics with 0 in place of T, and show that the semantics is equivalent in that the valid wff are the same. Then A is not CLQ-villid in the'O based semantics, so it is not CLQ valid in the T based semantics.) THEOREM 10. (Adequacy theorem for weak relevant logics). A .u., a thcotteJn 06 LQ -Li6 A .u., lQ-valid, wheJtc LQ.u., anLj weakeJt cx.tcI'l4-l011 06 BQ ott BQT. PROOF. Soundness is by the usual induction. earl i er results. Suppose - I- LQ A .
Proof of completeness assembles
Then A is not a theorem of LQI, by the coro 11ary to Theorem 4. Hence, by Theorem 8, A is not a theorem of CLQ*; and so A is not CLQ*-valid; that is, there is a valuation I onCLQ*m.s. M=(T,O,K,R,*, V) for which I(A,T)*'l. But 1) the m.s. flf is an LQ m.s. and 2) the valuation I is an LQ interpretation in M; hence A is not LQ valid. It remains to prove 1) and 2). ad (2). Apply the reduction: a < b iff a = b. Trivially, if a = b then a of type such that (A i 1\, V, 0,1) is a bounded distributive lattice satisfying the identities 'V'V x = x and "'(x V y) = 'VX 1\ -v If. (We refer to -v as the De Morgan negation.)t For a systematic study of these algebras see Ba 1 bes and Dwinger 1974 and Rasiowa 1974. Throughout this paper A denotes an arbitrary De Morgan algebra. For the notation we refer the reader to Balbes and Dwinger 1974 and Gratzer 1971. We let O(a,b) and 0Lat(a,b) denote respectively the principal De Morgan algebra congruence and the principal lattice congruence on A generated by the pair (a,b) in AxA. BIAl denotes the set of complemented elements of A, while C(A) consi sts of those elements a of A such tha t the Boo lean complement a' in A coincides with the De Morgan negation 'Va. It is clear that C{A) c B(A); and the equality holds iff A is a Boolean algebra. Finally we let Con-A to denote the congruence lattice of A and recall that it is a distributive algebraic lattice.
(2,2,1,0,0>
§
PRINCIPAL CONGRUENCES,
2.
In this section we shall give a characterization of principal congruences of a De Morgan algebra. The following Lemma - whose proof is straightforward - is crucial for the rest of the paper. LEMMA
2.1.
LeX:
a, b
°
E
A wUh
0..{mp.te • .{.6 ./>ubcUtteclilf .{.Meduub.te • .{.6 06 hught .{.6 utheJt 2
isa(a-complete)DBAiff (A,u,n,O) is a (c-complete) GBA.Also~,if<we have'a(a-complete) GBA (A,u,n, a> and define for IL, b, eEA, a o b = c iff IL U b - c and IL n b = 0, the new algebra (A, 0,0) is a (a-complete) DBA. For our proofs we use other algebraic systems: Ca4~na..t atgeb!I.iL6 (CA) andgenThey are systems (A, +, ~), where + is a binary operation and ~ an operation of countable rank. They are defined in 1.1 and 5.1 of Tarski 1949. eJuLUzed c.a4~nat atgeb!I.iL6 (GCA).
Let at: = (A, 0, 0) be a a-complete DBA. We define, as in 13.2 of Tarski 1949, the operation ~'- b means a ~ band b l a. We say that a is comparable with b iff a ~ b or b;(; a • if every bounded strictly increasing
THEOREM 1.7. Let a be a dam.i.l1ate.d -6 a.t..t-6 6Y the: 6oliowblg eondU--Lo 11-6 :
PODBA,
(XI
= (A, 0,0).
Let OL
M.G. SCHWARZE AND R. CHUAQUI
356
(a)
o< d
-
a, then theJl~ iA a dEA .6uch that
(b) A# .6a:t.iAMe6 the ccc, Then ot' l.: iA a GCA .6a:t.iA6y1ng:
(+) Folt eveJtlj a, b E A,
PROOF. Since nitely refining in then use 1. 6.
A# A#
b- a. By (a), there is an e such that c < c o e < d and a is comparable with c o e, But c o c ~ a contradicts the minimality of d, and a ~ co e contradicts the maxima 1ity of c. Thus ctd « b. • PROOF OF THE THEOREM. Using (xii) we immediately obtain from the that for every aEA there is a bEM such that a-b.
Lemma
We now prove that _ is finitely refining in A#. Let a, b, c E A# wi t h a 0 b _ c. Since c ~ a, there is an a' E A# such that a'';; c and a - a'. We have a'ob'=c for some b'. Thus. a'ob'-aob. Therefore. by 1.5 (6), b' _ b. Since we have shown that - is finitely refining in the ideal satisfied, we apply 1.6 and obtain that at /_ is a GCA.
A#
and (b) is
We now proceed to prove (+). Suppose b ~ a. Then by 1.8 there is a b ' E A# such that b _ b'. Thus b' ~ a. Again by 1.8, there is an a' such that a' ';;b' and a' - a. Therefore a/- = a'/- 0;;; b'/- = b/- . that
On the other hand, suppose a/- 0;;; b/-. Then there are a', b' E A ct' - a. b' - b and a'';; b: By 1.5 (3) b ~ a •• 2.-
REPRESENTATION THEOREMS.
such
a-ADDITIVE MEASUREMENT STRUCTURES
DEFINITION (i)
4>
2.1.
or-
ttj ILepILuent6
E
Let
iff:
= (A,
C/("
0,
0,;(:) be a PODBA.
357 We say that 4> .6bwng-
AN ,
(ii) (iii)
if a, if a,
bE
A and a.ob E A, then ep(aob) = 4>la) + ep(b) A, then a;(: b iff ep(b)'; 4>(a) ,
(tv)
if x
E wA
bE
and u£<w xi
E
A, then
¢(u..[<w
x) =
I
.tUbi<w 4> (x-i) •
We say that 0(, is l>btongttj ILepILuent:ab.te iff there is a ¢ which strongly represents oz.. • In order to obtain our first representation theorem we need a strengthening of 4.3 of Chuaqui 1979 b. THEOREM 2.2.
Let
TheILe il> an (-i) (U)
a
to be
and .6unMcient nOlL 0(,
be a CA. Then.the noUow..Lng con~on il> il>omolLph-ic t» a .6uba.tgeb!l.a. On 'X, :
neceUaJttj
a. E A .6 uch :tha.t
nOlL eVelLtj
b E A,
nOlL eveJttj b E A,
b.;
co
a ;
0"* b .;;; a., :theJte il> an
n
<W
.6uch.tha:t a .;;; n
b•
PROOF. The necessity of the condition is clear from Tarski 1949, 14.6. Suppose, now, that ot: satisfies the condition. We shall prove first, that 0(, is cardinally simple. Suppose b"* O. Then, by (i), b.;;; a. 00
From Tarski 1969,2.2 there is an XE wA such that b = L.< x . and )(.';;;a .{. W .{. .c for every ..[<W. By (ii) a';;;co x-i for every i<w. Thus, coa,;;;coL..[<w)(i= b; 'i .e. a = b. By Tarski 1949, 9.34, or, is cardinally simple.
00
00
00
Using a similar argument as in the proof of 4.3 of Chuaqui 1979 b, we obtain that if b.;;; a, then there is a nonnegative real number IL such that b = Jta.. Let, now, c be an arbitrary element of A. Then c.';;; co a and c. = L-i<W Y-i wi th Y-i';;; a for all -i < co , Therefore, c. = L..[<W IIL..[aJ = (L' an
FOIL eVelLY L. an n nOlL eveJttj -i < nand 1:-i,
oc.
< wand
il> llbtongly ILepILuent:able.
nOlL eveJttj i
a tj E nA
< W,
lluc.h :tha:t
358
M.G. SCHWARZE AND R. CHUAQUI
PROOF. From 1.8 we obtain that elL'/ - is a GCA which satisfies (+). We shall prove that (d) implies that its closure OL'/_ satisfies~hypothesis of 2.2 (For closures of GCA's see Tarsky 1949, Ch, 7.) Let ~ = (X'/- , bE B. Then b = };,< y.f- with yE wA• From (d) (1) we obtain an xEwxwA such .c W .(. # that };j<w Xij ~ "c and a 2 xi j for every c.! < ca , Let u E w A be such that
u.(.. - };.< x," j W .(.j Therefore, .(.
.(.
.;; 00
i < w, there is a
Then for each
y',';;u .•
b=};.< .(.
W
y./-=}; .... .(.
.(.""W
y'./-.;;};.< .(.
.(.
y'. - y..(. .(. W
such that
u./_=};.< .(.
.(.
W
};' (a) = l/J(ct/ - ). We have
into?i
or, by,
(1)
ep
E
(4)
if a,
bE
a
~ b
iff
x
E wA
if
E
A, then
ep
0
n
4>(aob) = l/J(aob/-) = 1/I(a/- + b/_) =
A, then (by(+)) b/-';; a/-
l/J(b/-)';; l/J (a/_)
iff
and u..(. < W x.(..
E
iff
4>(b)';; ep(a);
A, then
= l/J(ui<wxi/-) = l/J(ui<w
4>(ui <w x) because l/J
Defi ne
AN ;
(2) if a, bE A and a o b 1/I(ct/ -) + 1/I(b/-) = ep(a) + 4>(b) j (3)
•
lxJ-))
= .tubi<wl/J(xi/-)
= .tubi<w4>( xi
l,
is an isomorphism between CA's. •
2.3 remains true if we replace (2) by (2')
For every
b, c E A with
a ~ b,c, b ~ c
or
c
~ b.
The proof uses 4.1 of Chuaqui 1979 similarly as is used in 2.5 below. The next theorem is our main representation theorem. 16 or- if.! a f.J;(;Jt,[ctiy pOf.Jilive dominated ODBA,
THEOREM
2.5.
PROOF I
We shall prove first that
f.JtJr.ong.ty JtepJtuel1ta.b£.e.
0(,'/_
then
is a GCA and then prove
oc.. if.J it i so-
a-ADDITIVE MEASUREMENT STRUCTURES
359
n.
morphic to a subalgebra of In order to do this we shall prove a series of lemmas similar to Lemmas 1, 2, 3, 4 and Theorem 2 of Villegas 1964. The proofs are an adaptation of his proofs. We assume in all the lemmas that ~ is a strictly positive ODBA. LEMMA 2.6.
In
ot: Ls a f..:l:JUe.:te.tf pasa.'
CASE I. b = O. If there are finitely many atoms. it is clear that there is a measure. Hence, by 3.3. assume that x E wE' is the sequence of all atoms in E'.with x. Z" for every k<w • .c mk+ l -m k .(.+mk - r< Then, u- defines a functor in the following way: n F
M
~
T
M
6
M(n> (0,0>
)
( k,p>
,
~n(MJ >
( IMI,
6
I
where ~n(MJ ("I"'" "n) 1 iff M 1= ~("1"'" "n)' (n is the arity of (~, n > and denoted by n(~).) This functor (and therefore the formulas of iT) can be characterized by the following result. THEOREM 1. A 6unc.t0lt F: M ~ k > ----') M ~ ~,>o > io de61ned by a 601t,p mu£.a. ~ and" natwr.a.e. numbeJt n,.-t. e.,; F = F~ , .-t66 :the 60Uow- 0,0 >
M(
S~t
sex
wheJte io the ca..tegOlty 06 ./leU wah loced'ioomOltphiom!.l and:the E Me 601tget6ul nunc.tOft/.). We will sometimes denote E(M) and E(6) by IMI and 16/ respec-
tively.
Theorem 1 is implicit in Fraisse's course and appears explicitly in Sette and Sette 1978. In what follows, we use this theorem to give an algebraic characterization of interpretability. Atgeb4aic chCVtac.teJt.,
o
Q.~Yle
-
Let
iT
~1"'"
~t
.-tnteJtp!Letab~y -
= (m 1, ... , mt > and
~nteJt~etab~y
- We will work with four kinds
and i o be 1anguages T =( 111 , .,. , formulas of iT' Then, by a Quine-
interpretation we understand a functor T
fa : M. ,
(ii)
)
1S0
M
I
)
( IMI,
6
I
)
n•
TaM~-.-tnteJtp!LetabUdy-
M~
1S0
m1
~1
, (MJ,
... ,
m
t
~t
(M) > ,
'In addition to the hypothesis of (il suppose
FUNCTORIAL APPROACH TO INTERPRETABILITY there is given a formula q,U having only one free variable. pretation we understand a functor • T
rr : M.1S0
where
and M iso IMI
1u *" ~ .
1S0
m
m l q,l (M), ... , q,/(M) )
lu ,
( IMl
6I
61 u ,
MII u = {a ElM I : q,U 1M) lal
By a Tarski - inter-
M~
)
MI
1
• T
367
r
61 u is the restriction of 6
l} ,
• T
is the full subcategory of M~1S0 such that ME Ob(M iso)
iff
(iii) EquatLtq - ~nt~p4eta.b~q - Suppose that in addition to the hypotheses of (i) we have a formul a q, _ with two free vari ab1es , Then, an equality i nterpretation is a functor • T
so ---~)
I' = : M i
M~
1S0
MII----~
( IMI,
6
6" ,
----.+
1-1
m l q,l (M), ... ,
m
m.e.
where rJM) is the quotient of (IMI, q,ll (M),... , q,t 1M) ), wh.en this quotient exists, 6 is the projection of 6 in the obvious sense and M~ is the full • 1S0 m sUbcateg~ry of M ~so such that ME Ob(M ~so) iff the quotient (IMI, q,ll 1M), , ... ,
q,/IM)/q,=
exists.
(iv) CaM:e!.l~an - i.nt~p4eta.b~!d - Suppose that in addition to the (i) - hypotheses we have a fixed natural number d. Then, a Cartesian interpretation is a functor I' : M~ ------) c. 1 so
where m
~-l~(M) elm.
1 l
1S0
d
MII----~
( IMI ,
6 11----4
6 ,
1 I(ai , ... , ad)"'"
= q,~ .(.IM){a
M~
1
m,
.ml .mt q,l (M), ... , q,t (M I ) ,
m.
(a l,{. , ... , a/))
,,,., ad' ... , a
m. l,{.,
m.
... , a/)
We observe that each of these functors (i) - (iv) can be factored by means of two functors as follows
ANTONIO MARIO SETTE
368
• T
M.
1S0
r
a
M iso
----~)
'a'
M iso • T
where
f' (M i so)
0
1
is a subcategory of M i so
G
o' M i so
To see this, we take: (i) bil ity. (ii)
I' = I"
and
f'(M) = » the same as FF(ep) says about the elemen:s of M, F<j> says about the elements of r(M) modulo a "natural change" 1/ep in the truth-relation. Or. equivalently, that < r , {1/ep}ep > is a "semantical change" given in the world (i n F<j>' or, in other words, what
ANTONIO MARIO SETTE
374
M ;SO) accordi ng to the new s itua t ion created by F.
We will show that: THEOR EM 2. PROOF. (i'l1"'"
fine
EveJtlj modeL 06 a btaYl-6La:tLoi'l F de6-{.i'leJ.>
Let
i'lk>'
0
• T
r:
M iso
0
.... M iso
ai'!
",rtteJtpJte.ta-t.toi'l •
be a model for the translation F,
T
L), n(F(p",)) = m", and 0' = (mi,"" mil. Desuch that r ' : (1.1) =(11.11, F(Pl)(M), ••• , F ( PmL)(M»,
=(m , .. ·, m
l
T
0'
r' : M i so .... M i so
r'(6) = 6. r- is a functor and r'(M;so) = Mfs'o is a subcate90ry of :;so If N = ( 1.11, F (PI) 1M),... , F (PL) 1M)} and we define T from M fso to M i so 17* 1 17 L by T(N)=(r*(IM ),U (N), ... ,U (N», where for all (a , ... ,a )er* 1 m. 17'! m", (lMI).{., U .(.(N)(a ... , am ) = F(p.)(M) (b ... , b ,) iff ((b ... , b ,), ..{. m. m. l, l, 1, .{.
(a1 , ••• , am,)) e 171 and T(6)
.{.
r*(6), then
A.
T is a functor making the diagram
.(.
Set
... ~------E
commutative. iff (a
1,
In fact
Ir(M)1 = f*(!MI) = IT(M)I
and p.(f(M))(a , ... , a 1
A.
) = 1
m",
= 1 for all (b l, ... , b ,) such that ((b 1, ... ,b ,), m", 17'! m", m,[ iff U .(.(N)(a ... , a ) 1. For the other side,
F(p.) (M)(b , ... , b,)
.{.
... , a
))
m",
T(f'(6))/
E
1 17'!
.(.
= IT(6)1 =
l,
f*W, thus,
m,[
T(f'W)
r(6)
and
r s r- =f.
Tis
obviously dependent. We do not know what kinds of interpretations I' define a model for translation F, nor when a translation F has a model.
some
In what follows we will give some examples of interpretations. Ex. 1. All interpretatio~given in the beginning are examples of first order interpretations. Ex. 2.
1.1
E
Let T=(l'll'''''I'l/1.>'
Ob(M~so)' i.e.,
FUNCTORIAL APPROACH TO INTERPRETABILITY
375
and 1M ++ = {6 : 6 : 1M + -+ IMI}. Consider (): M -+ IMI++ such that ()(Cl) = Fa. where Fa.161 = 61a.). () is injective. Take T*(IMI) = ()(IMI) G. jMj++ and for
IMI _6_> INI. define ()(6) : T*(IM )
-+
(INI) by ()(6)(FCl)
Fo[Cl)'
=
T * : M~ the mapping IMI -+ 1/~"- [M) , so -+ E (M a,.so ) is a functor. Moreover. . where 1/'!(MJ = {«al' •••• a. ). (F •••• , F )J) is a natural transformation "n-t a.1 a.n . n. 1'1. "from 1r "- to 1r "- 0 T*. Thus by "code result" if I'" : M -: ... M? is defi ned m m ' so i so 1[M), ... , and M=( M, R , ... , mf< Rf< i ,
For each
IMj, Cl '" b
for all
j
M define
a binary relation '"
in
IMI
1
as follows:
if a, b
E
iff
= 1•••••
m. 0 Me equ1.vaient OVefL modw 06 the 60Jtm .po J.,o
PCd(dt)
or.. (!) r.Pi
~.
oveJt
§3. THE 'NUMBER' OF ADDITIONAL PREDICATES. THEOREM 3.1. -intl.{n-ite.
(a)
Let
Then theJte ewt
'" -It. e., and, -in 6ac.:t, S' ,(6 K J.,o PC (' DB occurs in 4>', and so belongs to R. SO 4>,0 B is as desired, except that it is PCd(&l) in the parameter B. But B is either Url'- ], which our definition of d -r ,e , always allows as a parameter, or else B= w, which can be dropped (since {w} is c4 -recursive). -
~ -FINITE AXIOMATIZABILITY USING ADDITIONAL PREDICATES.
§4.
In
§4
c.J
and §5,
=
(I~I,
E , url..Jl) Le., there are no
W.c
THEOREM 4.1. (a) Let t: be :tJr.a1t6d.-tve a.nd cJi = t+. 16 K J.A a.n R - d.a..M 06 (nMn-Ue modw, R E ledl, a.nd K J.A PC d (c4 ) (a.Uow{ng pMameteM), then K J.A PC (cJi) • PROOF I
The case with parameters follows at once from that without. So let = 3S, 1\ (~), ~ is the set of all S E le.til, and £:. is ~ -r.e •• Let £:. = {u : (3 bE 1"1) JL 1= 5' [a, b l l where 5' is .6. ' We now 0 have the hypotheses of Theorem 2.2 (to be applied below), taking 5(Jt) to be JtER, here for htl' there. and taking (rJi, K
= Mod
4> where 4>
.s»
Let K PU' be the theory (of admissible sets with urelements) in Barwise 1975 adjusted to say that the set of all urelements exists and is denoted by the indivi dua1 constant Ur. Let aell be
clt l , u*v)A\rl.dl(ur). A (u'jOv: u, vEur 1 \~e
can clearly assume that For 4>'
R,
ur lJi I
E
t: ,
we take: ,Ur,
r , R, u (u E UrJi
(f 1 A f
Af ) , 2 3 where the f{ areas follows: f 1 is simply K PU'. This is a recursive set of first order sentences in just E, Ur, having no fi ni te models, and so, by the an cestor of the theorem, can be replaced by a PC sentence. The EC (eli) - sentence f is: 2 (V) A (Vc: c is one of Ur, t, K, u(uE Url.il» A A.6. (t) A 3 (V,
E
»
ajV)
.6.
t
(R) IV)
R-
To define I' 3' use the notation of Theorem 2.2. There we said ot. and jJi I are disjoint, but since all formulas are sorted, it makes no difference. Thus ·the 4>* from 2.2 (for our 4» is a PC sentence about structures (roughly) of the form d (A; V, E ,Ur, R; R.f. (n< f». By inspection in §2, we see that 4>* = n 3 (Tl"") ~: on' wh,ere each un is of the form Ij xl x2 ... Ij Jtl JL 2 ... "Y n and in "Y n only quantifiers Ij x over 0'(, and bounded quantifiers Ij JL E * descends to the model (7C. +cA ,i.e. oc.+.A. 1= 4>* • Hence by Theorem 2.2, OL 1= 4>, as was to be shown. -
386
ROBERT VAUGHT
cA
THEOREM 4.1. (b) Su.ppolJe Then any PC (dit) lW-Uh paJUlmUVlIJ) .{.IJ PC(.4). d
cJIl= t+, and R E4. lwh. Nn (xoyo ... nOJIm
x yn) n
.6uc.h that .{.n any countable
(l)
R - .6ttr.uctutr.e
oz. , en, 1=
'UxO E A 3YO EA"x 1 E A 3 Y1 E A ... "n Nn
"-n
and only
.{.n
IxO ••• Yn) ;
and mOlLeOVeJl.,
In nac..t, the U.6ual pltOOn, e.g. the one "-n Vaught 1973, p. 10, an Svel'!On.{.u.6 TheOlLem c.onJ.>ttr.uc..tJ.> a w t N wh.{.ch -tJ.> eMily .6een to have the new pltopeJLty (2). n !HencenoJLth, "the Nn an 6.1" meaYlJ.> exac..t./'.y thou c.oYlJ.>ttr.uc..ted .{.n Vaught 1973. )
(A remark we won't use is that any 'J1 = 3 (SO ... ) ~ 'Ux O'" xk.(n) en where each en is "1:. 0 in ~ ", can be replaced by one like in 6.1. All these matters are different from but related to a proof on p. 21 of Vaught 1973 and especially to Gandy's Theorem (VI 2.2 of Barwise 1975).) 6.1 gives a "half-bounded" game. a (fully) bounded game. LEMMA wheJle 0
¢
6.2. Suppo.6e AO ~ Al
~
....
6.2, below, converts it, for special al> "-n
6.1, oz .{.;., c.ountable, a.nd lOLl = u A , n n N On 6.1, ot: 1= .{.n a.nd only .{.n n
Then nOlL the
PROOF. Clearly, (l) implies (3). Given (3), let the x-player play so as to exhaust A (as is easily seen to be possible). By the construction of the Nn (cf. Vaught 1973), at the end of the game one sees at once that (Jt, 1= • • In the right circunstances, below, using both 6.1 and 6.2, we shall be able to bound both x and y in the game. Moschovakis (for references see Barwise 1975 or Vaught 1973) originated the idea of using the closed games (like operation (A» to obtain reduction principles. In Vaught 1973, these ideas were used and extended to show the fo11 owi ng families satisfy the reduction principle: (4) c PC(tA)
if
wE
cd and cd has no ure1ements.
(5) c PCs ' c PC d' and also, over .
