Logic in Algebraic Form, Three Languages and Theories

LOGIC IN ALGEBRAIC FORM THREE LANGUAGES A N D THEORIES

WILLIAM CRAIG Uniuersity of California, Berkeley, Calif.,U.S.A.

1974

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ACKNOWLEDGMENTS

I owe perhaps most to the work of Charles M. Howard. but am also very greatly indebted to work on cylindric algebras, on polyadic algebras, and on Boolean algebras with operators. In addition, it is a pleasure to acknowledge valuable conversations with Daniel Gallin. Bill Hanf, Leon Henkin, Charles Howard, Ralph McKenzie, Donald Monk, Don Pigozzi, Alfred Tarski and Richard Thompson. Professors Henkin, Monk and Tarski also kindly made available to me a draft of their now published book on cylindric algebras. of which I made free use in ch. 4 and in parts of ch. 9. Large portions of the manuscript (roughly chs. 1-8) were read with great care by Ralph Seifert, who spotted many errors, would not permit vagueness, and made other useful suggestions. Similar remarks apply to Douglas J. Young, who read most of the remaining portions. I should like to thank Peter Eggenberger for his fine help in preparing the indices. Also, I should like to record my gratitude to Carrol Fluellen, Leslie Hausman, Jeanne Robinson, Dale Ogar, Ruthie Cephas, and Yulia Motofuji, who typed successive approximations with patience, accuracy, taste and judgment. T h e Miller Institute for Basic Research in Science enabled me to devote the entire academic year 1965- 1966 to research. During that year this investigation gathered needed momentum. Most of the results of ch. 9 and of ch. 10 were obtained, at least in rough form, during 1968- 1969, when I was on sabbatical leave from the University of California and also held a post-doctoral fellowship from the National Science Foundation of the United States. During other periods, grants GP-7578 and GP-26261 from this Foundation gave me valuable support.

vii

INTRODUCTION

The idea of using algebraic methods in the practice and theory of logic is quite old. Leibniz experimented with it repeatedly. Fundamental steps toward a comprehensive equational calculus of deductive His approach guided Peirce. reasoning were taken by Boole (see [4]). Schroder, and others, who aimed at improving Boole's work and at extending it from unary sets to other relations. Since then. the emphasis has changed from calculi to algebraic structures. In aim and emphasis, this book is close to the earlier tradition. As we shall argue in ch. 1, first-order logic with equality is concerned with certain set-theoretic operations. To overcome certain technical problems, and yet stay close to natural languages, Frege introduced quantifiers. Through this device he largely avoided dealing with these operations explicitly. In proof theory and, more recently, in semantics this device has been highly successful. Nevertheless, a more direct treatment of these operations is desirable. In addition, one should like to use algebraic methods. Regarding semantics, these desiderata seem to be largely satisfied by the first of the algebraic languages which we shall introduce. As will be seen in ch. 2 below, to each operation which is denoted by a formula of the language for first-order logic with eqJality there corresponds an operation which is denoted by a term of t5e algebraic language and which, roughly speaking, uniformly extends the given operation to a wider domain. Moreover, as will be seen in ch. 3, the two languages are model-theoretically equivalent. Regarding proof theory, these desiderata seem to be largely but. according to ch. 1, not always fully satisfied if m e carries out, as we shall, the following two-part procedure: First, one reduces problems of validity regarding the language for first-order logic with equality to problems of validity regarding an algebraic language; second, for the 1

2

INTRODUCTION

algebraic language concerned, one constructs an equational theory which is sound and complete. Each of our reductions involves the application of a process of algebraization to the class of those set-theoretic operations that are denoted by first-order formulas. Each also involves the selection of a set of generators for the resulting class of operations and the formation of an algebraic language whose primitive operations are the generators selected. We then obtain an effective mapping of the first-order formulas into terms of the algebraic language such that. with negligible exceptions. formulas having the same free variables denote identical operations if and only if their images under the mapping denote identical operations. Two rather natural processes of algebraization will be considered, the first yielding operations on sets of sequences of arbitrary finite length. and the second yielding operations on sets of sequences of length w. One set of generators will be selected for the class resulting from the first process, and two for the class resulting from the second process. Thus, three algebraic languages will result, two of which however differ only in the choice of primitives. The second of these three languages (in our order of presentation) can be obtained from the language for w-dimensional cylindric set algebras (see [15]) by adding symbols for a certain projection operation P and for that operation Q which is the conjugate of P , i.e., is induced by the converse of the relation (between sequences) which induces P . T h e operations definable in the third language are those definable in the second language, but the extra-Boolean primitives are now certain operations (Sa)and ( T a ) ,where (Sa)is a substitution operation of the language for w-dimensional polyadic set algebras (see [ 131) and ( T a ) is its conjugate. Operations on sets of sequences of arbitrary finite length which correspond to the primitive operations (on sets of sequences of length w) of the second language form the primitive operations of the first language. Thus the first and the second language are syntactically the same but have different, although related, interpretations. For each of these three languages we shall construct an equational theory which is sound and complete. T h e theory is equational (or algebraic) in the sense that terms are formed in the usual way from function symbols of the language and from individual variables, and

INTRODUCTION

3

that every theorem is an equality between terms. T h e theory is sound in the sense that each equality which can be derived is valid, i.e.. equates two terms which denote, for any given non-empty universe, the same operation. Finally, the theory is complete in the sense that all those equalities of the language which are valid can be derived. For each of these three theories, problems of validity in first-order logic with equality can therefore be reduced to problems of derivability in the theory. The value of this reduction has yet to be demonstrated. However, there are certain prima facie advantages. All variables of the equational theory are of one kind. Also, there are no variablebinding operators. Furthermore, in deriving equalities from axioms which are always themselves equalities, one can in each case proceed linearly, using small steps to vary the manner of denoting the given operation but never changing the operation itself. More precisely, to derive T = T' one can always proceed as follows: First produce T, at each later stage produce a term whose equality with the term produced at the preceding stage can be ascertained by substituting into one of the axioms, and at the final stage produce 7'. T h e axioms of these three theories are fairly simple. Those of the first theory were discovered by asking what was needed to make Q a suitable neat embedding (see [IS]), so that the neat embedding theorem from the theory of cylindric algebras could be applied.' Since it is complete, the second theory is of course an extension of the theory of w-dimensional cylindric algebras. It also turns out to be an extension of the first theory. Its additional axioms were obtained by asking what was needed to exploit the completeness of the first theory. This completeness in turn, and utilization of axioms for polyadic algebras with generalized diagonal elements, led to the axioms of the third theory. We regard the axioms of the first theory as most economical and those of the third as most unified. T h e third language and theory lend themselves readily to generalization, and will be treated accordingly. However, as will be seen in ch. 1 1 , cases then arise where completeness does not hold with respect to all equalities. In ch. 10, we prove completeness with respect to 'This was partly suggested by a related use of this theorem in Bernays [ I]. For a statement of the theorem, see[IS]. I t will not be used in the completeness proofgiven below, which is self-contained and more direct than our original proof. However, the proof below does not fully reflect our indebtedness to work in the theory of cylindric algebras.

4

INTRODUCTION

those equalities which involve only products of operations (Sa)and ( Ta).

Strictly speaking, only ch. 6 is algebraic, as this notion is now understood. There we shall characterize those models of our first equational theory which are isomorphic images of set algebras. We shall also give a rather simple characterization. up to isomorphism, of the algebras of those (non-logical) theories which are expressible in our first algebraic language.' In these algebras a certain distinction plays a r6le which is related to the rank or degree of a formula and which plays no r6le in what are known as the Lindenbaum-Tarski algebras. These algebras therefore may be useful for the study of the structures of first-order theories. The first half of this book (chs. 1-6) is thus concerned with operations on sets of sequences of varying finite length. In the second half (chs. 7- I I ), we deal with operations on sets of sequences of the same, primarily infinite. length. As has already been suggested, between the two halves there are many parallels. However, there are also some major differences. The first half contains an algebraic substitute for first-order logic which is more faithful to it than are those of the second half. Also. the model-theoretic and algebraic characterization in chs. 3 and 6 have no counterparts in the second half. On the other hand. t h e second half has the advantage of greater generality and of dealing with some problems of infinite quantification. This book is intended to be essentially self-contained. Each of the eleven chapters begins with a fairly informal description of the main trends or results. We also try to indicate there what parts of the chapter are needed elsewhere. The chapters concerning languages (chs. 1-3,7) form a unit which can be read independently of the rest. Only little acquaintance with them is required for reading the chapters concerning theories (chs. 4,5,8- 1 1). Finally, with one small exception, the chapter using algebraic notions (ch. 6) is not needed elsewhere. !These characterizations are adapted from the unpublished Ph.D. thesis of Charles M. Howard [161. His work has also greatly helped us in understanding the relationship of algebraic languages and theories of logic to the usual quantificationalones.

CHAPTER 1

SET-THEORETIC OPERATIONS FOR FIRST-ORDER LOGIC WITH EQUALITY

Assuming an arbitrary non-empty universe U to be given, we shall associate in this chapter with each formula (o of first-order logic with equality a certain set-theoretic operation j(o i. We shall also associate with each (o a specific way of forming j(oj, which starts from fairly simple set-theoretic operations JI / and uses substitution. none of the usual sysFor theoretical study of the operations tems for first-order logic with equality is wholly satisfactory. As we shall see, provability of a formula (o JI in the system does not always imply that / (o j = / JI j. As our first step toward constructing a more satisfactory theory we shall form a fairly simple set of operations i JI i from which all other operations jcp can be obtained by substitution. We conclude the chapter by comparing the choice of i ( o / as the operation which (o is to be regarded as denoting with other choices. We begin with some conventions. Some of these will be used immediately, others later on. Little or no explanation will be given of some standard notation and terminology, particularly from set theory. L e t f b e any function. We regardfas a set of ordered pairs ( r , s ) . We let domf be the domain offand ranfthe range off. For each s E dom f , we let & = f ( s ) =fs be the value of f for s. We use brace-colon notation for sets in the usual way. For example,

-

dom f = { s: for some r , ( r , s) E f } , f = { t : for some s E dom f , t = ( f ( s ) , s)) = {(f(s).s): s E d o m f } , and m n f = { f ( s ) :s E d o m f ) . If S is a set, we let ( f ( s ) : s E S) and ( f ( . ~ ) )be~ the ~ ~ function 5

6

SI I T H t O R f I I (

O P r R A I I O N S FOR FIRSl-ORDtR LOGIC W l r H E Q U A L I T Y [CH.

1

{ ( j ' ( s ) , , s )s: E do/,i.f.s E S } . Welet

f " = ( { . f ' ( s ) :s E S}: S & domf)

These conventions shall remain in force when other names are used forf'(s) or other expressions for the condition that s E S . An ordinal shall be the set of its predecessors. In particular, 0 shall be the empty set 4'. Until further notice we let a , /?,... be ordinals, i,.;. ... finite ordinals, and w the least infinite ordinal. For emphasis we shall sometimes state explicitly, for example, that 0 s i or that i < w . We consider an arbitrary set U . For any set T , we let

TU = {f:fis afunction, dom f = T , ran f

c

U}.

Thus, in particular, "U is the set of those sequericesf such that f is of order type CY and such that, for each p < a , the termfo of the sequence is an element of U . By our earlier conventions, if f E " U , then f = shall be a segment of ( j ; ) u < a .For each a' s a" s a , ( f B ) a r a B < d (.fo)B