" W—‘Tv ‘w" a... “'7 EXPRESSIVE COMPLETENESS Thesis for the Degree 0! 911. D. MICHIGAN STATE UMVERSITY Herbart Eu Hendry 1391626 ’ meets This is to certify that the thesis entitled IT "\ 751"» Q T) 1}: has been accepted towards fulfillment of the requirements for Date 0.169 LIBRARY Michigan State University ABSTRACT EXPRESSIVE COMPLETENESS by Herbert E. Hendry In this essay an attempt is made to explicate a concept of expressive completeness for first-order exten- sional languages. The explication is intended to fill a gap in our present understanding of such systems. We are often concerned to determine whether a system is consistent, is complete (in some definite sense), or has an independent set of axioms. These concepts as they are usually under- stood are all explained with reference to certain relevant features of statements of the system. But with regard to the terms of a system we have only a well-defined notion of independence. Certain symmetries between statements on the assertory side of a system and terms on its conceptual side suggest that we look for corresponding concepts of consis- tency and completeness. Expressive completeness, as it is here explained, is intended to be a candidate for the latter. The locus of this explication is an axiom system. Very roughly, a first-order extensional language is said to be expressively complete if it can define terms whose extensions exhaust the subclasses of, and relations on, its universe. The bulk of the work is devoted to refining and Justifying this account and to developing its consequences. Among the Herbert E. Hendry more interesting results are that for any finite universe an expressively complete first-order extensional language can be constructed, that no language is expressively complete if its universe is infinite, and that no language, whether its universe be finite or infinite, is complete if every— thing is in its universe. Expressive completeness is compared with a number of other completeness concepts. They are all found to differ in rather essential ways. All, save one, are obviously unsuitable to fill the above-mentioned gap in our under— standing. The exception is Tarski's notion of the complete— ness of concepts. A close examination of Tarski's develop— ment of this notion shows that although it isolates an important concept, it cannot be regarded as an acceptable analysis of expressive completeness. Relating these two concepts, it is shown that under certain minimal conditions any system that is expressively complete contains a set of sentences that is complete in the sense of Tarski. It is observed that the converse does not hold. That is, there are first-order extensional languages that are complete in the sense of Tarski but not expressively complete. EXPRESSIVE COMPLETENESS By Herbert E? Hendry A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Philosophy 1966 ACKNOWLEDGMENTS I am very much indebted to Professor G. J. Massey of Michigan State University. His careful reading and detailed criticism of this work have been the source of improvements in virtually every one of its sections. I am also indebted to Professor Robert Barrett of Washington University. It was in the course of conversations with him that the central idea for the work evolved. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . Section 1. INTRODUCTION 2. A CRITERION FOR ADEQUACY . . . 3. EXPLANATION AND JUSTIFICATION OF THE CRITERION . . . . . . A. AN INFORMAL SKETCH OF THE ANALYSIS. 5. ADEQUACY OF THE PROPOSED ANALYSIS 6. THE SEMANTICAL PRIMITIVE ./6i REFERRING EXPRESSIONS AND EXTENSIONS 8. ORDERED heads. 9. ABSOLUTE AND RELATIVE REFERRING EXPRESSIONS . . . . . . lO. PRIMITIVE DOMAINS AND PRIMITIVE SELECTION CLASSES. . . . . . . . . . . ll. AUXILIARY DOMAINS AND AUXILIARY SELECTION CLASSES. . . . . . . . . . l2. EXPRESSIVE COMPLETENESS 13. AN APPARENT ANOMALY. lA. RELATIVE EXPRESSIVE POWER. 15. THE ELIMINATION OF GENERAL TERMS l6. UNIVERSAL EXPRESSIVE COMPLETENESS iii Page ii 13 l7 19 2A 26 34 37 L42 147 149 514 62 67 Section Page 17. INFINITE PRIMITIVE DOMAINS . . . . . . 69 18. OTHER CONCEPTS OF COMPLETENESS. . . . . 72 19. TARSKI AND THE COMPLETENESS OF CONCEPTS. . 82 APPENDICES . . . . . . . . . . . . . . 90 List of Axioms . . . . . . . . . . 91 List of Definitions . . . . . . . . 92 List of Theorems . . . . . . . . . 9A BIBLIOGRAPHY. . . . . . . . . . . . . . 97 iv 1. INTRODUCTION There is, in one important respect, a striking similarity between the roles played by the terms and statements of an axiomatic language.1 We distinguish the primitive (undefined) terms of such a language from its derivative (defined) terms. Similarly, we distinguish between primitive statements (axioms or postulates) and derivative statements (theorems). These distinctions are vital for the study of axiomatic languages. For system- aticity is of the very nature of such languages; and these lI distinguish between axiomatic systems and axio- matic languages. An axiomatic language is one kind of axiomatic system. An axiomatic system, of course, is axiomatic. It is a system of signs or expressions governed by syntactical rules. An axiomatic system, how— ever, need not be interpreted. A language, and hence an axiomatic language, must be interpreted to the extent of having terms and statements. The fundamentum divisionis here is their mode of meaning. To be a term is to have an extension. To be a statement is to have truth value. The taxonomy of systems is presently not very well developed. Consequently, the above pronouncements must be regarded with considerable caution. My sole motive for including them is to make it clear that only such systems as are languages are under consideration. Cf. Church, sec. 07. distinctions point to two important sources of system~ aticity: definition and implication.2 The concepts of independence, consistency, and completeness have received much deserved attention in the field of axiomatics. They are well understood. Yet, ordinarily understood, these concepts are explained with reference to Statements. The parallel roles of terms and statements suggest that we look for corresponding concepts whose explanation would involve reference to terms. Indeed, we do not have to look far to find a corresponding 3 notion of independence. It is now commonplace to inquire whether a set of terms is independent, that is, whether any one of them is definable in terms of the others.“ But 2Frege wrote that the "aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another. . . . The further we pursue these enquiries [into the foundations of arithmetic], the fewer become the primitive truths to which we reduce every thing; and this simplification is in itself a goal worth pursuing." Frege, p. 2. But Frege does not tell us why simplification is worth pursuing. Perhaps, in line with the doctrines of neo—pragmatism, it is that simplicity (as a measure of systematicity) is in part constitutive of truth. 3For relevant literature by Padoa, Beth, Tarski, and Lindenbaum see Tarski(1). “The importance of independence should not be minim- ized. In the extreme case every accepted statement of an axiomatic language might be counted as one of its axioms. Here there is no systematicity. If, on this intuitive level, we may speak of degrees of independence, we might consider it to be a law that, other things being equal, systematicity varies directly in prOportion to degree of independence. corresponding concepts of consistency and completeness do not seem to have received careful attention. The primary aim of this essay is to provide an ex— plication of expressive completeness for first-order extensional languages. Expressive completeness, as it is here explicated, is intended to be a property a language has (or fails to have) in virtue of certain relevant features of its terms. If successful, then, the essay will constitute a significant first step towards filling a lacuna in semantic theory. 2. A CRITERION FOR ADEQUACY Antecedent to the explication of a concept it is desirable to have some relatively clear and objective criterion by which to determine whether the proposed explication is adequate. Subject to explanation and Justification in the next section the criterion adopted in this essay is the following. An explication of ex- pressive completeness will be considered adequate if, and only if, it fulfills the condition that: A language L is expressively complete if, and only if, theie is no language L such that (i) Lland L2 have the same universe of discourse, Iii) L1 and L2 have the same meaning base, and (iii) there is a term T2 of L2 and there is no term Tl such that Tl 1E defihable in L1 and Tl has the same meaning as 12. 1The criterion is that it fulfills this condition. The formulation of the condition might be regarded as a defintion, albeit a very poor one. Here I speak of a criterion for adequacy not in the sense of Tarski(l), Tarski(2) and Carnap(l) but in the sense of Carnap(S). In the latter we find Carnap writing of a similar formu- lation (for an explication of L—truth) that "it is an informal formulation of a condition which any proposed definition of L-truth must fulfill in order to be ade- quate as an explication for our explicandum." He adds that it has "merely an explanatory and heuristic func- tion." Carnap(S), p. 10. 3. EXPLANATION AND JUSTIFICATION OF THE CRITERION In reverse order, each of the three parts of the criterion will be explained and justified. Although a precise formulation of (iii) is diffi- cult, its intent is relatively clear. The point is that a language is expressively complete if any term added to its base would be synonymous with some term already de- finable in the language. Any term added to an express- ively complete language is redundant in the language. And to an expressively incomplete language there is always a term which could be added without redundancy. Were this part of the criterion violated, one of two absurd conse- quences would follow. Either (a) a language could at once be expressively complete and yet unable to formulate every- thing that there is to formulate about its subject matter, or (b) a language could be expressively incomplete even though it could formulate everything there is to formulate about its subject matter. By the very nature of the case, the sort of justifi- cation just outlined suffers from crudeness. Its crudeness issues from its lack of precision. But imprecision always accompanies criteria for an adequate explication. (This is especially true when the concept to be explicated—- expressive completeness in this instance-~is not firmly entrenched in pre-existent usage.) A demand for complete precision is a demand that the concept to be explicated be precise. Were this the case there would be no need for analysis. There are, however, several further difficulties with the formulation. But their resolution would demand the solution to a host of rather difficult problems. Among these are the problems associated with the nature of language and the disposition of definability. In con— nection with the first difficulty it will be noticed that the ensuing analysis relies on an uncritical quantifica— tion over first-order extensional languages. Thus it is committed to their existence. If it be asked what pre- cisely are the values of these variables, that is, what sort of thing is such a language, the honest reply is that there is no fully acceptable answer.1 I am inclined to lQuine's "Language is a social art" (Quine(5), p. ix.) and its ilk are of little help in this connection, nor, perhaps, are they intended to be. Yet even less pictur— esque accounts are only a little more satisfactory. Mates, for example, writes that "by 'language', in its most general sense, I wish to denote any aggregate of objects which are themselves meaningful or else are such that certain combin- ations of them are meaningful." Mates, p. 201. And Chomsky writes in the same vein that he considers "a language to be a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements." Chomsky, p. 13. But both of these understandings have the somewhat awkward consequence that the unit set of any sentence is a language and that any set of identify two languages only if (a) they have the same expressions, (b) they have the same meaningful expressions, and (c) a meaningful expression of one language has the same meaning in the other language.2 But entities that satisfy such conditions are not easily found. sentences, even from diverse languages, is a language. Carnap is a little more helpful. He writes that a language is a "system of signs, or rather of the habits of producing them, for the purposes of communicating with other persons, i.e., of influencing their actions, thoughts, etc." Carnap (l), p. 3. Elsewhere he gives a set—theoretical analysis of language. There a language is taken to be an ordered pair where a is the set of signs of the language and 2 is the set of its sentences. Contrary to the position of section 1, this account allows for uninterpreted languages. In the same place Carnap offers an analysis of an inter- preted language which seems to me to be more appropriate as an analysis of language. He construes an interpreted language as an ordered triple where a and 2 are as before and D is the relation which assigns values to the sentences of the language. Carnap(2), pp. 102 f. 2These conditions, of course, are not sufficient. Throughout the unaxiomatized portions of the analysis further conditions are supposed. For example, in section 1 it is supposed that languages have terms and statements, and in section 17 it is supposed that the alphabet of a language is countable. What conditions must be satisfied if a language is to be a first-order language is still another question. Un— fortunately, I cannot provide a precise answer. The only thing that seems clear is that quantification is restricted to "individual variables." I do not require that such languages have statement connectives, singular terms, or functors. Apparatus for quantification seems to be essential, but I think that good arguments to the contrary could probably be advanced. In any event we shall assume an infinite stock of variables and the apparatus for both universal and existen— tial quantification. For a discussion of first-order lan— guages and their classification see Church, sec. 30. What it means for a language to be extensional is explained in n. A of this section. The second difficulty is in part the quite general problem of dispositional concepts. But it is not merely that some languages leave undefined some of the terms that they could define. The more specific and, for present purposes, crucial difficulty is that there are languages that do not even have the apparatus for constructing de- finitions. Moreover, the terms that are said to be de- finable in such languages may not even occur among their well-formed expressions. One might say that a term is definable in such a language if its addition to the language would occasion a redundancy. But it is unclear what it means to say that a term is added to a language. It is for this reason that the above criterion for adequacy is formulated comparatively. It is designed to take into account the quite plausible claim that if there is a new term there is a new language.3 This whole matter is one that the analysis will have to skirt carefully. Its further consideration will be postponed for section 6. When it is said, as in (ii), that two languages have the same meaning base it is intended that they are both extensional, or both intensional, or etc.)4 The expression 30f. Church, p. A8, n. 111 and p. 50, n. 118. “Roughly, a language is extensional if its coexten- sive expressions can be interchanged salva veritate. A langmage is intensional if (1) it is not extensional, and its cointensive expressions can be interchanged salva verdgtate. For a fuller and more precise account see Carnap 5), secs. ll-l6. "or etc." is intended to take into account languages whose strongest meaning relation is such that they are neither extensional nor intensional. We might count Lewis' 5 equivalence in analytic meaning or Carnap's intensional isomorphism6 as such meaning relations. It is even con- ceivable that there be languages with other kinds of "ultra- intensional" bases. But it is presumed that an explication of expressive completeness for any of these sorts of languages would have to satisfy the above criterion for adequacy if it is to be acceptable. Thus, although the concept to be explicated in this essay is quite narrow in scope, the criterion for adequacy is quite general. The point of (ii), then, is this. We can compare two languages with respect to expressive completeness only if they are comparable. By the nature of the case, an intensional language will have terms for which an extensional language can provide no intensional synonym. Similarly, a hyper—intensional language will have terms for which an intensional language can provide no hyper- intensional synonym. There is, perhaps, a sense in which an intensional (or hyper-intensional) language is expres— sively more powerful than an extensional language. (This ‘ 5Lewis, pp. 245 f. 6Carnap(5), secs. 13-16. 10 added power, of course, has its cost in point of economy and ontology.) But further consideration of this topic would involve an extended study of the issues between extensionalism and intensionalism. It is perhaps now relatively clear what is intended by 'has the same meaning' as it occurs in (iii). Two terms have the same meaning if either they have the same extension and belong to some language or other which has an extensional base, or they have the same intension and belong to some language or other which has an intensional base, or etc.7 This is not intended to throw light on the difficult and interesting problem of synonymy for linguis- tic forms of ordinary language. But it has, I fear, con- sequences which would revolt the mildest of intension- alists. For example, in many languages for arithmetic the terms '5-1' and '6-2' are (as are other coextensive pairs of terms) interchangeable salva veritate. Such languages are, of course, extensional. It follows from the above understanding that they have the same meaning. 7I regard terms within the same language as having the same meaning if and only if they are interchangeable SEilva veritate. I find such terms semantically indis- tinmguishable. So to speak, terms which are indistin- guishable insofar as they affect the truth values of Statements in which they occur are indistinguishable in point of meaning. The above interlinguistic concept of Syncwwrw is intended as a natural extension of this int ralinguistic conception. 11 But the intensionalist will deny that they are synonymous and reject the account. He errs, however. His mistake is, I suspect, that he misidentifies terms and languages. In English the terms '5-1' and '6-2' are not interchangeable salva veritate. But this simply shows that we have a dif- ferent pair of terms and that these terms are not synonymous.8 Since they have the same extension (for English), it also shows that English is not an extensional language. The source of this confusion is, one might speculate, twofold. First, although we have only one pair of "types," we have two pairs of terms. That is, '5-1' and '6-2' are ambiguous. Second, we often do arithmetic in two languages, English and some one or another of several formal languages, never keeping clearly in mind just which language is being used. The rationale of (i) parallels that of (ii). We can compare two languages with respect to expressive complete— ness only if they are comparable. There is no basis for so comparing two languages unless their subject matter is the same. It may be anticipated, however, that there are conditions under which two languages offer interesting comparisons (with respect to expressive power) even though ‘ 8Occurrences of terms of the same language are counted as occurrences of the same term if they are of the same type and have the same meaning. Cf. Leonard, sec. 17.1. 12 their universes are nonidentical. Such comparisons will be considered in detail in section 1A. Despite the shortcomings of the criterion's formula- tion, it will prove to be serviceable. Moreover, the ensuing analysis should shed considerable light on the sources of its defects. A. AN INFORMAL SKETCH OF THE ANALYSIS The universe of discourse (or as we shall say the primitive domain) of a language is constituted by the things about which the language has something to say. This universe uniquely determines certain classes. First, it determines those classes which are members of its power set. These are the primitive selection classes of the language. It is among these classes that an absolute term for the language will find its extension. Second, the universe determines ordered n-ads of its members and thereby classes of such n—ads. A class of ordered n-ads whose members are all of the same degree is an auxiliary selection class of the language. It is among these classes that the extension of a relative term for the language will be found. A class which is either a primitive selection class or an auxiliary selection class is said to be a selection class of the language. Thus, the extension of any term for the language will be one of its selection classes. The following account of expressive completeness quite naturally suggests itself. A first-order extensional language is expressively complete if, and only if, it can define terms whose extensions exhaust its selection classes. 13 1A In the next section it will be shown that if the analysis just outlined can be effected, it will satisfy the criterion for adequacy. Subsequent sections fill in the details of the outline and develop some of its more impor- tant consequences. The ensuing analysis of expressive completeness is intended to be axiomatic. That is, its locus is an axio- matic system. Unfortunately, this system is set- theoretical. A more satisfactory analysis would be con- ducted with greater parsimony. It would be developed within a metalanguage whose structure does not drag along a commitment to the rather "unlovely" ontology of classes. The theory of virtual classes seems to offer an avenue of escape. But, because of special difficulties (which will not be discussed here) in formulating an adequate theory of relations within such a rubric, this avenue has not been pursued. Another misfortune of the metalanguage--a misfortune which besets all languages of its type——is that it is not known to be consistent. To steer clear of this latter dif- ficulity, I have tried to confine the exposition to set- theoretical concepts and principles whose credentials are, as much as can be hoped, of established repute. The metalanguage is many-sorted, but it is not type- theoretical. It employs three sorts of variables. The variables 'm', 'n', etc. are to take natural numbers as 15 their values. The variables 'L', 'Ll', etc. are to take first-order extensional languages as their values. And, finally, the variables 'x', 'y', 'z', 'u', 'v', 'w', 'xl', etc. are to take as their values the following sorts of things: (i) expressions of first—order exten- sional languages, (ii) members of the universes of such languages, and (iii) classes of these objects, classes of such classes, etc.1 The analysis will use but a single2 primitive seman- tical concept, that of being an extension. This is under— stood as a triadic relation obtaining between a referring expression, a class, and a language. (See sections 6 and 7.) It also employs a primitive syntactical concept, that of being a relative referring expression. *(See section 9.) Besides these a number of mathematical or logical concepts are used. These, however, are not formally examined. The analysis is built upon three axioms. Roughly, they are the following. (1) Something has at most one extension for a language (see section 7). (ii) A relative referring expression for a language has an extension for lHereafter, unless context dictates otherwise, 'lan- guage' is to be regarded as short for 'first—order exten- sional language'. Similar adjustments are required for other terms. For example, by 'expressively complete' will usually be intended 'expressively complete first—order extensional language'. 2Unless we count the use of 'L', 'L '. etc. as sur— rogate for a second semantical primitive. But, cf. sec. 7, n. l. 16 that language (see section 9). And (iii) the coordinates _of the members of the extension of a relative referring eXpression for a language are members of the extension of some absolute referring expression for the language (see section 10). In addition to these a number of math- ematical or logical principles are used without being formally examined. 5. ADEQUACY OF THE PROPOSED ANALYSIS If the analysis just outlined can be effected, it can be readily seen to satisfy the criterion for adequacy. First, if L1 is expressively complete in the above sense, then for any L if the primitive domain (i.e., 2 universe of discourse) of L is the same as that of L 2 there is no term of L2 for which Ll does not have a synonym. Suppose that L1 is expressively complete. Then 1’ the terms (or referring expressions) of L1 have extensions which exhaust its selection classes. Let L2 have the same 1' It follows that L1 and L2 have the same selection classes. But the extension of any term primitive domain as L for L2 is a selection class of L1' Hence, for any term of L2 there is a term of L1 is, there is no term of L2 for which Ll does not have a with which it is synonymous. That synonym. Second, if L1 is expressively incomplete, then such that the primitive domain of L2 is for there is some L2 the same as that of LI’ and there is a term of L2 which L has no synonym. Suppose that L is expressively l l incomplete. Then there is at least one of its selection classes (say) x which is the extension of none of its terms. Consider, then, L2 which is to be constructed in 17 18 the following way. L2 is to have but two terms in its base, T1 and T2. T1 is to be explained in such a way that (a) it is a universal term of L and (b) it is true of just those 2, things in the primitive domain of L 1‘ This guarantees that L1 and L2 have the same primitive domain. Thus x is also a selection class of L2. T2 can now be explained as having x as its extension. But L then, has no term which is a l, synonym of T Hence, L has the same primitive domain as 2' 2 L1 and a term for which Ll Subsequent sections fill in the details of the out- has no synonym. lined analysis. Hence, they provide an adequate explication of expressive completeness. 6. THE SEMANTICAL PRIMITIVE The sole semantical primitive concept of the analysis is that of being an extension of a referring expression for a language.1 Expressions of the sort 'x is an extension of y for L' will often be abbreviated by corresponding expressions of the sort 'Ext(L,x,y)'o The concept of being a referring expression can be explained directly in terms of the primitive. A referring expres- ngn is simply something that has an extension. It will thus prove useful to make reference to both of these notions in the extrasystematic explanation of the primi- tive. (Axioms will shortly guarantee that each referring expression has exactly one extension; hence, we are justi— fied in speaking of Eng extension of a referring expression.) The notion of a referring expression approximates quite closely the ordinary understanding of 'term'. Like terms, referring expressions have exactly one extension. And, like terms, referring eXpressions can be classified in traditional ways as being either general or singular and either relative or absolute. lUnless confusion threatens, we will often speak simply of the extension of a referring expression, omitting the strictly required reference to its language. 19 20 The extension of an absolute referring expression (general or singular) is to be understood as the class of all and only those things of which it is true (or to which it refers). Thus, the extension of a singular referring expression is always either a unit class or the null class. The extension of a general (absolute) referring expression may have any number of members. It is to be understood, of course, that a member of the extension of an absolute referring expression for a language is also a member of the primitive domain of the language. The extension of a relative referring expression is to be taken as a class of ordered n—ads. It will have as members all and only those ordered n—ads for which the referring expression holds true. It is to be understood that each member of the extension of an n—place (l such that (Ex)quv. The extension of the second is to be the class of all ordered pairs such that (Ex)vau. And the extension of the third is to be the class of all things u such that (Ex)quu. In general, if a circled numeral '(:)' occurs in a predicate each circled numeral '<:)' such that O1, the extension of an nrplace closed predicate, is the class of all ordered grads such that 3 u l. 20 O O, I O 0’ O O 0 E. The reason behind this second departure is that the . .u .u predicates of a language far outstrip its terms. If one attended solely to the terms of a language in assessing its expressive power, he would be woefully misled. Consider, for example, two languages, L1 and L2. Suppose that L1 counts only 'F' and 'G' among its terms. Suppose, further, that L is like L except for having the term 'H' where: 2 l Hx ++. Fx v Gx defines 'H' in L2. Clearly, L2 does not have an advantage over L1 in point of expressive power. The advantage of L if any, is its greater notational convenience. Thus, 2, in assessing the expressive power of a language we want to attend to both its terms and its predicates. Thus we will want to count ZF<:) v G(:> among the referring expressions of L1° The notion of a predicate affords a partial inroad on one of the earlier mentioned problems of definability. 3See Quine's Methods of Logic for a fuller treatment. Ibid. 23 We can safely say that a term is definable within a language if, and only if, the language has a referring expression which is synonymous with that term. 7. REFERRING EXPRESSIONS AND EXTENSIONS As anticipated, it is assumed that something has at most one extension for a language. That is, Al Ext(L,y,x).Ext(L,z,x).+ y=z. It was also anticipated that the concept of being a refer- ring expression would be explained directly in terms of the semantical primitive. A referring expression for a lan- guage is simply something which has an extension for that language. Thus, where 'RefExp(L,x)' abbreviates 'x is a referring expression for L'. D1 RefExp(L,x) ++ (Ey)Ext(L,y,x).l An almost immediate consequence of D1 and A1 is that a referring expression has exactly one extension, and con- versely. That is, Tl RefExp(L,x) ++ (Ey)[Ext(L,y,x).(z)(Ext(L,z,x) + z=y)]. Thus, we are justified in speaking of Lhe extension of a referring expression for a language. Accordingly, we con— textually introduce the functor 'ext'. Where 'ext(L,x)' 1It is tempting at this point to explain a language as something which has something with an extension. Thus (where 'L' abbreviates 'language') Lx ++ (Ey)(Ez)Ext(x,y,z). But rather than clarify the concept of language, this would only serve to emphasize the vagueness of our primitive. 2A 25 may be read as 'the extension of x for L': D2 ...ext(L,x)... ++ (EY)(Ext(L,y,x)....y...). In D2 '...y...' represents any sentence which contains at least one free occurrence of 'y' and no bound occurrences of 'x'; and '...ext(L,x)...' represents the result of re- placing each free occurrence of 'y' by 'ext(L,x)' in such sentences. Since we have from Dl that: RefExp(L,x) ++ (Ey)(EZ)(Ext(L,y,x).y=z), we have by D2 that: T2 RefExp(L,x) ++ (Ez)z=ext(L,x). D2 and T2 will later allow for a simple statement of other- wise cumbersome definitions and theorems. We will sometimes find it convenient to speak of the extensions of a language without explicit reference to the referring expressions of the language of which they are extensions. The justification for such talk is D3. Where 'Exten(L,x)' abbreviates 'x is an extension for L': D3 Exten(L,x) ++ (Ey)Ext(L,x,y). 8. ORDERED n—ads Although the proposed analysis will employ but a single semantical primitive, it will use several extra— semantical concepts which will not be formally examined. These concepts, germane though they be, are not peculiar to the present subject matter. They belong more properly to the province of mathematics or logic. In this section we digress to informally discuss one of the more important of these concepts, the concept of an ordered head. The expression '{xl,x2,...,x }' will be used to 1'1 refer to the class whose members are x1, x2, . . ., and xn. The expression '' will be used to refer to the ordered head whose first coordinate is x1, whose second coordinate is x2, . . ., and whose nth coordinate is xn. The motive for speaking of x1, x2, , and xn as coordinates rather than as members of should be obvious. Under standard accounts the members of an ordered head are not to be found among its coordinates. The primary aim of this section is to explain what might be meant by 'ordered nead' where '2' is a genuine variable. The standard accounts are of little avail; they end too soon. Rather than providing a definition of 'ordered nfad' they give us a recipe for constructing 26 27 infinitely many definitions, a definition of 'ordered 2-ad', a definition of 'ordered 3—ad', and so on, but not a defini— tion of 'ordered n—ad' where 'n' is a genuine variable. All too frequently discussions of this concept and related ones conclude with the words 'and so on'- Quine's discussion in Mathematical Logic is paradigmatic. In connection with the theory of relations he writes: Relations in the sense here considered are known, more particularly, as dyadic relations; they relate elements in pairs. The relation of giving (y gives 2 to w) or betweeness (y is between L and w on the other hand, is triadic; and the relation of paying (x pays 1 to z for w) is tetradic. But the theory of dyadic relations provides a convenient basis also for the treatment of such polyadic cases. A triadic relation among elements y, g, and w might be conceived as a dyadic relation borne by y to g;w. . . . Tetradic relations could be handled on the basis of triadic relations in a similar fashion, . . . Similarly for pentadic relations, hexadic ones, and so on. Thus Quine gives us a recipe for constructing definitions for 'n—adic relation' where '3' has a definite value but no definition of 'n-adic relation' where 'Q' is a variable. The difficulty in explaining 'ordered n-ad' is specifying what the values of 'n' are numbers of. For the number of coordinates of an ordered n-ad may be m (l is an ordered triad with but one 1 coordinate, its first, second, and third coordinates being lQuine(2), p. 201. 28 identical. It is sometimes said that n is the number of "positions” or "places" of an ordered n—ad. But this is simply to provide a name for the problem. For we want, then, to know what positions or places are. The following paragraphs (which rely heavily on standard accounts) provide a recursive definition of 'ordered head'and outline definitions of 'position' and 'coordinate'. First, we introduce the (binary) operator '< , >'. DA = {{x},{x,y}} This definition is due to Kuratowski. It can be shown to satisfy the condition that = +. x=u.y=v. Thus '' can be regarded as referring to the ordered dyad whose first coordinate is x and whose second coor- dinate is y, and we can define 'ordered 2—ad' (abbreviated 'Ord2-ad') by: DS Ord2-ad(x,y) ++ (Eu)(Ev)(x=.y={u,v}). It is to be noticed that being an ordered dyad (2-ad) and, shortly, being an ordered head (for some n) are here con- strued as binary relations obtaining between two classes.2 The reason for this is that an ordered n—ad (for some n) 2Thus, being an ordered . . .—ad is a ternary rela- tion obtaining between a number and two classes. When this relational character becomes important we will use such locations as 'is an ordered n—ad of' or 'is an ordered head relative to 29 is generally explained as an ordered dyad in accordance with the schema: S = >. 1’ n 1’ 2’ ’ n 2,..., Thus (e.g.) is an ordered dyad relative to {al,}, an ordered triad relative to {al,a2,} and an ordered tetrad relative to {al,a2,a3,au}. We have in S the makings of a recursive definition of 'ordered n-ad'. For, a few moments reflection on S in con- junction with D5 is enough to convince us that Ordn-ad(x,y).(Ez)(u=.v=yLHz}):+ Ordn+1-ad(u,v), and where 2.v=yU{z}). But the conjunction of these two truths yields the equiva- lence that where 2.v=yU{z}). We can then take D5 and D5' as jointly constituting a recursive definition for 'Ordn-ad'; for together they lay down a necessary and sufficient condition for determining whether one class is an ordered head of another. We turn now to a consideration of what might be meant when we speak of the positions and coordinates of an ordered n-ad. The coordinates of an ordered head relative to a class are, of course, simply the members of 30 that class. But we want to know more than this. Just as we want to be able to explain 'jth position of the ordered head x of y' so we want to be able to explain 'jth coordi- nate of the ordered grad x of y'. A few preliminary definitions will prove useful. The concept of being an ordered dyad has been explained as a binary relation. At this point it is necessary to make use of a related (absolute) concept. For this concept we adopt the term 'ordered pair' (abbreviated 'OrdPr'). It is to be explained as follows: D6 OrdPr(x) ++ (Ey)Ord2-ad(x,y). That is, an ordered pair is an ordered dyad of some class or other. Notice now that we can easily explain what is intended by the first coordinate of an ordered pair. D7 FirstCoord(x,y) ++ (Eu)(EV)(y=.x=u).3 Similarly we can explain what is to be understood by the second coordinate of an ordered pair. D8 SecndCoord(x,y) ++ (Eu)(Ev)(y=.x=v) One further definition is needed. We will understand by the ancestors of a class all of the entities generated from that class by the ancestral of classial membership. Thus, the ancestors of a class will include the class itself, its 3Strictly, this explains 'a first coordinate of'. But uniqueness is easily established. A similar remark also holds for D8. 31 members, members of its members, and so on. D9 Ancestor(x,y) ++ (z)[yez.(u)(v)(uez.veu.+ Vez).+ Xaz] Let A, , be an ordered n—ad of B, {al,a2,...,an}. Consider then the following m (m=n—l) ancestors of A.“ 2,...,an> Let the class whose members are these m ancestors of A be C. Notice that C's members are all ordered pairs. We can uniquely arrange these members of C in such a way that one succeeds another only if the former is the second coordinate of the latter. Above, C's members are exhibited as arranged by this rule. Clearly we can speak in this special sense of the first member of C, the second member of C, . . . , and the mth member of C. Now, we can easily explain what is meant by the first coordinate of A. It is the first coordinate of the first member of C. Similarly, the jth (l1) predicates and all relative terms are relative referring expressions, and nothing else is. It is, of course, assumed that all relative referring expres- sions have an extension. This assumption is made explicit in the axiom: A2 RelRefExp(L,x) + (Ey)Ext(L,y,x) where 'RelRefExp(L,x)' is short for 'x is a relative referring expression of L'- A2 connects the syntactical and semantical primitives. In conjunction with D1 it allows us to infer that: T3 RelRefExp(L,x) + RefExp(L,x), that is, that all relative referring expressions are refer- ring expressions. From T2 and T3 we have that: TA RelRefExp(L,x) + (Ey)y=ext(L,x) that is, something is identical with the extension of a relative referring expression for a language. In terms of the syntactical primitive we can explain the concept of being an absolute referring expression for a 3A 35 language. Where 'AbsRefExp(L,x)' abbreviates vx is an absolute referring expression for L', D10 AbsRefExp(L,x) ++.RefExp(L,x).mRelRefExp(L,x). That is, an absolute referring expression is a referring expression which is not a relative referring expression.l From D7 and T2 we have that: T5 AbsRefExp(L,x) + (Ey)y=ext(L,x). That is, something is identical with the extension of an absolute referring expression. And from T3 and D7 it follows that: T6 RefExp(L,x) ++. AbsRefExp(L,x) v RelRefExp(L,x), That is, something is a referring expression if, and only if, it is either an absolute or a relative referring expres- sion. From T6 and D1 it follows that: T7 (Ey)Ext(L,y,x) ++. AbsRefExp(L,x) v RelRefExp(L,x), That is, something has an extension if, and only if, it is either an absolute or relative referring expression. It might be thought that the syntactical primitive could be done away with. For the members of the extension of a relative referring expression are (as we shall later make explicit) all ordered n-ads. Thus, one might suppose, a relative referring expression can be explained as a refer- ring expression whose extension has only ordered n—ads as 1Where there is no danger of ambiguity the strictly required reference to the language for which a referring expression is a referring expression will be omitted. 36 members. This suggestion, however, is not viable; for, it has unwanted consequences. First, it would count some referring expressions which are ordinarily thought of as absolute as relative. Any term with a null extension would be relative. Second, there is no guarantee that some of the languages we are considering do not have ordered n—ads within their primitive domain. Thus, even on this score we would be forced to call absolute terms relative. lO. PRIMITIVE DOMAINS AND PRIMITIVE SELECTION CLASSES We associate two domains with a language, its primi— tive domain (more commonly, its universe of discourse) and its auxiliary domain. The primitive domain of a language is the class of all objects which are members of the exten- sion of any of the absolute referring expressions for the language.1 Thus, where 'PrimDom(L)' is short for 'the primitive domain of L', Dll PrimDom(L)=X(Ey)(AbsRefExp(L,y).xeext(L,y). As an immediate consequence we have that T8 x PrimDom(L) ++ (Ey)(AbsRefExp(L,y).x:ext(L,y). Perhaps a word of justification is needed. One might argue that the universe of a language has as its members all those objects which satisfy at least one place of at least one of the language's referring 11 am fully aware that this imposes restrictions under which we can say things like 'Let the universe of L be the class of F's'. One must first establish that each F is a member of the extension of some absolute referring expres- sion or other of L. There is no problem if L has a universal predicate whose extension is the class of F's. But there are difficult cases. G. J. Massey has suggested the follow- ing one. The universe of L is to be the set of natural numbers. L is to have variables, grouping indicators, quan— tifers, one connecti e (for conjunction), and one predicate, 'P' understood as ' is a prime'. Thus, only prime numbers are members of extensions for L. (And '(x)Px' is a true sentenCe of L.) But here I would want to say that we cannot Ln: the universe of L be the class of natural numbers. 37 38 expressions.2 This is correct, but it would be a mistake to conclude that 'RefExp' need replace 'AbsRefExp' in D8 and T8. For, it can easily be seen that if an object n satisfies the Lth place of an n—place referring expres- sion L of L, there is some absolute referring expression L' of L which has n as a member of its extension. If n=1, we can let L' be L. If n>l, then L is either (i) an n—place predicate or (ii) a relative term. (i) L is an n—place predicate. Construct L'l as follows. Replace all except the Lth numeral of L by distinct variables (not occurring in L); and replace the Lth circled numeral by '(:)'. Then let L' be any of the existential closures of L'l. L', then, is an absolute referring expression of L which has n as a member of its extension. (ii) L is a relative term. There is an n-place predicate L" which has the same extension as E. (Example: 2In The Structure 93 Appearance Goodman writes: ”In founding a system, we must not only choose the primitives but also determine the range of individual variables--the realm of individuals recognized by the system. . . . The individuals recognized include all that satisfy at least one place of at least one of these primitive predicates, and in addition . . . all sums of such individuals.” Goodman(2), pp. 85-86. (Goodman is here concerned with systems which incorporate the "logic" of the part-whole relation; this accounts for the addition of all sums of such individuals.) I would prefer this sort of treatment. But although satisfaction is an intuitively straightforward concept, it does impose rather difficult theoretical prob— lems. Cf. Tarski(l), p. 371, n. 15 and Tarski(2), pp. 190- I93. ".l' (“tiff") .. _. {1“ uni- ; ~'n - 1r. 39 if L is 'between', L" is '<:> is between <:) and <:)'.) But L' can be obtained from L" as it was obtained from L'l in (i). Thus L' is an absolute referring expression with n as a member of its extension.3 The effect of the above argument is that: T9 RelRefExp(L,x).yeext(L,x) .+ (En)(Eu) {Ordn-ad(y,u).(z)[2cu + (Ew)(AbsRefExp(L,w). zeext(L,w))]}. But this rather cumbersome statement is not derivable from the foregoing axioms. Let us supplement our axioms so that it is. We will assume, roughly, that the members of the extension of a relative referring expression are all ordered n—ads of some subclass or other of the primitive domain of its language. More precisely, A3 RelRefExp(L,x).y=ext(L,x):+ (En)(u)[uey + (Ev) (Ordn-ad(u,v).VcPrimDom(L))]. This is a rather prodigal axiom. And in a less inelegant formalization it would appear as a derivative statement. The difficulty here is that we have not made a fine enough analysis of semantical and syntactical distinctions. Such an account should probably find it advantageous to 3The argument requires that first-order extensional languages have the apparatus for universal quantification and at least as many variables as places in any of its referring expressions. 40 take as its primitive concept satisfaction understood as relation between an n-place referring expression and an "nfplace" sequence of objects. Being the extension of a referring expression would then be a derivative concept. But this strategy has difficulties of its own, and their resolution seems rather remote from the concerns of this essay. It should be noted in passing that D8 is not intended to correspond to the way we ordinarily determine the primitive domain of a language, Generally the universe of a language is most conveniently specified by a metalinguistic statement, e.g., 'The universe of dis- course for L is . . . .', or 'The values of L's variables are . . . .', or 'T is a universal term for L'. But this in no way conflicts with the definition. For the aim of the definition is to explain what the universe of a language is, not how we might most conveniently determine what it is.“ We shall speak of subclasses of the primitive domain of a language as primitive selection classes of the language. Thus (where 'PrimSel(L,x)' abbreviates 'x is a primitive selection class of L'): D12 PrimSel(L,x) ++ xcPrimDom(L). “See n. l of this section. 41 Obviously, the extension of any absolute referring expres— sion for a language is a primitive selection class of the language. That is, T10 AbsRefExp(L,y).x=ext(L,y) .+ PrimSel(L,x). But we cannot make the broader claim that the extension of a referring expression for a language is a primitive selection class of the language. For, generally, the extension of a relative referring expression is not a prim- itive selection class of its language. Nor do we have the converse of T10. The primitive selection classes of a language need not be exhausted by the extensions of its absolute referring expressions. When they are not so exhausted, we have seen, the language is expressively in- complete. But it would be premature to explain expressive completeness at this point, for not enough has been said about relative referring expressions. ll. AUXILIARY DOMAINS AND AUXILIARY SELECTION CLASSES Suppose we were to understand by the auxiliary domain (or AuxDom) of a language the class all of whose members are ordered n—ads of some one or another of the primitive selection classes of the language. That is, AuxDom(L)=x(EY)(En)(PrimSel(L,y).Ordn-ad(X,y)). As an immediate consequence we should then have that: (l) XeAuxDom(L) ++ (Ey)(En)(PrimSel(L,y).Ordn—ad(x,y)). We should also have that the extension of any relative referring expression for a language is a subclass of the auxiliary domain of the language. That is, (2) RelRefExp(L,x) + ext(L,x)cAuxDom(L). We would not have, however, that the subclasses of the auxiliary domain of a language are all extensions of rela- tive referring expressions for the language. There are two reasons. First, the language may fail to be expres- sively complete. Second, some of these subclasses cannot even be appropriately regarded as extensions of relative referring expressions. These are classes whose members (relative to the primitive selection classes of which they are ordered n—ads) do not have the same number of positions. That is, these are classes that are not "relations" of A2 “3 members of the primitive domain of the language. Conse- quently, the explanation of 'auxiliary selection class' cannot be modeled after the explanation of 'primitive selection class'. A better course is to first introduce the concept of an auxiliary selection class and, then, explain the auxiliary domain of a language as the class of all its auxiliary selection classes. The auxiliary selection classes of a language are those classes all of whose members are, for some n, ordered n—ads relative to some primitive selection class or other of the language. Or, where 'AuxSel(L,x)' is short for 'x is an auxiliary selection class for L', D13 AuxSel(L,x) ++ (En)(u)[u€x + (Ev)(PrimSel(L,v). Ordn—ad(u,v))]. In analogy to (2), then, we have as a theorem that: Tll RelRefExp(L,x) + AuxSel(L,ext(L,x)). We can now explain the auxiliary domain of a language as the class of all its auxiliary selection classes. That is, D14 AuxDom(L)=xAuxSel(L,x). The relation between an auxiliary selection class of a language and the auxiliary domain of the language does not correspond to the relation between primitive selec- tion classes and primitive domains. The former is the class membership relation; the latter is the relation of classial inclusion. AM As an immediate consequence of Dll we have that: T12 xeAuxDom(L) ++ AuxSel(L,x). Hence we have that: T13 RelRefExp(L,x) + ext(L,x)eAuxDom(L), that is, that the extension of a relative referring ex- pression for a language is a member of its auxiliary domain. We do not have, however, that the members of the auxiliary domain of a language (i.e., its auxiliary selection classes) are all extensions of some relative referring expression or other of the language. The reason, of course, is just that the language may be expres- sively incomplete. We will understand by a selection class of a language any class which is either a primitive or auxiliary selection class of the language. Thus, where 'Sel' is short for 'selection class', D15 Sel(L,x) ++. PrimSel(L,x) v AuxSel(L,x). We have as a theorem, then, that: TlA RefExp(L,x).y=ext(L,x) .+ Sel(L,y). That is, the extension of a referring expression for a language is a selection class for the language. The domain (abbreviated 'Dom') of a language can now be explained as the class whose members are the selection classes of the language. D16 Dom(L)=XSel(L,x). As an immediate consequence we have that: “5 T15 XeDom(L) ++ Sel(L,x). Thus we have that the extension of a referring expression for a language is a member of the domain of the language. That is, T16 RefExp(L,x).y=ext(L,x) .+ yeDom(L). Parallel to T14 and T16 we have, of course, that extensions for a language are selection classes for the language and members of its domain. That is, T17 Exten(L,x) + Sel(L,x), and T18 Exten(L,x) + xeDom(L). It will be observed that the converses of TlA, T16, T17, and T18 do not hold. It is just in this case that we want to say that a language is expressively incomplete. We are now in a position to explain expressive com— pleteness. But before doing so, let us note a few conse- quences of A3 and the chain of definitions leading to the introduction of 'Dom(L)'. One such consequence is that two languages with the same auxiliary domain have the same primitive domain. That is, T19 AuxDom(Ll)=AuxDom(L2) + PrimDom(Ll)=PrimDom(L2). The converse of T19 holds as a matter of set theory (and defintions). Hence, we have that T20 AuxDom(Ll)=AuxDom(L2) ++ PrimDom(Ll)=PrimDom(L2). We also have that languages with the same primitive domain have the same domain. A6 T21 PrimDom(Ll)=PrimDom(L2) + Dom(Ll)=Dom(L2) I have a strong suspicion, but as yet have been unable to prove, that the converse of T21 is a theorem. That is, that T22* Dom(Ll)=Dom(L2) + PrimDom(Ll)=PrimDom(L2) is a theorm. T22* can be shown to hold, however, provided L and L l 2 that both languages have finite primitive domains. The satisfy either of three conditions. The first is second is that no relation (i.e., set of ordered ngads) is a member of either of their primitive domains. The third is that neither has a class as a member of their primitive domain. The second condition is a consequence of the third and is, of course, weaker. The first condition is indepen- dent of the others. Under either of these conditions we also have that T23* PrimDom(Ll)=PrimDom(L2) ++ Dom(Ll)=Dom(L2). An asterisk following a theorem name is to be taken as a signal that something less than theoremhood is claimed for the formulas they precede. For T22* and T23* the theorems in question are conditionals with suitable formulations of either one of the three conditions or their alternation. But from the standpoint of this essay the third, and possibly strongest, condition isolates an interesting class of languages. In section 1“ a number of claims for such languages will be made. Since these claims cannot be made for languages which satisfy only the first or second, the antecedent of those theorems is to be some suitable formu— lation of the third condition. 12. EXPRESSIVE COMPLETENESS The explanation of expressive completeness is now quite simple. A language is expressively complete if its extensions exhaust its selection classes. That is, where 'ExpComp(L)' is short for 'L is expressively complete', D17 ExpComp(L) ++ (y)(Sel(L,y) + Exten(L,y)). We then have as theorems that: T24 ExpComp(L) ++ (y)(Sel(L,y) ++ Exten(L,y)), and T25 ExpComp(L) ++ (y)(yeDom(L) ++ Exten(L,y)). (We also have as corollaries of T17 and T18 the results of replacing their occurrence of 'Exten(L,y)' by occurrences of '(Ez)Ext(L,yz)'.) We have often had occasion to use the expression 'expressively incomplete.’ It was used in the sense that a language is expressively incomplete if it is not expres- sively complete. Thus, where 'EprnComp(L)' is short for 'L is expressively incomplete', D18 EprnComp(L) ++ mExpComp(L). Then, as theorems, perhaps too obvious to mention, we have: T26 EprnComp(L) ++ (Ey)(Sel(L,Y).NExten(L,y)), and T27 EprnComp(L) ++ (Ey)(ysDom(L).mExten(L,y)). A7 us (Again we have as corollaries the result of replacing 'Exten(L,y)' by '(Ez)Ext(L,y,z)'.) 13. AN APPARENT ANOMALY There is a consequence of the preceeding analysis which may appear to be anomalous. This apparent anomaly can be presented and dispelled informally. Consider, for example, a language whose primitive domain is mankind. Suppose that the language is expres- sively complete. None of the statements which can be formulated within this language should, from the vantage point of semantics, be surprising. It can say that Socrates married Xanthippe, that Socrates is a bachelor, that Xanthippe is a shrew, etc. What is surprising, perhaps, are some of the things which the language cannot formulate. It cannot say, for example, that Socrates drank hemlock, that Socrates lived in Egypt, that Socrates is mortal and that Socrates is not an elephant. The explana- tion is quite simple. These latter statements, normally understood, concern (among other things) not only men but also hemlock, Egypt, mortals and elephants. Each statement has in the extension of one of its terms something which is not a man and, hence, something which does not belong to the primitive domain of the language under consideration. The source of the apparent anomaly is now clear. Objects “9 50 within the primitive domain of a language may belong to classes which are not included in that domain.1 There are three points which tend to mitigate the force of the anomaly. Note, first, that we are not distressed by the fact that a formal language whose primitive domain is the class of all numbers cannot formulate such statements as '5 is blue', '3 is colorless', etc. Yet we would not, at least on this account, condemn such languages to expressive in- completeness. I am aware that there are those who would regard such statements as meaningless. But this is simply not the case. Take as a typical example Russell's familiar 'Laziness drinks procrastination'. Consider then the following argument. Only animate objects drink anything. Laziness is an abstract object. No abstract object is a concrete object. Animate objects are all concrete. Therefore, laziness does not drink procrastination. 1This paragraph requires a much more careful formu- lation than I am now able to give. For example, I do not mean that such a language could formulate a statement synonymous with 'Socrates is a bachelor' in the sense of Trexpressing the same proposition." That is, the synonymy is not intensional but rather extensional. But by the extensional synonymy of statements I do not understand mere agreement of truth value. I think a concept of extensional isomorphism would provide a good avenue of approach to the solution of this problem. This concept would be explained in a manner parallel to Carnap's notion of intensional isomorphism. Carnap(5), secs. lA-15. 51 The premises of this argument are meaningful. Further, they are true. Since the argument is valid, its conclu- sion is also true. But the conclusion is the negation of 'Laziness drinks procrastination'. Hence, the latter statement is false. Hence it is meaningful. We might speculate that what seems "funny” about statements of this type is that they are patently false. Given any such statement it seems that one can always construct an agru— ment whose premises are incontrovertibly true and whose conclusion is the negation of that statement. The above discussion must be regarded with a proper air of suspicion. For surely the semantical and logical paradoxes have their origin (in part) in an overeagerness to grant meaningfulness to linguistic forms. Yet the examples2 with which I have been concerned do not appear to be of the type which so readily lend themselves to the 2I would be hard put to specify principles which would isolate the cases I have in mind. It is not simply that they are patently false; for 'Cats are all dogs' is patently false and patently not a case in point. As further examples I would count Chomsky's 'Colorless green ideas sleep furiously', 'Sincerity admires John', and 'John frightens sincerity'. Chomsky, p. 15 and p. 42. Ryle's 'There exist prime numbers and Wednesdays and public opinions and navies' (Ryle, p. 23) is problematic; for if it is false, it is not patently false. Cf. White, c. IV. But I would definitely want to exclude Carnap's 'Pirots karulize elatically' (Carnap(3), p. 2) and Lewis Carroll's familiar 'Twas brillig, and the slithy toves did gyre and gimble in the wabe'. These latter examples contain pur- ported terms which are in themselves meaningless. Whether Hempel's 'The Absolute is perfect' (Hempel, p. 51) is in this category is, of course, a matter of some debate. 52 paradoxes. That is, none of the terms employed need be explained impredicatively. Second, it is only in an intensional sense that the above language cannot formulate the statements that Socrates drank hemlock, that Socrates lived in Egypt, etc. It can formulate, for example, the statement that Socrates is a man-who-drank-hemlock. And this statement is equiva- lent to the statement that Socrates drank hemlock. Clearly, if Socrates is a man-who-drank—hemlock, he drank hemlock. And if Socrates drank hemlock, since he is by hypothesis a man, he is a man-who-drank-hemlock. Thus, Socrates is a man-who—drank-hemlock if, and only if, he drank hemlock. This argument is easily generalized. Third, to say that a language such as the one contem- plated above is expressively incomplete is to condemn every language to expressive incompleteness. The source of the supposed expressive incompleteness was that things in the primitive domain of the language were members of classes which are not subclasses of that domain. An expressively complete language would have to have the universal class as its primitive domain. But, as will be shown in section 16, such a language cannot be expressively complete. If the reader is not convinced that the paradox is dispelled, all is not lost. He can take expressive com- pleteness to be what we shall shortly call 'universal ex- pressive completeness.’ There are two noteworthy 53 consequences of this move. The first has already been noted. No language will be expressively complete. The second consequence is that the criterion of adequacy formulated in section 2 and defended in section 3 must be rejected. 14. EXPRESSIVE POWER One outcome of the analysis of expressive complete- ness is that languages can be compared with respect to their expressive power. A language will be said to be expressively complete relative to another if the selection classes of the latter are all extensions of the former. Where 'ExpCompRel(Ll,L2)' is short for 'L1 is expressively complete relative to L2', D19 ExpCompRel(Ll,L2) ++ (x)(Sel(L2,x) + Exten(Ll,x)). It is a theorem, then, that: T28 ExpCompRel(Ll,L2).ExpCompRel(L2,L3) .+ ExpCompRel(Ll,L3). That is, the relation is transitive. If it is granted that (ELl)(EL2)(ExpComp(Ll).EprnComp(L2). Dom(Ll)=Dom(L2)),l the relation is readily seen to be nonsymmetrical and non— reflexive. It is provable that two languages which stand in this relation to one another are both expressively com- plete. That is, T29 ExpCompRel(Ll,L2).ExpCompRel(L2,Ll) .v. ExpComp(Ll).ExpComp(L2), lSuch languages are easily "constructed"; but this statement does not follow from our axioms. 54 55 The converse is not provable. But we do have that: T30 ExpCompRel(Ll,L2).ExpCompRel(L2,Ll) .++. ExpComp(Ll).ExpComp(L2).Dom(Ll)=Dom(L2). That is, languages with the same domain are expressively complete if, and only if, they are expressively complete relative to one another. Consequently, T31 ExpComp(L) ++ ExpCompRel(L,L). That is, a language is expressively complete if, and only if, it is expressively complete relative to itself. A language is said to be as expressively powerful as another if all of the extensions of the latter are also extensions of the former. Where 'AsExpPow(Ll,L2)' abbre- viates 'L1 is as expressively powerful as L2', D20 AsExpPow(Ll,L2) ++ (x)(Exten(L2,x) + Exten(Ll,x)). This relation is transitive and reflexive. That is, T32 AsExpPow(Ll,L2).AsExpPow(L2,L3) .+ AsExpPow(Ll,L3), and T33 AsExpPow(L,L). But the relation is nonsymmetrical.2 It is provable that if the domain of a language is included within the domain of an expressively complete language, then the former language is expressively complete if it is as expressively powerful as the latter. That is, 2That this is so is not a theorem; but languages can be constructed which show that 'AsExpPow' is neither sym- metrical nor asymmetrical. 56 T34 Dom(Ll)cDom(L2).ExpComp(L2).AsExpPow(Ll,L2) .+ ExpComp(Ll). A language is said to have greater expressive power than another just in case it is as expressively powerful as the other, and the other is not as expressively power- ful as it. Thus, where 'GrExpPow(Ll,L2)' abbreviates 'Ll has greater expressive power than L2', D21 GrExpPow(Ll,L2) ++. AsExpPow(Ll,L2).xAsExpPow(L2,Ll). This relation is transitive, irreflexive and asymmetrical. That is, T35 GrExpPow(Ll,L2).GrExpPow(L2,L3) .+ GrExpPow(Ll,L3), and T36 mGrExpPow(L,L), and T37 GrExpPow(Ll,L2) + mGrExpPow(L2,Ll). It is provable that a language whose domain includes the domain of a language with greater expressive power is expressively incomplete. That is, T38 Dom(Ll)aDom(L2).GrExpPow(Ll,L2) .+ EprnComp(L2). A language is said to have less expressive power than another if the other has greater expressive power than it. Thus, where 'LsExpPow(Ll,L2)' is short for 'L1 has less expressive power than L2', D22 LsExpPow(Ll,L2) ++ GrExpPow(L2,L1). This relation is also transitive, irreflexive and asym- metrical. That is, 57 T39 LsExpPow(L L2).LsExpPow(L2,L3) .+ LsExpPow(Ll,L3) 1’ and T40 mLsExpPow(L,L), and T41 LsExpPow(Ll,L2) + mLsExpPow(L2,Ll). Languages which have as much expressive power as one another are said to have equivalent expressive power. Where, 'EqupPow(Ll,L2)' abbreviates 'L1 and L2 have equivalent expressive power', D23 EqupPow(Ll,L2) ++. AsExpPow(Ll,L2).AsExpPow(L2,Ll). This relation is transitive, reflexive and symmetrical. Thus, T42 EqupPow(L L2).EqupPow(L2L3) .+ EqupPow(L1,L3), l’ and T43 EqupPow(L,L) and T44 EqupPow(Ll,L2) + EqupPow(L2,Ll). It is provable that if two languages with the same domain are equivalent in expressive power, then either both or neither are expressively complete (or incomplete)_ That is, T45 Dom(Ll)=Dom(L2).EqupPow(Ll,L2) .+. ExpComp(Ll) ++ ExpComp(L2), and T46 Dom(Ll)=Dom(L2).EqupPow(Ll,L2) .+. EprnComp(Ll) ++ EprnComp(L2). 58 We also have that if two languages are equivalent in expressive power, they have the same extensions, the same selection classes, and the same domain. That is, T47 EqupPow(Ll,L2) + (x)(Exten(Ll,x) ++ Exten(L2,x)), T48* EqupPow(Ll,L2) + (x)(Sel(Ll,x) ++ Sel(L2,x)), and T49* EqupPow(Ll,L2) + Dom(Ll)=Dom(L2).3 Of these, only the converse of T47 holds. Thus, T50* EqupPow(LlL2) ++ (x)(Exten(Ll,x) ++ Exten(L ,x)). Further, if a language is equivalent in expressive power to an expressively complete language, then it is itself expressively complete. T51* EqupPow(Ll,L2).ExpComp(L2) .+ ExpComp(Ll) Similarly, a language equivalent in expressive power to an expressively incomplete language is expressively in- complete. T52* EqupPow(Ll,L2).EprnComp(L2) .+ EprnComp(Ll). Two expressively complete languages are equivalent in expressive power if, and only if, they have the same selection classes and the same domain. That is, T53* ExpComp(Ll).ExpComp(L2) .+. EqupPow(Ll,L2) ++ (x)(Sel(Ll,x) ++ Sel(L2,x)), T54* ExpComp(Ll).ExpComp(L2) .+. EqupPow(Ll,L2) ++ Dom(Ll)=Dom(L2). The converses of T54* and T55* do not hold. 3See the last paragraph of section 11. 59 Given two languages, it may be the case that neither stands in any of the above relations to the other. For this reason it is desirable to be able to speak of a lan- guage as being expressively complete (or incomplete) with respect to a class of objects regardless of whether the class is identical with the primitive domain of the lan- guage. The remainder of this section is devoted to showing how this is possible. Just as we can speak of the domain of the universe of a language so we can speak of the domain of any class of objects. Here we give an analogous, if condensed, explana- tion of this broader concept. Where, 'dom(x)' is short for 'the domain of x', D24 dom(x)=y{ycx v (En) (u)[uey + (Ev)(v:x.0rdnead(u,v))1}. Then, a language is expressively complete with respect to a class (ExpComRes) just in case the extensions of the language exhaust the members of the domain of that class. That is, D25 ExpCompResp(L,x) ++ (y)(yedom(x) + Exten(L,y)). It is a theorem that: T55 ExpCompResp(L,PrimDom(L)) +v ExpComp(L). That is, a language is complete with respect to its primi— tive domain if, and only if, it is eXpressively complete. Also, a language is expressively complete if, and only if, it is expressively complete with respect to all of its primitive selection classes. That is, 60 T56 ExpComp(L) ++ (x)(PrimSel(L,x) + ExpCompResp(L,x)). And in general, T57 ExpCompResp(L,x) ++ (y)(y:x + ExpCompResp(L,y)). That is, a language is expressively complete with respect 1 I J? (I) to a class of objects if, and only if, it is expressiv complete with respect to all of its subclasses. It would perhaps be even more desirable to develcp a metrical concept of expressive power. Such a concept would assign a numerical value to each language; this value would be its degree of expressive power. A concept of this sort would allow for the comparison of any two languages with respect to their expressive capacity. But how this concept is to be explicated is by no means evident. Per- haps the most natural attack is to assign each language a simple proportion, the ratio of the number of its exten- sions to the number of its selection classes. But this is not viable; for any language with a non-null domain has an infinite number of auxiliary selection classes and, hence, an infinite number of selection classes. Further exploration of this material concept and the difficulties which are occasioned by its analysis fall out- side the scope of the present essay. We conclude by noting that there is a class with respect to which every language having at least one referring expression is expressively complete. 61 T58 (Ex)(L)[(Ey)RefExp(L,y) + ExpCompResp(L,x)]. The case in point is, of course, the null class. (The assumption that every language has at least one referring expression is rather modest. But to my knowledge it does not follow from the above axioms.) 15. THE ELIMINATION OF GENERAL TERMS In this section it is shown that for any language L with a finite primitive domain, there is a language L* which (a) has the same primitive domain as L, (b) is as expressively powerful as L, and (c) has no extralogical general terms.1 Let the primitive domain of L and L* be {al,...,an}. Thus, both L and L* have the same finite primitive domain. Let 'al', . . . , and 'an' be singular terms of L* whose respective extensions are the unit classes of al, . . . , and an. Let 'F' be any general term of L. Then 'F' is either an absolute or a relative term. (i) 'F' is an absolute term. The extension of 'F' is either the null class, a unit class of either a1, . . ., or an, (say {ai}), or an n—membered (l}), or one ,...,a 11 1k of L's mrmembered (l,...,}). If the extension of 'F' l...x ' can be explained as: n (EY)(y#y.xl=y. ... .xk=y). If the extension of 'F' is kai ,...,ai >}, then 'Fx ... can be explained as xl=ail . ... . x =a If the extension of 'F' is l l m m 1,..O’am>,000’}, then 'Fxl...xn' may be explained as {5)which the rudimentary language cannot seems properly to be ascribed to the superiority of a complete language over a fragmentary one. Is this, however, a proper account of the difference between a simple and a highly develOped language? And if it is, must we not be forced to say that for all we know every language is incomplete? A counterquestion is in order here: Incomplete with reference to what standard? Unless there exists a wider language of which a given symbolism is a part, we have no standard in relation to which it is incomplete. Further, even the fact that a symbolism L is part of another, L , does not necessarily make L incomplete, although it may be inadequate for certain purposes. The language of arithmetic can be said to be part of the language of real numbers; it lacks certain symbols and the rules for their usage. But although arithmetic is inadequate for certain purposes, e.g., for solving algebraic equations, it is not an incomplete arithmetic. NO parts of it are missing, as there would be from a sym- bolism which purported to be our arithmetic but which lacked the operation 4x4. Taken by itself it is the whole language. It is completely unlike a dictionary with missing pages. Any inadequacy which at a given moment a language comes to have is not due to incomplete- ness. The classification "incomplete" (and hence also the classification "complete") is not properly applicable 80 to a language. A symbolism which purports to be a language but which has missing parts can be called incomplete, but a language L does not become in- complete when it becomes a part of L2, because it does not purport to be L . It is a whole even though additions are made to it, since these addi— tions do not supply missing parts. To repine that, for all we know, every language may be incomplete is to indulge in the absurd complaint that a whole language is perhaps not a whole language. Further- more, it sounds as though a remedy may be needed, whereas there is no completing what is already a whoLe. We can add to it; but we cannot complete it. The kindest thing we can say is that it is suggestive. Some passages are more direct in their reference to expressive completeness. For example, within the context Of a general discussion of formal systems, Copi writes: When the formal system has been constructed, the question naturally arises as to whether or not it is adequate to the formulations of all propositions it is intended to express. If it is, it may be said to be 'expressively complete' with respect Ln that subject matter. We are here discussing what can be said in the system, not what can be proved. With respect to a given subject matter, a formal system is 'expressively complete' when it is possible to assign meanings to its undefined terms in such a way that every proposition about that subject matter can be expressed as a formula of the system.1 It should be noted that Copi does not explain what he understands by such obviously crucial terms as 'meaning', 'proposition', 'expresses a proposition', 'prOposition a system is intended to express', and 'subject matter'- He l“Ambrose, pp. 30-31. 15Copi, pp. 178—179. Author's emphasis. 81 does, however, explain that the functional completeness of various sets of statement connectives is one kind of 16 expressive completeness. Elsewhere, and along these same lines, we find Fraenkel and Bar-Hillel writing that the notion of notational (or expressive) nnn— pleteness with respect Ln n given subject matter deserves at least to be mentioned. Its meaning should be clear. As an illustration, let us only mention that the propositional calculus, based upon '3' and 'N' as the only connectives, is notationally complete with respect to the truth functions: in short, is truth functionally com- plete, since it can easily be shown that all truth functions are expressible on this base.17 When the claim is made that "its meaning should be clear" it is not intended that its meaning should be clarified but that it Ln clear. If that claim is correct, this whole essay becomes pointless. But I think that it is not. There is only one place in the literature known to me where we have an analysis of what might be thought of as expres- l8 sive completeness. The analysis is Tarski's. He speaks not of expressive completeness but of the completeness 9L concepts. In the next section we offer a brief account of his analysis. 161818., p. 192. l7Fraenkel, p. 295, n. 3. Author's emphasis. 18Tarski(l), "Some methodological investigations on the definability Of concepts," pp. 296-319. 19. TARSKI AND THE COMPLETENESS OF CONCEPTS In this section we Offer a brief explanation of Tarski's analysis of the completeness of concepts. It is argued that although his analysis isolates an important concept it does not provide an explication of expressive completeness. We conclude by showing that these two con- cepts, expressive completeness and completeness of con- cepts,are related in a rather definite way. Let n be an extralogical constant and let 8 be a set of such constants. Further, let p (v,8) be any Open sentence which has X as its only free variable and no extralogical constants other than those of 8. Then, (V)(v=c ++ ¢ (V,8))l is said to be a (possible) definition nL 3 Ln terms 9L L. Where Z is a set of sentences, p is said to be definable Ln terms 9L L nn the basis 9L t, if (1) n and each member of 8 occurs in some member or other of Z, and (2) some definition of n in terms of 8 is derivable from the sentences of E. lHere '++', etc. are used autonymously. Note that Tarski's account is directly applicable only to those sets of sentences containing these signs. 82 83 Where 21 and 22 are sets of sentences, let 8 be the 1 set of extralogical constants occurring in the members of 21 and let 82 be the set of extralogical constants occurring in the members of 2 Then, 21 Ln essentialLy richer than 2 2° 2 with respect to specific terms if (1) 22:21, and (2) there is some extralogical constant n such that ce8l, mce82, and p is not definable in terms Of 82 on the basis of 21. Finally, a set of sentences 2 Ln complete with l respect Ln its specific terms if there is no set of sen- is categorical, and 22 is essen- with respect to specific terms.2 tences 22 such that 22 tially richer than :1 Tarski does not argue the adequacy of his analysis. But categoricity is of great importance; and we must agree with Tarski when he writes that a "non-categorical set of sentences (especially if it is used as an axiom system for a deductive theory) does not give the impression of a closed and organic unity and does not seem to determine 2Very roughly, a set of sentences is categorical if any two normal interpretations of the set are isomorphic. For brief but careful formulations of this concept see Carnap(2), pp. 173—174, Church, pp. 317—332, and Mendelson, pp. 90-91. An interpretation is normal if it assigns the identity relation of its universe to ' ='. The isomorphism of two interpretations (not to be confused with an isomor- phism of their universes) can be explained as follows. Let . . , and c be the terms occurring in the members 0} a Cget of sentences S. Consider two interpretations Of S; I in the universe U and I* in the universe U*. Then I and I* are isomorphic interpretations of S if there is a function L which establishes an isomorphism between U and U* such that: for any c (1 < i < m), x, ., and Xn’ (i) if 01 is an absolute term, then Xeext(c:7 under I if and only if f(x)eext(ci ) under I*, and (ii) if c is a rela- tive term, then eext(c ) under I ii and only if cext(ci ) under 1* 84 precisely the meaning of the concepts contained in it."3 We must also agree with Carnap when he makes what seems to be the inverse claim that a categorical axiom system "specifies all the structural properties of its possible models."Ll There can be no doubt that Tarski's complete- ness of concepts is one (important) kind of completeness. But I do not think that it is the kind of completeness we want to speak of as expressive completeness. I have two reasons. First, Tarski's concept is applicable to uninter- preted sets of sentences. But sentences of such sets do not express any thing. Thus it seems imprOper to speak of sets of these sentences as being expressively complete. An Obvious, if inadequate, rejoinder is that sets of sen- tences which satisfy Tarski's conditions need only satisfy one further condition to be expressively complete. That condition is that they be interpreted. Thus, one might argue, if Tarski has not explicated expressive complete- ness, he has come very near to doing so. My second reason shows that this line of thought is mistaken. If we may speak loosely, and use the terminology Of Copi, a language (or theory) is not expressively complete unless it has the apparatus to express all the propositions about its subject matter. But Tarski has shown that there 3Tarski(l), p. 311. ”Carnap(2), p. 174. 85 are postulate sets for the real numbers which are complete 5 with respect to their specific terms. Relative to such systems, as is well known, there are real numbers which cannot be designated. If n is such a real number, then that (Ex)x=r is a proposition which cannot be expressed within the system. Thus the language system cannot be expressively complete. Hence, eXpressive completeness and completeness with respect to concepts are distinct. Let me remind the reader that Tarski's analysis is not being criticized as being inadequate or unimportant. My point is simply that it is not an adequate analysis Of expressive completeness. And, so far as I know, Tarski has never claimed that it is. It has been noted that the two analyses are con- cerned with different kinds of systems. Tarski is pri- marily concerned with nnLnn systems (though his concept is applicable to sets of sentences). Expressive complete- ness, however, was understood to be applicable only to language systems. And as was suggested earlier6 only those axiom systems which are interpreted are to be regarded as languages, and only those languages with axioms are to be regarded as axiom systems. We have seen that an interpreted axiom system which is complete with respect to its specific terms need not be 5Tarski(l), pp. 313-31u. 6Section 1. 86 expressively complete. It is quite natural at this point to inquire whether the converse holds, that is, whether an expressively complete axiom system is always complete with respect to its specific terms. It should be clear that this is not the case. Whether an axiom system is expressively complete is independent Of the character of its axioms.7 There is no logical restriction governing which of a system's sentences one selects as its axioms. A system's axioms may be as meager and uncategorical as one chooses. But even if the converse does not hold we can establish the following, somewhat related, claim. (1) If L is expressively complete, then there is an L' such that (i) L and L' are equivalent in expressive power, and (ii) there is a set x of true sentences of L' such that X is complete with respect to its specific terms. In order to establish this it will be convenient to make use of a theorem proved by Tarski.8 (2) If 2 is a monotransformable set of sentences, then 2 is complete with respect to its specific terms.9 Thus, to establish (1) it will suffice to show that: 70f. Sec. 1. 8Tarski(l), pp. 313-317. 9Roughly again, a set of sentences is monotrans- formable if there is at most one way of establishing an isomorphism between any two of its interpretations. 87 If L is expressively complete, then there is an L' such that (i) L and L' are equivalent in expressive power, and (ii) there is a set, E, of true sentences of L' such that 2 is monotransformable. Let us suppose that L is an expressively complete language. Then, by the reasoning of section 17, we know that L has a finite primitive domain. Let al, a2, . . . , and an be the n distinct members of that finite domain. Since L is expressively complete, L has referring expres- sions, Al, A2, . . . , and An, whose extensions are respectively {a1}, {a2}, . . . , and {an}. Let us now con- sider another language, L'. As will be shortly evident, L' is to have standard apparatus for quantification and truth—functional combination. The only terms of L' are to be '=', B , and Bn' '=' is to be understood as 1’ B2: having the ordered pairs , , , and 2 as the members of its extension. Further, the sentence '(x)x=x' is to be understood as true for L'. Thus, L and L' are to have the same primitive domain. Each B1 is to be explained in such a way that its extension is {a1}. Reasoning in the manner of section 15, we see that L' is expressively complete. Since L and L' have the same primitive domain and are both expressively complete, (by theorem 5 of section 14) (i) L and L' are equivalent in expressive power. It is clear that the following are true sentences of L'. 88 I (x)(le v B x v ... v an) 2 (Ex)[le . (y)(Bly + X=y)1 (Ex)[82x - (y)(82y + X=y)1 II . (Ex)[an . 1 (x)[le + (mB2x . mB3x . ... . man)] (x)[B2x + (mle . mB3x . ... . man)] III . (x)[an + (mle . NB2x . ... . mBn_lx)] Let X be the set of these sentences. Suppose, now, that we have two normal interpretations Of Z, I and 1*. If these interpretations render the sentences of S true, they are interpretations in nfmembered universes (or prim- itive domains), say, U and U* respectively. Let 81 be the subset of U which is assigned (as extension) to Bi under I. Similarly, let 8* be the subset of U* which is assigned 1 s to Bi under I . Clearly, Bi 1 is guaranteed by Group II above. Also, Bj=Bk if, and only szk’ and BJ.=Bk if, and only if, 8*j=8*k. This is guaranteed by Group III above. Think now Of bi as the as the sole member of 8* and 8* are unit sets. This if B sole member of 81 and b* It i i' is Obvious at this point that if a relation is to estab- lish an isomorphism between I and 1*, it can do so only by pairing bi with b*i. Thus there is only one way of establishing an isomorphism between I and I*. That is, 89 Z is monotransformable. Thus, (ii) There is a set, 2, of true sentences of L' such that E is monotransformable. This completes the proof of (3) and therewith the proof of (1). It is perhaps Of interest to Observe that just as the reasoning of section 15 shows how to construct an expressively complete language for a finite universe, so the above reasoning shows how to construct a complete theory for a finite universe.lo But there can be little comfort in this. For just as the language prescribed is cumbersome and unwieldy,so the theory prescribed is in- elegant and complex. It has, after all, more primitive terms and more primitive sentences than there are entities in its universe.ll loCompleteness, here, can be understood either as completeness with respect to specific terms or monotrans- formability. Tarski mentions as unsolved the problem whether only monotransformable sets Of sentences are complete with respect to their specific terms, i.e., whether the converse of (2) holds. 11We can, of course, take a conjunction Of sentences in Z and trivially reduce the number Of axioms to one. (Though this may require the addition of a new rule of inference.) A more genuine economy is effected simply by omitting any one (but not more than one) of the sentences in group III. APPENDICES [Einilv n. ALE-7U ...x: -x 9O A1 A2 A3 AXIOMS Ext(L,y,x).Ext(L,z,x).+ y=z RelRefExp(L,x) + (Ey)Ext(L,y,x) RelRefExp(L,x).y=ext(L,x):+ (En)(u)[uey + (Ev) (Ordn-ad(u,v).vCPrimDom(L))] 91 D1 D2 D3 D4 D5 D5' D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D20 D21 D22 DEFINITIONS RefExp(L,x) ++ (Ey)Ext(L,y,x) ...ext(L,x)... ++ (EY)(Ext(L,y,x)....y...) Exten(L,x) ++ (Ey)Ext(L,x,y) = {{x},{x,y}} Ord2-ad(x,y) ++ (Eu)(Ev)(x=.y={u,v}) Ordn—ad(x,y) ++ (Ex)(EY)(EZ)(Ordn-l-ad(x,y). u=.v=yu{z}) OrdPr(x) ++ (Ey)Ord2-ad(x,y) FirstCoord(x,y) ++ (Eu)(Ev)(y=.x=u) SecndCoord(x,y) ++ (Eu)(Ev)(y=.x=v) Ancestor(x,y) ++ (z)[yez.(u)(v)(uez.VEu.+ Vez).+ Xez] AbsRefExp(L,x) ++. RefExp(L,x).NRelRefExp(L.x) PrimDom(L)=X(Ey)(AbsRefExp(L,y).xeext(L,y)) PrimSel(L,x) ++ xcPrimDom(L) AuxSel(L,x) ++ (En)(u)[uex + (Ev)(PrimSel(L,v). Orda-ad(u,V))l AuxDom(L)=XAuxSel(L,x) Sel(L,x) ++. PrimSel(L,x) v AuxSel(L,x) Dom(L)=XSel(L,x) ExpComp(L) ++ (y)(Sel(L,y) + Exten(L,y)) ExplnComp(L) ++ «ExpComp(L) ExpCompRel(Ll,L2) ++ (x)(Sel(L2,x) + Exten(Ll,x)) AsExpPow(Ll,L2) ++ (x)(Exten(L2,x) + Exten(Ll,x)) GrExpPow(Ll,L2) ++. AsExpPow(Ll,L2).mAsExpPow(L2,Ll) LsExpPow(Ll,L2) ++ GrExpPow(L2,Ll) 92 D23 D24 D25 D26 93 EqupPow(Ll,L2) ++. AsExpPow(Ll,L2).AsExpPow(L2,Ll) dom(x)=y{ycx v (En)(Eu)[uey + (Ev)(m:x.Ordn—ad(u,v))]} ExpCompResp(L,x) ++ (y)(yedom(x) + Ext(L,y)) UniverExpComp(L) ++. ExpComp(L).(x)x£PrimDom(L) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 T22* T23* T24 T25 THEOREMS RefExp(L,x) e»(Ey)EExt(L,y,x).(z)(Ext(L,z,x) + z=y)] RefExp(L,x)‘++(Ez)z=ext(L,x) RelRefExp(L,x) + RefExp(L,x) RelRefExp(L,x) + (Ex)y=ext(L,x) AbsRefExp(L,x) + (Ey)y=ext(L,x) RefExp(L,x) ++. AbsRefExp(L,x) v RelRefExp(L,x) (Ey)Ext(L,y,x) ++. AbsRefExp(L,x) v RelRefExp(L,x) XePrimDom(L) ++ (Ey)(AbsRefExp(L,y).xeext(L,y)) RelRefExp(L,x).yeext(L,x) +. (En)(Eu){Ordn-ad(y,y). (z)[zeu + (Ew)(AbsRefExp(L,w).zEext(L,w)))]} AbsRefExp(L,y).x=ext(L,y) .+ PrimSel(L,x) RelRefExp(L,x) + AuxSel(L,ext(L,x)) XeAuxDom(L) ++ AuxSel(L,x) RelRefExp(L,x) + ext(L,x)eAuxDom(L) RefExp(L,x).y=ext(L,x) .+ Sel(L,y) XeDom(L) ++ Sel(L,x) RefExp(L,x).y=ext(L,x) .+ yeDom(L) Exten(L,x) + Sel(L,x) Exten(L,x) + XeDom(L) AuxDom(Ll)=AuxDom(L2) + PrimDom(Ll)=PrimDom(L2) AuxDom(Ll)=AuxDom(L2) ++ PrimDom(Ll)=PrimDom(L2) PrimDom(Ll)=PrimDom(L2) + Dom(Ll)=Dom(L2) Dom(Ll)=Dom(L2) + PrimDom(Ll)=PrimDom(L2) PrimDom(Ll)=PrimDom(L2) ++ Dom(Ll)=Dom(L2) ExpComp(L) ++ (y)(Sel(L,y) ++ Exten(L,y)) ExpComp(L) ++ (y)(yeDom(L) ++ Exten(L,y)) 94 T26 T27 T28 T29 T30 T31 T32 T33 T34 T35 T36 T37 T38 T39 T40 T41 T42 T43 T44 T45 95 EprnComp(L) ++ (Ey)(Sel(L,y).mExten(L,y)) EprnComp(L) ++ (Ey)(yeDom(L).mExten(L,y)) ExpCompRel(Ll,L2).ExpCompRel(Ll,L2) .+ ExpCompRel(Ll,L3) ExpCompRel(Ll,L2).ExpCompRel(L2,Ll) .+. ExpComp(Ll).ExpCOmp(L2) ExpCompRel(Ll,L2).ExpCompRel(L2,Ll) .++ ExpComp(Ll).ExpComp(L2).Dom(Ll)=Dom(L2) ExpComp(L) +» ExpCompRel(L,L) AsExpPow(Ll,L2).AsExpPow(L2,L ) .+ AsExpPow(Ll,L3) 3 AsExpPow(L,L) Dom(Ll)cDom(L2).ExpComp(L2).AsExpPow(Ll,L2) .+ ExpComp(Ll) GrExpPow(Ll,L2).GrExpPow(L2,L3) .+ GrnxpPow(Ll,L3) mGrExpPow(L,L) GrExpPow(Ll,L2)->%GrExpPow(L2,Ll) Dom(Ll)CDom(L2).GrExpPow(Ll,L2) .+ EprnComp(L2) LsExpPow(Ll,L2).LsExpPow(L2,L3) .+ LsExpPow(Ll,L3) mLsExpPow (L,L) LsExpPow(Ll,L2) + mLsExpPow(L2,Ll) Y. 1) P EqupPow(Ll,L2).EqupPow(L2,L3) . Equp ow(Ll,L3) EqupPow(L,L) EqupPow(Ll,L2) + EqupPOw(L2,Ll) Dom(Ll)=Dom(L2).EqupPow(Ll,L2) .+. ExpComp(Ll) ++ ExpComp(L2) T46 T47 T48* T49* T50* T51* T52* T53* T54* T55 T56 T57 T58 T59 96 Dom(Ll)=Dom(L2).EqupPow(Ll,L2) .+. ExplnComp(Ll) ++ EprnComp(L2) EqupPow(Ll,L2) + (x)(Exten(Ll,x) ++ Exten(L2,x)) EqupPow(Ll,L2) + (x)(Sel(Ll,x) ++ Sel(L2,x)) EqupPow(Ll,L2) + Dom(Ll)=Dom(L2) EqupPow(Ll,L2) ++ (x)(Exten(L2,x) «+ Exten(Ll,x)) EqupPOw(Ll,L2).ExpComp(L2) .+ ExpComp(Ll) EqupPow(Ll,L2).EprnComp(L2) .+ EprnComp(Ll) ExpComp(Ll).ExpCOmp(L2) .+. EqupPow(Ll,L2) ++ (x)(Se1(Ll,x) ++ Sel(L2,x) ExpComp(Ll).ExpComp(L2) .+. EqupPow(Ll,L2) ++ Dom(Ll)=Dom(L2) ExpCompResp(L,PrimDom(L)) ++ ExpComp(L) ExpComp(L) ++ (x)(PrimSel(L,x) + ExpCompResp(L,x)) ExpCompResp(L,x) ++ (y)(yex + ExpCompResp(L,y)) (Ex)(L)[(Ey)RefExp(L,y) + ExpCompResp(L,x)] mUniverExpComp(L) BIBLIOGRAPHY 97 BIBLIOGRAPHY Only those works which have been explicitly mentioned are cited. Ambrose, Alice. "The Problem of Linguistic Inadequacy," Philosophical Analysis, (ed.) Max Black. Bar-Hillel, Y. See Fraenkel, A., and Y. Bar—Hillel. Black, Max (ed.). Philosophical Analysis. Englewood Cliffs, New Jersey: Prentice-Hall, 1963. Carnap, Rudolph(1). Introduction to Semantics. Cambridge: Harvard University Press, 1942. (2). Introduction to Symbolic Logic. New York: Dover, 1958. (3). Logical Syntax Of Language. New York: Harcourt, Brace; and London: K. Paul, Trench, Trubner, 1937. (4). "Meaning Postulates," PhiloSOphical Studies 3 (1952), 65-73. Reprinted in Carnap(3). (5). Meaning and Necessity. Chicago: University of Chicago Press, 1947. Chomsky, Noam. Syntactic Structures. 'sGravenhage: Mouton and CO., 1957. Church, Alonzo. Introduction to Mathematical Logic. Vol. I. Princeton: Princeton University Press, 1956. Copi, Irving. Symbolic Logic. New York: The Macmillan Company, 1954- Fraenkal, A., and Y. Bar-Hillel. Foundations of Set Theory. Amsterdam: North-Holland, 1958. Frege,Gottlob. The Foundations of Arithmetic. 2d ed. Translated by J. L. Austin. New York: Harper and Brothers, 1953- Goodman, Nelson(l). "Condensation versus Simplification," Theoria, 27 (1961), 47-48. (2). The Structure of Appearance. Cambridge: Harvard University Press, 1951. 98 99 Leonard, Henry S. Principles of Right Reason. New York: Henry Holt and Company, 1957. Lewis, C. I. "The Modes of Meaning," Philosophy and Phenomenological Research, 4 (1944), 236-249. Reprinted in Linsky. Linsky, Leonard (ed.). Semantics and the Philosophy of Language. Urbana: University of Illinois Press, 1952. Mates, Benson. "Synonymity," University of California Publications in Philosophy, 25 (1950), 201-226. Reprinted in Linsky. Quine, W. V.(1). From a Logical Point of View. Cambridge: Harvard University Press, 1953. (2). Mathematical Logic. New York: Norton, 1940; rev. ed., Cambridge: Harvard University Press, 1951. (3). Methods of Logic. Rev. ed. New York: Henry Holt and Company, 1959. (4). "Reduction to a Dyadic Predicate," Journal of Symbolic Logic, 19 (1954), 180-182. (5). Word and Object. New York and London: Published jointly by the Technology Press of The Massachusetts Institute of Technology and John Wiley & Sons, Inc., 1960. Ryle, Gilbert. The Concept of Mind. London: Hutchinson's University Library, 1949. Tarski, Alfred(l). Logic, Semantics, Metamathematics. Oxford: Clarendon Press, 1956. (2). "The Semantic Conception of Truth and the Foundations of Semantics," Philos. and Phenom. Research, 4 (1944), 343-375. Reprinted in Linsky. TATE U ”Willi u! 293 WIIEIIIWIIIIW Ill! 5 3 O 8 5 14 7 5