Toward a Systematic Pragmatics

TOWARD A S Y S T E M A T I C PRAGMATICS BY

R. M. M A R T I N Defiartment of Philosophy, University of Pennsyluania, Philadelphia

1959

N O R T H - H 0LLAND P U B L I S H I N G C O M P A N Y AMSTERDAM

For E R N E S TNAGEL

my teacher and friend

INTRODUCTION The logical study of a language or of a language-system is usually one of three kinds. It is either syntactical, semantical, or $ragmatical. In syntax one is interested exclusively in the signs or expressions of the language and their interrelations. In semantics, which presupposes syntax and contains it as a part, one is concerned not only with expressions and their interrelations but also with the objects which the signs denote or designate or stand for in one way or another. Finally, in pragmatics, there is reference not only to the signs and what they denote but also to the speakers or users of the language. Just as all syntactical notions reappear as semantical ones, so perhaps do all semantical notions reappear as pragmatical ones. Or it may be rather that only syntactical notions need reappear in pragmatics. In fact, we shall have two kinds of pragmatics to consider, depending upon whether a semantics or only a syntax is presupposed. One kind, that presupposing a semantics, is presumably the most inclusive discipline devoted to the formal study of language. Syntax and semantics have undergone an intensive systematic development during the past few decades, to which philosophers and mathematicians have alike contributed. But no comparable development has taken place in pragmatics. Mathematicians have had little interest in the subject, and philosophical comment on pragmatics, valuable though it has been, has lacked the clarity and precision we have come to expect in exact syntactical and semantical investigations. In this book several tentative but systematic theories of pragmatics of a restricted kind are presented. Each theory is systematic in the sense that it is expressed within a formalized meta-language, the full structure of which is specified. Each meta-language is called a pragmatical meta-language and consists of either a syn-

XI1

INTRODUCTION

tactical or a semantical meta-language augmented in certain respects. Under investigation in each meta-language is a clear-cut objectlanguage, the pragmatical theory of which is to be formulated. Whether the object-language be a formalized or partially formalized, historically given natural language, or a fully formalized deductive system of the kind studied by mathematicians and formal logicians, is immaterial. But it is presupposed that the object-language is at least formalized to the extent of having its logical character specified. By this is meant that the primitives of the language must be given as well as a definition of ‘sentence’ and perhaps ‘term’ of that language. Also it must be specified whether the language is of first order, or of second order, or of still higher order. Perhaps also the non-logical axioms may be specified. But even if they are not the methods developed here should still be applicable. The pragmatical meta-languages formulated are thought to be of interest in providing partial explications of some important notions which have been neglected in the recent semantical literature, namely, notions connected with or akin to what has often been called the subjective intension of terms. According to J. N. Keynes, eg., the subjective intension or connotation of a name consists of “those properties which in the mind of any individual are associated with the name in such a way that they are normally called up in idea when the name is used” [Formal Logic, 2nd ed., 1887, p. 241. The subjective intension of a term is relative to the person who uses the term, and perhaps also to the time a t which it is used. What may for the moment be called the objective intension, on the other hand, is presumably not relative in these ways but is the same for all users a t all times. Perhaps objective intensions may be suitably defined in terms of subjective ones. Recently many theories of intension have been developed, notably those of Carnap and Church, but notions connected with subjective or relative intension have been almost wholly neglected. They have been dismissed as of merely psychological interest. Here we attempt to characterize some notions akin to those of subjective intension in a precise way and on the basis of what

INTRODUCTION

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appears to be a secure foundation in empirical science. If one approaches the problems of intension from such a point of view, the neglected notions of subjective or relative intension and allied notions may not be without significance. An exact or nearly exact characterization of them may help to get at the experiential roots of some of the complex notions which enter into meaning. Within the pragmatical meta-languages to be formulated a rather extensive theory of notions akin to that of subjective intension may be developed. These meta-languages provide what appears to be a first attempt a t formulating such a theory in a rigorous way. The point of view adopted throughout is wholly extensional. The pragmatical meta-languages formulated, in other words, are extensional meta-languages. Such intensional objects as propositions, properties, class-, relational, or individual concepts, are in no way taken as values for variables. Many important advantages accrue to extensional as over and against intensional meta-languages. We have spoken of semantics here primarily in the sense of the theory of denotation or reference, and we have spoken of subjective and objective intensions as being pragmatical in nature. But theories of absolute intension or meaning are also often regarded as a part of semantics. Absolute intensions, like objective ones, do not depend upon a specific user or time. Further, they are semantical rather than pragmatical. Usually theories of absolute intension are given only within intensional semantical meta-languages (containing propositions, properties or class-concepts, etc., as values for variables). But a theory of notions closely akin to those of absolute intension may be given within wholly extensional metalanguages. Such a theory we attempt to develop here. Although a good deal of systematic work has been done on the semantical theory of absolute intensions, no widely accepted or perhaps wholly satisfactory formulation of it has yet been given. Most formulations have lacked the clear-cut simplicity of syntax and of the semantical theory of designation. Here the theory of what is essentially a theory of absolute intensions is developed wholly within a very simple and restricted theory of designation. No new primitives are needed, nor are any axioms required other than those of the underlying theory of designation. Within a pragmatics

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INTRODUCTION

containing such a semantics as a part, then, we may explicitly compare and contrast absolute, objective, subjective, and perhaps still other kinds of intensions. In Chapter I the two-fold nature of pragmatics is discussed, as a discipline intimately bound up with logical analysis on the one hand and with the empirical study of language on the other. Different levels of pragmatical study are distinguished as well as several different kinds of pragmatical notions. The object-languages under consideration throughout are based upon Russell’s simplified theory of types. In Chapter 11, therefore, a brief exposition of systems based upon type theory is given. The formulation chosen is of an especially simple kind and well suited for the pragmatical purposes here. Because pragmatics contains either syntax or semantics as a part, we need also a formulation of the underlying syntax or semantics for systems based upon type theory. Because this syntax and semantics have been studied elsewhere in some detail, only a summary need be given here. In Chapter 111we turn to pragmatics proper. The first pragmatical theory to be formulated, which presupposes a semantics, is accommodated within a translational semantical meta-language of the usual kind (containing the object-language as a part). Because the object-language is based upon the theory of types, the translational pragmatical meta-language also incorporates a form of type theory. This pragmatical meta-language is formulated with some care, and various interesting notions may be defined within it. In Chapter IV some further notions of pragmatics are introduced. Also other kinds of theories are sketched, in particular a pragmatics based on inscriptions, a non-translational pragmatics, and one which presupposes a syntax only rather than a semantics. Finally, it is shown how these various frameworks may be extended so as to accommodate pragmatical theories of a more powerful kind in which human actions are taken account of. In Chapter v a notion of analytic truth is defined within the narrow confines of the underlying semantics. Several notions of Carnap’s L-senzadics are therewith definable, including notions of L-identity in terms of which absolute semantical intensions may be introduced. Absolute intensions may then be interrelated with the

INTRODUCTION

xv

various pragmatical notions. Some alternative theories of absolute intensions, notably those of Carnap and Church, are discussed very briefly. Because very little rigorous work on the foundations of pragmatics has been carried out to date, the formulations here are highly tentative and exploratory. Much work remains in order to render them more adequate in this way or that. Nonetheless it is hoped the material here will be of interest as a first step in helping to clarify the logical structure of this important part of the general theory of language. The author is indebted to Professor Rudolf Carnap for some critical comments and for many helpful suggestions. Also thanks are due, and herewith expressed, to the Research Committee of the University of Pennsylvania for clerical aid as well as for a Special Summer Research Grant in 1957. The author wishes to thank also the editors of the Journal of Symbolic Logic, the University of Chicago Press, Harvard University Press, the Athlone Press of the University of London, and PrenticeHall, Inc., who have very kindly granted permission to quote or borrow a few sentences from works published by them.

CHAPTER I

T H E N A T U R E OF PRAGMATICS

I n this chapter the nature and extent of pragmatics is discussed in somewhat general terms. Because the subject is comparatively new, there is still wide diversity of opinion regarding its scope and import and regarding its status as a science. Here various alternative conceptions of pragmatics are considered very briefly. None of these has been conclusively developed in great detail, so that the discussion is perforce very broad and tentative. Because of the extensive contributions Carnap has made to the study of language for philosophical or methodological purposes, let us first glance briefly in 3 A at Carnap’s conception of pragmatics. Carnap’s point of view is essentially intensional whereas the point of view here is wholly extensional. In 3 B therefore what appear to be some advantages of the extensional point of view are enumerated. In 3 C the connection between pragmatics and the philosophical tradition of pragmatism is discussed briefly. In 3 D various levels of pragmatical study are distinguished. Finally, in 9 E various fundamental relations of pragmatics are characterized in a somewhat preliminary way.

A. Carnap’s Conception of Pragmatics. “When we observe an application of language,” Carnap notes, “we observe an organism, usually a human being, producing a sound, mark, gesture, or the like as an expression in order to refer by it to something, eg., an object. Thus we may distinguish three factors involved: the speaker, the expression, and what is referred to, which we shall call the designaturn of the expression. (We say, e.g., that in German ‘Rhein’ designates the Rhine, and that the Rhine is the designatum of ‘Rhein’; likewise, the designatum of ‘rot’ is a certain property, namely the color red; the designatum of ‘kleiner’ is a certain relation, that of ‘Temperatur’ a certain physical function, etc.)

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T H E N A T U R E O F PRAGMATICS

“If we are analyzing a language,” he goes on, “then we are concerned, of course, with expressions. But we need not necessarily also deal with speakers and designata. Although these factors are present whenever language is used, we may abstract from one or both of them in what we intend to say about the language in question. Accordingly, we distinguish three fields of investigation of languages. If in an investigation explicit reference is made to the speaker, or, to put it in more general terms, to a user of a language, then we assign it to the field of pragmatics.. . If we abstract from the user of the language and analyze only the expressions and their designata, we are in the field of semantics. And if, finally, we abstract from the designata also and analyze only the relations between the expressions, we are in (logical) syntax. The whole science of language, consisting of the three parts mentioned, is called semiotic.” “Examples of pragmatical investigations,” Carnap continues, “are : a physiological analysis of the processes in the speaking organs and in the nervous system connected with speaking activities; a psychological analysis of the relations between speaking behavior and other behavior; a psychological study of the different connotations of one and the same word for different individuals; ethnological and sociological studies of the speaking habits and their differences in different tribes, different age groups, social strata; a study of the procedures applied by scientists in recording the results of experiments, etc. Semantics contains the theory of what is usually called the meaning of expressions, and hence the studies leading to the construction of a dictionary translating the object language into the metalanguage. . ..” 1 Carnap goes on to speak of descriptive semantics as the “description and analysis of the semantical features either of some particular historically given language, e.g., French, or of all historically given languages in general.” 2 Pure semantics, on the other hand, is concerned with a system of semantical rules “freely invented.” And a similar distinction is is made between pure and descriptive syntax. 1 R. CARNAP,Introduction to Semantics (Harvard University Press, Cambridge: 1942), pp. 8-10. 2 op. cit., p. 11.

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CARNAP’S

CONCEPTION OF PRAGMATICS

3

But no such distinction is relevant, according to Carnap, in discussing pragmatics. Descriptive semantics and syntax are indeed “based on” pragmatics, and “all knowledge in the field of descriptive semantics and descriptive syntax is based upon previous knowledge in pragmatics. Linguistics, in the widest sense, is that branch of science which contains all empirical investigation concerning languages. I t is the descriptive, empirical part of semiotic (of spoken or written languages); hence it consists of pragmatics, descriptive semantics, and descriptive syntax. But these three parts are not on the same level; pragmatics i s the basis for all of linguistics. . . . With respect to pure semantics and syntax the situation is different. These fields are independent of pragmatics.” 1 In a later writing, Carnap again says that “the analysis of meanings of expressions occurs in two fundamentally different forms. The first belongs to pragmatics, that is, the empirical investigation of historically given natural languages. This kind of analysis has long been carried out by linguists and philosophers, especially analytic philosophers. The second form was developed only recently in the field of symbolic logic; this form belongs to semantics . . . that is, the study of constructed language systems given by their rules.” 2 The conception of pragmatics put forward in the Introduction is in close agreement with that of Carnap. But there are some important differences which should be noted. In particular, pragmatics for Carnap is concerned only with natural language. There is, in other words, only descriptive pragmatics, not a pure pragmatics of formalized language-systems. But surely languagesystems, constructed for given scientific purposes, are used by scientists just as natural languages are, and hence may be subjected to pragmatical analysis. When a mathematician, for example, formulates and proves theorems of elementary arithmetic within a formal system, he is using the expressions of that system for given purposes. His official parlance is carried out within that system. 1

Op.cit., p . 13.

R. CARNAP,“Meaning and Synonymy in Natural Languages,” Philosophical Studies IV (1955) : 33-47. Reprinted in Meaning and Necessity, 2nd ed. (University of Chicago Press, Chicago: 1956), pp. 233-247. 2

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T H E N A T U R E OF PRAGMATICS

He may make some informal statements of a meta-mathematical kind here and there, or make explanatory or pedagogical comments on the side, but he is surely using the formal system quite as much as he might, on less technical occasions, use a natural language. In the subsequent pages, in fact, we shall be concerned almost entirely with pure pragmatics. We shall be able to define many pragmatical concepts relativized to a given language-system taken as objectlanguage. And this may be either a formalized language-system of a certain kind, or any fully or partially formalized natural language having a similar structure. Also Carnap’s conception of pragmatics is thoroughly intensional, as we have already noted. That is to say, for him the language in which the pragmatical theory of a given language is expressed is an intensional meta-language. Although Carnap has not formalized such a pragmatical meta-language to show its exact structure, it seems that such objects as propositions, properties, and individual concepts would in effect be admitted primitively as values for variables. “From a systematic point of view, the description of a[n] [object] language may well begin with the theory of intension and then build the theory of extension on its basis.”l In other words, intensional objects are presumably values for variables in the meta-language, and extensions are then defined or introduced contextually in some way or another. Thus the pragmatical metalanguage for Carnap presumably contains intensional entities such as propositions, and so on, as values for variables, as well as (presumably) the individuals, classes, relations, or functions which are designated by (some of) the primitives of the object-language. Whether such individuals, classes, etc., are primitively values for variables in the meta-language or are introduced in some other way, is not clear. But such objects must be values for variables if the theory of reference or designation is to be accommodated and the semantical concept of truth defined in the usual way. The pragmatical meta-language for Carnap would thus seem to contain, implicitly a t least, both extensions and intensions as values for variables. This is an important point and one which we shall have occasion to return to in a moment. 1 LOG.

cit., p. 34.

1, B]

EXTENSIONALITY

5

However this may be, one can surely agree whole-heartedly with Carnap that “it seems that a system of theoretical pragmatics is urgently needed, not only for psychology and linguistics, but also for analytical philosophy. . . . [Tlhe time seems ripe for attempts at constructing tentative outlines of pragmatical systems. Such an attempt may first be restricted to a small group of concepts (e.g., those of belief, assertion, and utterance) ; it may then be developed to include all those concepts needed for discussions in the theory of knowledge and the methodology of science.”

B. Extensionality. The point of view taken here, as has already been remarked, is wholly extensional. All the object-languages and all the semantical and pragmatical meta-languages to be discussed are extensional languages. Such languages are thought to possess important advantages over intensional languages. Because this point is somewhat controversial, let us note very briefly some of these advantages. In the first place, extensional meta-languages appear to be simpler in structure than intensional meta-languages of commensurate power. As we noted a moment ago, Carnap’s meta-languages appear to contain both intensions and extensions as values for variables. Extensional meta-languages, on the other hand, contain only extensions. These are therefore more economical ontologically. And because of this there is also greater simplicity in the axiomatic structure. We need axioms and rules in the meta-language characterizing only the extensional entities, and all axioms and rules governing intensions may be dropped. Also extensional languages have been constructed of sufficient power to contain practically all of mathematics and large areas of physical science. Most of the important language-systems which have been studied in the literature of mathematical logic are in fact extensional languages. Such languages have been shown to suffice for handling the most subtle and intricate inferences in mathematics and presumably physical science. There is therefore strong 1

R. CARNAP, “On Some Concepts of Pragmatics,” Philosophical Studies

VI (1955): 89-91, p. 91. Reprinted in M e a n i n g and Necessity, 2nd ed., p. 250.

T H E N A T U R E OF P R A G M A T I C S

6

presumption that they should also be able to provide the logical framework and to handle inferences elsewhere. Further, there appears to be little evidence that intensional notions are needed in order to formalize biology, including genetics, as the work of Woodger has shown. Of course one cannot be sure of this at the present time, but the presumption, if we mistake not, of biologists is in favor of extensional languages. Also in using only extensional languages and meta-languages, we avoid any need for justifying the introduction of “dubious” intensional entities. Of course, it may well turn out that such entities are indispensable for scientific and philosophical purposes. But, mindful of Occam’s razor, we prefer not to use them unless explicitly shown to be necessary. Carnap has recently said that he does “not think that there is any compelling reason for avoiding the use of an intensional language for science, because such a language can be completely translated into an extensional one. . . .” 1 If this is the case, then the converse translation can also be given and, by parity of reasoning, there should be no need of intensional languages for science. Thus translatability does not seem to provide a convincing argument in behalf of intensional languages. A further implicit argument on behalf of extensional languages will emerge below, when we see how various quasi-intensional notions of either a semantical or pragmatical kind may easily and naturally be introduced within a wholly extensional framework. To be sure, these arguments are not wholly conclusive. Much detailed work must be done before such an argument either way can be given. Meanwhile, it is of interest to press the extensional point of view as far as possible. And unfortunately very little work of this kind is being done by philosophers or methodologists for philosophical purposes.

C. Pragmatics and Pragmatism. Students of pragmatics are indebted to the writings of Charles Morris who, perhaps more than anyone else in recent years, has called attention to this im1

LOG. cit., p. 90.

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PRAGMATICS A N D PRAGMATISM

7

portant area of linguistic study. Let us therefore note briefly some of Morris’ comments concerning the nature of pragmatics. “The term ‘pragmatics’ has obviously been coined with reference to the term ‘pragmatism’,’’Morris has observed. 1 “It is a plausible view that the permanent significance of pragmatism lies in the fact that it has directed attention more closely to the relation of signs to their users than had previously been done and has assessed more profoundly than ever before the relevance of this relation in understanding intellectual activities. The term ‘pragmatics’ helps to signalize the significance of the achievements of Peirce, James, Dewey, and Mead within the field of semiotic. At the same time, ‘pragmatics’ as a specifically semiotical term must receive its own formulation. By ‘pragmatics’ is designated the science of the relation of signs to their interpreters, ‘Pragmatics’ must then be distinguished from ‘pragmatism’, and ‘pragmatical’ from ‘pragmatic’. Since most, if not all, signs have as their interpreters living organisms, it is a sufficiently accurate characterization of pragmatics to say that it deals with the biotic aspects of semiosis, that is, with all the psychological, biological, and sociological phenomena which occur in the functioning of signs. Pragmatics, too, has its pure and descriptive aspects; the first arises out of the attempt to develop a language in which to talk about the pragmatical dimension of semiosis; the latter is concerned with the application of this language to specific cases.” Note that Morris’ distinction between pure and descriptive pragmatics differs from that drawn above (in 3 A). The distinction there is determined by the kind of languages considered, pure pragmatics being concerned with language-systems, descriptive pragmatics (following Carnap) being concerned with natural languages. But for Morris, the distinction seems rather determined by the language or language-system in which the pragmatical theory is couched. A descriptive pragmatics for him would presumably be couched in a pragmatical meta-language as formulated for some specific language-system (as object-language). A pure 1

C. MORRIS,Foundations of the Theory of Signs (International Encyclo-

Pedia of Unified Science, Vol. I, No. 2, University of Chicago Press, Chicago:

1938),pp. 29-30 (italics added.)

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pragmatics would be more general, as applied to all object-languages or to all object-languages of such and a kind. I n the following pages, we shall be concerned with a pragmatics which is both pure and descriptive in the sense that it will be applicable to any specific object-language of such and such an infinitely large class. For Morris, pragmatics is essentially a behavioral discipline. . . [P]rugmatics is that portion of semiotic which deals with the origin, uses, and effects of signs within the behavior in which they occur.” Pragmatics presupposes both syntax (syntactics) and semantics. “When so conceived, pragmatics, semantics, and syntactics are all interpretable within a behaviorally oriented semiotic, syntac tics studying the ways in which signs are combined, semantics studying the signification of signs, . . . , pragmatics studying the origin, uses, and effects of signs within the total behavior of the interpreters of signs.” 1 Of course, we must be clear here as to just what is meant by ‘behavioral’, and all pragmatical study is no doubt behavioral in some sense or other. But it need not be behavioral necessarily in any fixed sense. In fact, the pragmatical meta-languages to be formulated below are neutral in this respect, in the sense that some of the primitive terms may perhaps be interpreted in some behavioral sense, but this is not required by the theory. And it may be that other interpretations will turn out to be more suitable. This is a matter we need not settle here. Our task here is solely to clarify the logical structure of pragmatics, more specifically, of certain pragmatical meta-languages. The interpretation is left open, in order to give the meta-languages different interpretations for different purposes. The kinship of the present treatment with traditional conceptions of pragmatism might therefore seem somewhat remote. Nonetheless, it lies clearly within that tradition and tries to provide the logical framework for some of the notionswhich within that tradition have remained somewhat obscure. I ‘ .

D. Different Levels of Pragmatics. The pragmatical analysis of a language-system, as we have already noted, may involve 1 C. MORRIS, Signs, Language, and Behavior (Prentice-Hall, New York: 1946), p. 219.

1,

Dl

DIFFERENT LEVELS O F PRAGMATICS

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logical, linguistic, psychological, physiological, and even social features. In pragmatics we take account not only of the syntactical and semantical features of the language but also of one or more of the following : the users of the language taken individually or severally or as members of social groups, the mental states or brainstates of the users as well as their activity or behavior as correlated with their use of language, the physical, biological, or social circumstances in which expressions of the language are used, the purposes for which they are used, etc. The careful study of many of these pragmatical features of language, which are not necessarily independent of each other, requires the help of empirical science. Several successive layers of pragmatical study can in fact be distinguished. In the first and probably simplest kind we should be concerned with certain relations between the expressions of a Ianguage and its users, relations such as acceptance, assertion, utterance, or even betie). A second kind of pragmatical study might take into account also the users’ actions and behavior as responses to linguistic stimuli. A third might also take into account various social features of language. Different approaches to pragmatics have in fact emphasized such divergent features. Ajdukiewicz, eg., has given an account of meaning in terms primarily of a relation of acceptance, a certain relation between the users of a language and its declarative sentences1 Kotarbinski, on the other hand, and Morris (as we have already noted) have concentrated interest upon the activities and behavior of people, in part as related to their use of language. 2 The study of actions or behavior as responses to linguistic stimuli would seem to presuppose some such relation as acceptance. For a person to perform an action on the basis of a sentence or set of sentences of a given language seems to presuppose that the sentence or set of sentences is accepted in an appropriate sense. (See (IV, H) below.) A pragmatics of actions would seem therefore to presuppose a pragmatics of acceptances. In a similar way, a pragmatics of acceptances wouId seem to be presupposed in a pragmatics in which C. AJDUKIEWICZ, “Sprache und Sinn,” Erkenntnis 4 (1934) : 1OC-138. See, e.g., T. KOTARBI~SKI, “ 0 Istocie i Zadaniach Metodologii Ogolnej,” Pweglad Filozoficzny 4 1 ( 1 938). 1

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T H E NATURE OF PRAGMATICS

social factors, purposes, feelings, etc., would be studied. A relation of acceptance will therefore play an important role in the sequel.1

E. Some Fundamental Relations. Out of the great multiplicity of important pragmatical relations to investigate, let us consider for the moment only three. We shall characterize these very roughly, and later will select two of them for more detailed study. Given a language-system L , we wish to consider what it might mean to say that a person X accepts at time t a sentence a of a language-system L. An experimental scientist, e.g., a worker in socio- or psycho-linguistics, might wish to characterize this phrase in terms of exact test conditions somewhat roughly as follows. Given a person X who speaks the language L at time t, and a sentence a of L , we can say that (1)

X accepts a at time t,

under the following circumstances. The experimenter asks X at time t whether a holds. After a small lapse of time X answers either ‘Yes’ or ‘No’. The lapse of time allowed for response is specified in advance by the experimenter. Also the circumstances of the experiment must be “normal” in some sense to be specified. And so on. If X answers ‘Yes’ to the question, (1) is said to hold. If X hesitates or fails to answer ‘Yes’, (1) fails to hold. Given any specific statement of the form (1) the experimenter can in principle decide whether it holds or not. In practice, he would no doubt also frequently regard statements of this form as holding or failing to hold without always subjecting it to experiment, by extrapolating from given data, eg., or by taking into account other observations or experiments. Another way of deciding whether sentences of the form (1) hold or fail to hold might be for the experimenter to observe whether X behaves as though he, X , regards a as true at time t. We need not consider the details as to how the experimenter would decide this in 1 The author is indebted in some respects t o Mr. GEORGE F. RIEMAN, Jr., for calling attention t o the importance of acceptances, in his “An Outline for a Pragmatical Epistemology,” an M.A. thesis a t the University of Chicago, 1952.

1,

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S O M E FUNDAMENTAL RELATIONS

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a concrete case. Perhaps one would wish to characterize ‘accepts’ in such behavioral terms rather than in terms of direct questioning. This is a matter we need not decide here. We need merely note that experimental meanings can be attached to ‘accepts’ in many different ways. Note that, on the first of these two characterizations, to say that X accepts a sentence a means essentially that X expresses a certain positive attitude toward a. Under the circumstances described, he, X , consciously takes a to be true or regards a as true when a is presented to him in an appropriate way. But X need not actually utter the sentence a , nor need he believe what a expresses (if belief is taken for the moment as somewhat stronger than acceptance). But he should presumably be willing to base his actions on a. If X accepts a at time t he then presumably accepts a as a basis for action. More specifically, suppose a is a sentence which is neither a logical theorem nor the negation of one. And suppose that it is not the case that X performs a certain kind of action without accepting a. Under these circumstances X accepts a at the time in question as a basis (necessary condition) for the given kind of action, so that this notion is perhaps definable, under suitable circumstances, in terms of ‘accepts’. (See (IV,H) below). In addition to relations of acceptance and acceptance for action, there is the important pragmatical relation of belief. Belief is often construed as a relation, involving no time factor, between a person and a proposition. For an explication of belief in this sense, propositions are needed as values for variables and this explication must be given therefore in an intensional meta-language. Efforts have also been made to characterize belief as a relation between persons and sentences, involving perhaps a time factor as well. Belief in this sense may be conscious or unconscious, and it may be explicitly verbalized or not. But whether belief in any sense can be characterized in exact experimental terms is not clear, and the scientific legitimacy of this notion might therefore be questioned. Perhaps belief is, in Carnap’s terms, a “theoretical construct”, so that an adequate characterization of it can be given only in a theoretical 1 The author wishes to thank Mr. WROEALDERSONfor calling his attention to the important notion of acceptance as a basis for action.

12

THE NATURE O F PRAGMATICS

language containing primitives which may be only indirectly characterized experimentally. If belief is taken as stronger than acceptance, then if X believes a t t what a expresses he also presumably accepts a at t , but not necessarily conversely. Following Carnap and Hempel, it is convenient to distinguish among ( 1 ) classificatory, (2) comparative, and (3) quantitative notions. 1 Note that acceptance, acceptance for action, and belief are classificatory. But we might also wish to consider a comparative notion of acceptance, according to which we could say that X accepts a at time tl with greater strength than he accepts b at t z . Perhaps also we could develop a scale or metric according to which we could distinguish degrees of acce$tance. But to contemplate either a comparative or quantitative notion of acceptance or other pragmatical notions would entail going more deeply into the experimental characterization of these notions than we can do for the moment. Here we shall be concerned only with classificatory notions, and hence with a narrow pragmatical theory of an especially simple kind. Of the three pragmatical relations described above, we shall consider here primarily that of acceptance. The characterization of ‘accepts’ given above in terms of direct questioning is very rough, and really no more than a series of hints as to how we should proceed if we were to give an exact account of it. There is in fact a vast multiplicity of meanings that can be given, as we have seen, and for specific purposes one might have reason to prefer one to another. For the purposes of the theories to be formulated below we shall require merely that the experimental characterization of ‘accepts’ be such as to satisfy the axioms laid down. A notion of acceptance not satisfying these axioms would not be suitable. But there is a vast number of meanings t o ‘accepts’ even with this restriction. The formulations below are strictly neutral as between these different meanings. Most of the pragmatical notions to be introduced are defined ultimately in terms of ‘accepts’ and the 1 See, e.g., C. G. HEMPEL, Fundamentals of Concept Formation in Empriical Science (International Encyclopedia of Unified Science, Vol. 11, No. 7, University of Chicago Press, Chicago: 1952), or R. CARNAP,Logical Foundations of Probability (University of Chicago Press, Chicago: 1950), pp. 8-1 1.

1,

El

SOME FUNDAMENTAL RELATIONS

13

exact meaning of them depends therefore upon the meaning of ‘accepts’ adopted. We do not present here strictly a theory of acceptances, but only the framework of one. More specifically, uninterpreted or partially interpreted pragmatical meta-languages will be presented rather than fully interpreted ones. Given an experimental meaning to ‘accepts’, each framework will yield a specific pragmatical theory. Philosophical methodology can provide the framework, but the help of the empirical scientist is needed to specify experimentally the exact meaning of ‘accepts’ needed for given purposes. 1

1 Cf. the comments concerning the interpretation of the primitives in J. C. C. “Outlines of a Formal Theory of Value, I,” by D. DAVIDSON, MCKINSEY,and P. SUPPES,Philosophy of Science 22 (1955) : 140-159.

C H A P T E R I1

TYPE-THEORETICAL SYSTEMS A N D S E M A N T I C AL S U B S T R U C T U R E The object-languages we consider are based upon the simplified theory of types. We shall first be concerned with a pragmatical meta-language containing a translation of the object-language. This pragmatical meta-language therefore is to be constructed also in accord with the theory of types, and certain class-theoretical notions are used fundamentally in the pragmatical part of the meta-language. But other forms of class or set theory could be used also. Later, in the discussion of non-translational pragmatics, no form of class theory is presupposed at all. The reader is presumably familiar with basic features of languages constructed on the basis of the theory of types, so that here we may rest content with only a brief sketch. In 5 A the primitive framework of the object-language T is given. Abstracts containing no free variables play an especially important role in the sequel, and these are introduced in 9 B. In 4 C the rules of the object-language are given, and in 5 D it is shown, following Wiener and Kuratowski, how relations and ordered couples may be accommodated. The pragmatical meta-language t o be constructed in Chapter 111 contains a semantical meta-language as a part. This semantical meta-language in turn contains a syntactical meta-language as a part. We must therefore formulate very roughly the syntactical and semantical meta-languages presupposed. Because these have been formulated in some detail elsewhere, we shall need only a brief summary. In 4 E the syntax presupposed is sketched, and in $5 F-G, a semantical meta-language (for the object-language T ) based on designation. A. Primitive Framework. The subdivision of variables, or of the entities over which they range, into types or levels is the leading idea of type theory. The variables of each type are to be regarded

11. A]

PRIMITIVE FRAMEWORK

15

as ranging over an appropriate kind of entity. The variables of the first or lowest type range over some given totality of objects. The variables of second type then range over classes of these objects. Variables of third type range over classes of classes of these objects. And so on. An exact specification of the entities taken as the fundamental objects need not be given. Once these objects are specified, the classes of them, classes of classes of them, and so on, are then determined. The variables of any type range only over the entities of that type, and no variables are admitted which range over all the entities considered. Some logicians think that the distinction of type has a certain “naturalness” about it and corresponds more or less roughly with intuition. Others find it somewhat ad hoc and cumbersome. However this may be, type theory has come to be regarded as one of the most important, presumably consistent, methods of providing a logical foundation for mathematics and natural science. The form of type theory used here is in essential respects that of Tarski. 1 The system to be formulated is called T , which may be outlined as follows. I n T , the only non-logical, primitive, relational constant is ‘E’, symbolizing the relation of membership. (Under logic is included here only the elementary theories of truth-functions, quantifiers, and identity. Hence ‘6’ is non-logical. But ‘=’, symbolizing identity, is taken as a logical primitive, together with suitable symbols for the truth-functions and quantifiers.) In addition T may contain primitive non-logical individual constants, class constants, or class constants of higher type. These may be thought of as designating fixed individuals, fixed classes of those individuals, classes of classes of them, and so on, respectively. The atomic formulae are therefore of the forms Ix E y’, See A. TARSKI, Der Wahrheitsbegriff in den Formalisierten Sprachen, Studia Philosophica I (1936): 261-405, esp. pp. 364-366 (also in TARSKI’S Logic, Semantics, Meta-Mathematics, tr. by J. H. Woodger (Clarendon Press, Oxford: 1956), pp. 152-278). Cf. also R. M. MARTIN, Truth and Denotation, A Study in Semantical Theory (University of Chicago Press, Chicago; University of Toronto Press, Toronto; and Routledge and Kegan Paul, London: 1958), pp. 151-159.

16

TYPE-THEORETICAL SYSTEMS A N D SEMANTICAL S U B S T R U C T U R E

where y is a variable or constant of type one higher than that of the variable or constant X, or ‘x

= y’,

where x and y are variables or constants of the same type. The primitive logical constants of T include ‘-’ and ‘v’, standing for logical negation and disjunction as is customary. The variables of each type may be arranged alphabetically, so that we can distinguish a first, a second, and so on. In T , we let the m-th variable of type n be written as

m times

The accents or superscripts indicate type level, and the subscripts serve to give the alphabetical position. For convenience one may write expressions such as (1) I

(2)

n,

xm.

‘n’ here is a meta-linguistic expression standing for a string of n superscripts, and ‘nz’for a string of m accents as subscripts. Expressions such as (2) are thus meta-linguistic expressions standing for variables of T . Occasionally we may write ‘ x z in place of (l), as the object-language variable stipulated by (2). This notational simplification will facilitate some statements below. Let the non-logical primitive individual or class constants of T be written IaI

..

I I I . .

I

, , , , . . . . I .

Occasionally we may use ‘a’: or ‘a’: in place of this in the obvious way. We may have either a finite or denumerably infinite number of primitive individual constants, but ordinarily we should have only a finite number of primitive constants of higher type. The universal quantifiers are to be expressed, as is usual, by I

n

(xWl)

p J

so that the total primitive vocabulary of T contains just ‘N’, ‘v’,

17

ABSTRACTS

11, BI

‘ E ’ , and ‘=’ as logical or non-logical constants, the variables x,1J, ‘d’,. . ., ‘x;’,‘xf, . . ., and so on, and the individual or class constants, ‘a:’, ‘a:’, . . ., a,2, , ‘a:’, and so on.

‘(’, ‘)’, t

I

€3. Abstracts. Because of the important role which abstracts and defined constants play in the pragmatics below, we take also a notation for abstraction as primitive. If is a formula of T , then ‘-----I

‘p 3------) rn will be regarded as standing for the class of all objects xk such that

-----. Expressions of this kind are called one-place abstracts, and terms then include all one-place abstracts, all variables, and all individual or class constants. More specifically, the terms of type 1 comprise just the variables and individual constants of type 1. But terms of type 2 comprise all variables and class constants of type 2, and in addition all abstracts of the form ‘x13-----’ n

where ‘-----’ is any formula of T perhaps containing ‘x: as a free variable. And so on. One-place abstracts are to be substituends for variables. This will be provided by the quantificational rules below. For explicitness, we define (by simultaneous recursion) the notions formwla of T and term of type n of T as follows.

‘xzis a term of type 2. A constant ‘a: is a term of type n. 3. If X is a formula and ‘xzis a variable, then ‘xk3X’ is a term n.

1. A variable

of typelz

+ 1.

4. If x$ and

n

+ 1, then ‘x$ 5. If

YE+‘ E

xk and y:

are respectively any terms of type n and y;+” is a formula. are terms of type n, then

‘xk = y:’

is a formula.

6. If X and Y are formulae, so are ‘,- X ’ and ‘ ( X v Y)’.

7. If X is a formula, and ‘x: formula.

a variable, then ‘ ( x $ ) X is a

18

TYPE-THEORETICAL SYSTEMS A N D SEMANTICAL S U B S T R U C T U R E

This definition gives in effect a summary of the primitive structure of T . The other truth-functional constants, ‘.’ for conjunction, ‘ 3 ’ for the material conditional, and ‘=’ for the biconditional, may be defined in the customary way. Also, the existential quantifiers ‘(Ex:)’ may be introduced as abbreviations for ‘-(x:)-’. Hereafter we let ‘ X and ‘ Y ’ ,possibly with primes or accents, stand for arbitrarily any formula of T . Also x‘:’ will be used to stand for any term of type n. (‘xz,on the other hand, stands only for a variable.) 1 may be introduced as follows. The null class of type n

+

‘A”+” abbreviates

‘4’3-

xy = xy’.

C. Rules. The rules of T may be sketched briefly as follows. We first have logical rules providing for the theories of truthfunctions, quantifiers, and identity. Those for the truth-functions (which may be omitted) include the Rule of Modus Ponens (MP). For the statement of the Rules of Quantification, we must consider the circumstances under which occurrences of variables and terms are said to be free in a formula X . An occurrence of a variable ‘xk’ in X is free in X provided it is not in a context of the form ‘ ( x l ) Y ’or ‘x”,Y’. Any occurrence of a primitive constant ‘a’: in X is free in X. Any occurrence of an abstract ‘xz3Y’in X is free in X provided every free occurrence of is a free occurrence of that any variable in Y other than variable in X . Let Frege’s sign of truth-assertion, ‘t’,be used in contexts such as ‘t X’to express that X is a theorem. As Rules of Quantification for T we have then the following.

‘xz

Q1. k (x:)X .I. Y , if Y differs from X only in containing free occurrences of a term of type n wherever there are free occurrences of the variable ‘xk’ in X.

( X v Y ) . 2 . X v (xz)Y , if there are no free occurrences Q2. I- (xk) of the variable ‘x:’ in X. The Rule of Generalization is as follows.

19

Gen. If I- X then k ( x z ) X . As Rules of Identity, we have the following. Idl.

k x > = .x:

Id2. k x: = x,” . X :3 : Y,where X contains at least one free occurrence of x$ and Y differs from X only in containing a free occurrence of x,“ in place of one free occurrence of x z in X. We also assume that distinct primitive constants stand for distinct entities. This is in effect provided by the following rules.

Id3a.

t

N

a;+’

= a,”+’, if m and

k are of unequal length.

If we have no primitive constants of type

> p but do have primitive

constants for types 5 p other than the first, then I d 3 a is to be construed accordingly. And ordinarily we should have only a finite number of such constants, as has already been remarked.

Id3b.

I-

N

a’, = a:, if m and k are of unequal length.

If we have an infinity of primitive individual constants, Id3b provides for an infinity of distinct individuals, and otherwise only a finite number of such. Also we have the Rule of Abstraction. Abst. I- x z E xz3Y . =. X , if ‘x,”’ is any variable not free in X and Y differs from X only in containing a free occurrence or free occurrences of ‘x,”’ in zero or more places where x z occurs freely in X. Note that all the above rules are regarded as logical rules, because they in no direct way characterize the non-logical primitive ‘ E ’ . Note that Abst, in particular, is a rule for the elimination of ‘ E ’ from certain contexts. The non-logical rules characterizing ‘ E ’ are as follows. First we have the R u l e of Extensionality.

Ext. k (x;”) (x:”) ((x:) (x: E x:+’ . =.X; E x:+’) .I . x:+’ = x:+’). If an infinity of primitive individual constants is available, there is no need for an A x i o m of I n f i n i t y . Otherwise we assume the existence of at least one infinite class of classes of individuals. The existence of infinite classes of higher types then follows. The Axiom of Infinity asserts that there exists at least one non-null

20

TYPE-THEORETICAL SYSTEMS A N D S E M A N T I C A L S U B S T R U C T U R E

class x; of classes of individuals, every member of which contains a proper subclass which is also a member of x:. Let 'x;+'

.=I.x t E x",")'.

c x*n+'' abbreviate ' ( x z ) ( x t E

To say that a class xi+' is a subclass of a class x:", is to saythat every member of xE+;" is a member of x;E+l. The Axiom of Infinity is then stipulated by the following rule. Inf.

I- (Ex;) ((Exf)xfE x;

4)).

x:

. (xf)(xfE xf .=I. (Ex:) ( x i E x i . x i

c

xt .

N

Finally, axioms of choice are assumed in the form of Russell's Multifilicative Axioms. These tell us that given any non-null class xF+2 of mutually exclusive classes (of type nf 1) there exists at least one class which contains one and only member in common with each non-null member of x;+'.

-

t (x;~+~)((Ex,"+')x,"+' E . (Ex;+')(Ex,"+')(x,"+' E .y+2 . p + 1 - x,"" . (Ex;)(x; E x;" . x? E x:")) . :3 : (EX:+')(X;+')(X~+' E x ; + ~ . (Ex.!J')xi E xi+' :3 : (Ex;) (xi)(xi E Mult.

%;+2

.

%.+I .=. . - . x2n = x ? ) ) ) .

Other non-logical rules may be given concerning the various primitive constants. Such rules will be appropriate to the particular subject-matter with which T deals. The distinction between the logical and non-logical rules of T , as drawn here, is an important one for the sequel. Some logicians would wish to include Ext, I n f , and Mult as logical rules. Here we regard as logical rules only those providing for the logic of truth-functions, quantifiers, identity, and abstraction, construing logic in a somewhat narrow sense.

D. Relations and Ordered Couples. 1 Note that T is formulated in such a way as to contain no variables for relations or functions, but only for classes, classes of classes, etc. Relations and hence functions can, however, be handled within T , using the device of Wiener and Kuratowski. The leading idea of this device is to regard relations as classes of ordered pairs of a certain kind. 1

Cf. Truth and Denotation, pp. 154-156.

11, Dl

ORDERED COUPLES

21

Thus, suppose we wish to gain the effect within T of a variable ‘R’ for dyadic relations between individuals, and to express that some individual x: bears R to 4. We form first the ordered pair of xi and xi as a class of two classes, one of them being the class whose only member is xi, the other being the class whose only members are xi and xi. Thus xt is such an ordered pair if and only if

(x;)(x? E x; :=: (xi)(xiE x; .=. xi = xi) .v . (xi)(xiE x; : =:xi = x: .v. .’s = xi)). Let

‘{x:}’ abbreviate ‘xi3xi = xi’ and

‘(xi,xi)’ abbreviate ‘xi3(xi= x,1 .v. x,1 = xi)’. Then

‘<x:,xi>’ may abbreviate ‘x;3(x; = {xi} .v. x; = {x:,~;})’. The effect of ‘R’as a variable can be achieved by using a variable I 41 x, of fourth type. Thus, to say that xi bears R to xi is to say that

‘<x:,x;>E x:’. Similarly, ordered couples of classes (of type n) may be introduced. Dyadic relations between classes of type n may then be handled by means of classes of type (n 3 ) . To say that x: bears a dyadic relation to xi is then to say that

+

<x,”,x,“> E xy3, using ‘x:+~’in effect as the relational variable. The treatment of relations of higher degree is as follows. The ordered triple of individuals xi, xi, and xi, e.g., may be identified with the ordered pair of pairs, the first of which is the ordered pair of xi with xi, and the second, of xi with xi. Let

‘<x:,xi,xi>’abbreviate ‘,(xi,xi>>’. The effect of a variable ‘R’for a triadic relation among individuals is then achieved by a variable of sixth type. To say that R holds

22

TYPE-THEORETICAL SYSTEMSAND SEMANTICAL SUBSTRUCTURE

among xi, xi, and xi is then to say that

And so on for relations of still higher degree. The treatment of relations between or among objects of higher type is similar. All the ordered pairs considered have been homogeneous in the sense that the members of the pairs, so to speak, are of the same logical type. But heterogeneous couples may also easily be introduced by correlating them uniquely with couples already introduced. This need not be shown in detail. We need note here merely that a completely general theory of ordered couples, triples, etc., of all kinds may easily be developed within T .

E. Syntax. To formulate in an exact way the syntax of T , we construct a syntactical meta-language for T as follows. This formalized nieta-language may be called M . 1 As primitive structural-descriptive names of the primitive symbols of T , we have in M ‘lp’ for ‘(’, ‘rp’ for ‘)’, ‘vee’ for ‘v’, ‘tilde’ for ‘ex’ for ‘x’,‘ay’ for ‘a’, ‘id’ for ‘=’, ‘ep’ for ‘e’, ‘invep’ for ‘3’, ‘ac’ for ‘”, and ‘subscr’ for 0’.Also the operation of concatenation, symbolized by ‘n’,is taken as undefined. We let ‘a’, ‘b’, etc., with or without numerical subscripts or accents, be the syntactical variables ranging over the expressions of T . Then ( a n b ) , read ‘a concatenated with b’, is the result of writing the expression a followed immediately by b. The underlying logic is the usual first-order logic with identity. In addition to the rules of the underlying logic, the following Syntactical Rules are needed. I-’,

S y n R 1 . t ~ l = pr f i . -1fi = vee . . . . . r p = subscr .

-

N

-

= vee.

... .

( E a ) ( E b ) l p= (anb) . SynR2. t (Ea)(Eb)subscr= (anb). N

. . . . -1fi

-

ac = subscr.

= subscr

.

-

rfi

( E a ) ( E b ) r p= (anb) . . . . .

Cf. Truth and Denotation, pp. 70-98, and TARSKI, op. cit., pp. 289-303. Also W. V. QUINE,Mathematical Logic (HarvardUniversity Press,Cambridge: 1951), pp. 291-305, and L. CHWISTEK,The Limits of Science (Kegan Paul, London : 1948), pp. 83- 100 and 162-1 9 1.

11, El

SYNTAX

23

S y n R 3 . i- (anb) = (cnd) .=. ( a = c . b = d :v: (Ee)(b= (end) . c [ane)) :v: ( E e ) ( a= (cne) . d = ( e n b ) ) ) .

=

(Each structural-descriptive name in the syntactical meta-language may also be called the structural description of the corresponding expression of T , and the result of concatenating structural-descriptive names is the structural description of the corresponding expression of T . For example, ‘( ~ n e x n a c n s u b s c y n y p n e x n a c n s u b ~ c y n i ~ n e x n a c n s u b s c y ) ’

is the structural description of ‘ ( x ‘ , ) ~ ’= , x‘,’.) (where S y n R 4 . If every sentence which results from ‘---a---’ ‘---a---’ is a formula of M containing ‘a’ as its only free variable) by replacing ‘a’ by a structural description is a theorem, then t- (a)---a---.

S y n R l and S y n R 2 are Rules of Distinctness. S y n R 3 is a R d e of Identity and S y n R 4 is a Rule of Infinite Induction. An exact syntactical characterization of T may be given within M on the basis of these rules. We need not develop this in detail, but will merely note a few definitions which will be useful later. We may let ‘a Bgn b’ abbreviate ‘((Ec)b= ( a n c ) .v. a = b)’, ‘ a Ends b’ abbreviate ‘((Ec)b= ( m a ) .v. a = b)’, and

‘ a Seg b’ abbreviate ‘(Ec)(aEnds c . c Bgn b)’. These define respectively the notions that the expression a begins (or is identical with) the expression b, ends (or is identical with) the expression b, or is a segment of (or is identical with) the expression b.

‘PS a’ abbreviates ‘ ( a = l p .v. a = YP .v. a = vee .v. a = tilde .v. a = ex .v. a = a y .v. a = i d .v. a = ep .v. a = invep .v. a = ac .v. a = subscr)’. ‘PS a’ reads ‘ a is a primitive sign of T’. ‘AcString a’ abbreviates ‘(b)(PSb . b Seg a :=I : b = ac)’. ‘SubscrString a’ abbreviates ‘(b)(PS b . b Seg a :I : b = subscr)’.

24

TYPE-THEORETICALSYSTEMS A N D SEMANTICAL SUBSTRUCTURE

a is a string of accents if and only if every primitive sign in a is ac. And similarly for a string of subscripts. ‘AtFmla a’ abbreviates ‘ ( E b )(Ec)( E d )(Ea’)(AcString b . Subscr String c . SubscrString d . (e = ex .v. e = a y ) . (a’ = ex .v. a’ = a y ) . ( a = (enbncnepna‘nbnacnd) .v. a = (enbncnidna’nbnd))’.

a is an atomic formula of T if and only if it consists of a variable or constant of given type followed by ‘e’ followed by a variable or constant of one higher type, or consists of a variable or constant of given type followed by ‘=’ followed by a variable or constant of the same type. That a is a variable may be defined as follows. ‘Vbl a’ abbreviates ‘(Eb)(Ec)(AcStringb . SubscrString c . a = (exnbnc))’. Similarly we may say that a is a primitive constant. ‘PrimCon a’ abbreviates ‘ ( E b )(Ec)(AcString b . SubscrString c . a (aynbnc))’.

=

We may let

‘ a Gen b’ abbreviate ‘(Ec)(Vbl c . a

=

(Zpncnrfinb))’.

a is a generalization of b if a consists of a variable enclosed by parentheses prefixed to b. We let

‘ ( a vee b)’ abbreviate ‘(Zqhanveenbnrp)’, ‘tilde a’ abbreviate ‘(tildena)’, ‘ ( ahrsh b)’ abbreviate ‘(tilde a vee b)’, ‘ ( adot b)’ abbreviate ‘tilde (tilde a vee tilde b)‘, ‘ ( a tripbar b)’ abbreviate ‘ ( ( ahrsh b) dot (b hrsh a))’, ‘ a qu b’ abbreviate ‘(lpnanrpnb)’, and

‘ a exisqu 6’ abbreviate ‘tilde a qu tilde b’. Also we let

‘ a M P b,c’ abbreviate ‘(c = (b hrsh a ) .v. b = (c lzrsh a))’.

11, El

25

SYNTAX

The definiendum here reads ‘ a may be reached from b and c by Modus Ponens’. Then ‘ a IC b,c’ abbreviates ‘ ( a M P b,c .v. a Gen b)’, so that a is an immediate consequence of b and c if a M P b,c or a Gen b. We may say that two variables or primitive constants or abstracts are of the same type as follows.

‘ a SmTp b’ abbreviates ‘ ( E c )( E d )(Ee)( E d )(Eb’) (Ec’) (Ed’) (AcString e . SubscrString c . SubscrString d . (u’ = ex .v. a’ = u y ) . (6’ = ex .v. b‘ = a y ) . ( ( a = (a’nenc) . b = (b’nend)) .v. (a = (exnencninzlepnc‘) . b = (b’nenacnd)) .v. (a = (a’nenacnc). b = (exnendninzlefinc’)) .v. (a = (exnencninvejmc’) . b = (exnendn invepnd’)))) ’. Then we may let ‘a Abst b,c’ abbreviate ‘(Ed)(Ee)(dSmTp e . Vbl d . (PrimCon . e = (a’ninvepnc)) . a = (enepndn invepnb)) ’. e .v. Vbl e .v. (Ea’)(Vbl a‘

‘ a Abst b,c’ reads ‘ a results from b and c by abstraction’. The notion of being a formula of T may be defined by Quine’s method of framed ingredients. We let , ‘ a FrIng b’ abbreviate ‘((Z4nrfinanZfinrp)Seg b .

-

(Zfinrfi) Seg a)’,

and ‘ a Prb c’ FrIng d)’.

abbreviate

‘ ( E d ) ( dBgn b . a FrIng d . c FrIng b .

-

c

The definienda here may be read respectively ‘ a is a framed ingredient of b’ and ‘ a is prior to c in b’. Then ‘Fmla a’ may abbreviate ‘ ( E b ) ( aFrIng b . (c)(c FrIng b :2 : AtFmla c .v. (Ed)(Ee)(dPrb c . e Prb c . (c = (tilde d ) .v. c = (d zlee e ) .v. c Gen d .v. c Abst d’e))))’.

26

T Y PE-TH E 0 R E T I C A L S Y S T E M S A N D S E M A N TI C A L S U B S T R U C T U R E

An expression a is an occurrence of b in c provided a begins c, b ends a , and (ansubscr) does not begin c.

‘ a Occ, b’ abbreviates ‘ ( aBgn c . b Ends a .

-

(ansubscr) Bgn c)’.

That a is a bound occurrence of a variable b in c may then be introduced as follows.

-

‘ a BOcc, b’ abbreviates ‘(Vbl b . a Occ, b . (Ea’)(Fmla a’ : (Eb’) (Ec’)(a = (bnb’) . (((b’nc’) Occ, (b qu a’) . (b qu a’) Ends c’) .v. ((b’nc’) Occ, (bninvefina’). ,- (bninvefina’)Ends c’))) .v. (aninvefin a’) Seg c))’. And a is a free occurrence of a variable b in c provided it is an occurrence of b in c but not a bound occurrence.

‘ a FrOccVbl, b’ abbreviates ‘(Vbl b . a Occ, b

. ,-a BOcc,

b)’.

We recall from fi C above that a n y occurrence of a constant b in c is a free occurrence of that constant in c. Hence we let

‘ a FrOccCon, b’ abbreviate ‘(Primcon b . a Occ, b)’. By a one-place abstract we mean an expression of the form (bninveflnc) where b is a variable and c a formula. ‘Abst a’ abbreviates ‘(Eb)(Ec)(Vblb . Fmla c . a

=

(bninvefinc))’.

We may then say that a is a free occurrence of the abstract b in c as follows. d . Fmla e . b = ‘ a FrOccAbst, b’ abbreviates ‘(Ed)(Ee)(Vbl (dninvefine) . a OCC,b . (a‘)(b’)(a’ FrOccVbl, b‘ . ,- b’ = d :2 : a‘ FrOccVbl, b’))’. By a term of T one means a variable or a primitive constant or an abstract. ‘Trm a’ abbreviates ‘(Vbl a .v. PrimCon a .v. Abst a)’. Also

b

‘ a FrOccTrm, b’ may abbreviate ‘ ( aFrOccVbl, b .v. a FrOccCon, a FrOccAbst, b)’.

.v.

11,

Ej

27

SYNTAX

Frequently we wish to speak of a term of type n, as above in

$9 B-C. This we do not define in general. We may, however, define successively ‘VbP a’ as ‘(Eb)(SubscrStringb . a = (exnacnb))’, ‘Vbl2 a’ as ‘(Eb)(SubscrStringb . a = (exnacnacnb))’, and so on. And similarly ‘PrimConl a’, ‘Primcon2 a’, and so on, may be defined. Also ‘Abst2 a’ may abbreviate ‘(Eb)(Ec)(Vbllb . Fmla c . a = ( b n invefmc))’, ‘Abst3 a’ may abbreviate ‘(Eb)(Ec)(VblZb . Fmla c . a = ( b n inve$nc))’, and so on. And then ‘Trml a’ may abbreviate ‘(Vbll a .v. PrimConl a)’, ‘Trm2 a’ may abbreviate ‘(Vb12 a .v. PrimCon2 a .v. Abst2 a)’, and so on. Also we let ‘a FV b’ abbreviate ‘(Ec)c FrOccVblb a’ and ‘Sent a’ abbreviate ‘(Fmla a

.

-

(Eb)b FV a)’.

The definienda here are read respectively ‘a is a free variable of b’ and ‘a is a sentence’. Preparatory to defining the notions of axiom and theorem, we need the following definitions. ‘a SF1: d’ abbreviates ‘ ( b SmTp c . (Ea’)(Eb’)(a’ FrOccVbld c . b‘ FrOccTrm, b . (e)(a’ = (em) . =. b‘ = ( e n b ) ) . (e)(d = (a‘ne) .=. a = (b‘ne)))’,

‘a SF?: d’ abbreviates ‘(Ee)(aFrIng e . (a’)(a’ FrIng e : 3 : a’ d .v. (Eb’)(b’ Pr, a’ . a’ SF1: b’)))’, and ‘a SF: d’ abbreviates ‘(a SF?: d . (c FV a

.=I. b = c))’.

=

28

TYPE-THEORETICAL SYSTEMS AND SEMANTICALSUBSTRUCTURE

The definienda of these definitions may read, respectively, ‘ a differs from d only in containing free occurrences of the term b in place of just one free occurrence of the variable c in d’, ‘a differs from d only in containing free occurrences of the term b in place of 0 or more free occurrences of the variable c in d’, and ‘a differs from d only in containing free occurrences of the term b wherever there are free occurrences of the variable c in d’. The primitive constants and the one-place abstracts containing no free variables will play an important role within the semantics below. Such expressions in fact will be taken as the only expressions of T which designate. Such expressions we shall call constants, more specifically, constants of such and such a type. Thus

-

‘Con1 a’ may abbreviate ‘Primcon1 a’, ‘Con2 a’ may abbreviate ‘(Primcon2 a .v. (Abst2 a . (Eb)bFV a))’, and so on. It may seem a little unusual to call abstracts containing no free variables ‘constants’, but they will behave essentially as such in the sequel and this terminology will be convenient. We may now let ‘TFAx a’ express that a is a truth-functional axiom of T , and ‘LogAx a’ may express that a is a TFAx or is stipulated by Q1, Q2, Abst, I d l , I d 2 , Id3a, or Id3b of 9 C . Also ‘DesAx a’ may express that a is a descriptive or non-logical axiom. Then ‘Ax a’ may abbreviate ‘(LogAxa .v. DesAx a)’, and ‘Thm a’ may abbreviate ‘ ( E b ) ( aFrIng b . (c)(cFrIng b :3 : Ax c .v. ( E d ) ( E e ) ( dPro c . e Prb c . c IC d,e)))’. ‘Ax a’ reads ‘ a is an axiom’ and ‘Thm a’ reads ‘ a is a theorem’. Similarly ‘LogThm a’, read ‘ a is a logical theorem’, may be defined by replacing ‘Ax c’ in the definiens of this last definition by ‘LogAxc’. We may also wish to say that an expression a is a proof of b. For this we may let

-

‘ a Proof b’ abbreviate ‘ ( b FrIng a . (c)(c FrIng a :2 : Ax c .v. ( E d ) ( E e ) ( dPr, c . e Pr, c . c IC d,e)) . (Ec)b Pr, c)’. (The last conjunct in the definiens here is not essential.)

11, F]

29

DESIGNATION

Finally a is a closzlre of b if and only if a contains no free variables and consists of a string of universal quantifiers prefixed to b. ‘QuantString a’ may abbreviate ‘(Eb)(a FrIng b . (c)(c FrIng b :3 : (Ed)(Vbl d . c = (Z$ndn@)) .v. (Ed)(Ee)(dPrb c . e Prb c . c = (W))’, and ‘ a Clsr b’ then abbreviates

d .a

=

‘(N (Ec)c FV a

. (Ed)(QuantString

(dnb))’.

This list of definitions gives us a fairly full survey of the formal syntax of T . Any syntactical theorems which may be needed may easily be proved upon the basis of SynR 1-SynR4 above using these definitions.

F. Designation. We now turn to a semantical meta-language for T based on designation containing the syntactical meta-language formulated above as a part. 1 The new primitive here is ‘Des’. Expressions such as ‘a Des xz’, where xk is any term of T , are to be significant. If ‘a:’, for example, then

xk is the constant

‘a Des a:’ reads ‘the expression a designates the individual a;’. Clearly if ‘u’ is taken as ‘(aynacnsubscYnsubsc~)’,then ‘ a Des a$’ is to hold. Also abstracts containing no free variables are to designate the correis a sentential function containing sponding classes. If ‘---x;---’ ‘xi’ as its only free variable, the expression ‘x:+--xi---’ designates the class of all classes x i such that ---x:---. The semantical meta-language here, to be called SM;,, contains not only the syntactical meta-language above as a part, but also (a translation of) T itself. Every primitive symbol of T reappears as a 1 See Truth and Delzotation, pp. 166-169, for a fuller development of what is essentially this meta-language.

30

TYPE-THEORETICAL SYSTEMSAND SEMANTICAL SUBSTRUCTURE

primitive symbol of the semantical meta-language. But in addition, as we have seen, the semantical meta-language contains structural descriptions of these symbols, concatenation, identity of expressions, and designation as primitive notions. The notions of formula, syntactical term, and translational term of SM;,,, may be defined simultaneously as follows. 1. Any variable or structural-descriptive term of M is a syntactical term of SM;,,. 2. Any variable or primitive constant of type n of T is a translational term (variable or constant) of type n of SM;,,. 3. If X is a formula of M , it is a formula of SM;,,. 4. If X is a formula of T , it is a (translational) formula of SM;,,. 5. If a is a syntactical term and x: is a translational term, then ‘a Des x l ’ is a formula of SM;,,. 6. If X and Y are formulae of SM;,,, so are ‘- X’and ‘(X v Y)’. 7. If X i s a formula of SM;,,, ‘x:’ is a translational variable, and a is an expressional variable, then ‘ ( x l ) X ’ and ‘(a)X’ are formulae of Shf;,,.

The rules of SMgedcomprise, in addition to those of the underlying logic, (translations of) the rules of T , and the syntactical rules S y n R 1 -SynR4. As semantical Rules of Designation, we have the following. DesR1. t a Des x l , where (i) x l is an abstract ‘x;-%X’ containing no free variables and ‘a’ is taken as the structural description of this abstract, for n 2 2, or (ii) x: is a primitive constant ,a,n ) and ‘a’ is taken as the structural description of this constant, forn 2 1 . DesR2. I- a Des x l .I. Con” a. DesR3. t a Des xk . a Des x: : 2 : x l = x:, if rn and k are of unequal length. DesR 1 stipulates that certain constants designate certain objects. DesR2 is a Restrictive Rule stipulating that if an expression designates any object of type n, then it is a constant of type n. In other words, constants (including abstracts containing no free variables) are the only expressions which designate. DesR3 is a Principle of

31

TRUTH

11, GI

Uniqueness stipulating that expressions designate at most one object. By DesR1, DesR2, and SynR4 we have that TF1.

I- (a)(Con" a . =. (ExL)a Des x:).

Also we have Princifiles of Univocality, one for each type level, as follows. TF2.

t (a)(Con" a

x: = x z ) ) ) , for rn and

.3.

(Ex",((a Des x z . (%:)(a Des x:

.D.

k unequal.

Also we have that distinct primitive constants designate distinct objects.

-

TF3. :3 :

I- PrimCon a . PrimCon b . a = b . a Des x z . b Des x: k unequal. N

x l = x:, for m and

G. Truth. In terms of 'Des' the semantical concept of truth for T may immediately be defined. In fact, we may let 'Tr a' abbreviate '(Sent a . (Eb)(Vbll b . (bninvepna) Des x$x: =

4))',

so that a is true if and only if a is a sentence and any vacuous abstract (bninvefina), where b is a variable of type 1 , designates the universal class of objects of type 1. This truth-predicate may be proved adequate in the sense that I- Tr a

.=. X,

where X is any sentence from the translational part of SM;,, and 'a' is its structural description. Thus T G l . t Tr a . =. X,if (etc.) As consequences of this we have the following. TG2. TG3. TG4. TG5. TG6.

I- Sent a :1 : Tr a .=. Tr (tilde a ) . I- Sent a . Sent b :1: Tr (a vee b) .=. (Tr a .v. Tr b). I- (Sent a . Ax a) .v. (Eb)(Axb . a Clsr b) :3 : Tr a. FTra.bGena : I : T r b . t T r a . Tr b . c M P a , b I: : Trc. N

32

TYPE-THEORETICAL SYSTEMS AND SEMANTICALSUBSTRUCTURB

-

TG7. I- Sent a . Thm a :I : Tr a. TG8. I- Sent a . 3 . (Tr a . Tr (tiZde a)). (Ea)(Sent a . Thm a . Thm (tilde a)). TG9. IN

Also we have the following theorems, which may be useful below. =X : :3 : Tr (anidnb). TGIO. k a Des XI. b Des x: . TG 1 1. I- Con" a . Con" b , Tr (anidnb) . Con"+' c :3 :Tr (anepnc hrsh bne9nc).

C H A P T E R I11

A PRAGMATICAL META-LANGUAGE We now turn to our main task, the construction of a pragmatical meta-language for T . The syntactical and semantical substructure has been sketched in the preceding chapter. The pragmatical metalanguage for T consists merely of a suitable extension of that framework, more specifically, of SM;,. In A the new primitive needed is discussed. The rules of the pragmatical meta-language are given in 3 B. In C a classification of the persons who use T is given. Some notions akin to those of subjective intension are introduced in 3 D, and some relations of co-intensiveness are defined in 3 E. In 3 F, the notion of a quasifiro#osition is considered briefly. Finally, in § G, various kinds of so-called total accefitance classes are introduced.

A. Acceptance. Of the three pragmatical relations discussed in D), we may select now for more explicit study a relation of acceptance. Later we shall see how the pragmatical meta-language may be extended to include relations of acceptance for action, and perhaps other relations, as well. A relation of acceptance will be taken as a pragmatical primitive in the meta-language now to be formulated. Let this be symbolized by ‘Acpt’. ‘Acpt’ is not to stand for a dyadic relation, because fundamental reference to the time t seems essential. Acceptance is rather to be triadic, and the atomic sentential functions involving ‘Acpt’ are thus of the form ‘ X Acpt a$, where ‘x’is a variable ranging over some given finite class of human beings, ‘a’ a variable ranging over the expressions of the object-language, and ‘t’ a variable ranging over appropriate slabs of time. ‘ X Acpt a$’ may be read ‘the human being X accepts the sentence a at time t’. (I,

34

A PRAGMATICAL META-LANGUAGE

Let ‘B’ be an additional primitive standing for a certain temporal relation between slabs of time. The pragmatical meta-language which arises from SM;,, by taking ‘Acpt’and ‘B’ as new primitives, then, we may call PMT. As variables of PMT, we need the two new styles of variables, ‘ X ’ , ‘Y’, etc., for human beings, and ‘t’, ‘tl’, ‘tz’, etc., for timeslabs. Altogether PMT contains, in addition to translations of the variables of T of all types, variables for the expressions of T , variables for the human beings being considered, and variables for time-slabs. The underlying logic of PMT must provide for quantification upon all kinds of variables admitted, and thus itself embodies a theory of types. But the classes over which the variables of higher type range are precisely those of T. No variables over classes of expressions are admitted, the underlying semantics being formulated without such, as we have seen. In addition to these new kinds of variables, we may of course admit constants designating specific human beings or specific times. Such constants will be useful in stating specific Acceptance Rules below. (The notions of formula and term of PMT may be defined by way of summary as follows. 1. If X is a formula of SM;,,, it is a formula of PMT. 2. If x: is a term of type n of SM;,,, it is a translational term of type n of PMT. 3. If ‘a’ is a syntactical term of M , it is a syntactical term of PMT. 4. If ‘t’ and ‘tl’ are variables or constants for time-slabs, then ‘t B tl’ is a formula of PMT. 5. If ‘X’ is a variable or constant for human beings, ‘t’ a variable or constant for time-slabs, and ‘a’ a syntactical term, then ‘ X Acpt a$’ is a formula of PMT. 6. If ‘X’ and ‘Y’ are variables or constants for human beings, ‘t’ and ‘tl’ variables or constants for time-slabs, ‘a’ and ‘b’ syntactic terms, and x: and y i translational terms of type n, then ‘X= Y’, ‘t = tl > , ‘ a = b’, and ‘x: = yz’ are formulae of PMT. 7. If X and Y are formulae of PMT, so are ‘- X ’ and ‘(Xv Y ) ’ .

84

ANALYTIC TRUTH AND ABSOLUTE INTENSIONS

tension. These we shall call absolute quasi-intensions. They are not strictly intensions at all in the sense of being proporties, class-, relational, or individual concepts, or the like, the semantical metalanguage here being wholly extensional. But quasi-intensions enable us to achieve some of the effect of genuine intensions and serve some of the purposes for which a theory of absolute intensions is designed, as we shall see. The absolute quasi-intension of a constant a of type n we may regard as the virtz4aZ class of L-identical constants of the same type.

' F AbsQuasiInt" a' abbreviates '(Con" a . F = b3b LIdCon" a ) ' , Note that the absolute quasi-intension of a is a virtual class of expressions here, so that this definition may also be given within a non-translational semantics (cf. (IV, E ) ) as well as within SM;,,. (An alternative, perhaps better, definition of ' F AbsQuasiInt" a' is given in 9 E below.) To give the absolute quasi-intension of a primitive or defined constant or of an abstract containing no free variables is in effect to give the constants L-identical with it. There is thus some kinship between this notion and traditional notions of intension. To give the intension of a term is in effect to give words or phrases which, to speak loosely, mean the same. Actually the absolute quasiintensions of abstracts, for example, are infinitely large, because one can always form a new L-identical abstract from 'x$X' merely by adding an analytic sentence to X as a conjunct. Similarly the absolute quasi-intension of a primitive individual constant consists of an infinity of expressions. But all such expressions are L-identical and therefore it seems reasonable and intuitively justifiable to regard the quasi-intension as the whole bundle of them, so to speak. The following theorems are immediate.

TCI. t F AbsQuasiInt" a .I. ( E b ) ( Fb . Con" b). TC2. t F AbsQuasiInt" a .G AbsQuasiInt" a :2 : F = G . TC3. k F AbsQuasiInt" a . F AbsQuasiInt" b :=) : a LIdCon" b. TCl is a law of existence, and TC2 and TC3 are principles of uniqueness.

v, cl

A B S 0 L U T E Q U A S 1-1 N T E N SIO N S

85

TC4. I- F AbsQuasiInt”+’ a . a Des x:+’ . F b :3 : b Des x:+‘. (Note that TC4 does not hold where a is a constant of type 1, because ‘a Des xi’ is not significant where a is a defined individual constant. Hence we have ‘n 1’ here rather than ‘n’.) In terms of absolute quasi-intensions a theory of absolute quasipropositions may be built up, similar to the way in which subjective quasi-propositions were introduced in (111, F). Consider the atomic sentence (1) ‘a: E a:’ of T. The absolute quasi-intension of this sentence should be uniquely determined by the absolute quasi-intensions of its constituent terms. This we may achieve by regarding absolute quasipropositions as virtual ordered couples of absolute quasi-intensions. Let us introduce a notation for virtual dyadic relations between virtual classes of expressions, similar to the way in which expressional abstracts were introduced in (111, F). Thus we let ‘F’ FG3XG” be defined as X , where F, G , etc., are any expressional abstracts containing no free variables. Expressions such as ‘FG3X stand then in effect for virtual relations between virtual classes of expressions. Also we may let ‘F’ 2 FG3XG” abbreviate ‘ N F’ FG3XG”, and ‘F (F’G’PXUF”G”3X) G’ abbreviate ‘ ( FF’G’3X G .v. F F”G”3XG)’. These definitions introduce notations, respectively, for the negution of a virtual dyadic relation between virtual classes of expressions, and for the logical suuTyt of two such relations. The absolute quasi-proposition corresponding to the sentence ( 1) we may regard as the virtual relation expressed by

+

‘FG3(FAbsQuasiIntl a: . G AbsQuasiInt2 at)’, where F and G are any expressional abstracts (containing no free variables) picked out in advance. The absolute quasi-proposition for (2)

a6

ANALYTIC T R U T H A N D A B S O L U T E I N T E N S I O N S

the negation of the sentence may be regarded as the negation of the virtual ordered couple expressed by (2). The absolute quasi-proposition for the disjunction of two atomic sentences may be regarded as the logical sum of the absolute quasi-propositions of the constituents. And so on, so that absolute quasi-propositions for sentences constructed from atomic ones by a finite number of applications of negation or disjunction may be introduced. Also we have absolute quasi-propositions for sentences containing suitably restricted quantifiers, essentially as in (111, F). Let us consider for the moment only atomic sentences. Thus we may suppose (where a is atomic)

‘FG3Y AbsQuasiProp a’ to be appropriately defined. As theorems we should then have the following. TC5. k FG3Y AbsQuasiProp a, where ‘a’ is taken as the struc(so that x l and xi+‘ tural of an atomic sentence ‘x;Ex;~+” are constants), and Y is ‘ ( F AbsQuasiInt” b .G AbsQuasiInt”+’ c)’ where ‘b’ and ‘c’ are taken respectively as the structural descriptions of x; and xi+‘. TC6. I- FG3X AbsQuasiProp a . F’G’3X AbsQuasiProp a :3 : F FG3XG . =. F F’G’aXG. Note again the definitions of this section, as well as those in the preceding sections of this chapter, may also be given within a nontranslational semantical meta-language.

D. Pragmatical and Absolute Quasi-Intensions.Let us return again for a moment to the pragmatical meta-language PMT, now that ‘Anlytc’ and derivative notions of L-semantics are available within the underlying semantics, whether translational or nontranslational. Consider again the class F of sentences of T under investigation. Now that ‘Anlytc’ is available, we have another category of users of T to consider, those who accept the analytic truths of T in F . Thus we may let

‘ X Acpt Anlytc,F,t’ abbreviate ‘(a)(Anlytca . F a : 2 :X Acpt a$)’.

V, D]

Q U A SI-IN T E N S I O N S

87

But in view of TA3 above, we have that

TD1. t X Acpt AnZytc,F,t .I. (a)(Fa . LogThm a :3 : X Acpt a$) although the converse of course does not hold without Godel’s completeness theorem. Let us consider now how the various types of quasi-intension are interrelated. We note that the subjective, intersubjective, intertemporal, or objective quasi-intension of a given constant can never within PMT coincide with the absolute quasi-intension of that constant, the former being classes of classes, the latter virtual classes of expressions. Within non-translational pragmatics, however, we have identified subjective and other kinds of quasiintensions with virtual classes of expressions. More particularly, if a is of type n, the virtual class of expressions which is the subjective quasi-intension of a (relative to X, F , and t) is a virtual class of expressions of type n + 1. So in non-translational pragmatics also the subjective quasi-intension of a given constant can never coincide with the absolute quasi-intension of that constant. And similarly for intersubjective, intertemporal, and objective quasiintensions. We do, however, have the following theorems within PMT concerning the interrelations of absolute with subjective and other kinds of quasi-intensions. TD1. t F AbsQuasiInt” a . xy+2 SubjQuasiInt a,X,F,t . X Acpt (anidnb),t . LogThm (anidnb) . F (anidnb) . b Des :I : F b .=. {xy} E x?+’, where ‘xy’ is taken as any primitive individual constant or any primitive or defined constant of higher type. TD2. I- F AbsQuasiInt” a . xq+’ IntSubjQuasiInt a,F,t . (X)X Acpt (anidnb),t . LogThm (anidnb) . F (anidnd) . b Des xy : 3 : F b .=. {xy} E xy+’, where (etc. as in TD1). TD3. I- F AbsQuasiInt” a . xy+’ IntTempQuasiInt a,X,F . ( t ) X Acpt (anidnb),t. LogThm (anidnb) . F (anidnb) , b Des $9 : 2 : F b .=. {xy} E where (etc.). TD4. t F AbsQuasiInt” a . x:+’ ObjQuasiInt a,F . (X)(t)XAcpt (anidnb),t . LogThm (anidnb) . F (anidnb) . b Des xy :2 : F b .=. {xy} E xff+’, where (etc.). 3

88

ANALYTIC TRUTH A N D ABSOLUTE INTENSIONS

Within non-translational pragmatics we should have rather the following theorem in place of T D 1.

TD5. t F AbsQuasiInt" a .G SubjQuasiInt" a,X,F,t . Con" b . X Acpt (anidnb),t .LogThm ( a ~ i d ~.F b )( a ~ i d n b.)Vbl" c :2 : F b .3. G (cninvefmcnidnb) . Analogous theorems may be given for 'IntSubjQuasiInt"', 'IntTemp QuasiInt"', and 'ObjQuasiInt"' as defined non-translationally. Also similar theorems may be given concerning total acceptance classes of various kinds.

E. Some Alternative Definitions. There are of course alternative ways of introducing absolute quasi-intensions within SM;,,. Rather than to regard the absolute quasi-intension of a constant as the virtual class of L-identical constants, we might construct a definition similar to that of 'SubjQuasiInt' but with no reference to 'Acpt' in the definiens. Thus we might let

'x:+~ AbsQuasiInt a' abbreviate '(Con" a .x$'-~ = 4'+'3(Eb) (Ec)(Ed)(Ee)(cDes ~ ' f " . c = (dninvepne) . b Sz e . Anlytc b))', taking 'Con"' in the sense of (11, E). According to this definition an absolute quasi-intension is a class of designated classes and therefore can be given only within SM;,,, but not within a non-translational meta-language. Concerning the notion just defined, we have obviously the following theorem.

T E 1. X"+2 1

c

t xT+2 SubjQuasiInt u,X,F,t . x;+'

AbsQuasiInt a :3 :

x;+2.

Within non-translational semantics the analogous definition is as follows.

' F AbsQuasiInt" a' abbreviates '(Con" a . F = b(Ec)(Ed)(Ee) (Con"+' b . b = (dninvepne) . c Sz e . Anlytc c))'. According to this, an absolute quasi-intension of a constant of type ?a is a virtual class of constants of next higher type, whose designata apply analytically, so to speak, to the object designated by the

V, F]

CONCLUDING COMMENTS

89

given constant. (‘Con”’ here likewise is taken in the sense of (11, E), rather than in that of 9 B above.) Likewise absolute quasi-propositions might be regarded not as virtual ordered couples of absolute quasi-intensions in the sense defined in tj C, but as virtual ordered couples of absolute quasiintensions in the sense just defined. Still another alternative would be to regard the absolute quasi-proposition corresponding to a sentence as the virtual classes of L-equivalent sentences. Thus

‘F AbsQuasiProp a’ might abbreviate ‘(Sent a . F

= b3b LEquiv a)’.

We need not decide here amongst these various alternatives, which anyhow are semantical rather than properly pragmatical. But there are some interesting relationships obtaining between or among the various alternative notions, and between or among these and the various pragmatical notions introduced above.

F. Some Concluding Comments. We should note that the theory of absolute intensions put forwardin $9 C-D above enjoys some important advantages over most alternative theories. In the first place, it is given within fully formalized semantical meta-languages. No primitives are required for the theory of analytic truth and allied notions other than those required for the ordinary semantical truth-concept. Further, the semantical meta-languages here are wholly extensional and are suitably restricted. 1 Alternative theories of intension, such as those of Carnap and Church, cannot be given within such meta-languages. Church’s theory 2, for example, is not only thoroughly intensionalist but involves an immensely rich ontology of objects, including not only intensions as values for variables, but intensions of intensions, intensions of these intensions, and so on ad infiniturn, together with Cf. Truth and Denotation, pp. 175-178 and 263-295. See A. CHURCH, “A Formulation of the Logic of Sense and Denotation,” in Structure, Method, and Meaning, Essays in Honor of Henry M . Sheffer (Liberal Arts Press, New York: 1951), pp. 3-24, and “The Need for Abstract Entities in Semantic Analysis,” Proceedings of the American Academy of Arts and Sciences 80 (1951): 100-112. See also the abstract by CHURCHin The Journal of Symbolic Logic XI (1946),p. 31. 1 2

90

ANALYTIC TRUTH AND ABSOLUTE INTENSIONS

suitable axioms characterizing them. Church says that “[tlo those who find forbidding the array of abstract entities and principles concerning them, I would say that the problems which give rise to the proposal are difficult and a simpler theory is not known to be possible.”l But also a simpler theory is not known to be impossible, and thus any presumption in favor of one rather than the other seems arbitrary. Church has not shown that simpler theories are impossible or even unlikely. And clearly, some of the purposes for which an intensionalist theory of absolute intension is designed may be served by the simple (extensionalist) theory put forward above. Not only is Church’s system thoroughly intensional, it requires primitives of such great complexity so as to accommodate not only substantially the whole of mathematics (in the sense in which Principia Mathematica does) but a vast new area of semantical theory concerning the staggering array of entities admitted. But the notion of analytic truth, and therewith the L-semantics based upon it, is essentially simple in structure. The meaning of ‘Anlytc’, in essentially the sense of Hilbert-Bernays and Tarski above, is clear-cut and obvious and an exact definition of it can be given in a very simple way, as we have seen. None of the complexities of Church’s theory are required for this purpose. And even if an important purpose could be exhibited for which such complexities seem needed, we might well prefer to modify the purpose or to attain it only partially than to accept the attendant complexities. A t any event, the simpler methods should surely be developed and explored as far as possible, and then cautiously extended if need be. Also semantics is augmented here by a pragmatics, which helps to compensate for the sparse modes of expression available. And, as we have seen above, important notions are definable within a systematic pragmatics, the use of which helps to render a complicated intensionalist semantics unnecessary. Carnap’s so-called method of extension and intension 2 is similar in aim to that of Church, but much more restricted both as to logical structure and ontology. Carnap’s theory is not formalized, however, and it is not easy to discern its precise character. He 1 2

“A Formulation of the Logic of Sense and Denotation,” p. 104. See Meaning and Necessity, esp. pp. 1-68 and pp. 145-172.

v, FI

CONCLUDING COMMENTS

91

envisages a “neutral” meta-language in which the individual and predicate constants of the object-language have both an “extension” and an “intension”. But what are the primitives of this neutral meta-language? What kinds of variables does it contain and what do they range over? What are its axioms and rules? Only in the light of precise answers to these questions can the exact meaning of ‘neutral’ here be determined. Meanwhile Carnap’s informal comments have been of value in helping to bring to light alternative possible ways of handling intensions, of formulating L-semantics, and the like. In both the theories mentioned, absolute intensions are identified with a non-linguistic kind of entity. In the theory put forward above, however, intensions are identified with (virtual) classes of expressions. Carnap himself has noted this as a possibility but has not developed it in any detail. “It is . . . possible,” he says, “to define in terms of classes certain entities which stand in a one-one correlation to properties or other intensions and therefore may represent them for many purposes. We defined earlier the Lequivalence class of a designator in [the language] S as the class of all designators in S L-equivalent to it.. .. It is easily seen that there is a one-one correlation between the L-equivalence classes in S and the intensions expressible in S. Therefore, the L-equivalence class of a designator in S may be taken as its intension or at least as a representative for its intension.” 1 Carnap’s L-equivalence classes correspond here with virtual classes of L-identical constants. The identification of intensions with such classes seems so simple and natural as to recommend itself above all other methods. And, indeed, it is curious that this method seems not to have been explicitly formulated in any detail heretofore at all. Quine has also hinted at an identification of meaning with classes of expressions. “Just what the meaning of an expression is what kind of object - is not yet clear,” he says, “but it is clear that, given a notion of meaning, we can explain the notion of synonymity easily as the relation between expressions that have the same meaning. Conversely also, given the relation of synonymity, it 1

Meaning and Necessity, p. 152.

92

ANALYTIC TRUTH A N D ABSOLUTE INTENSIONS

would be easy to derive [sic!] the notion of meaning in the following way: the meaning of an expression is the class of all the expressions synonymous with it. No doubt this second direction of construction is the more promising one. . .” 1 Of course Quine is talking here informally and of natural languages primarily rather than of language-systems, and it is not clear whether by ‘class’ he means actual class or virtual class. If we confine attention exclusively to language-systems, the relation of L-identity as introduced in § B above plays essentially the role in T which synonymity does within natural languages. Thus the notion of absolute intension introduced above seems in essential agreement with Quine’s notion of meaning. But Quine demands a behavioral definition of ‘synonymity’ for natural language whereas the definition of L-identity for T above is purely semantical, and, more particularly, is given within an extensional and very restricted kind of semantics. Quine makes no attempt to provide such a semantical definition of ‘synonymity’ for language-systems. 2 The identification of meaning or intension with classes of expressions also harks back to Russell, who suggests that a proposition might be ragarded as “the class of all sentences having the same significance as a given sentence.” 3 In terms of L-identity, two atomic sentences of T may be said to have the same significance if the two predicates have the same absolute quasi-intensions and the respective arguments do also. According to this view propositions are virtual classes of sentences rather than virtual ordered couples of absolute quasi-intensions. Consider the atomic sentence

The absolute quasi-intention of ‘a:’, which is the left component of the ordered couple constituting the absolute quasi-proposition (in the sense of § C) corresponding to (1) on the one view, would in 1

“Notes on Existence and Necessity,” Journal of Philosophy 40 (1943):

113--127, p. 120. 2 For some further comments about Quine’s views, see The Notion of -4nalytic Truth, Chapter 11, 5 F, Chapter IV,5 E, and Chapter v, 4 D. 3 See B. RUSSELL, An Inquiry into Meaning and Truth (Norton, New York: 1940), p. 209.

v, Fl

CONCLUDING COMMENTS

93

fact be the quasi-intension of xx, where

is any sentence which is a member of the proposition corresponding to (1) on the other Russellian view. Thus the two methods of handling propositions, although technically a little different, are essentially the same. And it would seem that almost anything that could be accomplished on the basis of one of these views, within the meta-languages here, could also be accomplished on the basis of the other, and conversely. (Cf. also the alternatives suggested in § E above.) We should note that the theory of intensions put forward here cannot handle intensions of intensions. But neither can the underlying semantics handle truths of truths, nor the pragmatics acceptances of acceptances. This provides no ground for thinking the theories of truth or acceptance inadequate. We can always gain a theory of the truth of truths of T by going into an appropriate metameta-language. The theory of intensions here is similar to that of truth in this respect. For a theory of intensions of intensions we may employ an appropriate meta-meta-language. Let us recall from (111, D) that the pragmatical quasi-intensions introduced there have members which the empirical scientist may identify or enumerate in specific cases. Note that the same is true of semantical quasi-intensions. The members of the absolute quasiintension of a given constant may be explicitly listed or enumerated. When we turn to the alternative theories of absolute intension, however, we meet with entities which within the theory have no identifiable parts or members or components. We are merely told, for example, that individual constants have “individual concepts” as their intensions. But what an individual concept is, or how one is distinguished from another, or how it is known when one has been met with, we are not told. Similar remarks apply to class-concepts, attributes, and so on. Such objects have no discernible parts or components or members or any other kind of object out of which they are generated or constructed or created in some way or another. Their kinship with metaphysical pseudo-entities is too close to use them as a basis for semantics or pragmatics, unless

94

ANALYTIC TRUTH AND ABSOLUTE INTENSIONS

simpler alternatives which avoid them have been shown impossible. 1 When a theory of absolute semantical intensions is supplemented with a theory of pragmatical intensions, as in the preceding chapters, much of the motivation for seeking an intensionalist semantics is lost. Many of the purposes for which an intensionalist semantics is designed may be served equally well by the pragmatics formulated here. We need not claim that all of the purposes may be so served - some of them may be somewhat suspect. At any event, enough surely has been said to show that the pragmatical approach is a fruitful one and one which should be more fully and carefully developed.

1 For some spirited comments along similar lines, see M. G. WHITE, o p . cit., passim (but cf. The Notion of Analytic Truth, Chapter v, 9 E), and A. J. AYER, “Meaning and Intentionality,” A tti del XII Congress0 Internaziolzale d i Filosofia, Relazioni Introduttive (Sansoni, Firenze: 1958), pp. 141-155, esp. pp. 145-146. See also QUINE’SFrom a Logical Point of View (Harvard University Press, Cambridge: 1953), esp. pp. 139-159.

APPENDIX

O N GOODMAN’S T H E O R Y OF PROJECTIBLE PREDICATES I n his Fact, Fiction, and Forecast 1, Goodman has sketched the beginnings of a theory of inductive inference. Unlike most theories connected with induction, which are either syntactical or semantical in character, this one involves pragmatical features fundamentally. Within it there is basic emphasis upon the actual inductions made in the past, as with Hume, upon actual inductive behavior, as it were. These past inductions serve as an explicit guide in framing principles of inductive inference, and such principles are justified to the extent that they conform with accepted inductive practice. In this Appendix we attempt to show that the foundations of Goodman’s theory may be formulated within a pragmatical metalanguage similar to those discussed above. Goodman’s objectlanguage does not appear to embody a theory of types. As objectlanguage, therefore, let us take any simple, applied first-order L 2 with or without identity, containing one or more primitive predicate constants each of specified degree. These may be thought of as being what Goodman calls “manifest” predicates. Also we may assume that some individual constants are available, either as primitives or defined, one for each individual in the fundamental domain of L . We may assume further that at least one such constant is available as a primitive. The underlying syntax is essentially that outlined in (11, E). Let the symbols of L be the variables ‘ x ’ , ‘x”,‘x”’, etc., the primitive manifest predicates ‘P’, ‘P”, ‘P‘”, etc., each of specified degree, 1 University of London, The Athlone Press: 1954 and Harvard University Press: Cambridge: 1955. The author wishes to thank Professor Goodman for some helpful comments concerning the material of this Appendix, although of course he is in no way responsible for its shortcomings. In essentially the sense of CHURCH,op. cit., p. 168 ff. See also Truth and Denotation, pp. 3 1-69.

96

APPENDIX

and ‘v’, ‘(’, ‘)’, and perhaps Also suitable one-place abstracts are to be available, so that ‘3’may be taken as an additional primitive. As structural-descriptive names for these symbols, we have then ‘ex’ for ‘x’,lac’ for ‘”, ‘pee’ for ‘P’, ‘@’for ‘(,, ‘rp’ for ‘vee’ for ‘v’, ‘tilde’ for ‘-’, ‘id’ for and ‘invep’ for ‘3’, essentially as above. Let the primitive individual constants of L be ‘a’, ‘a”, . . ., ‘a“ Defined individual constants may then be symbolized merely by using more accents. Let ‘ay’ be the additional structural-descriptive name needed for ‘a’. Italic letters ‘a’, ‘b’, etc., will be used as syntactical variables for the expressions of L. ‘n’ will symbolize the operation of concatenation, just as above. On this basis we know that the full syntax of L is forthcoming. Let ‘PredConOne a’ abbreviate that a is a one-place primitive predicate constant of L or a one-place abstract containing no free variables. Similarly, let ‘InCon a’ express that a is a primitive or defined individual constant and ‘Vbl a’ that a is a variable. Also ‘hrsh’ is to be defined here as above. To this syntax we add a pragmatics of acceptances, essentially as above. To the definitions concerning time, in (111, B), we now add the following. I-’,

‘=I.

I)’,

I=’,

’a*.’’.

‘tl U tz’ abbreviates ‘ ( t l B t z t l 0 tz))’.

.v.

tl P tz

.v.

( ( E t ) ( tP t l . t B tz) .

The definiendum here reads ‘tl is a time up to tz’, i.e., t l wholly precedes or is a part of t z or t l overlaps with a beginning slice of t2. We may now turn to Goodman’s theory, mirroring his definitions as closely as possible. For this we follow his actual text. “A hypothesis will be said to be actually firojected when it is adopted after some of its instances have been examined and determined to be true, and before the rest have been examined,” Goodman notes. 1 To say that a sentence a is “examined and determined to be true” at time t we take to mean merely that a is accepted a t t. (No reference to the semantical truth-concept here seems needed. Nor do we need a separate primitive for ‘examined’.) 1

Fact, Fiction, and Forecast, p. 90.

ON GOODMAN’S T H E O R Y O F P R O J E C T I B L E P R E D I C A T E S

97

If this identification is legitimate, the definition above may be given as follows. ‘ a AcProjSent t’ abbreviates ‘(Eb)(Ec)(PredConOneb . Vbl c . a = ( Z p n c n q h b n c ) . ( E X ) ( E t l ) ( E d ) ( t lB t . InCon d . X Acpt (bnd),tl . (t2)(tl B t 2 . t 2 U t :> : X Acpt (bnd),tz)) . X Acpt a,t . (e)(InCon e .I. X Acpt (bne),t) . ( t l ) ( tB t l .I X Acpt a,tl))’.

-

Note that a here must be a sentence of universal form ‘(x)Fx’, where F is a primitive or defined one-place predicate constant. Such instances of an hypothesis “as have already been determined to be true or false may be called respectively its positive and its negative instances or cases at that time. All the remaining cases are thus undetermined cases. If the hypothesis is to the effect that all so-and-sos are such-and-such, then the so-and-sos named in its positive cases constitute the evidence class for the hypothesis at that time, while the so-and-sos not named in either its positive or its negative cases constitute its projective class. A hypothesis for which there are some positive 01 negative cases up to a given time is said to be supflorted or to be violated at that time. . . . A hypothesis without any remaining undetermined cases is said to be exhausted.” 1 Within the pragmatical meta-language here we may let

‘ a PosInst b,t’ abbreviate ‘ ( E c )( E d )( E e )( E X )(PredConOne c . Vbl d . b = (Zpndnrpncnd) . InCon e . a = (cne) . (tl)(tl U t . 3 . X Acpt a,t1))’, ‘ a NegInst b,t’ abbreviate . . .(the same, but with ‘ X Acpt (tiZdena),tl’ in place of ‘ X Acpt a,tl’. . ., and

-

‘a UndetInst b,t’ abbreviate ‘(Ec)( E d )(Ee)(PredConOne c . Vbl d . InCon e . b = (Zfindnrflncnd) . a = ( m e ) . a NegInst b,t . a PosInst b,t)’. N

Note that Goodman’sdefinition of being an evidence class relative to a given hypothesis and to a given time presupposes a semantical relation of naming. None of the definitions given thus far have 1

Ibid., pp. 91-92.

98

APPENDIX

required this and hence may be given within a pragmatics presupposing only a syntax. But now a semantical notion is needed. Because the object-language does not embody a theory of types, we take a relation of multiple denotation rather than Des as a primitive. 1 Let this be symbolized by ‘Den’. (Concerning Den we assume suitable semantical rules. 2) We may then say that an individual constant a is a $roper name of an entity x if and only if an abstract (bninvepn bnidna), where b is a variable, bears Den to x. Thus

‘ a PrNm x’ abbreviates ‘(Incona . (Eb)(Vblb . (bninvepnbnidna) Den x))’.3 Then

‘ F EvidCls a,t’ may abbreviate ‘(Eb)(Ec)(Ed)(PredConOne b . PredConOne c . Vbl d . a = (1pAdAYpdpAbAdhrsh cndnrp) . (x)( F x : =: b Den x . (Eb’)(Ec’)(b’PosInst a,t . b‘ = (i?pnbnc’ hrsh C A C ’ A ~ ~ ). c’ PrNm x ) ) ) ’ . Instances of the definiendum here may be read ‘such and such a virtual class is an evidence class relative to a and t’. The notion of being a projective class may be defined similarly.

‘ F ProjCls a,t’ abbreviates ‘(Eb)(Ec)(Ed)(PredConOneb . PredConOne c . Vbl d . a = (lpndnrpn@nbnd hrsh CAdAYp) . (x)(Fx :E: b Den x . (Eb’)(Ec’)((b‘PosInst a,t .v. b’ NegInst a,t) . b’ = (@nbnc’hrsh c A C ’ A .~c‘~ PrNm ) x)))’. N

That a sentence a (of the appropriate form) is supported, violated, or exhausted at time t may be defined as follows.

‘a Sprtd t’ abbreviates ‘(Eb)b PosInst a$’, ‘a Vltd t’ abbreviates ‘(Eb)bNegInst a$, and

‘a Ehst t’ abbreviates ‘(Eb)(Ec)(PredConOneb . ,- (Ed)d UndetInst a$)’.

(@nCnYpnbnC) 1 2

3

See Truth and Denotation, pp. 99-142. Ibid., pp. 108-1 10. Ibid., pp. 213-226.

. Vbl c . a

=

O N GOODMAN’S THEORY O F PROJECTIBLE

PREDICATES

99

Two one-place predicate constants are co-extensive with each other if and only if every object denoted by one is denoted by the other and conversely. ‘ a CoExt b’ abbreviates ‘(PredConOne a . PredConOne b . @)(a Den x . =. b Den x ) ) ’ .

Goodman speaks indifferently of the actual projection of a

sentence of universal form (such as ‘(x)Fx’)as well as of a oneplace fwedicate. The actual projection of a predicate may be defined in terms of the actual projection of a sentence, ‘AcProjSent’, as follows.

‘ a AcProjPred t’ abbreviates ‘(Eb)(Ec)(Ed)(bAcProjSent t . Vbl c . b = (@AC+A~AC) . a CoExt d ) ’ . It may also be useful to refer to the person X who actually performs the projection at time t. Thus we may let

‘ X AcProjPred a$’ abbreviate ‘(Eb)(Ec)(Ed)(Vblc . d CoExt a . b = (@nCAr$AdAC) . X Acpt b,t . (Etl)(Ee)(tlB t . InCon e . X Acpt (dne),tl . (tz)(tl B t 2 . t 2 U t :2 : X Acpt (dne), t ) ) . (e’)(InCon e‘ .I. X Acpt (dne’),t) . (tl)(t B t i .=. X Acpt b,ti))’.

-

Note that these definitions are framed in such a way as to assure that the actual projection of a one-place predicate constant includes also, so to speak, the actual projection of all co-extensive one-place predicate constants. Goodman takes as a primitive relation “that obtaining between any two predicates such that the first is much better entrenched than the second.”l Suppose we are given two predicates, say ‘green’ and ‘grue’. Let us then “consult the record of past projections of the two predicates. Plainly ‘green’, as a veteran of earlier and many more projections than ‘grue’, has the more impressive biography. The predicate ‘green’, we may say, is much better entrenched than the predicate ‘grue’.” 2 Presumably we have under consideration at most a finite number 1 2

Fact, Fiction, and Forecast, p. 96. Ibid., p. 95.

100

APPENDIX

of persons and a finite number of times (and thus of momentary times). Hence we may suppose expressions such as (1)

‘NC ‘ X3(X AcProjPred a,t) = n’

as suitably defined by means of numerical quantifiers, for any given finite integer n. Also we may suppose that n

(2)

‘C NC ‘ X3(X AcProjPred a,ti)’ i=l

is suitably defined for any specific finite n. Expressions such as (1) may be read ‘the cardinal number of the (virtual) class of persons X who actually project the predicate a a t time t is n’. (2) then gives an expression for a suitable finite sum of such numbers. Let us think of n in (2) as being the number of momentary times wholly before some time t. Given any one-place predicate a, there is a finite number, which may be symbolized by an expression of the form ( 2 ) , which we may call the $rejection index of a at t. This we may introduce in context. We may let

‘;bi(a,t) = n’ abbreviate

‘(NC ‘ tp(Mom tl . t l B t ) = m .

m

i=l

NC ‘ X3(X AcProjPred a,tc) = n)’

for some suitable numeral ‘m’. Similarly we can suppose

‘@(a$) > pi(b,t)’ defined as

‘(pi(a,t)= n . fii(b,t) = m . n > m)’ for suitable ‘n’ and ‘m’ and where ‘n > m’ is defined merely by enumeration of cases up to n. Clearly a one-place predicate a is better entrenched at time t than a one-place predicate b if $;(a$) > pi(b,t). But is it much better entrenched? How much larger must pi(a,t) be to be much better entrenched ? Presumably some suitable numerical difference

O N GOODMAN’S T H E O R Y O F P R O J E C T I B L E P R E D I C A T E S

101

may be fixed here, depending upon various circumstances. 1 If this may be done satisfactorily, we may then suppose

‘ a MBEnt b,t’, read ‘the one-place predicate constant a is much better entrenched than the one-place predicate constant b at time t’, suitably defined. We go on now to Goodman’s definition of a projectible predicate relative to time t. These are one-place predicates which may be described roughly as follows. They are to be supported, not exhausted, and not violated at time t. Further, in Goodman’s own words, (i) “a projection is to be ruled out if it conflicts with the projection of a much better entrenched predicate,” 2 (ii) “[a] projected hypothesis with an ill-entrenched consequent is to be rejected if it conflicts with another hypothesis (1) that has the same antecedent and a much better entrenched consequent, and (2) that is either ( a ) both violated and supported or (b) neither,” 3 and (iii) “where a consequent could have been projected over the extension of a given antecedent by a hypothesis, any other hypothesis is illegitimate if it has nothing additional in its evidence class and yet uses a much less well entrenched antecedent to project the same consequent over other things”. 4 This third proviso “eliminates a projected hypothesis when some other hypothesis with the same consequent could have been projected, and the antecedent A of the original hypothesis ‘disagrees’ with the much better entrenched antecedent A’ of the other hypothesis in the following way: although among things to which the common consequent has been determined to apply, A applies only to those that A’ applies to, nevertheless A applies to some other things that A’ does not apply to.” 5 1 To avoid the difficulty of fixing a numerical measure here, Goodman takes ‘is much better entrenched than’ as a primitive. But there is still then the problem of framing suitable axioms characterizing it. In the present treatment the only axioms needed are those of the underlying pragmatics, including of course appropriate syntactical and semantical rules. 2 Fact, Fiction, and Forecast, p. 96. 3 Ibid., pp. 101-102. 4 Ibid., p. 104. 5 Ibid., p. 104.

102

APPENDIX

We let ‘a Drv F’ express that a is derivable from such and such a virtual class of formulae. 1 The definition of being projectible at time t we formulate as follows.

-

a Vltd t . a Ehst t : ‘ a Prjtble t’ abbreviates ‘(a Sprtd t . (Eb)(Ec)(PredConOneb . Vbl c . a = (lfincnrfinbnc) . (Ed)(d Drv e3(e = a .v. (Eb’)(b‘MBEnt b,t . e = (Zfincnrfinb’nc) . e Sprtd t . e Vltd t . e Ehst t)) . (tildend) Drv e3(e = a .v. (Eb’)(b’MBEnt b,t . e = (Zpncnrfinb‘nc) . e Sprtd t . e Vltd t . e Ehst t)) .v. (Eb)(Ec)(Ed)(PredConOne b . PredConOne c . Vbl d . a = (lfindnrfmlpnbnd hrsh cnd) . (Ee)(e Drv d’3(d’ = a .v. (Ec’) (c’ MBEnt c,t . d‘ = (lpndnrfidfinbnd hrsh c’nd) . (d’ Vltd t . d‘ Sprtd t :v: d‘ Vltd t . d‘ Sprtd t ) ) ). (tildene) Drv d’3(d’ = a .v. (Ec‘)(c’MBEnt c,t . d‘ = (ZfindnrfinZfinbnd hrsh c’nd) . (d’ Vltd t . d‘ Sprtd t :v: d‘ Vltd t . d’ Sprtd t ) ) ) ) ) . v . (Eb)(Ec)(Ed) (PredConOne b . PredConOne c . Vbl a . a = (ZfindnrfinZfinbndhrsh cnd) . (Ea’)(Eb’)(a’= (ZfindnrfinZfinb‘ndhrsh cnd) . b‘ MBEnt b,t . a’ Sprtd t . a’ Vltd t . a’Ehst t . (Ex)((Ec’)(EX)(c’ PrNm x . X Acpt (cAc’),~ . b Den x . b‘ Den x) . (x)((Ec’)(EX)(c’PrNm x . X Acpt (cnc’),t) . b’ Den x : 2: b Den x))))’. N

N

-

-

-

-

N

N

N

N

N

N

--

The three disjunctive clauses in the definiens are intended to correspond respectively with (i), (ii), and (iii) in the informal definition. As Goodman points out, some undesirable hypothesis are countenanced as projectible by this definition. What we need therefore is a more refined notion of degree of firojectibility. Goodman gives some suggestions which may well lead to such a notion. We need not examine these here, where we have been concerned to show merely that the basis of his theory can be accommodated within the pragmatical framework. The presumption is therefore that such further definitions as may be needed can also be given within this framework. If the contextual treatment of numbers used above in the definition of ‘MBEnt’ should turn out to be too restricted, some extension of the pragmatical meta-language must be made by introducing numbers (natural or real( ?)) as values for variables. f

See Truth and Denotation, p. 96.

O N GOODMAN’S T H E O R Y O F P R O J E C T I B L E

PREDICATES

103

It is not claimed that the formal definitions given here mirror adequately or with full correctness Goodman’s informal ones. But surely they approximate them to some extent, and such corrections as may be needed may presumably be given within the present framework. It would be of interest to develop Goodman’s theory further and thus to see how it connects with more standard treatments of induction, which after all are also designed to codify inductive behavior. We know that Carnap’s inductive logic, for example, may be formulated on the basis of the simple kind of semantics presupposed here. 1 This formulation may well be useful in studying the interrelations between Goodman’s theory and traditional forms of inductive inference.

1 See the author’s “A Formalization of Inductive Logic,” The Journal of Symbolic Logic, to appear.

INDEX Abstraction, Rule of 19 Abstracts 17 ff., 35, 77 Acceptance 9 ff., 33 ff. Actions 9, 71 ff. Ajdukiewicz, C. 9 Alderson, W. 11 n. Analytic truth 76 ff. Assertion 9, 70 Belief 9, 1 1 Bernays, P. 77n., 78 Beth, E. W. 78

Henkin, L. 78, 79 n. Hilbert, D. 77n., 78 Hiz, H. 71 n. Identity 19, 35 Infinity, Axiom of 20 Inscriptional pragmatics 65 f. Keynes, J. N. XII Kotarbifiski, T. 9, 71 Kuratowski, C. 20 L-equivalence 79 ff. L-identity 80 ff. L-semantics 79 ff.

Carnap, R. x I r , xv, 1 ff., 6, 7, 11, 12, 62, 78, 79 ff., 90 f. Church, A. XII, xv, 79n., 89f. Chwistek, L. 22 n. Co-intensiveness 50 ff. Concatenation 22

McKinsey, J. C . C. 13 n. Morris, C. 6 ff. Multiplicative axioms 20

Davidson, D. 13 n. Descriptions 81 ff. Designation 29 ff.

Nagel, E. v Non-translational pragmatics 66 ff. Normal acceptance patterns 40 ff.

Eagleson, C. 41 n. Extensionality, Rule of 19, 79 Extensional languages 5 f.

Ordered couples 20 ff., 53 ff.

Fitch, F. B. 63 n. Frege, G. 18 Godel, K. 78, 79, 87 Goodman, N. 65 n., 95 ff. Grzegorczyk, A. 71 n. Hempel, C. G. 12

Performance 71 ff. Pragmatical rules 37 ff., 69 f. Pragmatics, descriptive 3 f., 7 f. pure 3 f., 7 f . Pragmatism 6 ff. Projectible predicates 95 ff. Quantification, Rules of 18 f. Quasi-intensions 44 ff., 67 f., 83 ff. absolute 83 ff., 88 f .

INDEX

intersubjective 47 f. intertemporal 48 objective 49 subjective 44 ff., 67 f., 83 Quasi-propositions 53 ff., 85 f., 89 Quine, W. V. 22 n., 25, 65 n., 77 n., 78, 91 f. Rejection 43 f. Rieman, G. F. 10 n. Russell, B. 20, 81, 92 Scholz, H. 78 Structural descriptions 22 f . Suppes, P. 13 n.

Syntax 22 ff. Tarski, A. 15, 77n.. 79 Time 34, 36 f., 96 Total acceptance classes 57 ff. Truth 31 f. Types, theory of 14 ff. Understanding 61 ff. Utterance 9, 69 f. Wiener, N. 20 White, M. G. 75 n. Whitehead, A. N. 63 Woodger, J. H. 6, 15 n., 36

105

SPECIAL SYMBOLS

E

N

V

x;,

, ; a (z);,

etc. etc. etc.

3

(Ex;) Anfl

t L

{xz), etc. (,Y;,X;), etc. l p , vp, etc.

n

Bgn Ends Seg PS AcString SubscrString AtFmla Vbl PrimCon Gen hrsh dot

tvipbar 4u exisqu MP IC SmTp Abst FrIng

15,35 15 f., 35 16 16 16 16 16 17, 35, 77, 85 18 18 18 20 21 21 22 22 23 23 23 23 23 23 24 24 24 24 24 24 24 24 24 24 25 25 25 25

Pr Fmla occ BOcc FrOccVbl FrOccCon Abst FrOccAbst Trm FrOccTrm Vbll Vb12 Abst2 Abst3 Trml Trm2 FV Sent SF1 SF? SF Con1 Con2 TFAx LogAx DesAx AX Thrn LogThm Proof QuantString Clsr Des Tr Acpt B

25 25 26 26 26 26 26 26 26 26 27 27 27 27 27 27 27 27 27 27 27 28, 82 28 28 28 28 28 28 28 28 29 29 29 31 33 34

SPECIAL SYMBOLS

t , etc. 0 P

Mom F AcptNor tilde AcptNor vee AcptNorDisj Acpt M P Acpt TFLog Acpt I C Acpt Log Acpt Thnz Acpt Tr Rjctl Rjcta

s,"

SubjQuasiInt IntSubj QuasiInt IntTempQuasiInt ObjQuasiInt SubjCoInt IntSubjCoInt IntTempCoInt ObjCoInt SubjQuasiIntPart SubjQuasiCmprh SubjQuasiProp TSACThm TISACThm TITACThm TOACThm TSAC EqUiVlTSACThm EquivaTSACTh, UndsCornc,, UndScon

Utt Ass

34 36 36 36 39 40 40 41 41 41 42 42 43 43 44 44 44 45 f., 67, 83 48 48 49 50,67 51 51 51 52 52 54 57 57 57 57 58 59 59 62 62 69 70

Prfm j , g, etc.

LDet Acpt Actnl AcptActnz TwoPlAscPredCon Rlep R?ep Rep Var Anlytc LImp LEquiv LDisj LFls LIdCon

(4)

DefInCon DefSentDesDef AbsQuasiInt AbsQuasiProp Acpt Anlytc P, P', etc. U AcProjSent PosInst NegInst UndetInst PrNm EvidCls ProjCls Sprtd Vltd Ehst CoExt AcProjPred

Pi

MBEnt Prjtble

107 71 72 73 73 74 77 78 78 78 78 78 79 79 80 80 80 ff. 81 81 f. 82 84, 88 86, 89 86 95 96 97 97 97 97 98 98 98 98 98 98 99 99 100 101 102