The Pragmatics and Semiotics of Standard Languages 9780271073538

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THE PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

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THE PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

Albert M. Sweet

THE PENNSYLVANIA STATE UNIVERSITY PRESS University Park and London

Library of Congress Cataloging-in-Publication Data Sweet, Albert M. The pragmatics and semiotics of standard languages. Bibliography: p. Includes index. 1. Language and logic. 2. Pragmatics. 3. Semantics (Philosophy) 4. Logic, Symbolic and mathematical. 5. Semiotics. 6. Model theory. I. Title. P99.S937 1988 401.41 87-43 190 ISBN 0-27 1-00630-7 Copyright

O

1988 The Pennsylvania State University

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Contents

Acknowledgments Introduction Part I. PRAGMATIC FOUNDATIONS OF STANDARD LOGIC 1 Sentential Interpretations 2 Coherent Interpretations 3 Pragmatic Models of the Theory of Polyadic Algebras 4 Pragmatic Representability of Standard Formal Systems 5 Locally Standard Grammar Part 11. INTENDED MODEL THEORY 6 Pragmatic and Semantic Synonymy 7 Standard Pragmatics Extended 8 Standard Semiotic Systems 9 Potential Monomorphism 10 Standard Models 11 Realizations of the a-extension Relation 12 Conclusion: On Referential Meaning and Truth Notes References Index

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Acknowledgments

I am indebted to the Rutgers Research Council for a research fellowship permitting a year's uninterrupted work, without whose benefit this book cound not have been written. Thanks are also due to the Research Council for a grant in support of typing the manuscript. For helpful comments on various chapters or earlier versions thereof, I am indebted to Lucio Chiaraviglio, Robert Martin, Arthur Smullyan, John Bacon, Robert Weingard, Sarah Stebbins, Ernest LePore, James Carson, and Jaakko Hintikka. Responsibility for lingering errors is of course my own. I am grateful to my colleague, Michael Rohr, for patient tutoring on the word processor. Finally, there is a subtle category of indebtedness which may, roughly speaking, be called pedagogical. Most profoundly in this respect I am indebted to my father, who assured me, at the age of ten, that though truth is difficult, Albert Einstein himself was still devoted to its pursuit. To my mother and to the memory of my father, chemist and writer, this book is dedicated.

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Introduction

What's in the world that ink may character? This generalization of a question posed by Shakespeare reflects one of the central problems of philosophy, the nature of the relation between language and the world. This book uses elementary methods of formal logic to investigate the structure of this relation. The problem becomes more tractable if its scope is restricted to standard first order languages. Models of (= relational structures for) such languages may be thought of as representing possible worlds describable in such languages. How serious then is the loss of generality involved in the restriction to standard languages? This loss of generality is only apparent provided the deep grammatical structure of natural language can be represented as that of standard logic. We shall find substantial evidence that this representation can indeed be carried out, though in a local as distinguished from a global sense (which is elucidated in Chapter 5). It is commonplace to observe that what a word or sentence denotes or asserts reflects the intentions of the users of such expressions, as represented by characteristic forms of verbal behavior involving such expressions. What is theformal structure of this relation? It is clear that a complete answer to this question would allow the linguist, given sufficient data about verbal behavior, to reconstruct the relevant grammatical categories as well as their seman-

INTRODUCTION

tic analyses, and indeed to predict that particular expressions in a given text function as words or sentences. Our task is then to answer the above question in a manner which realizes this desideratum. We shall approach this problem by explicating the concept of an intended model, where the "intending" is represented by characteristic forms of verbal behavior which underlie the very grammar of the relevant text. From this viewpoint, our task is to describe the pragmatic foundations of standard grammar and logic, to apply thesefoundations to the task of distinguishing the intended models of standard theories, and finally to show that the modeltheoretic semantics thereby obtained is applicable to natural language, including the scientificfragment of natural language. It will be found illuminating to view the relation of theory to intended model in the light of the relation of sign to object signified, as conceived in the semiotic theory of C. S. Peirce. According to Peirce, the relation of sign to object is always mediated by a third relatum, called an interpretant; the signification relation is an irreducible triadic relation obtaining among sign, object, and interpretant. We shall investigate the structure of the signifcation relation so conceived, for the case in which the relevant signs are standard theories, the objects signifed by such signs are their intended models, and the interpretants of such signs are characteristic forms of verbal behavior of the users of the signs. In this introductory chapter I shall first undertake to clarify informally the idea of an intended model and the related concepts mentioned above, in the context of their use in the philosophic literature. Then I shall outline the formal development of the pragmatic foundations of standard logic and the theory of intended models, and their application to natural language.

1. INTENDED MODELS AND REFERENTIAL MEANING

1A. On Pragmatic Theory The concept of an intended model associates a semantic concept with a pragmatic one. It is only in this pragmatic context that the semantic concept of a model is in need of clarification. The con-

INTRODUCTION

cept of a model, considered in itself, we may usefully understand in the pristine sense which it possesses in the model theory of standard first order languages. The concept of intending enjoys no such clarity, however, and we shall undertake its explication by means of the pragmatic theory developed in this book. An informal clarification of this theory is undertaken in the present subsection, in the context of a discussion of pragmatic theory in general. The term 'pragmatics' was introduced by C. Morris [34] to refer to the study of the relation of signs to their users and contexts of use. Subsequently Carnap characterized pragmatics as the empirical investigation of natural language, including the descriptive semantics of natural language. So, for example, a linguist studying the German language may discover that the intended meaning of the word 'blau' is, for typical speakers of the language, the color blue. (Cf. [30], p. 4.) The manner in which a typical speaker of a language understands a given word may, Carnap suggests, be investigated as follows. The linguist may observe whether an object, either presented or described to the speaker, in association with a given word, elicits one of the three possible responses: affirmation, denial, or abstention. The response indicates whether the speaker takes the word to refer to the object. (Cf. [31], p. 235ff.) The linguist may also advance hypotheses about the dispositions of speakers to make one of the above three responses under analogous circumstances, and Carnap extends the concept of a speaker to include robots whose internal structure is known well enough to justify the framing of such hypotheses. Carnap defines the intension of a predicate, for a speaker, as the general condition which an object must fulfill in order for the speaker to ascribe the predicate to the object. The structure of such conditions of applicability is treated formally in standard pragmatic theory as developed in this book, but the term 'intension' is dispensed with altogether. Standard pragmatic theory extends Carnap's usage insofar as it treats the relation between formal languages and their idealized speakers, as well as the relation between natural languages and their real speakers. Indeed the treatment of the latter by standard pragmatics is based upon its treatment of the former. A pragmatic analysis of a formalized language, considered in relation to the verbal behavior of idealized speakers of the language, is developed by A. Grzegorczyk [34].

4

INTRODUCTION

In his semiotic theory, Peirce introduced the concept of an indexical sign, whose object depends upon the context in which the sign is interpreted; examples are such words as 'now' and '1'. (Cf. [16], 4.544.) Subsequently Bar-Hillel in [29] suggested that pragmatics concern itself exclusively with indexical expressions, and Montague in [15] developed this idea systematically. A pragmatic language is for Montague a language containing indexical expressions, whose structure is to be investigated by the intensional logician. Intensions are for Montague semantic entities upon which he bases his semantical analysis of pragmatic languages. Standard pragmatic theory, on the other hand, treats of intentions rather than intensions, and is not restricted to the study of languages containing indexicals. This theory is developed in the metalanguage of a language whose very structure is determined by the intentions of the users of the expressions of the language, as represented in their verbal behavior. The pragmatically induced structure of the language is that of standard first order logic. The local realizability of such formal languages within a natural language such as English accommodates the phenomenon of indexicals, without any need of the intensions required in Montague's treatment. The various approaches to pragmatics found in the philosophic literature, as illustrated by the above examples, may evidently be accommodated by Peirce's concept of the interpretant of a sign: we may understand pragmatic theory, in its most general sense, to be the theory of the role of interpretants in signification. This usage accommodates, for example, that of Bar-Hillel and Montague: the context of an indexical expression (for example, an act of pointing), insofar as it determines the object signified by the expression regarded as a sign, can be thought of as an interpretant of the expression. Evidently the present usage also accommodates that of Morris and Carnap outlined above: any characteristic form of verbal behavior which underlies the referential meaning of an expression can be thought of as an interpretant of the expression. Specializing the above interpretation of the term 'pragmatics', we may understand standard pragmatics to be the theory of the interpretants of the signs of standard first order languages. The immediate goal of standard pragmatics is that of recovering the algebraic structure of standard logic by means of a purely prag-

INTRODUCTION

matic construction (Chapters 1-4). The ultimate goal is that of providing the foundation for a semiotic theory of intended models and referential meaning, in which the relevant "intending" is represented by the very forms of verbal behavior which determine the grammar of the sentences whose referential meaning is in question (Chapters 6-12). Applicability of this theory to natural language, in a local as distinguished from a global sense, is made possible by the development of a theory of locally standard grammar, which is also founded upon standard pragmatics (Chapter 5).

1B. On the Concept of an Intended Model The concept of an intended model (realization, interpretation) of a theory, and the related concept of the intended reference of appropriate expressions, are familiar in the literature of the philosophy of science. (Cf. van Frassen [28], p. 66; Adams [I], p. 258, Suppes [36], pp. 1-9; Przelecki [35], p. 17; Carnap [30], p. 4.) An intended model of a scientific theory is, roughly speaking, a model of the theory which is distinguished by the intentions of the users of the theory, as represented in their scientific practice. We shall follow Suppes [37], p. 12, in the view that the meaning of the concept of a model in empirical science is essentially the same as in pure mathematics. We shall, moreover, extend this viewpoint to the case of natural language, by means of the theory of locally standard grammar developed in Chapter 5. Intended models of a specific theory are in practice characterized by requiring that some or all terms of the theory have a distinguished denotation which is described in antecedently understood terminology of the metalanguage of the theory. For example, E. Adams in [l] describes the intended models of a particular formalization of the theory of Newtonian mechanics by specifying that abpropriate terms of the theory refer to particles, mass, position, force, etc. The underlying idea is that the reference of the terms of the theory, so distinguished, reflect the intentions of the users of the theory as represented in their scientific practice. What is the structure of the intending relation so conceived? We shall undertake to answer this question in a general setting which includes the scientific fragment of natural language as a special case.

INTRODUCTION

As Montague [15]has pointed out, specification of a distinguished referential meaning for the terms of an empirical theory in a model of the theory, in the sense illustrated above, requires supplementing the usual resources of metamathematics with antecedently understood terms for empirical concepts, e.g., 'particle', 'mass', 'force', etc. The question then naturally arises: When is such a metalinguistic description of an intended model correct? When is such an intuitively described intended model of a theory really distinguished by the scientific practice of the users of the theory? This question can have a determinate sense only in the context of a general theory of the formal structure of the relevant "intending" relation. This fact can be appreciated in the light of the following example. Consider some standard formulation of Newtonian mechanics, for example, that of Montague in [15]. Let T be the theory so formalized and let m be the term of T whose intended reference is constrained by the metalinguistic stipulation that m refer to the magnitude mass. Then the referent of m in intended models of T is evidently velocity-independent, since T is the theory of Newtonian mechanics. But in the metalanguage of Tit may well be the case that a velocity-dependent magnitude is referred to by the term 'mass'. In this case the above specification of the intended interpretation of m might perhaps be emended to assert that m refers to velocity-independent mass or, more precisely but evidently not more illuminatingly, rest mass in the sense of relativity theory. At this point the metalinguistic description of the intended reference of the term m of the theory T appears to have become altogether heuristic in character. Let us refer to the problem illustrated in the above example as the incommensurability problem, invoking familiar Kuhnian terminology without drawing any Kuhnian conclusions. Reflections surrounding this problem have led H. Field in [lo]to conclude that the term 'mass' in Newtonian mechanics has no determinate reference at all. We shall draw no such conclusions. We shall suggest a straightforward Tarskian solution to the incommensurability problem in the context of the theory of intended models developed in this book. The incommensurability problem illustrates the need for developing this theory, in its most general form, independently of

INTRODUCTION

any empirical, perhaps incommensurable metalinguistic terminology which might be used to assert that a specified model of a theory is an intended model. Only in the context of such a general theory is there a determinate question of the correctness of an intuitive or empirical metalinguistic description of an intended model of a specific theory. There is a natural appeal in the idea of representing referential meaning model-theoretically. Indeed if this idea were not altogether natural, the celebrated theorems asserting the existence of non-standard models would never have been regarded as paradoxical. From this viewpoint the referential meaning of a theory (in standard formalization) may be identified with the variety of the theory, the class of its models. The completeness theorem of standard logic can then regarded as an existence theorem for the concept of referential meaning so understood: the referential meaning (= variety) of a theory is non-vacuous if and only if the theory is consistent in a finitistic, proof-theoretic sense. Traditional scruples about empirical significance may in this context be reconstructed as criteria governing referential precision, in a sense which may be illustrated as follows. Model-theoretic semantics as outlined above admits a partial uniqueness theorem: all finite models of a theory which are indistinguishable by the sentences of the language of the theory (which, that is, are elementarily equivalent) are isomorphic. Insofar as finite models are concerned, then, simple completions of a theory have unambiguous referential meaning, and the referential meaning of the terms of such completions are precise up to isomorphism. The importance of this fact is considered later in this subsection. The finite uniqueness property, however, does not hold in general; it is well known that there are elementarily equivalent models which are not isomorphic, whence there are complete theories with nonisomorphic models. Moreover, the theories whose interpretations are in practice the subject of inquiry in the philosophy of science are not in general complete. What kind of referential precision may such theories be expected to possess (assuming they admit standard, perhaps set-theoretic formalization and therefore the modeltheoretic semantics illustrated above), and how may such precision be recognized?

INTRODUCTION

These questions suggest a refinement of the model-theoretic semantics outlined above: the referential meaning of a theory might be identified with a distinguished subclass of its models. This approach has the promise of making the problem of obtaining appropriate representation theorems more tractable. All models in a distinguished subvariety of a theory might be isomorphic to models of the theory with important and interesting properties, even though this isomorphism property fails to hold for all models of the theory. How are such models to be distinguished? We shall consider the case in which they are distinguished by characteristic forms of verbal behavior of the users of the theory, so that such models may be said to underlie the intended referential meaning of the terms of the theory. We shall refer to such models as intended models. The task of intended model theory is then to describe the formal structure of the pragmatic determination of model-theoretic meaning, conceived in the above manner. We have seen that model-theoretic semantics admits a natural uniqueness theorem for finite models. The concept of an intended model suggests the possibility of extending this theorem to the general case. Under what conditions, then, may the elementarily equivalent intended models of a theory reasonably be said to be isomorphic? I shall call this the uniqueness problem of intended model theory. A satisfactory solution of this problem can be considered a criterion of adequacy for any theory of intended models. How should the principles of this theory be chosen? A desideratum of any proposed principle of intended model theory is that it generalize an associated principle of standard model theory in the following sense: the proposed principle should admit the limiting case in which all models are intended models, in which case the principle is a consequence of standard model theory. Insofar as intended model theory realizes this desideratum, it is consistent relative to standard model theory. The desideratum also provides a guiding principle in the development of intended model theory, as will be seen. Although solution of the uniqueness problem requires a departure from the realization of the above desideratum, it turns out that this departure is a minimal one. This fact is discussed in more detail in section 2. For the present it is instructive to consider the intuitive significance of the uniqueness prob-

INTRODUCTION

lem from the perspective of the pure model-theoretic representation of referential meaning. Let T be a categorical theory. A metalinguistic description of the denotations of the terms of Tin models of T is, from the viewpoint of the theory T itself, mere renaming. Distinctions among the (isomorphic) models of T may contribute to the clarification of the referential meaning of T i n some wider context, for example, an extension of T (with an extended vocabulary), but not to that of T itself considered independently of such contexts. For considered in itself, T may be said to have completely unambiguous meaning, represented equally well by any model of T. Indeed the terms of T may not even be said to possess the notorious property of vagueness, insofar as vagueness is regarded as a semantical and not an epistemological property. The concept of an intended model invites generalization of the concept of categoricity. We may say that the intended variety of a theory is potentially monomorphic if elementarily equivalent intended models of the theory are isomorphic, if, that is, the uniqueness problem is solved for the theory. If potential monomorphism is preserved by simple extension, then simple completions of a theory whose intended variety is potentially monomorphic are such that all their intended models are isomorphic. In view of the above discussion of categoricity, such completions allow extension of the semantics of intended reference to appropriate expressions, for example, predicates and individual constants: any of the isomorphic intended models of the completed theory may be used to define such reference. From the viewpoint of pragmatic theory, the case of complete theories represents a substantial idealization of actual theory construction. For one may expect pragmatically defined theories to be in practice axiomatizable, and consequently to encompass Godelian incomplete theories. Thus the criterion of completeness contemplated above indeed represents only an ideal limit of actual theory construction. There is nevertheless an important theoretical justification for the contemplation of this limit, as will be seen in subsection ID, and in Chapter 10. It should also be noted that uniqueness, to isomorphism, of the intended models of an incomplete theory is by no means impossible in the present theory; the

INTRODUCTION

conditions for such uniqueness, however, require further departure from the realization of the desideratum discussed above. Disambiguation of this kind, whose ramifications bear upon the explication of the idea of scientific realism outlined later in this subsection, is discussed in detail in Chapter 10. The manner in which the uniqueness problem may be solved is outlined in the following section. Before turning to this development, further clarification of the idea of an intended model may be gained by considering its relation to that of a standard model and to the concept of truth. The theory of models for a standard language provides a theory of truth for the language in a relative sense: the concept of truth in a model but not truth simpliciter is definable in this context. It is natural to attempt to introduce the concept of truth simpliciter for a standard language L by distinguishing an appropriate model of L (= relational structure for L ) relative to which such truth is defined. The intuitive idea is that any model of L represents a possible world describable in L, and among these is a distinguished model which represents the actual world, or fragment thereof, indexed for description in L. (Cf. van Fraassen [28], p. 343; Przelecki [35], p. 17.) On this view there is a distinguished model of L which represents the actual world, so to speak, up to describability in L, without misrepresenting that world. One might wish to define a realist semantics for L as a model-theoretic semantics for L in which such a model is satisfactorily distinguished. Let us say that such a model (if it exists) is a Frege model of L ; the reason for this terminology will become clear in the sequel. It is customary to say that the natural numbers under the successor operation constitute the standard model of Peano arithmetic. Similarly the real numbers under the familiar operations and relations are said to constitute the standard model of the theory of complete ordered fields. (We continue to assume a first order formulation of such theories.) Standard models of specific theories, understood in the sense illustrated by these examples, might perhaps be considered to be Frege models by those to whom the above realist terminology is congenial. But realists might in fact differ about the Frege models of the language of real number theory; for models of non-standard analysis (more precisely, reducts of such models

INTRODUCTION

to the language of real number theory) might be considered better candidates for Frege models of this language than standard models of real number theory, by realists who believe in infinitessimals. This question is discussed in detail in Chapters 8 and 10; for the present I will compare the concepts of standard models, Frege models, and intended models. In this comparison, use of the realist ideas underlying the concept of a Frege model will not involve the begging of any philosophic questions. We have seen that standard models of a theory may but need not be regarded as Frege models of the language of the theory. This fact is illustrated not only by the example of real number theory considered above but also by the case of false theories which are thought to admit a standard model. A standard model of a false theory is clearly not a Frege model of the language of the theory, in which only true sentences hold. Under what conditions do Frege models and standard models exist, and if both exist, when do they coincide? We shall approach these questions in the context of intended model theory. And we shall consider the conditions under which Frege models and standard models, when they exist, are intended models. The present discussion bears upon the interesting treatment of intended models of Przelecki [35]. According to Przelecki (p. 17), the truth concept for a language L can be defined by choosing: from all models of L (all fragments of reality that L can speak about) its proper or intended model (that fragment of reality which L does speak about). It is clear that Przelecki here refers to what we have called a Frege model. He continues (p. 18): The factors, pragmatical in nature, which decide what a given language actually speaks about do not determine its proper model in a unique way. This observation of Przelecki also reflects the ideas we have associated with the concept of a Frege model: such models are indeed pragmatically determined, sometimes not uniquely. But we shall see that under appropriate conditions the pragmatic determination of Frege models is unique to isomorphism.

INTRODUCTION

In the above discussion we have seen reason to distinguish between intended models and Frege models. But we are apparently confronted by conflicting intuitions. On the one hand, the intentions of the users of a language determine the referential meanings of its terms, in virtue of which sentences of the language are true or false. On the other hand, false theories as well as true ones may evidently be said to have intended models; such theories are false simpliciter, but of course they are true in their intended models. How are these intuitions to be reconciled? How may the intentions of the speakers of a language distinguish the Frege models of a language and simultaneously distinguish the intended models of a specific false theory of the language? This question is answered in Chapters 10 and 12. In this context, as will be seen, the incommensurability problem may be resolved.

1C. On Peirce's Semiotic Theory The theory of intended models, which underlies the model-theoretic semantics outlined in the previous subsection, is developed in the spirit of Peirce's semiotic theory. In this subsection I shall discuss the relevant ideas of Peirce's semiotic theory, and then outline the manner in which these ideas are developed in the theory of intended models. In the common idiom the word 'sun', or a picture of the sun, or perhaps an idea of the sun, may be said to be a sign of the sun. Now Peirce asks us to consider also that a sunflower, in turning toward the sun, is also a sign of the sun. In these diverse contexts, does the word 'sign' have any uniform meaning? This question has a purely formal or structural interest as well as a philosophic one. Its philosophic interest lies of course in its possible relevance to the problems of semantic theory. But simply to phrase the semantic problem as a special case of the problem of signification, conceived in such generality, has strong appeal. What is the structure of the sign relation conceived in such generality? Peirce himself approached this problem taxonomically; he developed an elaborate system of classification of the various forms of signs. He viewed this enterprise as a formal one, defin-

INTRODUCTION

ing logic itself as the formal or quasi-necessary theory of signs. Semiotic theory so conceived is to be developed by abstraction from the characters of observed signs. Peirce concluded that "a sign is anything which determines something else (its interpretant) to refer to an object to which itself refers (its object)." (Cf. [16], 2.303.) The signification relation is an irreducible triadic relation obtaining among sign, object, and interpretant. In this relation a sign is associated with its object in virtue of the mediation of its interpretant. The interpretant is another sign of the same object, another "representamen" which "stands for" the same object relative to yet another interpretant, whence an endless sequence of interpretants is generated. What is the relation of "referring" or "standing for" to which Peirce appeals in explaining the sign relation? It is customary to say that the semantic parameters of a theory refer to their values in a model of the theory, and we have seen that intended reference (of the semantic parameters of pragmatically constructed theories) can be thought of as reference mediated by characteristic forms of verbal behavior which represent the relevant intentions. We have also seen that, for Peirce, signification is reference mediated by an interpretant. It is then but a short step to identify signification (restricted to expressions of an appropriate language, i.e., to symbols in Peirce's sense) with intended reference. This step requires identification of the relevant interpretants and intentions, but the former are evidently the observable manifestations of the latter, as, for example, pointing represents an intention to denote. Thus intended reference may evidently be identified with signification, and in our investigation of the former we shall be guided by Peirce's theory of the latter. We have seen that in explaining the sign relation Peirce appealed to the relation of referring, whose meaning is taken for granted. On the other hand, in view of the otherwise quite general and abstract character of Peirce's semiotic observations, it is tempting to contemplate their postulational reconstruction. Any such development must avoid principles which assert the existence of Grelling-like signs such as the following. Let

INTRODUCTION

mean that the sign x, relative to the interpretant y, signifies the object z. Here we have represented the sign relation as a binary function from sign-interpretant pairs to objects. (Cf. [16], 5.448n.) Now let y be a distinguished interpretant and x a distinguished sign such that for all z s(x, y ) = z iff s(z,y ) # z. Relative to the interpretant y, the sign x signifies those objects which do not signify themselves. It follows that s(x,y) = x iff s(x,y) # x whence there can be no such sign x and interpretant y as arguments of the signification function. The task of carrying out a postulational development of semiotic theory based on Peirce's work is as inviting as it is difficult to realize. Such a development would illuminate the formal structure of the mediating function of the interpretant in signification, surely the most profound and original of Peirce's semiotic insights. Let us tentatively sketch the outlines of such a development; the result will provide guiding principles for that part of semiotic theory whose postulational development we shall undertake, the theory of intended models of standard theories regarded as signs. In turn the latter theory may suggest refinements of the more general postulates. A formal postulate on the signification function which may be abstracted from Peirce's semiotic observations is the following uniqueness principle (cf. [16], 5.196, 5.448n): If s(x,y) = z and s ( x t , y ) = z', then z = z'. If two signs have the same interpretant, then they signify the same object. This postulate can be thought of as articulating what it means for a sign to be a pure "representamen" in an irreducibly triadic relation. For if, contrary to (I), an interpretant were associated with different objects of different signs, there would appear to be an intrinsic dyadic relation between these signs and their objects, independent of the effect of the interpretant. A sign of this type would appear to be more than a representamen of its object, whose place could perhaps not be taken by any interpretant. Prin(1)

INTRODUCTION

ciple (1) is a strong postulate, to which I shall return. This postulate clearly rules out the existence of the paradoxical sign discussed above. Another postulate which may be abstracted from the semiotic observations of Peirce which were outlined above, an existence postulate on interpretants, is as follows (cf. [16], 2.303): (2)

If s(x,y) = z, then for some y'

#

y, s(y,yl) = z.

The endless sequence of interpretants which may be associated with a sign in accordance with this postulate, Peirce calls the "entire general intended interpretant" of the sign. Postulate (2) expresses perhaps the most intriguing of Peirce's semiotic principles. Other postulates on the signification relation may similarly be abstracted from the semiotic observations of Peirce, and though scholars may differ about the accuracy of attributing all of them to Peirce himself, there is good reason to conclude that there exists implicitly in the work of Peirce such a postulational approach to semiotic theory. For example, postulate (2) asserts that there is no "last sign" in the sequence of interpretants developing the meaning of a given sign. Now Peirce also held that there is no first sign in this development; this latter property suggests an obvious analogue of postulate (2). Peirce also asserts the irreflexivity of sign and object at [16] 5.287, thereby denying the possibility of selfreference. We shall not follow him in this. Irreflexivity of sign and interpretant would allow simplification of postulate (2) above; one may reasonably surmise that this was his intent. The question naturally arises whether pursuing semiotic investigations at this level of generality may throw light on the problem of meaning and interpretation in linguistic theory, and in the philosophy of science. Could such investigations, for example, enable us to substitute the membership predicate of a particular set theory for the variable x in the context s(x,y ) , in a manner which is useful for the philosophy of mathematics? This kind of question is addressed in Chapter 11. Could such investigations enable us to substitute for the variable x in this context the term for mass in Newtonian mechanics (in an appropriate formulation), in a manner which would throw light on the incommensurability problem mentioned above? This kind of question is addressed in Chapter 12.

INTRODUCTION

Finally, could such investigations enable us to substitute an arbitrary sentence of English for the variable x in the above context, in a manner which would throw light on the semantics of English? This kind of question is addressed in Chapter 5. Our approach to these questions may be outlined as follows. We shall not undertake to develop semiotic theory at the level of generality indicated above, despite the natural appeal of such a program. We shall instead develop semiotic theory at a more restricted level of generality, where the signs are standard first order theories, their interpretants are characteristic forms of verbal behavior of their idealized users, and their objects are models which, in virtue of the verbal behavior of the users of the theories, may be said to be intended models. This analysis extends to the sentences and to the constants, both logical and non-logical, of such theories. If the metalanguage in which this investigation is carried out is restricted to the resources of metamathematics, we shall speak of the pure theory of intended models. Otherwise we shall speak of the applied theory. The perspective of the applied theory is then evidently that of the theory of interpretation in Davidson's sense [33], which will be considered in subsection ID. For the present we note that intended model theory is evidently more general than Davidson's interpretation theory, in the sense that the meaning of non-finitely axiomatizable theories is treated in the former. The perspective of the applied theory of intended models is also that of the theory of Frege models, associated in the previous subsection with the idea of scientific realism. The uniqueness problem, on the other hand, admits of solution within the pure theory of intended models. In order to develop intended model theory as a semiotic theory in the spirit of Peirce, it is inappropriate to assume a standard language already given. For to be a sign in Peirce's sense requires an interpretant, whence we must prove, not assume, that the "signs" of the relevant language are indeed signs in virtue of the efficacy of appropriate interpretants. We shall approach this problem by showing, first, that the very structure of standard logic can, in an appropriate sense, be pragmatically recovered. Accordingly, the theory of intended models presupposes the pragmatic foundations of standard logic, in which the algebraic structure of standard logic is recovered by means of a purely pragmatic construction. Models

INTRODUCTION

of the relevant pragmatically constructed algebras of formulas are homomorphic images of such algebras, so that not only the grammar and logic of the formulas of the algebra but also the structure of its very models is recovered pragmatically. The triad consisting of (i) an algebra of formulas as described above, (ii) the pragmatic foundations of the algebra, and (iii) an appropriately distinguished model of the algebra, can therefore be thought of as constituting a semiotic triad in the Peircean sense outlined above: the algebra of formulas signifies the model relative to the interpretant represented by the associated pragmatics. For if the object of a sign relative to some interpretant depends upon a particular parsing of the sign, then one should expect that this parsing should itself be fixed by the interpretant. In the above triad this is vouchsafed by the pragmatic determination of the structure of the algebra of formulas. Moreover, the structure of the object signified by the sign as well as that of the sign itself is determined by the associated interpretant. And there can be little doubt that Peirce himself would welcome the result that in this case the relation between sign and object is that of homomorphism; this is just to say that the relation is iconic. Evidence for the appropriateness of this Peircean terminology goes beyond the above discussion, and reflects logical developments unknown to Peirce but to which his semiotic ideas seem remarkably relevant. In order to see this, the connection between the model-theoretic terminology of this and subsection 1B should be noted. In subsection 1B we spoke of models in the standard model-theoretic sense; in this subsection we have spoken of algebraic models of algebras of formulas which represent the structure of standard languages. That this ambiguity is a harmless one is shown in Chapter 6: there is a natural correspondence between such models which reveals the fact that corresponding models represent the same intuitive structure. In light of this fact the algebraic models of the semiotic triads discussed above may be thought of as models in the standard sense. Such triads represent a special case admitted by the theory of intended models, that in which intended models are uniquely distinguished by the relevant pragmatics. If we consider the general case, the uniqueness problem of model-theoretic semantics, dis-

INTRODUCTION

cussed in subsection lB, reappears. But this problem becomes tractable in the context of intended model theory, in virtue of its semiotic treatment of the intending relation. This will be seen in Chapter 9. Roughly speaking, since the relation between a standard theory and its intended models is taken to be the signification relation, the interpretant of this signification, represented by the pragmatics of the theory, determines the uniqueness properties of the class of intended models of the theory. In this context the signification relation is represented by a function resembling the general signification function discussed above. Given a theory and its pragmatics as arguments of this function, the function selects from among the models of the theory those models which may be said to be intended models in virtue of the verbal behavior represented by the pragmatics. Under appropriate circumstances, elementary equivalent intended models of the theory are isomorphic, whence the uniqueness problem of model-theoretic semantics admits of solution in the sense of subsection 1B. Underlying this solution is the Peircean principle that signification is always mediated by an interpretant. The semiotic principles (1) and (2), which were introduced above in order to suggest the implicit postulational character of Peirce's semiotic theory, find realization in the theory of intended models, though in a restricted sense. Principle (I), when restricted to linguistic signs, or symbols, may be understood to assert that pragmatically synonymous expressions are semantically synonymous. The relevant synonymy relations may be explicated in the theory of intended models in such a way that, for appropriate signs of a pragmatically constructed language (e.g., sentences, predicates, and individual constants), pragmatic synonymy implies semantic synonymy. This is shown in Chapter 7. An analogue of Peirce's principle (1) holds also in the case of theories which are synonymous in the sense of the theory of definition. The semantical counterpart of the definitional synonymy of two theories is coalescence of the varieties of the theories. These concepts are generalized in the theory of intended models in such a way that pragmatically synonymous theories have coalescent intended varieties. Pragmatically synonymous theories, that is, are semantically synonymous in a sense which realizes principle (1). This idea is developed in Chapter 8.

INTRODUCTION

Predictive or observational equivalence may also be regarded as a form of pragmatic synonymy obtaining among theories. The question whether there is a corresponding form of referential or model-theoretic synonymy is addressed in Chapter 12. Reflection on the semiotic principle (1) in light of this problem suggests a possible counter-example to this principle. Peirce's semiotic principle (2) asserts, roughly speaking, that the object signified by a sign is revealed by the development of the sign in a succession of interpretants. (Cf. [16], 5.594.) An analogue of this principle is realized in the theory of intended models, in the following sense. In this theory, under appropriate circumstances, the identity

can be interpreted to assert that the predicate x of a language whose structure is determined by the pragmatics y signifies the relation z (up to isomorphism). If y' is a pragmatics which is an extension of the pragmatics y (in a sense which is elucidated in Chapter 9), then s(x,y'), the object signified by x relative toy', is a refinement of s(x,y) (in a sense which is also elucidated in Chapter 9). In this way, the relation signified by the predicate x may be revealed in increasing detail by a succession of interpretants of x. As signification is refined in the succession of interpretants of a sign generated by a scientific intelligence, so belief converges on agreement in the long run of scientific inquiry. (Cf. [16], 5.407-8, 5.589.) Peirce's appealing principle of long-run agreement is quite strong, and the question naturally arises whether it may be given a natural foundation. We shall conclude this subsection with a brief discussion of the manner in which such a foundation may be provided within standard pragmatic theory. Standard pragmatic theory contains an intersubjectivity principle which asserts that, under certain conditions, there is interpersonal uniformity with respect to the possible visible manifestation of belief constituted by the acceptance of a sentence. If such acceptance actually represents belief, then the intersubjectivity principle of standard pragmatics can be considered to express an analogue of the convergence of belief appropriate to the long run of scientific inquiry envisaged by Peirce. That the concept of acceptance as developed in standard pragmatic theory does approximate the

INTRODUCTION

concept of belief is argued in subsection ID. Moreover, the intersubjectivity principle of standard pragmatic theory need not be postulated explicitly; it is a consequence of the postulates of standard pragmatic theory, none of which places any constraint on the differences between any two speakers. Taken together, however, these postulates constrain the differences among speakers just sufficiently to establish the intersubjectivity principle required for the pragmatic foundations of standard logic. The idealized character of the verbal behavior described by these postulates reflects the ideal of scientific inquiry appropriate to the long run. In this way Peirce's principle of long-run agreement, represented as the intersubjectivity principle outlined above, may be given a foundation within standard pragmatic theory.

ID. On Davidson's Semantic Program Intended model theory, when applied to a natural language by means of the theory of locally standard grammar, provides a local model-theoretic semantics for the language. This semantic theory treats the relation obtaining among the logical forms, or deep grammatical structures, of the sentences of the language, their associated meanings, and the linguistic dispositions of speakers of the language which determine those grammatical forms and meanings. The relevant linguistic dispositions may be said to represent beliefs of the speakers who possess them, and to underlie the meanings of sentences expressing those beliefs. Davidson in [33] stresses the inseparability of beliefs and meanings which are associated in this manner. He suggests that interpretation theory be developed in analogy with decision theory, which by appropriate axiomatization of preference orderings generates simultaneous scales of measurement for degree of belief as well as degree of preference. Consequently he suggests that the theory of interpretation develop a simultaneous axiomatization of belief and meaning. The present theory can be thought of as making a contribution to this program in semantics and doxastics. The viewpoint of the present theory also accommodates Davidson's application of Tarski's theory of truth to the theory of meaning. (Cf. [32].) The

INTRODUCTION

purpose of this subsection is to outline the development of these ideas. In this discussion we will first consider the idealized case represented by the theory of intended models of standard languages. Then we will consider the application of this theory to natural language by means of the theory of locally standard grammar. Davidson's theory treats of belief and standard pragmatic theory treats of acceptance. Acceptance of a sentence is evidently to be distinguished from belief that the sentence is true. The distinction may perhaps be marked by requiring that belief in the truth of a sentence entails belief in the logical consequences of the sentence, whereas the corresponding property does not apply to acceptance. Standard pragmatics develops an idealized concept of coherent acceptance; in this context, acceptance of a sentence entails acceptance of its consequences, a fact which suggests that coherent acceptance approximates belief. This approximation in any case appears reasonable in the present context, as will be seen. In standard pragmatic theory the algebraic structure of any standard first order language may be recovered by means of a purely pragmatic construction, in which the relevant beliefs of the speakers of the language are represented by coherent patterns of pragmatic valuation (accept, reject, or withhold) of the expressions over the alphabet of the language. In the context of algebraic logic, this fact provides a representation theorem for the concept of referential meaning. For it is a consequence of a general principle of algebraic logic that the representing algebra of formulas is homomorphic to a model (an algebraic structure which represents the concept of a model in the standard model-theoretic sense). Any such model represents the possibility of assigning a referential meaning to each sentence of the algebra; this meaning can be taken to be the recursion underlying the truth value of the sentence in the model. Thus in the context of the pragmatic axiomatization of belief (represented as coherent acceptance) underlying the associated algebra of formulas, the existence of a model homomorphic to the algebra guarantees that the relevant beliefs are indeed beliefs in the truths of sentences whose grammatical and logical structure is fixed by the coherent pragmatic valuations representing those beliefs. The existence of such a model therefore associates referential meaning with belief as a consequence of the axiomatization,

22

INTRODUCTION

and this can be taken to be the desired representation theorem for the concept of referential meaning. In general there are many such models, and so it remains to find an appropriate uniqueness relation corresponding to the above representation of referential meaning. In considering the analogy with decision theory, Davidson anticipated this problem. (Cf. [33], p. 315.) The theory of intended models suggests a uniqueness theorem for the above representation of referential meaning which appears in keeping with Davidson's decision-theoretic analogy. The appropriate uniqueness relation is elementary equivalence of intended models (called potential monomorphism in subsection 1B). The conditions under which such a relation obtains are outlined in section 2. When such conditions obtain, simple completion of the relevant pragmatically constructed standard theory (as outlined in subsection 1B) may fix unique reference of the semantic parameters of the theory, up to isomorphism. It is true that a substantial idealization of actual theory construction may be involved in the contemplation of complete theories, but this is in keeping with Davidson's analogy between interpretation theory and utility theory. In the latter, the typical requirements on preference orderings for the measurement of utilities also involve an enormous idealization of observed preferences. The latter idealization proceeds, Davidson points out, by expanding the domain of preferences whose structure is to be explained by utility theory. Davidson suggests an analogous approach to interpretation theory: it should proceed by expanding the domain of verbal behavior (for example, occasions of a speaker holding a sentence to be true) whose underlying structure is to be explained. (Cf. [33], p. 314.) This behavior, in the theory of intended models, is represented by acceptances and rejections of sentences whose very grammar is determined by such behavior. In this way we are assured that it is not some other grammatical parsing which underlies the associated meanings and beliefs. The above ideas apply to natural language in virtue of the pragmatic theory of locally standard grammar, developed in Chapter 5. This theory represents the deep structure of English as that of standard first order grammar, though in a local as distinguished from a global sense. The relevance of this theory to Davidson's semantic program for English may be outlined as follows.

INTRODUCTION

Davidson in [32] has argued persuasively that a recursive truth definition for a language which satisfies Tarski's Adequacy Condition provides an acceptable theory of meaning for the language, a theory which has the merit of representing the meaning of each sentence of the language as a function of the meanings of its words, in virtue of the recursion defining truth for the sentence. Davidson's suggestion for carrying out his semantic program is also appealing: we should seek the truth conditions of sentences in canonical form and regard those conditions as applying also to corresponding sentences not in canonical form. What, then, are the appropriate canonical forms, and what is the nature of their correspondence with the remaining sentences of the language? The theory of locally standard grammar suggests an answer to both these questions. The desired canonical forms of sentences are stylistic variants (called standard variants) which represent their locally standard formalization. The correspondence of such standard variants to the remaining sentences of the language is represented by a relation of pragmatic synonymy. Standard model-theoretic definitions of truth, and in particular the Tarski definition of truth, apply to standard variants, and the possibility of identifying the truth conditions of arbitrary sentences with the truth conditions of their standard variants thereby suggests itself. This identification amounts to the assumption that pragmatically synonymous sentences are semantically synonymous, provided, following Davidson, we identify meaning and truth conditions. This assumption is but another application of the Peircean principle (1) of subsection 1C.

2. Standard Pragmatics and Semiotics Section 1 has surveyed the intuitive ideas and semantical problems which motivate the development of the pragmatic foundations of standard logic and the theory of intended models. In this section we shall outline the formal development of these theories. The pragmatic foundations of standard logic and their application to the grammar of natural language are outlined in subsection 2A. The theory of intended models and its application to semantic theory are outlined in subsection 2B.

INTRODUCTION

2A. Pragmatic Foundations of Standard logic It is shown in Part I that every consistent standard formal system is representable by a pragmatically constructed polyadic Boolean algebra of formulas. Such algebras represent the structure of standard first order logic. The construction is completely determined by the verbal behavior of the users of the formulas of the algebra, and in particular by their dispositions to assign one of the pragmatic values accept, or reject, or withhold to arbitrary expressions over the alphabet from which the formulas of the algebra are constructed. The morphology and grammar of the formulas of the algebra, as well as their logical (polyadic Boolean) structure, are completely determined by such pragmatic valuations. The primitive ideas of standard pragmatics are a set L (of expressions), a set U (of idealized users of the expressions in L), a set V (of pragmatic values which may be assigned to the expressions of L by members of U), a set C (of conditions under which such pragmatic valuations may be performed), a dyadic relation Ion C, and a distinguished element b of C. The set V is postulated to contain three elements which represent, intuitively, the pragmatic values accept, reject, and withhold. The primitive ideas of standard pragmatics are defined implicitly by the definientia of a sequence of five definitions, beginning in Chapter 1 with the definition of a pragmatic interpretation, and concluding in Chapter 2 with the definition of a coherent interpretation. The pragmatic valuations of the expressions of L which are represented by a coherent interpretation induce the structure of a locally finite polyadic Boolean algebra of countably infinite degree on a pragmatically distinguished subset of L, with respect to a congruence relation which is also pragmatically constructed. The expressions in the domain of the algebra may be parsed as standard predicate formulas whose morphology and grammar are also determined by the pragmatic valuations which are represented by the associated coherent interpretation. Moreover, every consistent standard formal system is representable by a pragmatically constructed polyadic algebra of formulas of the kind outlined above. The precise statement and proof of these propositions is reached in Chapters 3 and 4, and may be referred to as the fundamental theorem of standard pragmatics.

INTRODUCTION

Standard pragmatics as developed and applied in this book is to be distinguished from the pragmatics of natural language, in which meaning may vary with contexts represented as indices or points of reference, such as speakers or times (cf. Montague [15], p. 98). In this book, we wish to investigate the manner in which the signs of a standard language are understood intersubjectively by their idealized users. This investigation embodies a study of standard languages within a metalanguage in which standard modeltheoretic resources are supplemented with pragmatic ones, without altering the former. In this context we may ask: How are the intended models of a standard theory distinguished by the pragmatics of the theory? In a similar way, standard pragmatic theory is to be distinguished from the "pragmaticized" development of semiotic theory of Martin [14], in which syntax itself, as well as semantics, is relativized to language-users. Standard pragmatic theory employs no such relativization. As indicated above, although none of the postulates of standard pragmatic theory places any constraint on the differences between any two language-users, taken together these postulates constrain the differences among languageusers just sufficiently to secure the pragmatic foundations of standard logic in the intersubjective sense required for their intended applications. Standard pragmatic theory, although clearly to be distinguished from the pragmatics of natural language in the Montague tradition, is nonetheless applicable to the grammar of natural language, and in particular to the phenomenon of indexicals which led to the development of the pragmatics of natural language. In the context of standard pragmatic theory we shall develop a grammatical theory which represents the deep structure of English as that of standard first order grammar, though in a local as distinguished from a global sense. The deep structure of a fragment of English will be called locally standard if each sentence of the fragment is an element of a standard neighborhood -a set of sentences which admit standard formalization in roughly the following sense: each sentence of the neighborhood is pragmatically synonymous with a sentence of a distinguished subneighborhood, which is the set of closed formulas of a pragmatically constructed polyadic Boolean algebra of for-

INTRODUCTION

mulas. This algebra may be regarded as a standard "chart" of the associated standard neighborhood. Locally standard English is defined as that fragment of English which may be covered by standard neighborhoods. The lexicon of this fragment includes indexical~,adverbs, and attributive adjectives, and a locally standard analysis of the deep structure of such grammatical categories is thereby achieved. This analysis suggests the possibility of identifying the grammar underlying deduction with the grammar underlying language learning. The locally standard analysis also suggests the possibility that the structure of the scientific fragment of a natural language is represented locally in the language at large.

28. Intended Model Theory In Part I1 a theory of intended models is developed as an extension of standard model theory, using standard pragmatics to represent the "intending". In order to outline the basic ideas involved, let Ln be a pragmatically constructed standard language, where II is the pragmatics which determines the structure of Ln in accordance with the fundamental theorem of standard pragmatics. The set T of closed formulas of Ln which are pragmatically congruent to some tautological formula is defined to be the distinguished theory of Ln. The congruence relation is so constructed that the sentences of T may be thought of as the sentences of Ln which are accepted intersubjectively by the idealized users of Ln under the distinguished condition of pragmatic valuation. As is shown in Part I, every consistent standard theory is representable as the distinguished theory of a pragmatically constructed language of the form Ln. The theory of intended models develops the idea that the intended models of T may be distinguished by means of the very pragmatics which determines the logical structure of T. The basic idea of intended model theory is that of a selection function a which assigns to each system of pragmatics ll of an appropriate kind a set u(n) of models of the distinguished theory T of the language Ln determined by the pragmatics II. Intuitively, the elements of u(ll) are the intended models of T, where the "intending" is represented by the pragmatics II which determines the

INTRODUCTION

grammatical and logical structure of Ln. The motivation is derived from Peirce's semiotic theory: the intended variety a ( n ) of T may be regarded as the object signified by T regarded as a sign, relative to the pragmatics II as interpretant. The selection function a is defined implicitly in Chapter 8 by seven postulates, all but one of which are generalizations of principles of standard model theory, in the sense outlined in section 1. It is a desideratum of intended model theory that its principles, so far as is possible, have this property. Departure from complete realization of this desideratum is required in order to resolve the uniqueness problem, as was indicated in section 1. In order to solve this problem for a wide class of languages, however, it proves sufficient to relax commitment to the desideratum only in the case of a very restricted kind of language, called basic languages. It is shown, by means of an embedding theorem of Tarski, that if a language of the form Ln contains a basic sublanguage, in an appropriate sense, then any two elementary equivalent intended models of the distinguished theory of Ln are isomorphic (cf. Chapter 9). The principles governing the relevant sublanguage relation are in complete accordance with the above desideratum. This relation embodies a reconstruction of the idea that the referential precision of the sublanguage of Ln is preserved by extension to Ln. Such extension is called a-extension. The theory of intended models admits various realizations of the a-extension relation. It is shown, for example, that definitional extension constitutes such a realization. Another realization of the relation of a-extension is the reducibility relation on theories, introduced in a sense which unifies model-theoretic and definitional ideas regarding reducibility. A further realization of the a-extension relation, introduced in Chapter 11, may perhaps be regarded as expressing the grain of truth in the various formulations of the verifiability principle of meaning. But this realization, called minimal extension, does not represent a criterion of meaningfulness; it constitutes a sufficient but not a necessary condition under which the referential precision of a theory is preserved by extension of the theory. The intuitive idea is that theories which the positivist might consider metaphysical are, in the present theory, not meaningless but perhaps difficult to disambiguate.

28

INTRODUCTION

The selection function of intended model theory which is implicitly defined by the postulates of Chapter 8 is not uniquely determined by those postulates. Stronger postulates are considered, but we leave open the question whether the postulates of Chapter 8 may be strengthened in a plausible way so that a unique selection function is thereby defined. In Chapter 10 one of these postulates is strengthened in a way which facilitates investigation of the relation between intended models and standard models of certain theories. But the strengthened postulate is not a generalization of principles of standard model theory, in the sense outlined above. By means of this postulate it is shown that, for pragmatically constructed languages Ln of an appropriate kind, the intended models of the distinguished theory of Ln are standard models of that theory. Even if no concept of standard model exists for the distinguished theory of a language Ln, the concept of standard reference of pragmatically distinguished signs of Ln may under appropriate conditions be introduced. When such conditions obtain, the intended reference of pragmatically distinguished signs of Ln may coincide with their standard reference. Among the applications of intended model theory to the philosaphy of science are the following. In Chapter 10 an explication (though not a justification) of the idea of scientific realism is proposed. This approach admits a uniform general (though in practice profoundly difficult) method of settling questions about the correct realist interpretation of scientific theories. In Chapter 12 the present theory is applied to the problem of referential indeterminacy which has been associated with scientific revolutions. In this context the semantics of theory change associated with scientific revolutions is accommodated in a manner which involves no indeterminacy of reference beyond that associated with the Lowenheim-Skolem-Tarski theorem. Also in Chapter 12 the present theory is applied to the problem of the apparent synonymy of observationally equivalent theories. It is concluded that observationally equivalent theories are not referentially synonymous in any natural sense analogous to the paradigm conventionalist sense in which theories of alternative scales of measurement are referentially synonymous. This conclusion depends upon the model-theoretic

INTRODUCTION

representation of referential meaning outlined above, but the naturalness of this representation is one of the fundamental themes of this book. The central theme of this book may be summarized as follows. Standard logic admits pragmatic foundations upon which a theory of intended models may be built as an extension of standard model theory; this theory is applicable to natural language, including its scientific fragment, by virtue of a theory of locally standard grammar whose development is also founded upon standard pragmatics. The local model-theoretic semantics thereby obtained has the virtue of generality as well as the relative simplicity and elegance associated with standard model theory. Such apparently non-standard phenomena as indexicals, for example, when considered locally (that is, in sufficiently small linguistic neighborhoods), admit standard formalization and therefore standard model-theoretic analysis. From this perspective it appears plausible that the structure of the scientific fragment of a natural language such as English is represented locally in the language at large. The most difficult task in the standard (set-theoretical) axiomatization of scientific theories is presented by the case of quantum mechanics, but this is not known to be impossible. Even if God plays dice, it is not known that he is, at the level of deep semantical analysis, a polyglot.

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Sentential Interpretations

The first step in the pragmatic determination of standard linguistic structure is taken in this chapter. The primitive ideas of standard pragmatics are introduced and the concept of a pragmatic interpretation is defined. Pragmatic interpretations provide a preliminary step in the implicit definition of the primitive ideas of standard pragmatics, and in the pragmatic determination of standard linguistic structure. A specialization of the concept of a pragmatic interpretation is then introduced, the concept of proto-sentential interpretation, which underlies the Boolean structure of pragmatically determined polyadic algebras of formulas. This chapter concludes by introducing the concept of a sentential interpretation, which provides an illustration of the manner in which the algebraic structure of standard logic may be recovered by means of a purely pragmatic construction. The primitive ideas of standard pragmatics are a set L (of expressions), a set U (of idealized users of the expressions of L), a set V (of pragmatic values which may be assigned to the expressions of L by their users), a set C (of conditions under which such pragmatic valuations may be made), a dyadic relation I on C, and a distinguished condition b of C. The primitive ideas of standard pragmatics will be implicitly defined by the definientia of a

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PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

sequence of five definitions, beginning in this chapter with the definition of a pragmatic interpretation and concluding in Chapter 2 with the definition of a coherent interpretation. The pragmatic valuations of the expressions of L represented by a coherent interpretation induce the structure of a polyadic Boolean algebra on a pragmatically distinguished subset of L, with respect to a pragmatically determined congruence relation. Both the domain of the algebra and the structure of the congruence relation are determined by the pragmatic valuations represented by the coherent interpretation. The expressions in the domain of the algebra may be parsed as standard predicate formulas whose morphology and grammar are also pragmatically determined in the sense outlined above. Moreover, every consistent standard formal system is representable as a pragmatically constructed polyadic algebra of formulas of the kind outlined above. The precise statement and proof of these propositions is re&hed in Chapters 3 and 4. We turn now to the first step in the development of this program. In order to simplify the exposition, we introduce the following existence condition for the primitive ideas of standard pragmatics. We shall say that the existence condition (for the primitive terms of standard pragmatics) holds iff L is the set of all finite sequences over a nonempty finite set Lo, U is a nonempty set, V = (0,1,2), and (C,s) is a Boolean algebra (that is a distributive complemented lattice, in which we let A, v, and - be the meet, join, and complementation operations, respectively). Under the existence condition, the set Lo may be referred to as the alphabet of the set L of expressions. The elements 0,1,2 of the set V may be understood to represent the pragmatic values reject, accept, and withhold, respectively.' If c,cf E C, then c A c' is the condition that both c and c' obtain, and C is the condition that c does not obtain. Let co = c A C? for some c E C. We shall say that co is the impossible condition, that c, = Co is the necessary condition, and that elements of C which are distinct from co are proper conditions of pragmatic valuation. The distinguished condition of pragmatic valuation b E C may be regarded intuitively as representing the total evidence possessed by the users of L for valuing expressions of L in the set V. A condition c 5 b of C may be taken to be the outcome of an experiment (in an idealized sense explained in Chap-

SENTENTIAL INTERPRETATIONS

ter 7) which has actually been performed by some user of L. Expressions in L which are distinct from the empty expression will be called proper expressions of L. The idea of pragmatic valuations of expressions of L by the language-users in U, under conditions in C, may be expressed as follows. Let D = vUXC, the set of all functions from the Cartesian product of U and C into V, and let II be a function from L into D. Intuitively, if II (e)(u, c) = v, then the user u, under the condition c, is disposed to perform the valuation v with respect to the expression e. If II (e) (u, c) f 2 then, intuitively, the user u regards the condition c as germane to the acceptance or rejection of the expression e. If II(e) (u, c) = 2 then, intuitively, the user u does not regard the condition c as germane to the acceptance or rejection of the expression e, perhaps because e is not a well-formed sentence of the language of u, in which case no condition would be germane to the acceptance or rejection of e. These ideas suggest the possibility of recovering the grammatical as well as the logical structure of the language of the users of L by means of a function II which assigns to each expression e of L an appropriate set of pragmatic valuing dispositions II (e) E D. We are therefore led to the following definition, which provides a preliminary step in the pragmatic construction of standard linguistic structure, and in the implicit definition of the primitive ideas of standard pragmatics.

Definition 1 II is a pragmatic interpretation (of L relative to U, V, C, I, and b) iff the existence condition holds and II is a mapsuch that: ping from L into vUXC I. II (e) (u, c0) = 2 for all e E L and u E U. 11. II(e)(u,b) # 2 for some e E L and u E U. 111. If n ( e ) ( u , c , ) # 2, then n ( e ) ( u , c ) = II(e)(u,c,), for all e E L, u E U, and proper c E C. IV. If II(e) (u, c) = 1 = II(e)(u, c'), then II(e) (u, c v c') = 1, for all e E L, u E U, and c,cl E C. Clause I of Definition I asserts that the impossible condition is not germane to any expression of L. Clause I1 of Dl asserts that the distinguished condition of pragmatic valuation is germane to some expression. Clause I11 of Dl asserts that if the necessary con-

36

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

dition of valuation is germane to an expression, then this valuation is preserved under all proper conditions. It is a consequence of clauses I and I1 that b # co. We should also expect that a pragmatic interpretation assigns the value 2 to the empty expression, for every user under every condition, and that the analogue of clause IV for the value 0 holds. These requirements need not be postulated explicitly, since they are provable in the case of reasonable specializations of the concept of a pragmatic interpretation, as will be seen. Clause IV of Dl asserts that if s is accepted under a condition c' as well as c, then s is accepted under the disjunction c v c' of these conditions. For example, if the sentence 'a is warm' is accepted under the condition that butter melts near a given object, as well as under the condition that a column of mercury of a specific description near that object has at least a certain length, then according to IV the sentence is accepted under the disjunction of these conditions. It is a consequence of IV that if s is accepted under both c and C, then s is accepted under cvC = c l , the necessary condition. The importance of this fact will be seen below. It is a common intuition that establishing public conditions of acceptability for a sentence provides a means of endowing the sentence with intersubjective meaning. The investigation of this yhenomenon is undertaken within standard pragmatic theory. The theory of intersubjective meaning depends essentially upon the pragmatic foundations of standard linguistic structure outlined at the beginning of this chapter. The development of these foundations requires specialization of the concept of a pragmatic interpretation introduced in Definition 1. Before considering the relevant specialization, it is appropriate to note that pragmatic interpretations accommodate the concept of stimulus meaning, in a sense generalized from that of Quine [17]. We may define the affirmative stimulus meaning of an arbitrary expression e of L, for user u E U, relative to a pragmatic interpretation II, as the set ( c E C: II(e) (u, c) = 1). The corresponding negative stimulus meaning is defined by replacing the pragmatic value 1 with the value 0.Then we may define the stimulus meaning Cfi(e) of expression e, for user u, relative to the pragmatic interpretation II, to be the ordered pair of affirmative and negative stimulus meanings of e for u. It is by no means as-

SENTENTIAL INTERPRETATIONS

sumed that the stimulus meaning of an expression is to be identified with its entire meaning; for we wish to investigate, in Part 11, the relation between pragmatically and semantically characterized meanings. At this point we may consider the relation between stimulus meaning and pragmatic synonymy, in the following sense. If e and e' are expressions of L and ll is a pragmatic interpretation, then we define En(e, e' ) iff e and e' are interchangeable in all expressions of L, salvo valore re II. If En(e, e'), then we shall say that e and e' are pragmatically synonymous, relative to II; for in this case each user of L regards e and e' as interchangeable in all expressions of L , under each condition of pragmatic valuation. It follows that: (1.1)

If En(e,ef), then Ch(e) = Cb(e' ),

for each user u in U. That is, pragmatically synonymous expressions have the same stimulus meaning for each user in U. In Chapter 7 we shall consider a concept of intersubjective stimulus meaning; an analogue of (1.1) will be seen to hold for intersubjective stimulus meaning. In Chapter 8 we shall consider analogues of (1.1) in which pragmatic synonymy of distinguished expressions of L implies their semantic synonymy. We now return to the pragmatic determination of standard linguistic structure. The next step in this program is provided by the concept of a proto-sentential interpretation, whose definition is facilitated by the following conventions. For n = 0,1,2 we define functions dnfrom U x C into V as follows. For all u E U and c E C: dn(u,c) =

n if c is proper 2 otherwise

Thus d2 is the constant function in D = vUXC whose value is always 2. Intuitively, d2is the uniform set of dispositions of users of L to perform the valuation withhold under any condition. If e and e' are expressions of L, then we shall understand the expression ee' to be the concatenate of e and e' in the usual sense: ee' is the expression consisting first of the elements of e followed by the elements of e'.

38

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

Definition 2 II is a proto-sentential interpretation2iff II is a pragmatic interpretation and there are expressions & and of L such that, for all s,sl E L; u,ul E U; and c E C:

-

I. If II(s) f d2, then II(&s-s) = do. 11. If II(s) d2, then II(s-s) = d2 = I I ( -s-s). III. ~f ~ ( s ) ( u , = c )2 = ~ ( s ~ ) ( u +, c~)( & s s ~ ) ( uand , c )c # c,, then either II (&sst) (u,c ) = II (&sst)(u,E ) , or for some c' < C, ~ ( s ) ( u , = c ~o )or ~ ( S ~ ) ( U , C = ' ) 0. IV. If II(s)(u,c)= 1 = II(s')(ut,c),then for some u" E U, II(&ss')(u",c)= 1 . V. If II ( &s-S) = do = II ( &st-st ) ,then for all arguments ( u,c ) E U x C, the values of II ( - s ) and II (&ss') are (partially) fixed by the values of II(s) and II(st)according to the tables:

+

y 2

-

0

2

(fl)

If the expressions & and are as described in the definiens are associated pseudoof D2, then we shall say that & and conjunction and pseudo-negation signs determined by I I . The intuitive idea is that further conditions on a proto-sentential interpretation will distinguish genuine conjunction and negation signs in the language of the users of L. The motivation for Definition 2 may be clarified if we consider the set S=

-

(S E L:

-

II(&s-s) = do)

where & and are associated pseudo-conjunction and pseudonegation signs determined by a proto-sentential interpretation I I . Intuitively, S is a set of sentences in which conjunctions and negations are formed in accordance with the conventions of Polish notation. The justification of this intuition, which requires appropriate specialization of the concept of a proto-sentential interpretation, is obtained in Chapter 4, where it is shown that every consistent standard formal system is representable by a pragmatically constructed polyadic Boolean algebra of formulas, such that the closed for-

SENTENTIAL INTERPRETATIONS

mulas of the algebra are the expressions of a set S defined pragmatically as above. The motivation for the definition of S is the fact that, if &s-s is a formula in Polish notation, where & is the conjunction sign and is the negation sign, then s is also a formula. For suppose otherwise. Then s has the form sls2,where sl is the shortest formula beginning s, so that sl is the first argument of & and also the argument of in &s-s. Then s2must contain a binary connective one of whose arguments contains -sl as a part. But then s 2 cannot end the formula &s-s. Thus if &s-s is a sentence (closed formula) in Polish notation, then so is s. Thus if s is indeed a sentence, then this fact may be reflected in the logical knowledge, or competence, of the users of L, in virtue of which s is a member of the set S defined as above. To the clarification of this idea we now turn. The context will always make clear the identity of the pragmatic interpretation which determines a set of the form S according to the above definition; consequently we shall avoid notational devices such as subscripts on 'S' which would indicate this determination explicitly. A similar remark applies to all pragmatically determined syntactic categories in the sequel. The valuation tables of clause VI of definition D2 are like Kleene's strong three-valued tables (cf. [12], p. 334), except for the case in which both s and s' are valued 2. This departure is motivated by the fact that the values in the set V = (0,1,2) are essentially pragmatic values. In particular, the value 2 for withhold does not in any sense represent a value-gap. For example, if s' = -s, then &ss' should evidently be rejected even if judgment is withheld regarding s and thus st. In general, if II(s)(u, c) = 2 = II(st)(u, c), then it should not be required that II(&ssl)(u,c) = 2. A further example illustrating this fact is as follows. Let s assert that a specific individual is a mammal, and let s' assert that this individual lays eggs. Then it might be the case that &ss' is rejected by the users of L under all proper conditions, including conditions which are not germane to s and s'. The non-functionality represented in the pragmatic valuation table for & accommodates examples such as these. This non-functionality makes the pragmatic determination of Boolean structure more difficult than it would otherwise be, but the task

-

-

40

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

can be accomplished. The non-functionality in the valuation table for & is further clarified by proposition (1.2) below. Let II be a proto-sentential interpretation which determines associated pseudo-conjunction and pseudo-negation signs & and -. By Dl(I), D2(I), and D2(II), II(&s-s) = do for some expression s such that II(s-s) = d2 = II(-s-s). Thus & and are distinct expressions, and & is a proper expression of L. By D2(VI), 11(-&s-s) = d l , so that is also a proper expression of L. By D2(II) and D2(I), S is nonempty. If s,sfE S and II(s)(u,c) = 2 = II (sf)(u, c), then as noted above, II ( &ss' ) (u, c) is not uniquely determined by the table for &: at the argument (u, c), II(&ssf) may be either 0 or 2. The meaning of this non-functionality in the valuation table for &, and also the motivation for the remaining clauses of Definition 2, are clarified by the following consequences of D2.

-

-

(1.2)

If II(s)(u,c) # 2 # II(s)(u',c), then II(s)(u,c) = ~(s)(u',c).

Proof. On the hypothesis of (1.2), suppose that II (s)(u, c) = 0 but II (s)(u', C) = 1. Then II ( -s) (u, c) = 1 by D2(V), so that for some u", II (&s-s) (u", c) = 1, by D2(IV). But II (s)# d2by hypothesis, whence II(&s-s) = do by D2(I), a contradiction. (1.2) expresses an intersubjectivity property of the valuations of the users of L, which depends essentially upon clause IV of D2. Intuitively, clause IV describes a kind of information-processing capacity of the idealized users of L. The justification for contemplating such idealization is considered in detail at the end of Chapter 2. If U is a unit set, then clause IV becomes redundant; in this limiting case the information-processing principle expressed by clause IV need not be explicitly postulated as part of the pragmatic foundations of standard linguistic structure. In the general case, clause IV suffices for all the intersubjectivity principles required for these foundations; for example, propositions (3.1) and (3.3) of Chapter 3, as well as (1.2). To the extent that the theory of locally standard grammar developed in Chapter 5 is adequate to natural language, the information-processing principle expressed by clause IV represents the entire social or intersubjective dimension of natural language. The idealization underlying clause IV is especially

SENTENTIAL INTERPRETATIONS

plausible in the case of scientific languages. We shall return to this topic at the end of Chapter 2. In virtue of proposition (1.2) we may define the core of a proto-sentential interpretation II to be the mapping a from L x C onto V such that: a(e,c) =

2 if II(e)(u,c) = 2 for all u E U. O(1) if II(e)(u,c) = O(1) for some u E U

for all e E L and c E C. The core of a proto-sentential interpretation ll represents the structure of II which is invariant under transformations on II, regarded as a set of ordered quadruples (e, u, c, v) such that II (e)(u, c) = v, which leave e and c unchanged. The pragmatic values v # 2 are preserved under such transformations. It is this invariant property of appropriate pragmatic interpretations which, as we shall see, underlies the interpretation of certain expressions of L as signs whose significance may be uniformly grasped by their users. It is an obvious consequence of the above definition that, if a is the core of ll and II (e) = ll (e' ), then a (e, c) = T (e', c) for all c E C. The existence of the core of a proto-sentential interpretation is a consequence of proposition (1.2), which as we have seen depends upon the information-processing principle, clause IV, of Definition 3. Clause IV is also required to show that the core of a proto-sentential interpretation satisfies a valuation table for expressions of the form &ss' like that of clause V, as will be seen. The significance of the non-functionality in the table for & of clause V may be clarified if we define:

where a is the core of a proto-sentential interpretation n. Intuitively, N is the set of sentences accepted intersubjectively under the necessary condition of valuation. By clause I of D2, N E S; and by clause I11 of D l , s E N iff a(s,c) = 1 for all proper conditions c. For appropriate specializations of the proto-sentential interpretation II, N may be regarded as the set of analytic sentences of the language induced over L, as will be seen.

42

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

Under this interpretation of the set N, the significance of the &-table of Definition 2 is clarified by the following proposition: (1.2)*

Let II(s)(u,c) = 2 = II(sl)(u,c). Then II(&sst)(u,c) = 2, or -&ssl E N, or for some c' < c, II(s)(u,c') = 0 or n ( s l ) ( u , c f ) = o.

Proof. On the hypothesis of (1.2)*, let II(&ss')(u,c) # 2. Then by clause V, II (&ssl)(u, c) = 0.If c = cl then -&ssl E N. Let c f el, so that clause 111 applies. Assume that the last disjunct of the consequent of (1.2)* fails to hold. Then by 111, II(&ssl)(u,c) = II(&ss')(u,E). As shown above, II(&ss')(u,c) = 0. Then n(-&SSO(U,C) = 1 = n(-&ssl)(u,e) = II(-&SSO(U,C v E), by Dl(1V). Then -&ss' E N, as was to be shown. Intuitively, proposition (1.2)* asserts that the pragmatic value withhold is preserved under conjunction, provided the conjunction is not regarded by the users of L as analytically false, and provided that rejection of the conjunction is not resolvable into rejection of one of its conjuncts. In virtue of proposition (1.2)* so understood, the patterns of pragmatic valuation represented by the tables of D2, though not strictly functional, are conditionally functional in a sense which suggests that these three-valued tables constitute a natural generalization of the tables restricted to (0,l). Thus (1.2)* may be viewed as describing the nature of the "defect" in the functionality of the &-table of D2. This "defect" reflects the properties of the pragmatic value withhold (= 2) which underlie the departure from strict functionality in the generalization of the two-valued parts of these tables to the three-valued case appropriate to pragmatic theory. We shall return to this topic in Chapter 7. As indicated above, appropriate specializations of the concept of a proto-sentential interpretation, called coherent interpretations, induce a polyadic Boolean algebra of standard predicate formulas over L, such that the closed formulas of the algebra are expressions of a set S defined pragmatically as above. In this case, the expression & may be recognized to be the conjunction sign and the expression may be recognized to be the negation sign of the formulas of the algebra. Before turning to the concept of a coherent interpretation, it is instructive to make a brief digression in order to con-

-

SENTENTIAL INTERPRETATIONS

sider weaker specializations of proto-sentential interpretations, called sentential interpretations, which determine the expected Boolean structure on S directly, not necessarily as a subalgebra of a wider algebra of formulas. In this way we may illustrate in the present chapter the manner in which standard linguistic structure may be pragmatically determined. It will be shown in Chapter 4 that coherent interpretations are sentential interpretations, and it is important to note that the concept of a sentential interpretation is not explicitly required for the pragmatic foundations of standard linguistic structure.

Definition 3 II is a sentential interpretation iff II is a protosentential interpretation which determines associated pseudo-conjunction and pseudo-negation signs & and such that, for all s, s t , s" E S, e E L, u E U, and proper c E C:

-

I. If II(&s-sl) (u,c) = 0 = II(&s~-s~~), then II(&s-sV)(u,c) = 0. 11. If II(&s-sl)(u,c) = 0, then II(&&ss"-&s's")(u,c) = 0. 111. En (s,- -s) . IV. En (&ss',&sls). V. En (&s&sls",&&ss's" ). VI. En (s,&ss) . VII. If e is a concatenate of the expressions & or -, then II(e) = d2.

VIII. n(&-S--s) = do = n(&&ssf-&ssr). IX. If s = &e, then e = s1s2for some s, ,s2 E S. X. If s = -e, then e E S. XI. s is not a proper initial segment of st. XII. &ssl # -s".

-

Throughout the remainder of this chapter, let & and be associated pseudo-conjunction and pseudo-negation signs determined by a sentential interpretation II. By clause VIII of D3, if s and s' are in S, then so are &ssl and -s. Let So be the set of expressions of S which do not begin with & or -. Then by clauses IX and X, S is generated from So by the operations of prefixing & and -. Then by clauses XI and XII, S is freely generated from So by these operations. By clause VII, &, -, A S, where A is the

44

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

empty expression. Also by clause VII, in view of D2(I), II(A) = d2. The following consequences (1.3)-(1.5) of Definition 3 hold for all s,sl E S. For perspicacity in referring to expressions of S, we introduce the following definitions: "(s) & (s')" for "&ss'", and (s)"for "-s", where s,st E S. We shall omit brackets introduced by such definitions, when this convenience is unambiguous.

"-

If II (s)(u, C) = 2 = II (st)(u, C) , then II (s&sl) (u, c) = 2 or II (s& -st) ( u, C) = 2. Proof. On the hypothesis of (1.3), let II(s&sl)(u,c) # 2. Then by D2(VI), II (s&sl)(u, c ) = 0, so that II ( (s&sl))(u, c ) = 1. Now suppose that II(s&-sl)(u,c) # 2. Since II(-sl)(u,c) = 2, II (s& -sl) (u, C) = 0, by D2(VI). Then II((s&s) & (st&s)( u, c) = 0, by D3(II), so that II ((s& s)&- (s&st)(u, c) = 0, by D3(1V). Since II ( (s&sl)(u, c) = 1, ll (s&s)(u, c) = 0 = II(s) (u, c), against the hypothesis of (1.3).

-

-

-

According to (1.3), for all arguments (u, c) E U x C, either the table for & of D2 fixes uniquely the value of II(s&sl) from the values of II (s)and ll (s' ), or else the table fixes uniquely the value of II (s & -st) from the values of II (s)and II( -sl). By means of (1.3) it may be shown that:

Proof. The consequent of (1.4) is true in general if II (s)(u, c) = 0. If II (s)(u, c ) = 1, then by the hypothesis of (1.4), II ( -st) (u, c) = 0, so that II (st)(u, C) = 1 = ll (s&sl)( u , c). Now let II(s) (u, c) = 2. Then either II ( -st) (u, c) = 0 or II ( -st) (u, c ) = 2, by D2(VI). In the former case, II(sl)(u, c) = 1, so that II (s&sl)(u, c ) = 2. In the latter case, II (st)(u, c) = 2. Then II (s&s)(u, c) = 2, for otherwise, by (1.3), II (s&-s') (u, c) = 2, against the hypothesis of (1.4). Let "s -- s'" abbreviate (1.4) it follows that:

If II(s

-

"-(s& -st) & - (st&-s) ". Then from

s') (u,c) = 1, then II(s)(u, c) = H ( s l )(u,c).

A sentential interpretation II induces a Boolean structure on the set S determined by II, in the following sense. The core a of ll

SENTENTIAL INTERPRETATIONS

determines a natural equivalence relation R, on S , which is defined, for all s,s' E S :

R,(s,st) iff a ( s

-

s f , b )= 1.

-

That R, is an equivalence relation on S is a consequence of D3(I). By ( 1 . 3 , if R, (s,s' ), then a(s,b ) = ~ ( s b' ,) . Let & be the binary operation on S defined such that &(s,st)= &ss'. Let = be the unary operation on S defined such that ( s )= -s. It is proved in Chapter 4 that: If lS is a sentential interpretation, then (S,&,= ) is a Boolean algebra with respect to R,, where & is the meet operation and is the complementation operation of the algebra.

=

It is also shown in Chapter 4 that the set S of (1.6) is the set of closed formulas of a pragmatically constructed polyadic Boolean algebra of standard predicate formulas, and that the algebra of (1.6)is a Boolean subalgebra of this algebra, provided II is a coherent interpretation. To the theory of coherent interpretations we now turn.

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2 Coherent Interpretations

In this chapter the theory of proto-sentential interpretations is extended so that the sentences determined by appropriate specializations of proto-sentential interpretations may be shown to be the closed formulas of a standard predicate language. The relevant specializations of proto-sentential interpretations are called coherent interpretations. This terminology is motivated by the fact that coherent interpretations are defined with respect to the concept of a coherent belief function, in the sense of subjective probability theory. In this chapter we shall define the concept of a coherent interpretation, and develop the properties of coherent interpretations which are required for the pragmatic determination of standard linguistic structure in subsequent chapters. The first step in the specialization of proto-sentential interpretations to coherent interpretations employs the concept of a lexical interpretation. In the definition of a lexical interpretation it is useful to have a notation for expressing the result of concatenating an expression with itself a specified number of times. For this purpose we define recursively, for any expression e of L: le = e, and ne = ( n - l)ee, the concatenate of ( n - 1)e with e. The definition of a lexical interpretation is as follows.

48

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

Definition 4 II is a lexical interpretation' iff II is a protosentential interpretation and there are associated pseudo-conjunction and pseudo-negation signs & and determined by II such that, for some expression 3 of L, the set

-

I = ( i E L: n ( 3 i & p-p) = do for some p E L; no initial segment of &, -, or 3 is a final segment of i; and neither &, -, nor 3 occurs in i )

is infinite, and for all e, e', e" E L and i,j E I: I. If e is a concatenate of elements of I U (&,-,a) or e has the form e'&, ef-, ef3, e'&ie", or ef-ie", than II(e) = dZ. 11. If II(3i&eni-eni) = do and m # n, then II(3i&emi-emi) = d2. 111. If II(3ie) = do, then II(3e) = d2. IV. If II(&e-e) = do, then every occurrence of i in e is part of an occurrence in e of an expression of the form 3ip, for some proper expression p of L. Intuitively, 3 is a symbol for existential quantification, and the expressions in the set I are individual variables in the language of the idealized users of L. These intuitions are justified in Chapter 4, where it is shown that for appropriate specializations of the concept of a lexical interpretation, every consistent standard formal system is representable by a pragmatically constructed polyadic Boolean algebra of standard predicate formulas, whose individual variables constitute a set of the form I defined pragmatically as above. The motivation for the definition of the set I may be illustrated as follows. If the expression 3i&p-p displayed in the definition of I is a formula in Polish notation whose logical constants are the signs &, -, and 3, for conjunction, negation, and existential quantification, respectively, then i is a variable. For suppose otherwise. Then i =je where j is a variable. Then e&p-p is a formula, but not an atomic formula since it contains &. Now &, -, or 3 cannot begin i, by the last clause of the definiens. Then e begins either &, -, or 3 and ends i, against the second clause of the definiens. Thus i is indeed a variable, on the assumption that 3i&p-p is a formula in Polish notation and our intuitions about

COHERENT INTERPRETATIONS

-, and 3 are correct. This assumption holds if II is a coherent interpretation, as will be seen. Let the expressions &, -, and 3 be as described in the definiens of D4. Then we shall say that these expressions are associated pseudo-conjunction, pseudo-negation, and pseudo-quantification signs determined by II. By definition of the set I, these signs are not members of I. By clause V of D4, the empty expression is not in I. That 3 is a proper expression, distinct from & and -, will be shown below. By clause I of D4, II(&A-A) = do, where A is the empty expression. Thus by D2(I), II(A) = d2. The concept of a lexical interpretation permits the definition of further grammatical categories which, intuitively, are employed in the language of the users of L. In the definition of these categories, &, -, and 3 are understood to be associated pseudoconjunction, pseudo-negation, and pseudo-quantification signs determined by a lexical interpretation II. We then define, for each positive integer n, the set:

&,

(Pn)* = (FE L: for some i E I, II(3i&Fni-Fni) = do, F does not occur in &, -, or 3, and no initial segment of &, -, 3, or i occurs in F) We shall refer to the expressions in (Pn)* as pseudo-predicates; the motivation for this terminology is as follows. Consider the Polish notation in the discussion of the set I above, and let this notation contain no function symbols. Let the displayed expression 3i&Fni-Fni in the definition of (Pn)* be a closed formula in this notation, where i is an individual variable. Then F is either a predicate of degree n or an expression of the form Gal . . .am, where G is a predicate and a , , . . . ,am are individual constants. For by hypothesis, Fni is a formula. Then Fni is an atomic formula. For F cannot begin &, -, or 3 , by the second hypothesis on F, and neither &, -, nor 3 can begin F, by the third hypothesis on E Then Fni has the form Gal . . .amni, where G is a predicate and al,. . .,am are individual constants, since the displayed formula is closed and i cannot occur in E If m = o, then F = G is a predicate of degree n. Otherwise F = Gal . . .a,,,. In accordance with the terminology introduced above, F is a pseudo-predicate. The concept of a pseudo-predicate provides a means whereby, in the pragmatic

50

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

determination of standard linguistic structure, predicates and individual constants are characterized. This characterization takes the following form. We first define:

K = ( a E L: II(&Fna-Fna) = do for some F E (Pn)* and no expression of I is a proper final segment of a ) Intuitively, the expressions in the set K are individual constants in the language of the idealized users of L . This ~ intuition is justified in Chapters 3 and 4. The motivation for the definition of the set K may be illustrated as follows. Consider the Polish notation in the discussion of the set (Pn)*.If the displayed expression &Fna-Fna in the definition of K is a closed formula in this notation and F is a pseudo-predicate, then a is an individual constant. For otherwise a is an individual variable, whence the displayed expression is not a closed formula. Thus a is indeed an individual constant, on the above assumptions about the expression &Fna-Fna. These assumptions are justified in Chapters 3 and 4, where it is shown that every consistent standard formal system is representable by a pragmatically constructed polyadic algebra of formulas whose constants represent the expressions of a set of the form K defined pragmatically as above. By means of the sets (Pn)*and K associated with a lexical interpretation, we may define3 the set P n = (FE (Pn)*:no expression of K occurs in F ] . Intuitively, the expressions of Pn are predicates of degree n. This intuition is justified in Chapters 3 and 4, in the manner indicated above for the set I of individual variables. The set K may be empty, but the set P = P' U p2 U . . . is not empty, as is shown after Definition 5 below. The expressions &, -, and 3 are not in P, since they are not in any set of the form (Pn)*. By clause I of Definition 4, &, -, 3 $! K. By clause I1 of D4, sets of the form (Pn)*are disjoint, whence sets of the form Pn are disjoint. The sets (Pn)*and I are disjoint. For suppose F E (Pn)*n I. Then for some i E I, II(3i&Fni-Fni) = do, where the displayed expression contains only &, -, 3, and i E I, against D4(I).

COHERENT INTERPRETATIONS

It follows that the sets P and I are disjoint. That K is disjoint from I and also from P is shown after Definition 5 below. The empty expression is not in P, else an expression of the form gi&ni-ni, for some i E I , is valued do by I I , against D4(I). That the empty expression is not in K is shown after D5 below. If the sets (&,-,3),I,P,K are associated with a lexical interpretation II as above, then we shall refer to the quadruple ((&,- , 3 ) , I,P,K) as a pseudo-morphology determined by II. We wish to consider the conditions under which pseudo-morphologies are standard morphologies. This purpose is served by the following definition. Definition 5 II is a standard interpretation iff II is a lexical interpretation which determines a pseudo-morphology ((&, ,3 ) , I,P,K) such that for all F,G,tl,. . .,tn,p,q,e,el,el,. . . ,en, f l , . . ., fm E L:

-

I. I f F € P n a n d t l , ..., t,EIUK,thenforsomeeEL,II(e& Ftl . . .tn-Ftl . . .t n )= do. 11. If II(e&p-p) = do = II (el&q-q), then for some e" E L, II(e"&&pq-&pq) = do. 111. If II(e&p-p) = do, then II (e&-p- - p ) = do. IV. If II(e&p-p) = do and i E I , then II(e&gip-3ip) = do. V . If II(e&p-p) = do, then one of the following conditions holds: ( a ) p = F t l . . .tn for some F E Pn and t l , . . .,tn E I U K (b) p = &qr and II(el&q-q) = do = II(eM&r-r)for some q,r,e',e" E L. (c) p = -q and ll(el&q-q) = do for some q,el E L. (d) p = 3iq and II(el&q-q) = do for some q,el E L and i E I.

VI. If el ,..., en,fl ,...,f m ~ I U K U P U(&,-,a) andel .. .en, f l ...fm, then n = m, el = fl ,..., and en = fm. VII. If F E Pn and G E Pm, then II(eFGe') = d2. The motivation for Definition 5 may be clarified as follows. Let II be a standard interpretation and let Q = ( p E L: II(e&p-p) = do for some e E L )

52

PRAGMATICS AND SEMIOTICS OF STANDARD LANGUAGES

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where & and are as described in Definition 5. From clauses I-V of D5 and the definition of Q it follows immediately that:

(2.1)

Q is the smallest subset of L such that: (a)Ft l . . . t n E Q , i f F E P n a n d t l,..., t , , E I U K . (b) &pq,-p, 3ip E Q, if p,q E Q and i E I.

Intuitively, the expressions in Q are standard predicate formulas in Polish notation. This intuition is justified in Chapters 3 and 4, where it is shown that every consistent standard formal system is representable by a pragmatically constructed polyadic algebra of formulas whose domain is a set of the form Q defined pragmatically as above. The motivation for the definition of the set Q is like that for the definition of the set S in Chapter 1. It was shown in Chapter 1 that if &p-p is a formula in Polish notation, where & is the conjunction and the negation sign, then p is also a formula. The same reasoning may be used to show that, on the same assumption about & and -, if e&p-p is a formula in Polish notation, then so is p. For by hypothesis, & begins a formula. The supposition that p is not a formula implies, by the reasoning of Chapter 1, a contradiction: p = p1p 2 where pz does and does not contain a binary connective, one of whose arguments contains -p as a part. Thus if p is a formula, then this fact may be reflected in the verbal behavior of the users of L, in virtue of which p is a member of the set Q defined pragmatically as above. To the clarification of this idea we now proceed. It might be thought that it is necessary to require that the expressions e, e', and e" in clauses I-V of Definition 5 and in the definition of Q be sequences of quantifiers, that is, sequences of expressions of the form 3i for i E I. That it is not necessary to complicate these definitions by adding this requirement is shown in Chapter 3, where by means of appropriate specializations of standard interpretations, clauses I-V and the definition of Q are proven to hold in the more perspicuous forms just described. As seen above, if the displayed expression e&p-p in the definition of the set Q is a formula in Polish notation, then so is p. The motivation for the definition of the set Q would then be obvious, if e&p-p were a formula in Polish notation. But if ll is a coher-

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COHERENT INTERPRETATIONS

ent interpretation and II(e&p-p) = do, then e&p-p is indeed a formula in Polish notation, as will be shown in Chapter 3. Before defining the concept of a coherent interpretation, we shall first consider some consequences of Definition 5 which are required for