what is the role of model theory in the study of meaning?

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Journal of Pragmatics 17 (1992) 5 1 l-522 North-Holland 511 What is the role of model theory in the study of meaning? Klaus Robering 1. Model theory and critics One of the most important events in the development of modern linguistics was the break-through of model-theoretic semantics in the 1970s which is so intimately connected with the works of Richard Montague (1974). But ever since its first applications to linguistics, the adequacy of model theory for the study of natural language semantics has been doubted (cf. Jardine 1975; Potts 1975 - and with special reference to Montague’s version of that approach - Potts 1976), and although model theory is a well-established subdiscipline of mathematical logic, its proper role for the foundations of logic is as much debated as its adequacy in the study of natural language meaning. Constructivists (Kamlah and Lorenzen 1967: 1933195; Martin-Liif 1984: 3) and finitists (Hilbert and Bernays 1934: 125-131), as well as logicians who, like Hinst (1982) conceive of logical rules as based on pragmatic principles, are more often sceptical about the explanatory value of model theory as an account of meaning. Admittedly, also these critics may use model-theoretic means as technical tools in their formal work, and there is something like a constructive model theory. But what is at issue here is not the technical apparatus of model theory but its status as an explanatory account to meaning. When the eminent logician A. Mostowski (1965), in his lectures on the development of mathematical logic from 1930 to 1964, opens his lecture on the theory of models1 with the statement, ‘the modern form of semantics is the theory of models’, this seems to represent a problem as settled which is still being discussed. This is even more true for linguistics than for logic and metamathematics. So, Bickhard and Campbell’s (B&C’s) foundational article on the application Correspondence to: K. Robering, Institut fur Linguistik, Technische Universitat Berlin, Sekr. TEL 6, Ernst-Reuter-Platz 7, D-W 1000 Berlin 10, Germany. 1 Mostowski (1965: 119-131). See e.g. Chang (1974) for supplementary material about the history of model theory. An excellent introduction to the model theoretic approach to natural language semantics is given by Bach (1989). 0378-2166/92/$05.00 0 1992 - Elsevier Science Publishers. All rights reserved

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Page 1: What is the role of model theory in the study of meaning?

Journal of Pragmatics 17 (1992) 5 1 l-522

North-Holland

511

What is the role of model theory in the study of meaning?

Klaus Robering

1. Model theory and critics

One of the most important events in the development of modern linguistics was the break-through of model-theoretic semantics in the 1970s which is so intimately connected with the works of Richard Montague (1974). But ever since its first applications to linguistics, the adequacy of model theory for the study of natural language semantics has been doubted (cf. Jardine 1975; Potts 1975 - and with special reference to Montague’s version of that approach - Potts 1976), and although model theory is a well-established subdiscipline of mathematical logic, its proper role for the foundations of logic is as much debated as its adequacy in the study of natural language meaning. Constructivists (Kamlah and Lorenzen 1967: 1933195; Martin-Liif 1984: 3) and finitists (Hilbert and Bernays 1934: 125-131), as well as logicians who, like Hinst (1982) conceive of logical rules as based on pragmatic principles, are more often sceptical about the explanatory value of model theory as an account of meaning. Admittedly, also these critics may use model-theoretic means as technical tools in their formal work, and there is something like a constructive model theory. But what is at issue here is not the technical apparatus of model theory but its status as an explanatory account to meaning. When the eminent logician A. Mostowski (1965), in his lectures on the development of mathematical logic from 1930 to 1964, opens his lecture on the theory of models1 with the statement, ‘the modern form of semantics is the theory of models’, this seems to represent a problem as settled which is still being discussed.

This is even more true for linguistics than for logic and metamathematics. So, Bickhard and Campbell’s (B&C’s) foundational article on the application

Correspondence to: K. Robering, Institut fur Linguistik, Technische Universitat Berlin, Sekr. TEL 6, Ernst-Reuter-Platz 7, D-W 1000 Berlin 10, Germany. 1 Mostowski (1965: 119-131). See e.g. Chang (1974) for supplementary material about the history of model theory. An excellent introduction to the model theoretic approach to natural

language semantics is given by Bach (1989).

0378-2166/92/$05.00 0 1992 - Elsevier Science Publishers. All rights reserved

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512 K. Robering 1 Model theory and meaning

of model theory to linguistics (this issue) should be highly welcome to every semanticist who is interested in the proper foundations of the discipline. All practitioners of model-theoretic semantics are furthermore challenged by B&C’s article to reflect on what they are really doing when they define models for natural language fragments. B&C’s article contains (1) an investigation into the semiotic foundations of model-theoretic semantics, (2) the develop- ment of their own ‘interactionist’ alternative to the model-theoretic approach, (3) an analysis of how model theory’s basic errors infect the whole building of grammar, the global architecture of the system of syntactic categories as well as the grammatical analysis of details like propositional attitude constructions or the analysis of pronouns and anaphora, and (4) interactionist proposals toward the solution of the aforementioned problems. Because it is impossible to discuss all of these, I shall concentrate on (I) and (3).

But let me conclude these introductory remarks with a short comment on (4). Whether one likes it or not, Montague’s model-theoretic approach to natural language semantics has set the standards of discussion in that field. One of its essential procedures is to illustrate its usefulness and adequacy by means of example: i.e., by establishing a fragment of a natural language and providing a model-theoretic semantics for it by defining the concepts of truth (in a model) and consequence either directly for the expressions of the expressions of the fragment, or by a translation into a formal language for which, in turn, such definitions are given. B&C, however, remain content to outline interactionist revisions of the model-theoretic procedure, and to merely sketch connections to recent developments in formal semantics (e.g., to the logic of demonstratives, anadic logic, and the logic of action-theoretic concepts). Furthermore, they argue for a more ‘algebraic’ approach to seman- tics, such as provided, for example, by Tarksi’s theory of cylindric or Halmos’s theory of polyadic algebras. z But all this, certainly, should be put

together into a unified presentation of an illustrative fragment with algebraic semantics that exemplifies the merits of their framework. Their sketchy outline leaves too many questions unanswered. What does the system of interactionist semantics as an integral whole - including interactionist theories of modality, intentionality, ability, action, etc. - look like? What kind of algebraic system should be used for semantic representation? Henkin et al. (1985: 263327 1) give a large bibliographic list of different algebraic logics, which includes algebraic versions of tense, modal, and generalized quantifier logics, besides others. But what the interactionist needs for his or her semantics is an integrated system embracing all these algebras as its parts. B&C give some hints to the literature, but they do not list the basic operations of their system and do not state axioms governing these operations. Further-

* Note that there are also strong links between model theory itself and algebra (and topology);

cf. the classic monograph by Rasiowa and Sikorski (1963).

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K. Robering 1 Model theory and meaning 513

more, most of the more common logics used for the sake of semantic representation are higher-order systems. On the other hand, very little seems to be known about higher-order algebras (though there is, of course, the well- developed combinatory logic, which is another kind of algebraic approach to logic); Henkin et al. (1985: 271) mention just one single contribution to the theory of higher-order polyadic algebras.

2. Is model theory a mathematical theory of meaning?

B&C in their article argue that model theory is not an adequate approach to the phenomenon of natural language meaning. Their argumentation, however, seems to rest on the assumption that model theory is a subdiscipline of mathematics which deals with meaning just in the same sense as, say, arithmetic, as a branch of mathematics, deals with the natural numbers. This assumption is highly dubious, as I shall now explain.

As conceived from a broader perspective, linguistics is a branch of semio- tics, i.e. the general theory of signs. One strong tradition in semiotics views languages as special kinds of correlations between two sorts of entities: expressions on the one hand, and meanings on the other. 3 Languages are only one species of such correlations, codes (e.g., the Morse code), signalling systems, artificial languages (programming languages, languages of logic, etc.) being other species of the same genus. One of the major tasks of semiotics is to describe and classify such semiotic correlations4 Given a class E of expressions and a class M of meanings (whatever entities these may be), there are quite a lot of different set-theoretic relations R (c E x M) correlating expressions and meanings. But not all of these relations are (have been, or will be) employed by sign users to send out, exchange, or pick up information. Thus, a semiotic correlation (such as a language) always is a correlation for a sign user or a community of sign users. Consequently, only two subsets of Pot (E x M) (the power-set of E x M) are of any semiotic interest: (1) the subset of those relations which are actually used in a community, and (2) the subset of those correlations which might have been used by a community (i.e., which are not used for accidents of history only, and which are not ruled out as means of communication by principled reasons). 5 The linguistic instantiations

3 The other great tradition insists that it is a fallacy to argue from the significance of expressions

to the conclusion that there are special entities - ‘meanings’ - which are signified by expressions.

The meaning of an expression, according to this tradition, cannot be an entity or an object but must consist of something else, e.g., the rule which governs the use of the expression (to quote just

one possibility). 4 An analysis of different types of semiotic correlations in terms of causal and epistemic notions is given by Posner (1990: 6-12).

5 As e.g. that the expressions in the domain of a semiotic correlation have to be of finite size and that the number of its basic expressions should be finite, too; cf., for example Davidson’s

(1984: 3-15) celebrated ‘Learnability Constraint’.

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of these concepts are the notions of an actual language I E L,(C) of

community C, and a possible language P EL, (C) for community C. 6 Clearly, the actual languages of a community form a subset of the possible ones, i.e., La (c) E L, (C’) G Pot (Ex A4). Following tradition, let us call an expression- meaning-pair s = (e, m) of an actual or possible language I a ‘sign’ of I (sEI).

Now, B&C’s negative arguments in the first part of their paper aim at demonstrating that (1) no possible language I’ (EL, (C)) for C could reason- ably belong to a species of semiotic correlations which B&C call ‘encodings’, but that (2) model theory just views natural languages as such encodings. B&C define encodings (encoding systems) as semiotic correlations 1 (EPot (Ex A4)) which are such that, in order for an agent A to have command of 1. A has to know (at least) for each sign s = (P, m) of I both the expression e of s, and its meaning m. Take as an example the Morse code. To have command of the code, one must implicitly know the patterns of dots and strokes which make up the Morse alphabet, and know the characters of the Latin alphabet which those patterns stand for. (Furthermore, one must know which pattern stands for which character.)

There are two time-honored problems with respect to the notion of an actual language I of community C, and part of B&C’s argument (as I understand it) is that these problems are unsolvable if languages are encod- ings. These are the two problems: (Pl) How is it possible that I becomes a means of communication for C? (P2) How does a new member of C (for instance a child) acquire knowledge of I? - Some signs of 1 may be definable in terms of other signs; but each (non-circular) chain of definitions has to stop somewhere. The best one can hope for is to arrive at a stock B of basic signs of I, by means of which all the other signs of 1 are definable.7 Now, take (P2)

6 LO (c) is meant to be the set of actual, L, (c) the set of possible languages for C. Think of multilingual communities if the plural sounds strange to you. I abstract in this and in the

following from matters of e.g. diachronic. diatopic. diastratic. etc. variations of language.

although it is known that they are not without semantic relevance. ’ To circumvent this problem in their interactionist approach, B&C adopt a version of the

doctrine of implicit definitions. There is a model-theoretic notion of implicit definability (Boolos

and Jeffrey 1974: 245-249) according to which an expression R is implicitly definable from

expressions Q1. , Q,,, (different from R) in theory TH if any two models of TH with the same domain which agree in their assignments of entities to Q1. , Q,,, also agree in what they assign to R. This notion is of no help in explaining how the basic signs of B are acquired by a learner. which fact is most obvious in the case of first order systems. where (by Beth’s definability theorem) implicit definability (in the sense explained) coincides with explicit definability. What B&C seem

to envisage is a version of the doctrine of implicit definability according to which the meanings of the basic signs from B mutually determine each other by simultaneously fulfilling a set A of

conditions (axioms). But this doctrine is incoherent: A (first-order) axiom system (e.g.. that of group theory) does not implicitly define the basic expressions of the language in which it is formulated (in the example given: does not attach a meaning to the circle "0". which is intended

to denote the group operation) but explicitly detines a second-order relation (in the example: that between a domain G and a binary operation 0 on G such that G is a group with respect to 0).

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K. Robering / Model rheory and meaning 515

first. Someone who knows the restriction 1, of I to B as its basic vocabulary may acquire the entire language 1 if he or she is provided with definitions of the non-basic signs. But how does he or she learn the restriction Is of I? The languages I and IB being encodings, he or she has to know the meanings of the signs of B in order to have full command of I8 ! Of course, there is the possibility of explaining to her or him the signs of B in a non-circular manner by means of another semiotic correlation r. But then the same questions which have been asked for the original 1 arise for /‘. Or consider the case (P2) for the whole community C. The members of C may arrive at an agreement about the definitional expansion I of f, if their semiotic practices agree on lB, i.e., if they use the expressions of I, according to the meanings enlisted in B. But how do they achieve such an agreement on l,? - There have been several proposals toward a solution of these problems (cf. Lewis 1969; Schiffer 1972; von Kutschera 1975, 1983; Bennett 1976; Harman 1977), and the difficulties and foundational problems connected with the model-theoretic treatment of common knowledge are aptly discussed by Barwise (1989). But since B&C do not treat these proposals, I shall not discuss them either; let us now turn instead to the alleged consequences for the model-theoretic approach to natural language meaning.

B&C’s refutation of model theory presupposes that model theory adheres to semiotic tradition in constructing languages as semiotic correlations. This, however, cannot be true! Take as an example Montague’s (1974: 247-270) famous PTQ-analysis of the quantificational system of English (PTQ is the acronym of the title of the relevant article by Montague). English expressions are semantically analyzed by translating them into IL (Intensional Logic - Montague’s version of an intensional type theory), which is a formal language with model-theoretic semantics. Adopting B&C’s point of view, we should expect the meaning of an English expression from the PTQ-fragment - such as unicorn, for example - to be an entity which is model-theoretically associated with its IL-translation - unicorn’ (this is a mnemotechical metalinguistic name for a certain constant of IL). But there is no such entity. What is done in PTQ (and generally in model-theoretic semantics) is to provide definitions of notions like ‘model’, ‘truth’/‘satisfaction in a model’, ‘consequence’, etc. A model X (of IL) has as one of its components a function F that assigns to each constant a (of IL) a set F(a), which is set-theoretically constructed out of other components of X. This funtion F serves as the starting point in a recursive definition of a ternary function I/ 11 which assigns to each formula l3 (of IL), each model 3, and each 2f-assignment g* a set-theoretic entity 11 l3 ll’,g

s An assignment does just that for the variables which F does for the constants: it assigns to

each variable an entity that is provided by the model. For example, dividual variables are assigned individuals from the domain of the model. There is, however, a certain asymmetry (in Montague’s IL) between the constants and the variables, which has to do with the extension-intension-

distinction and does not matter here.

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516 K. Robering / Model rheory and meaning

(for details (cf. Montague 1974: 258-259)): the ‘intension’ of !3 in I with respect to g. Thus, there is a complex ternary function (i.e., a 4-ary relation) instead of a binary correlation. The question ‘What (entity) is the meaning (intension) of the expression unicorn?’ simply misses its point here! There is no such entity, or alternatively: there are quite a number of them ~ one in each

model. Consequently, the aforementioned two problems (Pl) and (P2) with the

notion of a language in community are not problems of model theory itself. Model theory does not claim to single out from the totality of all possible models for an object language I that one distinguished element which provides the ‘real’ or ‘actual’ meanings of l’s expressions. Problems (PI) and (P2), however, raise questions about the real, distinguished, or ‘intended’ model of an object language. Now, the notion of an intended model is not a technical

notion of model theory; it presupposes that the language in question is an interpreted one: that its expressions have been given meanings in advance. This interpretation then distinguishes the intended model (and the ones isomorphic to it) from the rest. For example, the interpretation which requires the expressions of first-order arithmetic to deal with the natural numbers distinguishes the standard model ‘% = (IN, 0,‘, + , .) from the non-standard9 ones. But note, the intended model W is singled out by means of contentual arithmetical notions like ‘number’, ‘zero’, ‘successor’, etc., not by means of formal model-theoretic notions. If there is an intended model 21, for object language 1, it seems to be best to call those entities m which are associated with the expressions of I by a non-intended model 8 ‘quasi-meanings’ (relative to b). Model theory treats meaning and quasi-meaning on a par; non- intended models are just as good models as the intended one. lo

But why consider such strange things as quasi-meanings at all? I consider it one of the merits of B&C’s article to challenge this question. Several answers seem to be possible. A model - so one might say ~ represents a possible situation in which the object language might have been used, and the model’s quasi-meanings are the objects which the expressions would have signified in this situation. This conception is used in a recent text-book by Chierchia and McConnell-Ginet (1990: 78-79) to motivate the concept of a model; Chierchia and McConnell-Ginet’s way of thinking about models is clearly inspired by situation semantics (Barwise and Perry 1983). But the very same conception of models as situation may be found even before the rise of situation semantics (e.g., in Tennant 1978: 28). Another, more traditional answer is to view

9 Cf. Boolos and Jeffrey (1974: 191-195) for a discussion of non-standard models of arithmetic. lo There are additional problems for the distinction between the intended model and the non- intended ones as soon as intensional contexts are considered. A set is certainly an entity which can

reasonably be considered as the extension of a (monadic) predicate. But functions which assign sets to possible worlds are reasonable as intensions only. if one believes in possible worlds. I shall not discuss possible worlds semantics or its interactionist alternative; cf. van Benthem (1985) for a

survey of recent developments in the field of intensional logic.

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K. Robering / Model theory and meaning 517

models and their quasi-meanings as tools for the investigation of the conse- quence relation of the object language (Jardine 1975: 235). A widespread misunderstanding of model-theoretic semantics which seems to have influ- enced B&C is summed up in the popular description of formal semantics as ‘concerned with the truth-conditions of sentences’. This may be a correct description of, for example, Davidson’s version of formal semantics (among others), but the model-theoretic approach is more general and employs not the concept of truth but the relation of being true in a model, as explained above (Montague 1974: 188; Davidson 1984: 68-69). As concerns the applica-

tion of model theory to linguistics, l1 this relation is a step toward an

explication of the consequence relation in terms of models. I2 The proper object of the application of model theory to linguistics seems to me to be the consequence relation. True, the model-theoretic analysis of the consequence relation has its problems as well (Etchemendy 1988a, b), but this is quite another story.

3. Providing expressions with their ‘full operative powers’

Let me turn now to B&C’s critical analysis of the juncture of model theory and categorial grammar, and to their suggestions towards an interactionist reform of this approach. There are two aspects of categorial grammars to be distinguished: (1) the syntactic machinery of categories and cancellation rules,13 and (2) the use of categorial systems in semantic analysis. As I understand B&C, the purely formal apparatus of categorial grammar is (roughly) in order as it stands, but the way in which the machinery is put to work has to be revised. First of all, expressions of the various categories (especially the sentences) have to be provided with their ‘full operative powers’. This suggests that the categories in model-theoretic frameworks lack part of that power. Now, what does this mean? A rather common criticism of all varieties of formal semantics is that they are ‘static’, at best adequate for the ‘descriptive’ or ‘assertive’ uses of language, and that they neglect such

I1 There are of course other applications of model theory, for example, to combinatorics or to

algebra. Some rather profound theorems of group theory were proved by the Russian mathemati-

cian A.I. Malcev in the 1940s with help of a model-theoretic analysis of the formulation of group theory - and use of the compactness theorem (Mostowski 1965: 121). An application of model

theory to a problem of combinatorics may be found in Boolos and Jeffrey (1974: 260-266). As considered from applications like these, model theory is perhaps more aptly defined as the theory

of the relationships between (formal) languages and algebraic structures. Surely, this will be the mathematician’s favorite description of the subject.

I* The characterization of this relation by means of models is only one of several ways to reach at an analysis of the notion of a consequence relation - others being the syntactic procedure by

definition of a derivability relation and the methods of truth valuations. I3 Cf. Moortgat (1988: esp. ch. 1) for a recent survey of several systems of categorial grammar.

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518 K. Robering / Model theory and meaning

phenomena as questioning, ordering, wishing, evaluating, etc. These frame- works are, in other words, unable to account for those kinds of discourse which do not describe some state of affairs but rather operate on the background context and serve to change it.

I am not sure whether these criticisms hit their target. It is a quite common procedure (though, perhaps, not a correct one; see below) to distinguish between two dimensions of meaning. For example, utterances of the sentences (1) and (2) below share the same semantic unit - the ‘content’14 or ‘sense’ which is expressed by them ~ but differ in the ‘force’ (Germ.: Krqfc) with which this is expressed (Frege 19 1 S/l 9: 62 and 144).

(1) 1st die Sonne grol3er als der Mond? (2) Die Sonne ist griiber als der Mond.

A proposition (a Fregean ‘thought’; Germ.: Gedanke) is expressed by (2) with assertive force (“behauptende Kraft”, Frege (1918/1919: 63) but with interro- gative force by (1). The same distinction is drawn, for example, by H.S. Leonard (1959: 179): “An utterance expresses a concern and indicates a topic of concern”. Utterances of the sentences (1) and (2) share their indicated topic but differ with respect to the concerns expressed. Now, if contents or topics are independent of force (insensitive to concerns), there is a natural partition of the theory of meaning into a subtheory of content (semantics proper) and a theory of force, which is more often subsumed under pragmatics (e.g., speech act theory as the theory of illocutionary forces). Furthermore, force is a concept which is only relevant to those syntactic categories (of sentences of different types) whose expressions are normally uttered as independent units in discourse, whereas content is connected with all categorematic expressions. The force of a complex expression does not depend on the forces of its components, whereas its content is a function of the contents of its syntactic parts (Frege 1918/19: 144). l5 The recursive machinery of categorial grammar is thus only needed in a theory of content.

If this is really true, then the revision of categorial grammar needed in order to take account of the forces is rather obvious. Let s be the category of

I4 There is a tradition in semantics, which goes back at least to Frege, and to which also Tarski and Carnap, among others, have contributed, of defining the content of a sentence as the set of all its (non-trivial) consequences. The application of model theory to linguistics as viewed as a theory of the consequence relation may thus also be characterized as a theory of content. Note, that the notion of content just explained is independent of the conception of meanings as entities, cf. fn. 3). I5 This is correct only if there are no ‘illocutionary connectives’, which combine expressions which have been already marked for a special force. Such modes of combining expressions are

excluded, for example, in von Kutschera’s (1983: 10) action theoretic semantics, but are admitted in the illocutionary logic of Searle and Vanderveken. Zaefferer (1984: 120). too, classifies the German particle schon as a monadic illocutionary connective, which forms a rhetorical question out of another one.

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K. Robering 1 Model theory and meaning 519

‘unforced’ sentences, i.e. of the sentences which are neutral to distinctions of force; such expressions are often called ‘sentence radicals’. Let furthermore,

Y , . . . , f” be the categories of full, ‘forced’ sentences. Then f’is (1 < j < m) are the categories of the illocutionary force indicating devices, used for the syntactic marking of special forces. Of course, model-theoretic work begins just now. What kinds of entities correspond to expressions of kindf’ (1 < j < m)? What is the appropriate generalization of the notion of consequence? What kinds of functions are to be used to interpret expressions of categoryf’ls, etc.?16 The deviation from standard model-theoretic procedure needed to take account of illocutionary forces is not very dramatic if force and content are separable aspects of meaning.

The severe distinction between force and content, however, is not without problems of its own. Is it really true that, for example, the assertive-interroga- tive-distinction is just a matter of communicative functions of utterances rather than of content? Suppose that there are questions as genuine semantic units besides propositions. An interrogative sentence may be syntactically embedded in a declarative one (indirect questions). If the interrogative sen- tence in this construction has the same meaning - namely, a certain question - as it has as an independent clause, then the propositional content of the entire construction does depend on a non-proposition (if a form of compositionality is still assumed). If this,17 or something like this, is true, then there are no good reasons for the above bipartition of the theory of meaning into a subtheory of content and a subtheory of force (Belnap 1990).

Thus, there may indeed be good arguments for rethinking the distinction between force and content as well as the boundary between semantics and pragmatics. But does this mean that we have to abandon the model-theoretic paradigm? Not at all - I think. What is needed in any case (i.e., whatever will turn out to be the correct solution to the force-content-problem) is a richer theory of context and context change. But such a theory is compatible with the model-theoretic procedure, as should be evident from the work of Kaplan, Bennett, and others on indexicals, and from Stalnaker’s (1978) approach to formal pragmatics. Suppose that we are working with models which assign quasi-meanings to expressions relative to contexts of a set C. Take as an example a proposition with commissive force (e.g. ‘I promise to submit my paper before the deadline’). The point of a corresponding utterance would (roughly) be to change the class of those propositions which the speaker is committed to make true by his future actions.18 In order to take account of

I6 Cf. Zaefferer (1984: 15-26) for some possible answers to these questions. I7 Another, less well-known problem for the severe distinction between force and content besides that of (embedded) questions is the ‘paradox of free-choice permission’ (Kamp 1973,

1978). I8 Descriptions like this are incompatible with the set-theoretic construction of propositions (or quasi-propositions in models) out of possible worlds, times, and truth-values. A function (from

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520 K. Robering ! Model rheory and meaning

this, the contexts c of C could be enriched by a context-coordinate cKcs,, which is the set of all propositions that the speaker cs of context c is committed to make true at a time t’ later than the time cT of the context. An utterance in context c of a sentence in the commissive mood expressing proposition P would thus result in the addition of P to the context coordinate cKcs). Of course, the single coordinate CKcs, would not suffice if we consider the whole range of forces. A matrix (S,, J of “commitment-slates” (Hamblin 1971: 136) is to be introduced, which fixes for each participant t of a context (speaker, hearers, bystanders, etc.) which attitude a t should bear on which propositions (or other kinds of content, e.g. propositional functions). Correspondingly, addition of a proposition to a certain commitment-slate is only one possibility of changing such a slate and thereby the context. Other sorts of operations are possible (negating an item of a slate, wiping it out, making a minimal revision of a slate in order to account for new data, etc.). Now, B&C will certainly respond that just this is the interactionist’s envisaged revision of model-theoretic categorial grammar. An utterance (according to their view) must contain devices which identify, ‘within the overall webs of knowledge relevancies (function indications) that collectively constitute the situation convention, that spot (or those spots) that are to be the focal object(s) of operative transformations’, which are effected by the utterance. This seems to be nothing else than the identification of the relevant-commitment slate. Secondly, an utterance must employ a device to identify ‘the transformation to be performed on that (those) focal point(s)‘. This, again, seems to be nothing else than the identification of the operation to be performed on the commitment-slate, which has been determined in the first step. If this interpretation of B&C’s proposal is correct, then the interactionist suggests something which semanticists working in the model-theoretic frame- work have already begun to do.

References

Bach, Emmon. 1989. Informal lectures on formal semantics. Albany, NY: State University of New York Press.

Barwise, Jon, 1989. On the model theory of common knowledge. In : Jon Barwise, The situation in

logic, 201-220. Stanford, CA: Center for the Study of Language and Information.

Barwise, Jon and John Perry, 1983. Situations and attitudes. Cambridge, MA: MIT Press. Belnap, Jr., Nuel D., 1990. Declaratives are not enough. Philosophical studies 59: l-30.

Bennett, Jonathan, 1976. Linguistic behaviour. Cambridge: Cambridge University Press. van Benthem, Johan, 1985. A manual of intensional logic. Stanford, CA: Center for the Study of

Language and Information. [2nd ed. 1988.1

world-time-pairs to truth-values) is individuated by what values it takes at what arguments. If a

function assigns a special truth-value to a world-time-pair, it does so no matter how the inhabitants of that world act and could not do otherwise. Pragmatics ~ B&C are completely right

here - needs some other conception of propositions than the possible worlds construction.

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