Truth-theoretic Semantics and Its Limits

Kirk LudwigPhilosophy Department

Indiana University

1. Introduction

My aim in this paper is to render an account of the goals and methods of truth-theoretic semantics and to discuss its limitations. Let me say at the outset that although the project I describe is, I believe, in conformity with the main idea that Davidson had, my aim in this paper is not Davidson exegesis. I want to get on the table a certain approach to providing a compositional semantics for natural languages which I think does about as much along the relevant lines as any theory could, without getting bogged down in exegetical questions.

I want to begin by distinguishing at least two different projects in the theory of meaning. The first I will call giving a theory of meaning, by which I mean a general account of the nature of linguistic meaning, what it is for words and sentences to be meaningful, what it is for particular words and expressions to mean what they do, and how this is bound up with our use of them. The second project is to give what I will call a meaning theory, as opposed to a theory of meaning. A meaning theory, as I will be using the expression, is a theory for a particular language which admits of a division into a (finite number of) semantical primitives and (a possible infinity of) complex expressions, which can be stated in a finite form and which in a sense to be made clear enables us to specify for each of the sentences of the language what it means. As a partial, further specification of what I have in mind by this, it will be a requirement on a meaning theory, in the sense I have in mind, that knowledge of it put one in a position to understand any potential utterance of a sentence of the language on the basis of understanding the contained semantical primitives and their manner of combination in the sentence. We can call this a compositional meaning theory.

The two projects are not unconnected. There is now a long tradition in the philosophy of language of trying to shed light on the theory of meaning by way of reflection on how to construct and to confirm meaning theories for particular languages. I will come back to this later.

2. Meaning theories.

The most straightforward way to provide a meaning theory would seem to be to provide an axiom for each primitive expression in a language L, which (in some straightforward sense) gives its meaning, from the totality of which one could derive formally a specification of the meaning of any sentence of the language relative to a context of utterance—a specification, moreover, grasp of which enables us to understand the sentence, while the mode of derivation enables us to understand how our understanding of the sentence rests on our understanding of its significant parts and their mode of combination.

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A target theorem (for a declarative sentence) could take the following form:

(M) For any speaker s, and time t, means in L, taken relative to s at t, that p.

We allow of course ‘p’ to be replaced by an open sentence containing as free variables ‘s’ and ‘t’.

So far, so good. Now we have divergence between at least two different approaches.

First, a natural thought is to take ‘that p’, when ‘p’ is replaced by a closed sentence, to be a referring term, and see the goal as being to formulate a theory that enables us to associate with each sentence of L and each context of utterance a referring term of the form ‘that p’ so as to enable us to understand the sentence as uttered in that context. Now let us give these referents a uniform name: propositions. Here this means no more than whatever referents will serve the job specified.

We want to be able to derive M-theorems for each sentence from axioms attaching to its primitive components. This amounts to associating a referring term with sentence which is derived from axioms governing its parts. Thus, it is natural to take the axioms governing its primitive components to be relating its parts to objects in such a way that we are in a position to understand those parts, and proposition to be a structured complex of those parts, the structure paralleling in some way the mode of combination of the primitive expression in the object language sentence. In this way we make provision for the use of the power of quantificational logical in deriving theorems from axioms.

Without being concerned for the moment with how the technical details are to be worked out, the question arises what role the assignment of objects to expressions and sentences in the language is doing in meeting the primary goal of the meaning theory, as specified above.

The answer is that associating entities with every expression in the language is, first, not necessary in order to provide a meaning theory, in the relevant sense, for a language, and, second, not sufficient.

I will take up first the claim that it is not necessary, by introducing the second approach, which is to formulate a meaning theory for a language L (focusing for the moment on its declarative sentences) in terms of a certain body of knowledge we can have about a truth theory for the language (I will use ‘truth theory’ in a way parallel to my use of ‘meaning theory’, and ‘theory of truth’ in a manner parallel to my use of ‘theory of meaning’). This will put us into a position to see why the association of entities with every expression in a language in the manner envisioned above is not sufficient to meet the desiderata on an adequate meaning theory for a language.

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3. A simple model in a recursive reference theory

But how is it possible exactly? I begin with a simple model borrowed from the first pages “Truth and Meaning.”

Davidson begins that essay by noting that on the face of it assigning entities to expressions which we call their meanings does not help us understand how to understand complex expressions on the basis of their significant parts. Give just (1) and (2), it looks as if the interpretation in (3) is open to us. If we add that concatenation means instantiation (4), we are still no better off.

(1) ‘Theatetus’ means Theatetus.(2) ‘sits’ means the property of sitting. (3) Theatetus sits = Theatetus the property of sitting.(4) Concatenation means the relation of instantiating.(5) Theatetus sits = Theatetus the relation of instantiating the property of sitting.

The moral is that we need more than assignments of entities to expressions, we need a rule that tells us how to go from those to what a complex expression, in a different semantic category, such as a sentence, means.

As an illustration, Davidson introduces a reference theory for a fragment of a language. I’ll vary the example a bit with an eye to bringing out some further lessons.

Consider an object language fragment of French consisting of singular terms ‘Marie’, ‘Jean’, and any concatenation of ‘La mère de’ with a singular term. Call this infinite fragment ‘L’. It is clear that merely assigning entities, say, referents, to each of these doesn’t tell us what the referent is of every expression. Suppose then we take up Frege’s suggestion and assign a function to ‘La mère de’, and give a rule, say that

[r1] the referent of the concatenation of ‘La mère de’ with a referring term t is the value of the function ‘La mere de’ refers to given the referent of t as its argument.

This doesn’t help either, because while it does correctly give the referent, it doesn’t do it in a way that makes it scrutable to us. What we need to do is, obviously, to say what sort of mapping the function effects, and the way to do that is to say that it is a function that maps persons onto their mothers. Then we can restate our rule:

[r2] the referent of the concatenation of ‘La mère de’ with a referring term t is the value of the function that maps people onto their mothers given the referent of t as its argument, i.e., the mother of the referent of t.

Here the contrast between [r1] and [r2] shows that the assignment of the function is doing no work at all. We might as well drop out the middleman and say:

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[r3] the referent of the concatenation of ‘La mère de’ with a referring term t is the mother of the referent of t.

There are three points to make about this. The first is that in explaining the function of a complex expression on the basis of its parts one must give a rule. The second is that simply correctly giving the referent of every expression in the language doesn’t ipso facto tell us, in the sense in which we are interested in it, what the referent of every expression in the language is. The third is that the thing we want can be attained without assigning an entity to every expression in our language.

Now I want to make a further use of this simple theory by showing how we can use it to give a meaning theory, and not just a reference theory, for L. First, let me state the theory.

[REF]1. ‘Marie’ refers in L to Mary2. ‘Jean’ refers in L to John 3. For any singular term t in L, ‘La mère de’ + t refers in L to the mother of what t refers

to in L.

The rules of inference are as follows:

(I) From (3), any instance may be inferred.(II) From any sentences of the form ‘t refers in L to y’ and any sentence of the form

S(what t refers to in L), S(y) may be inferred.

A canonical reference theorem of [REF] is any theorem derived from (1)-(3) using (I) and (II) which is an instance of ‘t refers to y,’ in which ‘y’ is replaced by a meta-language referring term that does not include ‘refers to’. Such a proof we will call a canonical proof. Next I state an uncontroversial criterion of adequacy for the reference theory, namely, what I will call Convention R.

Convention R: An adequate reference theory for L entails every sentence of the form ‘t refers to x’, where ‘t’ is replaced by a structural description of a singular term of L and ‘x’ is replaced by a term in the meta-language that translates it.

What knowledge about [REF] would enable us to know that it meets Convention R? Let us stipulate that in each base axiom the term used on the right of ‘refers to’ translates the one mentioned on the left, and in (3) ‘the mother of’ in the meta-language translates ‘La mère de’ in L. Given this, each canonical reference theorem the meta-language expression used on the right of ‘refers to’ translates the expression mentioned on the left. I will say that any theory of reference which satisfies this requirement on its axioms satisfies Convention A. That [REF] satisfies Convention A, given rules (I)-(II), suffices for it to satisfy Convention R.

The step to seeing how to transform this into a meaning theory is accomplished by restating Convention R.

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Convention R: An adequate reference theory for L entails every sentence of the form ‘t refers to x’ for which the corresponding instance of ‘t means x’ is true.

At this point, given that [REF] satisfies Convention A, and, hence, Convention R, we can add another valid rule of inference:

(III) From a canonical reference theorem of the form ‘t refers in L to y’ infer ‘t means in L y’.

We will call any theorem so derived a canonical meaning theorem.

What this shows is that we can use [REF] to arrive at a specification of the meaning of each referring expression in the language. This does not make [REF] a meaning theory for L, however. It is just a reference theory. It doesn’t state anything itself about what expressions in the object language mean, as opposed to what they refer to. But if it is the right sort of theory, and we know that it is, and understand it, then we can use it to specify what each referring term means.

Adjusting for the shift from a whole language to a fragment containing only referring terms, we can say that a meaning theory for such a fragment should be a body of knowledge that puts one in a position to understand any referring term of the language on the basis of understanding the contained semantical primitives and their manner of combination in the term. With this in mind, we will identify the meaning theory for L with the body of knowledge we need to have about [REF] that puts us in a position to use it for this purpose.

In particular, if we know [K-REF]

(i) the axioms of [REF], as stated in (1)-(3), (ii) what each axiom of [REF] states and that each states what it does,(iii) in base axioms the term used on the right of ‘refers in L to’ translates the one

mentioned on the left, and in (3) ‘the mother of’ in the meta-language translates ‘La mère de’ in L (that is, that [REF] satisfies Convention A),

(iv) rules of inference (I) and (II),(v) what a canonical reference theorem is, and(vi) rule of inference (III).

then, we are in a position to know for any expression of L what it means in this sense: we are thereby in a position to know for an arbitrary singular term t of L its corresponding canonical meaning theorem. In addition, because we know that [REF] satisfies Convention A, and because a canonical proof draws only on the content of the axioms, we can see in the proof of a canonical reference theorem how each expression contributes to fixing its referent in virtue of what it means and its mode of combination with other terms, and, hence, how our understanding of the expression rests on understanding its significant parts and mode of combination. Thus, we can treat [K-REF] as a meaning theory for L.

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4. Truth-theoretic semantics

By now, it is clear what the strategy is. Out of an extensional theory for a language, we can squeeze the elixir of meaning, if it has certain properties, and we know that it does. Let us then see how the extension to the whole of a language is effected. For illustration, I introduce a simple axiomatic truth theory for a language L without quantifiers or context sensitivity, and whose sentences are all declarative sentences.

Truth theory [TRU]

1. ‘Claudine’ refers to Claudine2. ‘Robert’ refers to Robert3. For any name N, N + ‘dort’ is true in L iff what N refers to is sleeping.4. For any names N1, N2, N1 + ‘aime’ + N2 is true in L iff what N1 refers to loves what N2 refers to.5. For any sentence S, ‘Ce n'est pas le cas que’ + S is true in L iff it is not the case that S is true in L.6. For any sentences S1, S2, S1 + ‘et’+ S2 is true in L iff S1 is true in L and S2 is true in L.

‘N’ ranges over names and ‘S’, ‘S1’ and ‘S2’ over sentences of the object language.

As rules of inference, we allow:

Universal Instantiation (UI): from a universally quantified sentence any instance may be inferred.Substitution (S): from any sentences of the form ‘t refers to y’ and any sentence of the form S(what t refers to in L), S(y) may be inferred. Replacement (R): for any sentences x, y, S(x), S(y) may be inferred from x + ‘iff’ + y and S(x).

Our criterion of adequacy for the truth theory is Tarski’s Convention T, which requires that

The truth theory entails all instances of the schema (T)

(T) is true in L iff p

in which is replaced by a structural description of an object language sentence as composed out of its significant parts and ‘p’ is replaced by a metalanguage sentence that translates it.

A canonical truth theorem of [TRU] is any theorem derived from the axioms using only (UI), (S), and (R) whose last line is of the form (T) and in which no semantic vocabulary of the metalanguage remains (i.e., ‘is true in L’). We call such a proof a canonical proof. It is clear that a canonical proof draws only on the content of the axioms in proving a canonical theorem.

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What knowledge about [TRU] would enable us to know that it meets Convention T? We stipulate that

(i) in each reference axiom the name used on the right of ‘refers to’ translates the name mentioned on the left

(ii) in each predicate axiom the predicate used in the meta-language in giving truth conditions for the object language sentence translates the object language predicate.

(iii) in each recursive axiom the logical connective used in the meta-language to give the truth conditions for the object language sentence translates the logical connective in the object language

We will say that if the theory meets this condition, it meets Convention A. We will say that a truth theory that meets Convention A is interpretive. It is clear that given the rules of inference and that [TRU] meets Convention A, for each sentence of the object language its canonical theorem is such that the metalanguage sentence on the right hand side translates the object language sentence for which it is used to give truth conditions, and, hence, that [TRU] satisfies Convention T.

The step to seeing how to transform this into a meaning theory is accomplished by restating Convention T.

The truth theory entails all instances of the schema (T)

(T) is true in L iff p(M) means in L that p

such that the corresponding instance of schema (M) is true.

At this point, given that [TRU] satisfies Convention A, and, hence, Convention T, we can add another valid rule of inference:

[Transference] From a canonical truth theorem of the form ‘ is true in L iff p’ infer means in L that p.

We will call any theorem so derived a canonical meaning theorem.

What this shows is that we can use [TRU] to arrive at a specification of the meaning of each sentence of the object language. This does not, however, make [TRU] a meaning theory for the object language. It is just a truth theory. It doesn’t state anything itself about what sentences in the object language mean, as opposed to what conditions under which they are true. But if it is the right sort of theory, and we know that it is, and understand it, then we can use it to specify what each sentence means.

We characterized a meaning theory as a body of knowledge that puts one in a position to understand any sentence of a language on the basis of understanding the contained semantical primitives and their manner of combination in the sentence. With this in mind,

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we will identify the meaning theory for the object language with what body of knowledge we need to have about [TRU] that puts us in a position to use it for this purpose.

In particular, if we know [K-TRU]

(i) What the axioms of [TRU] are, as stated in (1)-(6).(ii) What each axiom states and of each that it states what it does.(iii) That [TRU] is interpretive, i.e., meets Convention A.(ii) The rules of inference UI, S, R, and T.(v) What a canonical truth theorem is.

This puts us in a position to infer for each sentence of the object language a meta-language sentence that explicitly states what the object language sentence means, on the basis of a proof that traces out, at each step, the contribution of each object language expression to fixing the truth conditions of the sentence to which it contributes, on the basis of reference or truth conditions given using a term the same in meaning with it. We can thus see what the contribution is of each semantical primitive to the interpretive truth conditions of the sentences in which it occurs on the basis of what it means. Thus, we can treat the body of knowledge characterized by [K-TRU] as a meaning theory for L.

That completes the basic account of the goals and methods of truth-theoretic semantics. I close this section of the paper with four remarks.

First, it is clear that on this way of understanding the project of truth-theoretic semantics, the truth theory is not identified with a meaning theory. The meaning theory is rather a body of knowledge about the truth theory. It is not an objection to the project then that the truth theory itself cannot do the duty of a meaning theory.

Second, it is clear that the project is not to replace the investigation of meaning with the investigation of truth on the grounds that the concept of meaning is too confused for using a properly scientific investigation of language, but rather the pursuit of a traditional project by means of a certain kind of indirection.

Third, it is clear that it is not the goal of truth-theoretic semantics, as here conceived, to effect a kind of reduction of meaning to truth conditions, in any sense.

Fourth, the approach shows how to achieve the aims of a compositional meaning with no more ontological resources than are required for the theory of reference, and, hence, that the introduction of entities to assign to every significant expression of the language is not necessary in order to give a meaning theory for a language.

5. Vagueness and the semantic paradoxes

The formulation of the meaning theory as a body of knowledge about the truth theory provides a simple way of incorporating semantically incomplete expressions (those for

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which the rules of use do not fully specify its application conditions) or semantically incoherent expressions (those whose rules of use allow the derivation of contradictions). The meaning theory can be true even if not all the axioms of the truth theory are. Thus, while the Liar Sentence (L) gives us the theorem (TL), the meaning theory only asserts (ML).

(L) L is not true(TL) ‘L is not true’ is true iff L is not true L is true iff L is not true.(ML) ‘L is not true' means that L is not true.

(TL) is necessarily false, but (ML) is true. The same thing goes for vague predicates such as ‘bald’. If Boderline is a borderline case for ‘bald’, the truth theory gives us

(TB) ‘Borderline is bald’ is true iff Borderline is bald.(MB) ‘Borderline is bald’ means that Borderline is bad.

(TB) is neither true nor false, but (MB) is straightforwardly true.

6. Quantifiers and Context Sensitivity

I want to touch briefly on the complications introduced by quantifiers and context sensitivity.

(a) Quantifiers. The extension to a language including quantifiers is generally straightforward. We move from talking about truth conditions to satisfaction conditions. But one point about quantifier clauses is worth drawing attention to. Using functions from variables to objects as satisfiers, we can give a standard clause for a universal quantifier as follows (where f is a v-variant of f iff f is like f except at most in what it assigns to v) :

(Q) For any function f, predicate P, variable v, f satisfies ‘(Every’ + v +’)’ + P iff every v-variant f of f is such that f satisfies P.

The device we use in the metalanguage to carry out the recursion is a restricted quantifier. So in stating what it is to meet Convention A for a (context insensitive) language with quantifiers, we can’t require that the metalanguage quantifier translate the object language quantifier. What we require instead is that the quantificational determiner used in the metalanguage translate the quantification determiner in the object language.1 For a

1 Alternatively, we might make use of a suggestion David Wiggins once made, to reformulate the quantifier clauses along the following lines:

(Q’) For any function f, predicate P, variable v, f satisfies ‘(Every’ + v +’)’ + P iff every y is such that, the v-variant f of f such that f(v) = y, is such that f satisfies P.

Then we can simply require that the metalanguage quantifier translate the object language quantifier. This extends to restricted quantifiers in the object language straight forwardly (where f is v/P1-variant of f iff f is like f except at most in what it assigns to v and f satisfies P1).

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restricted quantifier, we give the clause (where f is v/P1-variant of f iff f is like f except at most in what it assigns to v and f satisfies P1):

(QR) For any function f, predicate P1, P2, variable v, f satisfies ‘(Every’ + v + ‘:’ + P1 + ‘)’ + P2 iff every v/P1-variant f of f such that f(v) = y, is such that f satisfies P2.

Again, we require that the main quantification determiner in the metalanguage translate the quantificational determiner in the object language.

(b) Context Sensitivity. Context sensitivity requires a modification of the form of the theory and also of Convention T and Convention A. For illustration, consider the following two axioms:

A1. For any speaker s, time t, for any x, ‘je’ refers to x relative to s at t iff x = s.A2. For any speaker s, time t, name N, N + ‘dort’ is true relative to s at t in L iff what N refers to is sleeping at t.

To restate Convention T for a context sensitive language, we use our second formulation with a relativized truth and relativized meaning predicate:

The truth theory must entail all instances of the schema (T)

(T) For any speaker s, time t, is true relative to s at t in L iff p(M) For any speaker s, time t, means relative to s at t in L that p

such that the corresponding instance of schema (M) is true.

Examples of each are:

For any speaker s, time t, ‘Je dort’ is true relative to s at t in L iff s is sleeping at t.For any speaker s, time t, ‘Je dort’ means relative to s at t in L that s is sleeping at t.

Restating Convention A is more complicated, so I won’t go into it in detail, but the basic idea is to require for referring terms that are context sensitive that the reference clause give use the meaning rule for the expression in determining its referent relative to context, that is, the rule that expresses competence in the use of the term. For sensitive predicates, we can introduce a requirement similar to the one expressed in our extension of Convention, where we add an extra argument place to the meaning predicate for the satisfier.

One important point to note about this is that once we have moved to a context sensitive language, the temptation to say that the work of the truth theory in the sort of meaning

(QR) For any function f, predicate P1, P2, variable v, f satisfies ‘(Every’ + v + ‘:’ + P1 + ‘)’ + P2 iff every y is such that, the v/P1-variant f of f such that f(v) = y, is such that f satisfies P2.

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theory sketched could be done by a translation theory disappears, because it clear that the relation between the object language sentence and the metalanguage sentence that provides relativized truth conditions is not that of translation.

5. Non-declaratives

A traditional difficulty for the program of truth-theoretic semantics is that there are many sentences of natural languages which are not true or false, such as imperatives and interrogatives. One need not, however, go to the heroic effort of trying to show that all sentences of the language admit of truth evaluability in order to extend the program to sentential moods besides the indicative. We just need to recognize a general notion of linguistic fulfillment condition for sentences of which truth is one, and a conceptually central, variety. Fulfilment conditions for non-delcaratives (exclamatives, like ‘Ouch’ or ‘Congratulations’ aside) can be given recursively in terms of truth conditions. We say, for example,

For all atomic , is fulfilled(s,t,u) iff if is declarative, then is true(s,t);if is an imperative, then is obeyed(s,t,u);if is an interrogative, then is answered(s,t,u).

I here introduce an additional relativization to a speech act performed using --this turns out to be important also in the general treatment of demonstratives. And I understand the sentence to be universally quantified with respect to ‘s’, ‘t’, and ‘u’. Recursive clauses are given using the fulfillment predicate, and the clause for atomic sentences (or formulas in the general case) invoked after we’ve burrowed down to the atomic sentences. This allows us to handle mixed-mood sentence such as ‘If you go to the store, buy some milk’ straightforwardly. We then give obedience and response conditions for imperatives and interrogatives (both subsumed under the heading ‘compliance conditions) recursively in terms of truth. To illustrate, for imperatives would use a clause of the following form:

For any imperative , is obeyed(s,t,u) iff A(S,t,u) makes it the case that core() is true(S,t,u) with the intention of fulfilling u.

Here core(x) is function from the imperative to its declarative core (which may be an open sentence in certain cases) . For example, core(‘Put on your hat’) = ‘You will put on your hat’. Since there are different kinds of interrogatives (yes-no, why, how, wh-) whose forms call for different kinds of responses, the clause for interrogatives would have a number of subclauses. I illustrate the clause for wh-interrogatives.

For all wh-interrogatives , is answered(s,t,u) iff A(s,t,u) makes it the case that [there is a : is a completion of core()]([you will say ] is true(s,t,u) with the intention of fulfilling u).

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A(s,t,u) is a function from speaker, time and speech act to an intended audience. (We could easily require in addition that the respondent give a correct response, or, alternatively, a response which he believes to be correct.)

We generalize Convention T, once again:

A fulfillment theory T is adequate if it is formally correct and entails all instances of

(F) is fulfilled(s,t,u) iff p

in which is replaced by a structural description of an object language sentence such that (1) if is a declarative sentence, the corresponding M-sentence is true, and (2) if is an interrogative or imperative, the corresponding R-sentence is true.

The R-sentence for ‘Put on your hat’ is

‘Put on your hat’ directs(s,t,u) that A(s,t,u) puts on his hat at t.

The R-sentence for ‘What time is it?’ is

‘What time is it?’ directs(s,t,u) that A(s,t,u) is to make it the case that he utter a completion of core(‘What time is it?’).

And so on.

The extension to a full theory with quantification and satisfaction is straightforward. We introduce three satisfaction relations, one for declaratives (satisfies), one for imperatives (i-satisfies), and one for interrogatives (q-satisfies), and define the latter two recursively in terms of the first. The form of the quantifier clauses does not change at all. This allows us to handle such sentences as ‘Invest every penny you earn’ ([For every x: x is a penny and you earn x](invest x)), in which the quantifier ‘every penny’ takes wide scope over the imperative mood marker.

Thus, truth-theoretic semantics can be extended in a straightforward way to non-declaratives which admit of a bivalent evaluation by an extension of the basic idea of the approach.

This discussion should further highlight the distance between (now) fulfillment-theoretic semantics and a recursive translation theory, which does not give you any articulation of the conditions under which an utterance of an imperative or interrogative is fulfilled.

7. Is the relevant body of knowledge really adequate?

What I have hoped to do up to this point is

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(1) to explain how truth-theoretic semantics is to be understood as a pursuit of a perfectly traditional project, that of constructing meaning theories (in the sense articulated at the outset) for particular languages;(2) to show that it is mistake to suppose its work can be done by a translation theory;(3) to show that it can very plausibly respond to the problem of vague predicates and the semantic paradoxes precisely because it does not identify the meaning theory with the truth theory but rather with a body of information about it; and(4) to that it can be extended in a straightforward way to non-declaratives without falsifying how these sentences actually function in the language.

Now I want to return take up two important questions about it. The first, which I take up in this section, is

(i) whether the body of knowledge I have claimed is sufficient really is so,

and I’ll say why a question arises about this. The second, which I take up in the next section, is

(ii) whether, even if it is sufficient in some straightforward sense, and granting that it is not equivalent in any straightforward sense to a recursive translation theory, there might not yet be a sense in which the illumination of what object language expressions and sentences mean rests in part essentially not upon propositional knowledge but upon non-propositional understanding of the metalanguage, that is, antecedent competence in expressions known to be systematically related in meaning to expressions in the object language.

With respect to the first of these questions, this issue, put to me by Miguel Hoeltje, is whether what is stated in [K-TRU] (repeated here) suffices to understand the truth theory.

[K-TRU]

(i) What the axioms of [TRU] are, as stated in (1)-(6).(ii) What each axiom states and of each that it states what it does.(iii) That [TRU] is interpretive, i.e., meets Convention A.(iv) The rules of inference UI, S, R, and T.(v) What a canonical truth theorem is.

The meaning theory involves three levels of languages. There is the object language. There is the language of the truth theory, the metalanguage, and then there is the language in which the meaning theory is stated, the meta-meta-language. The object was in part to state something sufficient to understand the metalanguage. This is important because (a) both the canonical truth theorems and canonical meaning theorems are in the language of the truth theory and (b) the canonical proofs, which are to illuminate the compositional structure of object language sentences, are stated in the meta-metalanguage.

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The question that Hoeltje raises is whether (ii) in [K-TRU], which is intended to turn the trick, really does so. Let’s take an example of the sort of knowledge that gives us, and I’ll vary the language of the truth theory from English to Serbian (in the Cyrillic alphabet) to bring out the point.

‘ , За свако име н н⁀“dort” је истинито у л акко оно нашта н реферира’ means that for any name N, N + ‘dort’ is true in L iff what N refers to is спава

sleeping.

The worry is this: how are we supposed to know anything about the relevant syntactic/semantic structure of axiom, and hence of the metalanguage sentence, from basically getting a statement of its meaning as a whole? And if we don’t know that, how can this give us knowledge of the metalanguage, i.e., the language of the truth theory, which we grant is essential to using it in the way intended.

As a first remark about whether what is stated in [K-TRU] is sufficient, it is worth noting that if we have an explicit statement of the form

A means that p

for each of the axioms of the truth theory, then we can state a truth theory in the meta-metalanguage for the object language. We would of course also have to introduce inference rules corresponding to those for the metalanguage and define a canonical truth theorem and canonical meaning theorem for the meta-metalanguage. This is straightforward, but isn’t yet included in what we would know in knowing the axioms or in knowing what else is stated in [K-TRU]. We could of course state the rules mentioned in (iv) in the meta-metalanguage. But they would still be rules for the truth theory in the metalanguage and not for the truth theory in the meta-metalanguage. What we could add is that we know that if (UI) is a valid rule of inference for the metalanguage then (UI*) (the corresponding rule for the meta-metalanguage) is a valid rule of inference for the meta-metalanguage, and so on, and that if T is a canonical truth theorem for the truth theory stated in the metalanguage, then the theorem of the truth theory in the meta-metalanguage which is derived by invoking corresponding rules on corresponding axioms in a corresponding order is a canonical truth theorem in the truth theory statement in the meta-metalanguage, and so on. This looks like it would do the trick. So, I want to say just knowledge stated in (ii) really does get us a long way, so far that a bit more clearly gets us the rest of the way.

However, this isn’t how I envisioned it going. So let me return to the original idea, which I think has not been made explicit enough. First of all, we run proofs on the truth theory as a syntactic object. So we should state the rules in the meta-metalanguage, something not made explicit in the first run through. Furthermore, the rules are stated in terms of syntactic/semantic categories that apply to metalanguage terms, so it presupposes a recursive syntax for the metalanguage which sorts terms into the categories necessary the description of the structure of sentences in metalanguage in terms of the types of semantically primitive expressions in the language and how sentences are systematically built up out of them. Armed with this information, we will be able to parse the structure of

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the axioms of the truth theory, and given the knowledge stated in (ii), see how to interpret connectives, determiners, predicates and so on. In fact, we have omitted to include in the statement of the truth theory a statement of the corresponding information about the object language. That should be included in the statement of the truth theory, and bundled into (ii). So what we need to make explicit that we had not so far is that we also know a lot about the syntactic structure of the metalanguage, and that the truth theory should itself include a recursive syntax for the object language.

For example, if we know that in

, За свако име н н⁀“dort” је истинито у л акко оно нашта н реферираспава

‘ ’ is a variable, that ‘ ’ is a logical connective, that ‘ ’ is a restricted н акко За свако имеquantifier, that enclosing an expression in '“' and '”' forms a name, that for metalinguistic variables v and u, v + ‘⁀’ + u is a restricted quantifier, and how the sentence is constructed out of its parts, which will determine scope relations, etc., we’re in a position to interpret basically each word in it given what it means as a whole.

I conclude that this challenge can be met, and I take [K-TRU] to be understood to be so amended.

8. The limits of truth-theoretic semantics

Let me return to the second of the questions I mentioned, namely,

(ii) whether there might not yet be a sense in which on this approach the illumination of what object language expressions and sentences mean rests in part essentially upon not propositional knowledge but upon non-propositional understanding of the metalanguage and ultimately of the meta-metalanguage, that is, antecedent competence in expressions known to be systematically related in meaning to expressions in the object language.

I think the answer to this question is ‘yes’, and that it helps to bring out something of the limits of truth-theoretic semantics, as it has been explicated here.

Let’s grant knowledge of the language of the truth theory and think about how what we know about it helps us to use it to interpret object language expressions and to gain insight into the logico-semantic structure of object language sentences. First of all, Convention A enables us to read off from axioms what object language expressions contribute in context to the meaning of a sentence. But the knowledge it gives us of both the content of the object language expression and its semantic role is clearly relative to our prior grasp on the corresponding content of the metalanguage expression and its semantic role. Let’s take five sorts of axioms as examples.

A1. For any speaker s, time t, for any x, ‘je’ refers(s,t) to x iff x = s.

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A2. For any function f, speaker s, time t, referring term N, f satisfies(s,t) N + ‘dort’ in L iff what N refers to is sleeping at t.A3. For any function f, speaker s, time t, variable v, f satisfies(s,t) v + ‘dort’ in L iff f(v) is sleeping at t.A4. For any function f, sentences S1, S2, f satisfies(s,t) S1 + ‘et’+ S2 iff f satisfies S1 in L and f satisfies S2 in L.A5. For any function f, predicate P, variable v, f satisfies ‘(Chaque’ + v +’)’ + P iff every v-variant f of f is such that f satisfies P.

A1 is the most informative of these. Antecedent understanding of ‘refers’ tells us what the category is, and (given Conv A) we know that the clause expresses a rule of meaning for determining the referent of the expression without using an expression the same in meaning with it. But in the case of A2-A5, it is clear that understanding the content and semantic role of the object language expressions depends on our antecedent understanding of the metalanguage sentence used to give satisfaction conditions together with knowledge that the theory meets Convention A. Similarly, our understanding of the contribution of each expression to fixing interpretive truth conditions of a sentence on the basis of its meaning, as exhibited in how its axiom enters into a canonical proof of a canonical theorem for it, likewise relies on our understanding of the metalanguage expression used to give satisfaction conditions for it and our knowledge that the theory satisfies Convention A.

There is a sense, then, in which the theory shows something about the object language, in light of our understanding of the metalanguage and knowledge that the theory meets Convention A, that the theory does not say. This, I think, turns out to be inescapable in giving a meaning theory of the kind we have set as our goal here. I’ll say more in support of this in a moment. This is not to say that the theory together with knowledge that it meets Conv A is not informative, of course, but only to say that the mode by which we gain information about the object language rests on prior understanding of metalanguage terms and what information Conv A gives us about their relation to object language terms for which reference and satisfaction conditions are being given.

But if all of this is right, if the illumination the theory provides does rest essentially in part on antecedent understanding of a language, then in what sense have we stated something in [K-TRU] that puts us in a position to understand any potential utterance of an object language sentence and to understand its compositional semantic structure? Here of course we have in mind what we mean by propositional knowledge, where we think of it as something that is essentially independent of knowledge of any language. But in a perfectly ordinary sense of body of knowledge, [K-TRU] does state a body of knowledge that suffices. Perhaps the trouble here has to do with the particular notion of propositional knowledge that underlies the worry.

I want to take this up in connection with the final output of the theory, which consists in explicit statements about what object language expressions mean relative to contextual parameters. For I think this is the crux of the issue, how these sentences convey to us what an object language sentence means relative to a context. Consider (M) for example.

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(M) For any speaker s, time t, ‘Je dort’ means(s,t) in L that s is sleeping at t.

Surely here we have got something that simply states what the sentence means. This at least does not rely on antecedent understanding of the metalanguage sentence and knowledge that it interprets the object language sentence. But in fact it does. We have not escaped reliance on antecedent understanding of the metalanguage.

It is easier to see this by instantiating (M) to a particular speaker and time, KL at T.

‘Je dort’ means(KL, T) in L that KL is sleeping at t.

You of course understand the sentence used in the complement. And given that you understand it and you know that

‘x means(s,t) in L that p’ is true in English iff x taken relative to s at t in L translates ‘p’ in English.

you are in a position to understand the object language sentence taken relative to KL and T. But suppose that you did not understand the complement sentence. Then it is clear that you would not understand ‘Je dort’ taken relative to KL and T.

Now, to this it might be said:

“Yes, but in that case you would not understand the proposition expressed by the sentence, and so of course we would not expect you to be in a position to understand the object language sentence. This does not show at all that what is stated is not sufficient. And it does not show that the illumination comes only in the manner you sketch, i.e., via understanding of the complement sentence and knowledge that it translates the object language sentence. Of course we have to understand the complement sentence to understand what the sentence states. But that is so for any claim, whether it is about meaning or anything else.”

9. Back to propositions

This brings me back to propositions finally as the referents of sentential complement clauses whatever they may be in such contexts. Let us take sentential complements of the form ‘that p’ to contribute a proposition to the second argument position in ‘s in L means y’ (for simplicity I drop relativization to context). What kind of singular term is ‘that p’? We seem to have three options.

1. it is a description (construed as a quantifier)2. it is a referring term that merely introduces an object into the proposition3. it is a term that refers via a Fregean mode of presentation

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My claim is this: no matter which of these options we take, it turns out that it is a true sentence of the form ‘x in L means y’ enables us to understand ‘x’ only if we can construct from ‘y’ a metalanguage sentence that we understand and understand to be the same in meaning as ‘x’.

Option 1. ‘that p’ is a description. What is the form of the description? The basic option is to take advantage of the fact that the sentence ‘p’ expresses the proposition, and to treat ‘that p’ as having the form ‘the proposition expressed by ‘p’ in English’. Alternatively, we could treat it as a context sensitive self-referential description: ‘the proposition the speaker of this token ‘p’ expresses with it’.

means in L the proposition expressed by ‘p’ in English means in L the proposition the speaker of this token ‘p’ expresses with it.

It is clear, however, that this would not put someone in a position to understand the object language sentence unless he understand the mentioned sentence or the mentioned sentence as used by the speaker.

Option 2. ‘that p’ contributes only its referent to what is said. We may still compatibly with this take the referent to be determined by a description. We could think of it functioning like Kaplan’s dthat(the F).

means in L dthat(the proposition expressed by ‘p’ in English)

But then what is said clearly doesn’t put one in a position to understand independently of understanding ‘p’. For the same thing could be said using a directly referring term. Let us name the proposition expressed ‘Bob’. We say the same thing then in saying:

means in L Bob

Option 3. The mysterious Fregean mode of presentation. Frege would have held that ‘p’ in the context following ‘means that’ has an indirect sense that is a mode of presentation of its customary sense, i.e., the proposition expressed by ‘p’. Of course, one mode of presentation might simply be ‘the proposition expressed by ‘p’ in English’. But it is clear already that this will not give us understanding of the object language sentence independently of understanding ‘p’ in English. What other mode of presentation might we appeal to? I venture to say that no one has any earthly idea. But whatever it is, we know that it has to meet a certain constraint, a constraint which I think cannot be met. The constraint is that the mode of presentation must be such that one cannot present to oneself the proposition via that mode of presentation without grasping it. For otherwise, could grasp the proposition expressed by ‘ means in L that p’ without understanding .

It is tolerably clear, I think, that as a matter of fact the way we are clued into the meaning of an object language sentence via an M-theorem for it is by way of our understanding the metalanguage sentence and the requirement that it be the same in meaning as, along the relevant dimension, relative to context, the object language sentence. And it seems clear

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that anything you put in the second argument place in that construction will not put you in a position to understand the object language sentence except insofar as it at least codes for such a sentence. The difficulty is that understanding a sentence is not at bottom a matter of standing in a relation to an abstract object and associating it with a sentence. It is a matter of knowing how to use it for certain purposes. To convey what a sentence means we can rely on antecedent understanding of terms appropriately related in meaning, or we can explain the purpose of the sentence in the kind of activity that is central to its linguistic meaning. The second of these, however, will not take the form of relating a sentence to any object.

Admittedly it is difficult to give an in principle argument against the possibility in the absence of a clearer target than Sinn as a mode of presentation of an object. So in part I intend this to be a challenge.

One could, of course, insist on a very abstract reading of mode of presentation, and allow that the mode of presentation of a proposition may simply consist in our understanding a sentence and knowing it is relevantly related to what another sentence means. But this would not be a vindication that the way these sentences work depends essentially on our understand the sentence in the complement in its language and that it interprets the sentence mentioned on the left.

This point extends to the various ways we have of trying to indicate the meanings of subsentential expressions by assigning them properties and relations and functions. To make meaning scrutable by assignment of an object to an expression, we must use an expression that at least codes for an antecedently understood expression which is understood to interpret the express whose meaning we are given. This is transparent in for example the claim that ‘rouge’ in French means the property of being red, or ‘aime’ in French means the relation of loving.

The conclusion to reach is the following:

1. The assignment of entities to expressions in pursuit of a meaning theory for a natural language is neither necessary nor sufficient. It is not necessary because the same goals can be achieved by the method of truth-theoretic semantics. It is not sufficient because assigning the correct entities along does not put us in a position to understand object language expressions. We must assign entities using terms that code for expressions we understand and understand to interpret object language expressions to which meaning entities are being assigned.

2. A theory of meaning designed to issue in theorems of the form ‘ means in L that p’ does its work inevitably in part by way of showing us something about what object language expressions mean by way of our antecedent understanding of metalanguage terms understood to interpret them. There is, thus, a limit to the depth of the illumination of meaning in the object language we can expect. To get greater illumination, we have to move to a different form of account. (This is obviously connected with the dispute between

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Davidson and Dummett over whether a modest theory of meaning is adequate to our philosophical purposes.)

10. The theory of meaning

Let me then conclude by returning to the question of the relation between the project of giving a meaning theory for a language and the project that I called giving a theory of meaning. It should be clear now that, and why, giving a meaning theory is not enough for giving a theory of meaning. It basically illuminates by reliance upon prior non-propositional grasp of meaning. Davidson, I think, was well aware of this, and hoped to bridge the gap between a meaning theory and a theory of meaning by an account of how a meaning theory could be confirmed for a speaker on the basis of evidence that did not presuppose anything about a speaker’s words or any detailed knowledge of his attitudes. This is a way of connecting up the structure articulated in a meaning theory with the use of words by speakers of the language for which it is theory. Something like this is clearly required. Davidson’s fundamental assumption about radical interpretation was that a correct meaning theory could be recovered ultimately from purely behavior evidence and what we can know about agents and speakers a priori. I won’t argue for it here, but I think this assumption is mistaken.

This still leaves us with the task of relating language, words and expressions, ultimately sentences, to the uses to which they are put for the purposes for which they are designed. What this requires is a theory of communicative institutions and the forms of collective agency that underlie them. If we cannot hope for a reduction to behavioral responses to the environment, then we must look to relate meaning to the attitudes people have in using words and expressions in communicative contexts. This will inevitably rest on an antecedent understanding of how we are able to represent the world in thought. But it holds out the hope of breaking out of the circle of semantic concepts, which the fixation on propositions as the central theoretical tool in the theory of meaning does not.

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