6.9—
Tarski's Semantics

Tarski's work on truth, and Davidson's claim that Tarski's technique can also yield an account of meaning, have garnered a great deal of attention, and have been met with sharply polarized reactions. A great deal

of the literature discussing Tarski's work has been devoted to the problem of deciding just what Tarski's theory does —what kind of theory it is, and what it is a theory of . It is thus probably wise to begin by presenting some of the essentials of Tarski's account. Tarski wrote three papers that are of central importance to his work on truth: "The Concept of Truth in Formalized Languages" (1956b), "The Establishment of Scientific Semantics" (1956c), and "The Semantic Conception of Truth and the Foundations of Semantics" (1944). The first of these papers begins with the following synopsis: "The present article is almost wholly devoted to a single problem—the definition of truth . The task is to construct—with reference to a given language—a materially adequate and formally correct definition of the term 'true sentence' " (Tarski 1956b: 152, italics in original).

Tarski's project is thus one of providing a "definition" of truth that is "materially adequate" and "formally correct." This terminology requires some comment. It is somewhat controversial what Tarski meant by 'definition' and what he achieved in this regard. For the term 'definition' has acquired a specialized usage in metamathematics that implies something at once weaker and stronger than some more ordinary uses of the word. If one constructs a set theoretic model M(D) of a mathematical domain D , then a concept C in D is said to be "defined by" the set-theoretic construction that corresponds to it in the model.^[8] Thus, for example, in the Principia Mathematica, numbers are said to be "defined" by sets; but it is quite controversial whether this result really has any consequences with respect to the nature of numbers. So a logician's expressed desire to provide a "definition" of truth may easily turn out to be merely a desire to provide a model-theoretic construction characterizing truth in a language. However, Tarski does say things that indicate that he may be interested in more than this. He says, for example, that it is his wish to provide a "definition" that corresponds as closely as possible to familiar uses of the word. And what he explicitly cites as the familiar use of the word is the "classical" conception of truth in which truth is understood informally to consist in correspondence to reality. This will have direct consequences for the conditions Tarski believes to be relevant to the "material adequacy" of a truth definition.

Before moving on to these conditions, however, it is important to note that Tarski does not aim to supply any general definition of truth-in-any-language-whatsoever. Rather, truth-definitions are relativized to languages. The stated reason for this is that the "same sentence" can appear in different languages, and may be true in one, false in another, and

meaningless in a third. Tarski is thus taking "sentences" or "expressions" to be defined in terms of concatenations of graphemes or phonemes, an assumption that arguably is not completely unproblematic. So the search for a "definition of truth" is really a search for the conditions that must be met by a definition of truth relative to any language L . One such condition is "material adequacy," by which Tarski means, informally, that the truth-theory for L, T(L) , should have the consequence that, for any sentence S in L, T(L) assigns S the value TRUE iff what is asserted by S is true. Tarski suggests that the constraint that truth theories have such biconditionals as consequences be formulated in terms of a schema, which he calls convention T :

(T) X is true, if and only if, p ,

where p is a schematic letter to be replaced by a sentence of L and X is a schematic letter to be replaced by a ("structure revealing") name of the sentence that replaces p . He writes that "we shall call a definition of truth 'adequate' if all these equivalences follow from it" (Tarski [1956b] 1985: 50). Tarski refers to this conception of truth as "the semantic conception of truth" (ibid., 51), the point being that truth is defined in terms of relationships between expressions and states of affairs in the world (hence a semantic relationship), rather than being defined syntactically in terms of derivability from formally specified axioms. (This point, often glossed over today, was perhaps the most significant feature of Tarski's approach in the climate in which it was first propounded.)

The issue of "formal correctness" is driven by several concerns Tarski raises with respect to classes of languages that are not subject to the kind of definition he desires. First, he believes that languages can be characterized in the desired fashion only if they are "exactly specified," in the sense that in "specifying the structure of a language we refer exclusively to the form of the expressions involved" (ibid., 52). This excludes languages that involve lexical ambiguity and elements that are dependent upon pragmatics or context, such as demonstratives and indexicals. Second, he points out that certain classes of languages—languages that he calls "semantically closed"—are inconsistent because they are prone to the generation of paradoxes such as the antinomy of the liar. Languages are said to be "semantically closed" if they contain the resources for naming expressions occurring within the language, for applying the term 'true' to sentences in the language, and for stating the truth conditions

of the language (ibid., 53). The concept of truth, Tarski claims, is not definable for semantically closed languages.

It is perhaps obvious that these observations lead to the conclusion that truth is not definable for natural languages, since these are lexically ambiguous, employ demonstratives and indexicals, and have resources for referring to their own elements and making truth-assertions about them. Tarski embraces this conclusion, though other writers have since attempted to treat these features in a way that avoids Tarski's negative result. More easily overlooked is the fact that the linguistic features that interest Tarski include things like axioms and theorems, which play a large role in logic and mathematics, and are strongly connected with the notion of truth in those domains, yet are notably absent (not to mention irrelevant to empirical truth) in natural languages. This would be highly problematic if Tarski's stated aim was to provide a general "definition" of truth, but is perhaps innocuous so long as one is carefully attentive to the fact that what he is about is providing a model-theoretic characterization of truth for those languages for which this might be done.^[9]

The definition of truth is constructed out of a more basic notion of satisfaction . Satisfaction is a relation that obtains between any objects and a special class of expressions called "sentential functions," which are expressions such as "X is white" or "X is greater than Y ." (Sentential functions are differentiated from sentences in that they may contain free variables.) Intuitively, an object O satisfies a sentential function F if replacing the variable in F with the name of O results in a true sentence. This, however, will not serve as a definition of "satisfaction" for Tarski's purposes, as his aim is to define "truth." And so he employs another strategy—that of "defining" satisfaction for a language L in an extensional fashion. In order to accurately represent Tarski here, I shall cite his own text:

To obtain a definition of satisfaction we have rather to apply again a recursive procedure. We indicate which objects satisfy the simplest sentential functions; and then we state the conditions under which given objects satisfy a compound function—assuming that we know which objects satisfy the simpler functions from which the compound one has been constructed. (Tarski [1956b] 1985: 56, emphasis added)

From this definition of satisfaction for sentential functions, one follows for sentences (functions in which there are no unbound variables). Sentences are either satisfied by all objects (in which case they are true) or

else they are satisfied by no objects (in which case they are false). This, indeed, provides a definition of truth: "Hence we arrive at a definition of truth and falsehood simply by saying that a sentence is true if it is satisfied by all objects, and false otherwise " (ibid.).

Here we have a general schema for talking about truth in a language L , given that L falls within the specified class of languages. It is a schema for truth-definitions rather than a general truth-definition because the two more basic semantic notions of naming (or, to employ Field's [1972] useful paraphrase "primitive denotation") and satisfaction, are defined for expressions only relative to a language.

Given Tarski's desire for the introduction of all semantic terms only by definition, it is important to be attentive to the way in which satisfaction and primitive denotation are treated in Tarski's articles. For the "definition" of 'satisfaction' for a language L consists merely in (a ) providing a mapping from simple functions to the sets of objects that satisfy them, and (b ) providing a recursive rule for producing such a mapping for complex functions, given the values of the simple functions. And similarly, one may assume that the "definition" that would be given for the relation of primitive denotation would simply be a mapping from a class of expressions to a set of objects. These are "definitions" in the mathematician's sense of exactly specifying the function performed in settheoretic terms. But they are surely not "definitions" in the sense of explaining what satisfaction or designation consist in . This has led to some criticisms of the scope of Tarski's accomplishment, some of which (Field 1972 and Blackburn 1984) I shall allude to in developing a more general analysis of the problems with the notion of pure semantics. These and other concerns cast some doubt upon whether Tarskian semantics in fact provides the "pure semantics" desired by the critic of the Semiotic Analysis.

6.9.1—
A Nonconventional Analysis?

First, it is by no means clear that the analysis presented by Tarski renders the semantics of languages essentially nonconventional. Tarski says that we "indicate which objects satisfy the simplest sentential functions" (Tarski [1956b] 1985: 56, emphasis added). But in the context in which he is speaking, this "indication" can be interpreted in either of two ways, both of which are plausibly interpreted in conventional terms . On the one hand, one might wish to supply a semantic analysis of an existing formal language (say, Hilbert's geometry). In this case, one is ap-

proaching an existing public language game that is conventionally established. The ability to "indicate" the objects that satisfy the sentential functions in such a language game by no means shows that the relationship of satisfaction is essentially nonconventional. Making the mapping from expressions to interpretations explicit in no way implies that the preexisting system is nonconventional. And indeed the way in which the mapping is indicated in the formal model is itself conventional.

Alternatively, one may be defining a new language game de novo, and hence stipulating its semantic assignments. Here there is no preexisting convention-laden public language game. But in doing this one is necessarily defining a convention for semantic interpretation. Doing so by no means shows there is an independent stratum of meaning or even satisfaction that obtains apart from the conventions established by the theorist. At best, if there is a preexisting set of markers, there are infinite numbers of mappings between that set and sets of objects. And mappings do, indeed, exist independent of mapping conventions. But a mapping, per se, is not a semantic relationship. Nor does the existence of mappings that are independent of conventions establish the existence of semantic relations that are independent of conventions. Semantic assignments are represented by mappings and involve mappings, but mappings are not themselves semantic.

6.9.2—
The Conventionality of the Markers

Tarski has also made an illicit move in assuming that "sentences" and "expressions" that constitute the domain of the mapping can be defined in terms of concatenations of graphemes or phonemes, and the pure semanticist would be wrong in concluding that this amounts to a nonconventional definition. There are at least three problems here. First, as argued above, markers and counters are conventional in character. Thus, while it may be right to say that the same physical patterns may get concatenated in more than one language, it does not follow that the same complex markers are employed, nor that identical strings of markers in two languages are the same sentence. Second, the marker kinds themselves are underdetermined by physical pattern and are essentially conventional. Third, if sentences are defined in terms specific to their mode of representation, it is not clear how one is to account for the fact that the same sentence can be both spoken and written, and can potentially be represented in other modalities (e.g., Morse code, ASCII coding, etc.) as well. As an idealization, Tarski's move is permissible within certain

bounds; as a real definition, it seems inadmissible. This seems to undercut the pure semanticist's claim that Tarski's semantics is free from conventional taint. Even if we agree that the mapping from expressions to objects is nonconventional, the overall language is still conventional because the domain of expressions is conventionally established.

6.9.3—
Field's Argument

In a justly famous article, Hartry Field (1972) undertakes an extensive examination of Tarski's theory of truth. Field argues that Tarski succeeded in reducing truth to what Field calls "primitive denotation," but failed to define primitive denotation in nonsemantic terms. And thus, in Field's view, the remaining project in semantics for naturalists such as himself is to provide a nonsemantic account of primitive denotation. The crux of Field's argument is that merely extensional characterization of semantic notions such as denotation or satisfaction, while adequate for model-theoretic purposes, does not constitute a genuine reduction of semantic terms, any more than we may produce a genuine reduction of the notion of valence that proceeded by saying

("E ) ("N ) (E has valence nºE is potassium and n is +1, or . . . or E is sulphur and n is -2). (Field 1972: 363)

There seem to be at least two problems with merely extensional characterizations, on Field's view. First, they do not reduce semantic properties to nonsemantic properties in the sense of "reduction" employed in the sciences and relevant to the incorporation of semantics within the project of physicalism. Second, they seem to license unfortunate would-be "reductions": "By similar standards of reduction, one might prove that witchcraft is compatible with physicalism, as long as witches cast only a finite number of spells: for then 'cast a spell' can be defined without use of any of the terms of witchcraft theory, merely by listing all the witch-and-victim pairs" (ibid., 369).

Field seems right in his claim that Tarski's extensionally based account of his primitive semantic properties fails to yield any robust account of their nature. What Field directly argues is that Tarski's characterizations do not yield a reduction of these properties in terms that demonstrate compatibility with physicalism, but we shall see below in Blackburn's criticisms that this point can be generalized beyond Field's physicalistic agenda as well.

6.9.4—
Blackburn's Argument

Simon Blackburn (1984) argues that Tarski in fact gives no definition of any semantic notions, but merely describes a "neutral core" that "connects together truth, reference, and satisfaction" but "gives us no theory of how to break into this circle; that is, of how to describe what it is about a population which makes it true that any of their words or sentences deserve such semantic descriptions" (Blackburn 1984: 270). Blackburn's chapter on Tarski and truth presents a number of insights that are not easily separated. But one important observation he makes is that the specific character of Tarski's characterizations of the semantic notions renders them ill suited to serve as definitions. In particular, there are two problematic features of these characterizations: their extensional character and their relativization to a language. First, in giving a list-description of, say, names in language L and their denotations, one does nothing to explain what the property is that is being characterized. A list-description tells you what objects are named by what terms, given that you know that the property characterized by the mapping is supposed to be naming in a particular language, but it tells you nothing about naming per se. One can make use of these lists only if one also knows that they are descriptions of how L -speakers use this set of expressions as names, and hence we have no real definition here (see ibid., 268-269). Second, the definition of, say, 'satisfies' for L1 is completely different from the definition of that same word (or a corresponding word) relative to L2 . The satisfaction relation is provided merely in terms of extensional characterization for particular languages . It is defined differently for each language individually, because there is a different mapping of expressions onto objects in each language, and there is no overarching notion of satisfaction apart from those relativized to particular languages. If satisfaction were really defined extensionally (indeed, even if it were fully accounted for in extensional terms), it would seem to be the case that there is no property or function called "satisfaction" common to L1 and L2 , but rather it would be more accurate to speak of separate notions of satisfaction-for-L1 and satisfaction-for-L2 . This, Blackburn observes, is a problem for Tarski's account. For although Tarski is surely right in relativizing truth to a language,

it does not follow that there is nothing in common to . . . truth as expressed in English sentences, and as expressed in those of any other language whatsoever. Reflection upon the application of an abstract semantic system to any actual population shows that there must be. (ibid., 270)

In other words, there is clearly something in common to notions such as truth or satisfaction across languages. But list-accounts for individual languages do not provide any indication of this common feature. Hence Tarski's analysis does not do an adequate job of "defining" the semantic properties.

I believe that this part of Blackburn's analysis is quite right. For our purposes, however, there is a certain aspect of Blackburn's approach that cannot be simply accepted without some justification. For when Blackburn says that Tarski does not tell us how to "break into the circle" of truth, reference, and satisfaction, he glosses this by saying that it gives us no theory "of how to describe what it is about a population which makes it true that any of their words or sentences deserve such semantic descriptions" (ibid., 270). Blackburn explicitly rejects the idea that one can separate a purely semantic account from a pragmatic account that ties a purely abstract language to the actual practices of a community (ibid., 269). This is, of course, very much in accord with what I wish to argue in this chapter. But by the same token, it is the very point which the fictional critic of this chapter wishes to contest. So the most we are really permitted to take from Blackburn here is the conclusion that Tarski's analysis does not provide a definition of the semantic terminology in nonsemantic terms (except perhaps in the model-theoretic sense of "definition"). What we are not licensed to conclude from Blackburn's arguments is the more robust thesis that the notions of satisfaction and primitive denotation presented by Tarski do not constitute notions that are legitimately semantical, yet do not have conventional elements.

At best, we might be able to make the following argument towards that conclusion on the basis of Blackburn's considerations. We might regard Tarski's "definitions" in one of the following two ways: (1) as attempts to give accounts of familiar semantical notions in nonsemantic terms, or (2) as stipulative definitions of how he is going to use those terms. If we interpret the definitions as stipulative in character, Blackburn's observations are enough to show that "denotation" and "satisfaction" thus defined are not really semantical notions at all, but merely model-theoretic counterparts of semantical notions. If we interpret Tarski in the first way, Blackburn's arguments show that Tarski has not successfully reduced the familiar semantical notions, but Blackburn has not shown that these notions are not "pure" in the sense of containing no conventional (or "pragmatic") element. This will require a further original consideration of the import of Tarski's work.