Davidson and Tarski
Davidson makes a specific suggestion as to how the output of radical interpretation should be codified as a formal theory of meaning. To understand this will require a brief account of the connection between Davidson and Tarski’s work on truth.
Davidson introduces the idea of a theory of meaning without giving a clear idea of its purpose. He makes two comments on the subject. One is that knowledge of such a theory would suffice for understanding [Davidson 1984: 125]. The other is that it is a necessary condition for languages to be learnable that a constructive or compositional account of the language could be given [ibid: 3]. But, even taken together, these do not explain how provision of a theory of meaning helps the philosophical enterprise of clarifying linguistic and mental content. (Davidson does not say explicitly, for example, that speakers have implicit knowledge of a theory of meaning which explains their ability.) This issue of just what the theory is supposed to be a theory of, and what (if any) its explanatory role is, is an important one. As I will describe, it is one of the key foci of McDowell’s debate with Dummett.
Davidson gives a clearer account of his motivation for the particular structure of the theory he advocates. In particular, he is more explicit in his reasons why such a theory should be extensional. Theories of meaning of the form: s means m - where m refers to a meaning of a word or sentence - have proved to be of little use in showing how the meaning of parts of a sentence structurally determine the meaning of the whole. Things can be improved by modifying the theory’s structure to be: s means that p, where p stands for a sentence. But this still leaves the problem that ‘wrestling with the logic of the apparently non-extensional ‘means that’ we will encounter problems as hard as, or perhaps identical with, the problems our theory is out to solve.’ [ibid: 22] The solution is to realise that what matters for such a theory is not the nature of the connection between s and p but that the right s and p are connected:
The theory will have done its work if it provides, for every sentence s in the language under study, a matching sentence (to replace ‘p’) that, in some way yet to be made clear, ‘gives the meaning’ of s. One obvious candidate for matching sentence is just s itself, if the object language is contained in the meta-language; otherwise a translation of s in the meta-language. As a final bold step, let us try treating the position occupied by ‘p’ extensionally: to implement this, sweep away the obscure ‘means that’, provide the sentence that replaces ‘p’ with a proper sentential connective, and supply the description that replaces ‘s’ with its own predicate. The plausible result is
(T) s is T if and only if p. [ibid: 23]
Further reflection suggests that, if this is to serve as an interpretation, the appropriate predicate for T is truth. We want the sentence s to be true if and only if p. (McDowell gives a strikingly similar account of the role of truth in a meaning theory in ‘Meaning, communication and knowledge’ section 2 [McDowell 1980: 119-122; 1998a: 31-34].)
The proposed theoretical schema has the further advantage (and motivation) that it dovetails with Tarski’s account of truth. Tarski’s account is pressed into service to show how the meanings of sentences are constructed from the meanings of words (which are themselves abstracted from the meanings of sentences). Davidson’s use of Tarski inverts its normal explanatory priority. Tarski assumes that facts about meaning can be presupposed in the task of giving an extensional definition of truth in a language. By contrast, Davidson suggests that truth is a suitably primitive, transparent and unitary notion to shed light on meaning [Davidson 1984: 134]. With this change of emphasis, Davidson can then borrow Tarski’s technical machinery to articulate the structure of a given language. I will now outline Tarski’s semantic conception of truth.
There continues to be considerable disagreement about the philosophical significance of Tarski’s work on truth, of just what (if any) light it sheds on truth’s nature [cf Davidson 1990; McDowell 1878; 1998a: 132-154]. But it is uncontentous that Tarski provides a logical model of how the truth condition of any sentence of a formal (first order) language can be derived from a more basic set of axioms. One way of thinking about Tarski’s semantic conception of truth is to consider two much simpler models than the one which is useful for Davidson. So consider truth theories for each of the following:
1. A language with a finite number of sentences.
2. A language with a finite number of whole atomic sentences and truth functional connectives which can be used to build an unlimited number of more complex, molecular sentences from them.
3. A language whose building blocks are sub-sentential predicates, truth functional connectives and quantifiers: Frege’s first order logic.
Take the first. At its simplest, a Tarskian truth theory could be given for a language with a finite number of atomic whole sentences with no connectives for building further molecular sentences. The aim of a Tarskian truth theory is to specify the truth conditions of any sentence in the language. In this case, the truth theory would simply be a list of instances of the T schema: s is T if and only if p. Each sentence named on the left hand side of an instance of the T schema would be paired with a sentence in English, for example, which stated the condition under which it was true. Since the aim of a Tarskian theory is to spell out truth conditions, we can help ourselves to the fact that the sentence on the right hand side translates or interprets the named sentence on the left. Armed with the list, and given any from the finite number of atomic sentences, one could look up the condition under which it was true.
Note that each instance of the T schema is written in a metalanguage. It contains the name of an object language sentence - usually achieved just by putting inverted commas round a sentence - and a condition spelling out its truth condition, in the form of a sentence of the metalanguage. The object language sentence is thus named or mentioned and the sentence of the metalanguage spelling out its worldly truth condition is used to give its truth condition.
Now consider the second case. A further degree of complexity is introduced by adding truth functional connectives to the object language. A simple list will no longer serve as a truth theory because there is no limit to the number of complex sentences that can be formed simply by conjoining, for example, one atomic sentence with others, or even perhaps different tokens of the same sentence type. The resultant complex sentence can then be further conjoined with other atomic or complex sentences. Note that the truth condition of the final result will still be determined by a complex function of the truth conditions of the component atomic sentences and thus derivable from them. But since the process of conjoining sentences can in principle be repeated without limit, a general method of spelling out the truth conditions of arbitrary sentences of the object language will have to be recursive. It will have to be repeatedly applicable so that the output of one operation can feed in as the input to the next application until the constituent atomic sentences are reached. So a Tarskian truth theory for this language will have to contain both a list of the atomic sentences with a specification of each of their truth conditions and also some general rules specifying how the truth condition of a complex sentence, say, ‘A and B’ is a function of ‘A’ and ‘B’ (where A and B may themselves be complex sentences needing further analysis). The last element is given by the truth table for each of the connectives. So ‘A and B’ is true iff ‘A’ is true and ‘B’ is true. ‘A or B’ is true iff ‘A’ is true or ‘B’ is true.
For a simple language with the form of propositional or sentential logic, a Tarskian truth theory is still quite simple. It contains axioms giving the truth conditions of the basic or atomic sentences of its object language and iterable rules specifying the truth conditions of complex combinations of atomic sentences using connectives. Given the specification of the truth conditions of the basic sentences and given general rules for the truth conditions of complex sentences that can be built up from them, the truth conditions of arbitrarily complex sentences can be calculated. Given a complex sentence, one can analyse it as a function of the truth conditions of its component parts which can in turn be decomposed eventually to reach atomic sentences whose truth conditions are listed.
The third kind of case introduces a yet further level of complexity. In this case, the component parts of whole sentences need not themselves be whole sentences. Frege’s predicate logic (the logic of one, two or many place predicates and relations together with the universal and existential quantifiers) is just such an example. Because predicate logic allows for whole (‘closed’) sentences to be built up from sub-sentential units (‘open sentences’) which are thus themselves neither true nor false, a truth theory for it requires, in addition to what has gone before, the technical notion of satisfaction. The open sentence Fx, unbound by any quantifier, is neither true nor false. It says something like: ‘...is F’. Suppose ‘F’ is replaced by ‘red’, then ‘... is red’ is neither true nor false by itself. But Tarski suggests we can say that it is satisfied by some object providing that that object is red. Similarly the open sentence ‘x loves y’ is satisfied by an ordered pair of objects if the first loves the second.
Given this intuitive idea of the satisfaction of predicates by objects (in fact by ordered sequences of objects to cope with multi-place relations), Tarski then goes on to define separately the conditions under which existentially and universally quantified closed sentences are satisfied by sequences of objects. Finally he defines the conditions under which any closed sentence is true in terms of satisfaction. It turns out that a closed sentence is true if it is satisfied by all sequences of objects.
Tarski needs to ensure that that the definition of truth in terms of satisfaction has the right consequences for the truth of universally and existentially quantified sentences. Thus in the case of a universally quantified sentence it should be true just in case everything is as it says. Taking the example of ("x4)( x4 is round) - where the subscripts allow for many variables in a sentence - Richard Kirkham puts the point in this way:
Taski ensures this by setting two conditions that must be met for a sequence S, to satisfy a universally quanitified sentence such as ‘("x4)( x4 is round)’:
1. S must satisfy the open sentence that would be created by deleting the quantifier. So in this case it must satisfy ‘x4 is round’. Thus whatever object S has in the fourth place must be round.
2. This same sentence must also be satisfied by every sequence that is just like S except that it has a different object in the fourth place. [Kirkham 1992: 156]
Since
[F]or every object in the world, there is some sequence just like S except that it has just that object in the fourth place. So, since condition (2) tells us that all of these sequences must satisfy ‘x4 is round’, the condition says, in effect, that everything in the world other than the object in the fourth place of S must be round. Between them (1) and (2) are saying that every object must be round. [Kirkham 1992: 156]
Thus ‘("x4)( x4 is round)’ is satisfied by sequence S if and only if every object is round. A consequence of this definition is that if S satisfies the sentence then so does every other sequence because, again, those sequences satisfy the sentence only if every object is round.
The definition of the satisfaction of an existentially quantified sentence is given in the following related way.
An expression of the form ‘($xk)f’ is satisfied by a sequence S if and only if some sequence differing from S in at most the kth place satisfies f.
But… for every object there is some sequence just like S except that it has that object in the kth place. So the conclusion will be met when and only when something in the world fs. Notice here too that if one sequence satisfies the existential claim, they all do. [Kirkham 1992: 157]
Thus both universally and existentially quantified sentences are satisfied by one sequence if and only if they are satisfied by every sequence. With these definitions in place Tarski is able to stipulate that a sentence of either kind is true providing it is satisfied by every sequence.
Even this brief sketch of the general shape of a Tarskian truth theory for a language with the same logical structure as Fregean predicate logic should reveal in outline the modest philosophical role (if less modest technical role) of satisfaction. It serves as part of the internal workings of the theory so that correct instances of the T schema can be derived for each whole (ie closed) sentence of the object language. The basic axioms of the theory concern the satisfaction of predicates by objects rather than truth of sentences. But although Tarski provides a definition of truth in terms of satisfaction, the primary purpose of a truth theory is to allow the derivation of instances of the T schema for complex sentences in the object language.
With this brief summary in place, Davidson’s use of Tarski should now be clearer. The Tarskian machinery provides a model of how the truth condition of any sentence in a language might be determined by a finite list of axioms (which give the meaning of individual words) and rules of combination (which give the grammar). Setting out a Tarskian truth theory for a language such as English articulates the structure of that language.
Of course, this use of Tarski depends both on it being possible to regiment natural language in the same way as artificial or formal languages, and there are notorious obstacles to this project. (Perhaps the most familiar, and one on which Davidson himself worked, is the analysis of reported speech. For recent discussion see [Dodd 2000: 19-48].) It also depends on finding a way to overcome the following result of inverting the explanatory priority of truth and meaning, and using merely an extensional connective in the theory.
As the quotation in the previous section makes explicit, at the heart of his meaning theory Davidson replaces the intensional connective ‘means that’ with the extensional form ‘s is true if and only if p’. Clearly, however, the fact that the truth values of the left and right hand side of this conditional agree does not in itself ensure that the right hand side provides an interpretation of the sentence mentioned on the left. In Tarski’s use of the T schema, it can simply be assumed or stipulated that the right hand side provides an interpretation by being the same sentence as, or a translation of, the sentence mentioned on the left. (Tarski helps himself to facts about meaning to show how to determine the conditions under which a truth predicate applies to sentences of the object language.) But Davidson has to earn the right to that claim. His suggestion is that instances of the T schema should not be thought of as interpretative in themselves [Davidson 1984: 61]. Rather, it is the fact that each instance can be derived from an overall theory for the language which also allows the derivation of many other instances of the T schema with the right matching of truth values, which is interpretative. Only if the theory systematically matches correctly words on the left and right hand side of T schema instances will it have a chance of generating all and only true instances. But if it does, that is very strong evidence for a match of meaning. Instances of the T schema play a role within the larger deductive structure of the meaning theory of a whole language.
I can now summarise Davidson’s project of outlining a formal theory of meaning. On the twin assumptions that we have a clearer antecedent understanding of the concept truth than we have of meaning and also that natural language has a structure that can be regimented, a formal theory of meaning which allows the derivation of instances of the T schema (‘s’ is true iff p) represents a language in a finite axiomitised theory.
Davidson, D. (1984) Inquiries into Truth and Interpretation, Oxford: Oxford University Press
Davidson, D. (1990) ‘The Structure and Content of Truth’ (The Dewey Lectures 1989), Journal of Philosophy 87: 279-328.
Kirkham, R.L. (1992) Theories of truth, Cambridge, Mass.: MIT Press
McDowell, J. (1998a) Meaning knowledge and reality, Cambridge, Mass.: Harvard University Press