Tarski's definition of truth for finitary languages

Question

Recall that in Tarski's semantic conception of truth a T-sentence is an equivalence of the form:

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

where x is the name of a sentence of the object-language, and p is its translation in the meta-language. Schema (T), however, only provides a partial definition of the predicate true. So, to define truth we need the concept of satisfaction as defined by Tarski in his paper of 1933.

My question concerns the definition of the predicate true for finitary languages of sentential logic. In other words, I have a finite number of sentences, and no predicate whatsoever.

In that case, can truth be defined as the logical product of all instances of schema (T) without resorting to the concept of satisfaction? (I understand that satisfaction also applies to finitary languages, but my question is can we do without?)

Answer 1

The answer is a simple yes. If you have only finitely many sentences, then you can just list them in your definition. It would be something like:

x is true iff (x=x1 & p1) or (x=x2 & p2) or ... or (x=xn & pn)

where x1..xn are sentence names and p1..pn are sentences.

To see that, recall why the notion of satisfaction was needed in the first place. If the language has infinitely many sentences, then the definition of truth has to involve something like recursion. But truth cannot be directly defined recursively, since truth applies only to complete sentences, and sentences have parts which aren't necessarily sentences themselves. Satisfaction, however, can be defined recursively, since it applies to sentence parts as well. But if your language is finite, then you don't need recursion.

Here's what Tarski says about this, right after presenting Schema T in "The Concept of Truth in Formalized Languages", pp. 188-189:

If the language investigated contained a finite number of sentences fixed from the beginning, and if we could enumerate all these sentences, then the problem of the construction of a correct definition of truth would present no difficulties. ... But the situation is not like this. Whenever a language contains infinitely many sentences, the definition constructed automatically according to the above scheme would have to consist of infinitely many words, and such sentences cannot be formulated either in the metalanguage or in any other language.

Answer 2

The relevant passage concerning this is the following:

"While the words 'designates,' 'satisfies,' and 'defines' express relations (between certain expressions and the objects 'referred to' by these expressions), the word 'true' is of a different logical nature: it expresses a property (or denotes a class) of certain expressions, viz, of sentences. However, it is easily seen that all the formulations which were given earlier and which aimed to explain the meaning of this word ... referred not only to sentences themselves, but also to objects 'talked about' by these sentences, or possibly to 'states of affairs' described by them. And, moreover, it turns out that the simplest and the most natural way of obtaining an exact definition of truth is one which involves the use of other semantic notions, e.g., the notion of satisfaction. It is for these reasons that we count the concept of truth which is discussed here among the concepts of semantics, and the problem of defining truth proves to be closely related to the more general problem of setting up the foundations of theoretical semantics." (Alfred Tarski, The Semantic Conception of Truth)

An example which Tarski gives to illustrate the nature of the notion of satisfaction is the following:

"The sentence 'snow is white' is true if, and only if, snow is white"

Satisfaction involves a relation between a sentence and the sentence's referent, or as Tarski puts it, the "objects 'talked about' by these sentences," and this relation is unavoidable. Independently of whatever logical formulation is used, i.e. whether it amounts to a finite lists of sentences, whether it involves sentential logic or quantification; regardless of the formulation, the most basic elements of any metalanguage will always consist of atomic formulas whose truth value depends on this relation. The reason for this is that it is essential to what language is.

On the one hand, there is reality whose nature and existence is independent from our knowing anything about it, and on the other hand, there is the various means of representing such reality in such a way that it is conducive to understanding and communication. Language is one such means, and it's very reason for being is to serve in this representative capacity, so without the relation which underlies the notion of satisfaction, language is nothing at all. It makes no sense to speak of the truth of a sentence by itself independent of what it represents, because it would be reduced to a useless sequence of symbols. In the same way, it doesn't make sense to speak of reality as true independent of any perception, description, understanding or representation, because truth by its very nature is the aptitude of such representations to accurately portray reality. This is precisely what Tarski was saying when he said that truth "expresses a property (or denotes a class) of certain expressions, viz, of sentences."

Jamin Asay writes:

"Tarski’s interest was never in replacing our ordinary conception of truth with the kind of definitions he offers. Rather, Tarski’s definitions work in conjunction with our ordinary conception of truth. We know that Tarski’s definitions are successful only if they are materially adequate, in which case they entail all the T-sentences. For the T-sentences to provide an independent check on Tarski’s definitions, they must be expressing important facts about our ordinary conception of truth (or, at least, about the truth conditions of our sentences), and not vacuous logical truths." (Jamin Asay, "Tarski and Primitivism About Truth")

Therefore, the notion of satisfaction is essential to any definition of truth because it refers to the aptitude of sentences to fulfill their essential purpose. And no, truth cannot be defined as the logical product of all instances of a schema without resorting to the concept of satisfaction, because the notion of satisfaction is required to compose any such schema at the atomic level.

Answer 3

The answer depends (1) on the requirements that one sets on the definition (of truth), and (2) on one's preferred (informal) concept of truth.

Tarski posed two requirements on the definition of truth, "material" adequacy and "formal" adequacy. Material adequacy is about the extension of the truth predicate. Truth for all the correct sentences, and only for the correct sentences, should be entailed by the definition of truth. It is here that Schema T, which is Tarski's formalization of Aristotle's definition of truth, is invoked.

We should like our definition to do justice to the intuitions which adhere to the classical Aristotelian conception of truth - intuitions which find their expression in the well known words of Aristotle's Metaphysics: To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true. ("The Semantic Conception of Truth", "The Meaning of the Term 'True'")

Formal adequacy is about the choice of terms in the definition of truth. By this Tarski meant aspects of the concept of truth other than Schema T. Tarski himself deferred to the "ordinary" concept of truth, which he identified with viewing truth as a kind of "correspondence" between words and world. Tarski's "semantic" concept of truth, which ties together the concepts truth, reference and satisfaction, is Tarski's formal interpretation of the informal idea of word - world correspondence.

However, it is easily seen that all the formulations which were given earlier and which aimed to explain the meaning of [the word 'truth'] referred not only to sentences themselves, but also to objects "talked about" by these sentences, or possibly to "states of affairs" described by them. And, moreover, it turns out that the simplest and the most natural way of obtaining an exact definition of truth is one which involves the use of other semantic notions, e.g., the notion of satisfaction. (ibid, "Truth as a Semantic Concept")

For a finitary language, defining truth as a Schema T conjunction is possible, and it will be trivially materially adequate. But can it be formally adequate? Perhaps so, if one is (unlike Tarski himself) a minimalist regarding the concept of truth. We need not prejudge here whether the concept of satisfaction is a necessary part of the concept of truth. What we cannot avoid is this: if the (informal) concept of truth, for any language, contains anything of essence besides Schema T, then a definition of truth in terms of Schema T alone will not be formally adequate.