What Was Tarski’s Thesis about Logical Truth? And Is It True?

Let me refer with *Tarski’s Thesis* to the strongest thesis that Tarski postulated asserting the extensional equivalence between a certain pretheoretical concept of logical truth and his technical concept of logical truth. In this talk I make explicit a few theses about logical truth that sound Tarskian somehow, including one that most deserves the name *Tarski’s Thesis*. And for each of these theses I claim that either it is presumably true but too weak, or it is too strong and false, or it is appropriately strong but also false.

One thesis that in my view is probably true, but is not the strongest thesis Tarski postulated, is

(T1) A sentence of a classical propositional/quantificational language is logically true in the relevant pretheoretical sense iff it is true in all classical propositional/quantificational models which (re)interpret its constants (other than its classical propositional/quantificational logical constants).

(T1) talks about a very restricted set of logical constants, while Tarski clearly did not just have in mind this particular set. A second thesis is

(T2) A sentence of a formal language which possibly extends a classical propositional/quantificational language with new logical constants which are propositional connectives, quantifiers or predicates is logically true in the pretheoretical sense iff it is true in all classical propositional/quantificational models which (re)interpret its constants (other than its logical constants).

The problem with (T2) is that, no matter how one understands the notion of truth in a model that appears in its formulation, and given a natural choice of logical constants, it is obviously false, and it is pretty absurd to think that Tarski might have had something like this in mind. Elementary considerations show that classical propositional or quantificational models are not appropriate for a theory of the logical properties of the modal logical constants, such as ‘necessarily’. A third thesis, that is perhaps true but again weak in a sense, and at any rate not Tarskian, is

(T3) A sentence of a classical propositional/quantificational/modal language is logically true in the pretheoretical sense iff it is true in all propositional/quantificational/Kripke models which (re)interpret its constants (other than its classical propositional/quantificational/modal logical constants).

(T3) talks about a very restricted set of (non-modal) logical constants, and it talks about a non-Tarskian notion of model. Here is a fourth thesis:

(T4) For every formal languageLwhich possibly extends a classical propositional/quantificational language with new propositional connectives, quantifiers and predicates, there is a peculiar classCof models for the language and a notion of truth in a model for L and C such that: a sentenceSofLis logically true in the pretheoretical sense iffSis true in all models inC.

This cannot have been Tarski’s Thesis, for many reasons. For one thing, (T4) is too abstract. For another, he clearly saw his thesis as open to refutation, while (T4) can be easily shown to be true. (Nevertheless, the truth of (T4) indicates that in some sense the abstract model-theoretic approach is especially suited to the task of characterizing relativized notions of logical truth.) In my view, which can be historically supported, Tarski’s Thesis was this:

(T5) A sentence of a formal language which possibly extends a classical propositional/quantificational language with newextensionallogical constants which are propositional connectives, quantifiers and predicates is logically true in the pretheoretical sense iff it is true in all classical propositional/quantificational models which (re)interpret its constants (other than its (extensional) logical constants)

(for a certain intuitive notion of extensionality that I characterize straightforwardly).

Is (T5) true? It is certainly not obviously false. And to a good extent it seems to underlie the practice of using classical models in the model theory of languages having as logical constants generalized quantifiers, the predicate of identity, and other constants denoting notions invariant under permutations of the quantificational domain. But I note that (T5) is false, under very reasonable assumptions which are familiar from standard logical practice. The first assumption is that a monadic predicate ‘E’ meaning “exists” is a logical constant. It may be noted that it is a logical constant both under “pragmatic” conceptions of logical constanthood and under Tarskian “permutationist” conceptions. (It denotes a notion invariant under permutations of actual domains. This notion is also invariant under homomorphisms of actual domains, in the sense recently defined by Feferman.) The second assumption is that ‘E’ is to be satisfied by an object at a world if that object exists at that world. ‘E’ can be given a simple satisfaction clause in the definition of satisfaction in a classical quantificational model, very much like the clause for identity: a model *M* plus valuation *v* satisfies E(*x*) if *v*(*x*) exists. ‘E’ thus understood is an extensional predicate.

The third assumption is that a quantifier ‘(forall *x*)’ with the following intuitive semantics is a logical constant: the propositional contribution of ‘(forall *x*)’ to a proposition expressed by a sentence containing ‘(forall *x*)’ gets specified when one simply specifies a quantificational domain, given purely in extension, which constitutes the range of ‘(forall *x*)’ specifies the set {you, me}, and that is enough.

This notion of quantificational propositional content arguably underlies the classical quantificational model conception of the quantifiers, and is reflected in one standard semantics for quantified modal logic as follows: This notion of quantificational propositional content arguably underlies the classical quantificational model conception of the quantifiers, and is reflected in one standard semantics for quantified modal logic as follows: (forall *x*)*A* is said to be satisfied at a world *w* by a valuation *v* of the variables with objects of a previously given domain *D* iff every valuation *u* of the variables with objects of *D* which differs from *v* at most at ‘*x*’ satisfies *A* at *w*.

Given these assumptions, it is not difficult to exhibit sentences which are model-theoretic logical truths in the sense of (T5), but are not logically true in a presumably relevant pretheoretical sense. For example, the sentence

(*) (forallx)E(x)

is a model-theoretic logical truth in the sense of (T5): no matter what classical quantificational model we choose, (*) will be true in the model (classical models have actual domains). But (*) can be used to express contingent propositions, such as the proposition that you and I exist. Provided we take necessity as a mark of pretheoretical logical truth (as Tarski apparently did not care to do, but most philosophers do), it follows that (*) is not logically true.

Posted by Greg Restall at April 13, 2004 09:23 AMComments

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