The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

5
Tarski's Thesis and the Ontology of
Mathematics

CHARLES CHIHARA


1. WHAT IS TARSKI'S THESIS?

According to some philosophers, there is a widespread belief that model theory provides us with an accurate conceptual analysis of the intuitive, pre-analytic notions of logical consequence and logical truth, at least when the sentences to which these notions are to be applied are those of certain formalized languages.1 To undermine this belief is the principal goal of John Etchemendy's book. Vann McGee has continued this attack in his article ( McGee 1992). In this paper, I shall assess some of the objections of these two philosophers, focusing especially on Etchemendy's 'contingency argument', which raises questions about the existence and nature of mathematical objects.

The thesis that a sentence of a formalized language is logically valid (or logically true) iff it is true in every model is called by McGee 'Tarski's thesis' ( McGee 1992, 273). Actually, McGee also applies the term to the thesis that a sentence of a formalized language is logically consistent iff it is true in at least one model. The term makes reference to the great Berkeley logician, even though it is admitted that Tarski's notion of model differs from the notion we use today, because ' Alfred Tarski is credited with the analysis of logical validity as truth in every model' ( McGee 1992, 273).

Etchemendy also attacks a third version of 'Tarski's thesis', according to which, if Γ is a set of Sentences of a formalized language, and φ is a sentence of this language, then φ is an intuitive logical consequence of Γ iff φ is a model-theoretic consequence of Γ ( Etchemendy 1990, 4-5). I shall refer to this last statement of Tarski's thesis as 'the third version', 'the first version' being restricted to the one involving logical truth and validity, and 'the second version' referring to the one involving logical consistency and satisfiability. All three forms of Tarski's thesis assert an extensional identity between an intuitive notion and a model-theoretic notion in a way that

____________________
1
Such a claim is made many times in ( Etchemendy 1990).

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