Some Sober Conceptions of Mathematical Truth
Résumé
It is not sufficient to supply an instance of Tarski's schema, ""p" is true if and only if p" for a certain statement in order to get a definition of truth for this statement and thus fix a truth-condition for it. A definition of the truth of a statement x of a language L is a bi-conditional whose two members are two statements of a meta-language L'. Tarski's schema simply suggests that a definition of truth for a certain segment x of a language L consists in a statement of the form: "n(x) is true if and only if t(x)", where "n(x)" is the name of x in L' and "t(x)" is a function t : S -> S' (S and S' being the sets of the statements respectively of L end L') which associates to x the statement of L' expressed by the same sentence as that which expresses x in L. In order to get a definition of truth for x and thus fix a truth-condition for it, one has thus to specify the function t. A conception of truth for a certain class X of mathematical statements is a general condition imposed on the truth-conditions for the statements of this class. It is advanced when the nature of the function t is specified for the statements belonging to X. It is sober when there is no need to appeal to a controversial ontology in order to describe the conditions under which the statement t(x) is assertible. Four sober conceptions of truth are presented and discussed.
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