Mathematicians who claim to have proved \(S\) but who do not provide the proof (perhaps because it will not fit in the margin) pose a double problem for the future: the mathematical problem of whether a valid proof of \(S\) can be constructed, and the historical problem of whether knowledge of such a proof can be attributed to the original mathematician. To justify his pre-theoretic definition of “logical consequence”, A. Tarski [“The establishment of scientific semantics” (1936)] needed to prove, in effect, that (1) \(\Gamma \models \sigma \to \square\) (All the sentences of \(\Gamma\) are true \(\to\sigma\) is true). J. Etchemendy [The concept of logical consequence (1990; Zbl 0743.03002), p. 86] took Tarski’s proof to have been to suppose that \(\Gamma \models \sigma\) and that all the sentences of \(\Gamma\) are true and that \(\sigma\) is false, and show that they are inconsistent; but, Etchemendy objected, such a proof shows only that \(\square(\Gamma\models\sigma\to\) (All sentences of \(\Gamma\) are true \(\to \sigma\) is true)), which does not in turn imply (1). This is Tarski’s fallacy, a case of \(\square p \to\;q/ \therefore p\to \square q\).
(1) can be proved for standard first-order logic using the completeness theorem; but (1) holds also for standard higher-order logic, where the completeness theorem is not available. The author seeks, and finds, a deeper proof, requiring explication of the notion of “formal” as invariance under isomorphic structures (and showing that Etchemendy’s conception of semantics as interpretational is inadequate); and that this explication provides a natural delimitation of the logical terms of a language (as those whose evaluations commute with all isomorphisms of domains). Thus Tarski need not have committed “Tarski’s fallacy”; the author does not explicitly treat the historical problem of whether Tarski did. The paper concludes with replies to some well-chosen questions and objections.