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## Model TheoryDuring the 1930s and 1940s Tarski maintained a strict distinction between mathematics and metamathematics. Because of this, he was still in 1938 reluctant to accept that a set of formal axioms could serve to define the class of structures which satisfy them—as for example the class of rings consists of the structures that satisfy the axioms of ring theory. But mathematical developments put him under pressure to change his mind. By 1950 he was ready to embrace what we now know as model theory, and he devoted the early 1950s to setting up the basics of the theory. In the course of this work, Tarski rejigged his old truth definition, so that instead of defining ' mostly use just the separate recursive clauses of the definition, for example that satisfies ∀ in does guarantee that the relation ' The model-theoretic truth definition uses an adaptation of the idea of satisfaction that Tarski introduced in his 1933 truth definition and exploited in the 1936 paper. If you apply that model-theoretic adaptation to the 1936 definition of logical consequence, you get is a logical consequence of of where now Tarski's 1953 essay [20] seems to have had some unintended consequences among philosophers. A number of people conflated the 1936 definition with the 1953 one, and called both of them the 'model-theoretic definition of logical consequence'. I think the conflation is unfortunate, because the question we discussed in 1.1.1 above, about analytical relations between meanings, is one of the most important questions addressed in the 1936 definition, but it is meaningless for the languages of first-order model theory. Later, during the 1980s, the 'model-theoretic definition of logical consequence' attracted the attention of some philosophers who reassessed it as a contribution to conceptual analysis. Peter in his invitation to me (1) referred to a 'defence of model theory, as far as the foundations of logic are concerned'. I think I'll give this a miss. To me, model theory is a way of addressing certain kinds of question in mathematics, chiefly but not exclusively in geometry, algebra and number theory. The main link to foundations of logic is that some techniques of model theory made their way into axiomatic set theory around 1960 and continue to have an influence in large cardinal theory. |

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