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Model Theory

During 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 'φ is a true sentence of the language L' it defined 'φ is a sentence true in the structure M for the language L', where now L is a formal language whose nonlogical symbols have no meaning and the structure M is used to assign meanings to these symbols. This new truth definition is known as the 'model-theoretic truth definition'. You can find it in standard textbooks of model theory. But in practice model theorists

mostly use just the separate recursive clauses of the definition, for example that a¯

satisfies ∀yφ(x¯, y) in M if and only if for every element b of M, a¯ b satisfies φ(x¯, y)

in M. These clauses are all older than Tarski's work. The definition as a whole

does guarantee that the relation 'φ is true in the structure M' is set-theoretically definable, though today most logicians would reckon that this is intuitively obvious. Occasionally it's useful to know that the definition can be written as a set-theoretic formula of a particular form.

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 T if and only if every model of T is a model

of φ (3)

where now φ is a sentence and T a set of sentences, in a language whose nonlogical symbols are meaningless. It happens that the righthand clause of (3) is a relation that appears very often in model theory, so it would be useful to have a name for it. On the basis of the facts above, Tarski in 1953 [20, p. 8] proposed reading the relation as 'φ is a logical consequence of T '. Model theorists have tended to follow Tarski's lead and pronounce the relation as 'T entails φ' or 'φ is a consequence of T '. The use of the name has nothing to do with any interest in the concept of logical consequence itself.

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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