Truth in a Structure
The core semantic notions in first order logic, notably that of satisfaction of a set of formulae by an interpretation, structure or model, are sometimes attributed to Alfred Tarski. However, though Tarski undoubtedly played a key role in the establishment of model theory, its not so clear whether this key concept is properly attributable to him.
The Feferman's on Tarski on Truth
What they said

In their biography of Alfred Tarski, Anita and Solomon Feferman observe that Tarki's theory of "truth" comes in two flavours. The first, published in Polish in 1933 and later translated, was directed toward philosophical audiences, and talks about defining truth in some language using an appropriate metalanguage. The second was not published until the late 1950's and was directed primarily to a logical audience. This is concerned with the relative notion of truth in a structure, the starting point for the subject of model theory.

The Feferman's claim that Tarski had already "arrived at" this latter notion by 1930.

Some Doubts

Now I find this puzzling, for in 1930 Gödel completed his doctoral dissertation, published later that year under the title "The Completeness of the axioms of the functional calculus of logic", which proved, in modern parlance, the completeness of first order logic. The notions of validity and satisfaction used in but not introduced by this work were not materially distinct from the notion which Tarski is said to have "arrived at" by 1930, but they were not then novel. Gödel refers the reader to Hilbert and Ackerman, 1928, but the essential ideas go back considerably further than that.

For example, in 1915, Löwenheim published a paper "On possibilities in the calculus of relatives" which Van Heijenoort describes thus:

Löwenheim's paper deals with problems connected with validity, in different domains, of formulas of the first order predicate calculus...

Well, strictly speaking this isn't accurate, since it really isn't about the first order predicate calculus, it is as its title suggests about the calculus of relations. However, the difference is largely cosmetic.

In his introductory definitions Löwenheim uses but does not consider it necessary to explain the notion of satisfaction. The concept corresponding to a valid first order sentence he introduces under the name "identical equation" which he does not find it necessary to define or describe (though his intention is clear in context).

Here my familiarity with the literature, such as it is, peters out. It appears that the notions of satisfaction in a structure, and validity, gradually evolved along with the evolution of logical languages toward our modern conception of first order logic. The differences along the route from first order expressions in the relational calculus to the present day conception of first order logic, are almost entirely notational, and no substantive changes have therefore been necessary in the underlying semantic concepts.

So what is it that we are to suppose Tarski arrived at by 1930?


There is some interesting historical material relevent to this in Richard Zach's doctoral dissertation and in material which he wrote with others for a new history of modern logic, both available from his web page.

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