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Model-Theoretic Semantics

So far, nothing that I've mentioned is directly to do with semantics, i.e. the study of meanings. Tarski called his truth definition the 'semantic definition of truth', most probably because of a formal similarity with what Kotarbin´ski had called 'semantic definitions'. In his truth paper [19, p. 193f.] he listed some notions that he called 'semantic': denotation, definability, truth. The notion 'meaning' was not in his list, and this is certainly not an accident.

During the 1960s a number of papers appeared that were about extending model theory from non-modal formal languages to modal ones. Some people described this as giving 'model-theoretic semantics' for modal logics. I suppose that originally 'giving a semantics' meant giving a model theory that would allow one to talk in a concrete and precise way about truth and satisfaction of modal formulas. But a subtle shift started to take place. In a standard model for modal logic, each relation symbol has an 'intension', which is a function taking each possible world to a set that is the extension of the relation symbol in that world. You can think of extensions as references, and intensions as meanings—though a lot of people have criticised these analogies. So you can think of a model for the modal logic as assigning to each meaningful expression of the language an intension that represents the 'meaning' of that expression. Around 1970 Richard Montague adapted all these notions to the study of fragments of natural languages, building on earlier work of Rudolf Carnap. From that date onwards it became common to refer to Montague-style model theories of natural language as 'model-theoretic semantics'. (Though Barbara Partee, a pioneer in this area, describes her field as 'formal semantics'.) From the mid 1970s onwards, the people who did model-theoretic semantics were mostly linguists or philosophers of language. The earlier model-theoretic semantics had been done mostly by philosophical logicians, and almost never by model theorists.

Model-theoretic semantics is useless for lexicography—you learn nothing about the meaning of the Greek noun skindapsós by being told that its intension maps every possible world to the set of all the things in that world that fit the description skindapsós. But it comes into its own for describing how the meaning of a compound phrase depends on the meanings of its constituents. Earlier we illustrated how the clauses of Tarski's truth definition tell us what things satisfy a compound formula, in terms of what things satisfy its immediate subformulas. Tarski had one clause for each logical operator: the logical operators →, ¬, ∀ etc., are all of them expressions whose meaning is explained by saying how the meaning of a compound formed by means of them depends on the meanings of the constituent expressions. In modal logic and its variants we add to those logical constants other expressions like 'necessarily', 'believes', 'until'. Formal semanticists push the boat out and apply similar machinery to 'himself', 'hardly ever' and 'so much as' (for example).

Model-theoretic semantics and the model-theoretic definition of logical consequence were always completely separate. You might reckon that there is a link, because both of them are involved with giving meanings. But there are major differences. First, in studying logical consequence we are only concerned with the meaning of one expression; model-theoretic semantics aims to get a purchase on language as a whole. Second, Tarski always assumed that the expression 'logical consequence of' was not in the formal language L; it was an expression of the metatheory. Of course one can put it into the object language, but Tarski himself avoided doing this, because he had proved that languages containing enough of their own metatheory generate contradictions. So a person who wants to add 'logical consequence of' to the object language has the extra task of proving that the resulting language is still consistent. And third, the aim with logical consequence was to give a definition of it, under suitable constraints. Model-theoretic semantics doesn't give definitions, it gives truth-conditions.

So it was curious to read the introduction to Peter Schroeder-Heister's [14] and find him claiming that 'classical model-theoretic semantics' makes various assumptions about how logical consequence should be defined. I assumed at first that he was using 'model-theoretic semantics' as a name for the model-theoretic definition of logical consequence. But then almost at once he talks about model-theoretic treatment of the logical operators, and that really is in the realm of model-theoretic semantics. Well, it's not good history but it's an intriguing question all the same. Could there be a theory that helpfully combines definition of metatheoretic notions with the techniques of model-theoretic semantics? What problems would it run into? What constraints should it aim to observe? What kinds of new result could we expect from combining the two things? I think it's clear that Peter himself doesn't want to go down this road, but somebody else might. (Maybe somebody already has, in which case I give them my apologies and best wishes.)

 
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