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Model Theory

时间:2013-08-16 16:49:53  来源:  作者:

http://plato.stanford.edu/entries/model-theory/

Model Theory
First published Sat Nov 10, 2001; substantive revision Wed Jul 17, 2013
Model theory began with the study of formal languages and their interpretations, and of the kinds of classification that a particular formal language can make. Mainstream model theory is now a sophisticated branch of mathematics (see the entry on first-order model theory). But in a broader sense, model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Alfred Tarski's truth definition as a paradigm. In this broader sense, model theory meets philosophy at several points, for example in the theory of logical consequence and in the semantics of natural languages.

1. Basic notions of model theory
2. Model-theoretic definition
3. Model-theoretic consequence
4. Expressive strength
5. Models and modelling
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1. Basic notions of model theory

Sometimes we write or speak a sentence S that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that S comes to express a true or false statement, we are said to interpret S, and the added information is called an interpretation of S. If the interpretation I happens to make S state something true, we say that I is a model of S, or that I satisfies S, in symbols ‘I ⊨ S’. Another way of saying that I is a model of S is to say that S is true in I, and so we have the notion of model-theoretic truth, which is truth in a particular interpretation. But one should remember that the statement ‘S is true in I’ is just a paraphrase of ‘S, when interpreted as in I, is true’; so model-theoretic truth is parasitic on plain ordinary truth, and we can always paraphrase it away.

For example I might say

He is killing all of them,
and offer the interpretation that ‘he’ is Alfonso Arblaster of 35 The Crescent, Beetleford, and that ‘them’ are the pigeons in his loft. This interpretation explains (a) what objects some expressions refer to, and (b) what classes some quantifiers range over. (In this example there is one quantifier: ‘all of them’). Interpretations that consist of items (a) and (b) appear very often in model theory, and they are known as structures. Particular kinds of model theory use particular kinds of structure; for example mathematical model theory tends to use so-called first-order structures, model theory of modal logics uses Kripke structures, and so on.

The structure I in the previous paragraph involves one fixed object and one fixed class. Since we described the structure today, the class is the class of pigeons in Alfonso's loft today, not those that will come tomorrow to replace them. If Alfonso Arblaster kills all the pigeons in his loft today, then I satisfies the quoted sentence today but won't satisfy it tomorrow, because Alfonso can't kill the same pigeons twice over. Depending on what you want to use model theory for, you may be happy to evaluate sentences today (the default time), or you may want to record how they are satisfied at one time and not at another. In the latter case you can relativise the notion of model and write ‘I ⊨ tS’ to mean that I is a model of S at time t. The same applies to places, or to anything else that might be picked up by other implicit indexical features in the sentence. For example if you believe in possible worlds, you can index ⊨ by the possible world where the sentence is to be evaluated. Apart from using set theory, model theory is completely agnostic about what kinds of thing exist.

Note that the objects and classes in a structure carry labels that steer them to the right expressions in the sentence. These labels are an essential part of the structure.

If the same class is used to interpret all quantifiers, the class is called the domain or universe of the structure. But sometimes there are quantifiers ranging over different classes. For example if I say

One of those thingummy diseases is killing all the birds.
you will look for an interpretation that assigns a class of diseases to ‘those thingummy diseases’ and a class of birds to ‘the birds’. Interpretations that give two or more classes for different quantifiers to range over are said to be many-sorted, and the classes are sometimes called the sorts.

The ideas above can still be useful if we start with a sentence S that does say something either true or false without needing further interpretation. (Model theorists say that such a sentence is fully interpreted.) For example we can consider misinterpretations I of a fully interpreted sentence S. A misinterpretation of S that makes it true is known as a nonstandard or unintended model of S. The branch of mathematics called nonstandard analysis is based on nonstandard models of mathematical statements about the real or complex number systems; see Section 4 below.

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