ONT Re: Model Theory
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Note 8
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| 1. Introduction
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| 1.2. Model Theory for Sentential Logic (cont.)
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| We are now ready to build a bridge between the language $S$ and its models,
| with the definition of the truth of a sentence in a model. We shall express
| the fact that a sentence p is true in a model A succinctly by the special
| notation:
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| A |= p.
|
| The relation A |= p is defined as follows:
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| 1.2.3. [Definition of A |= p, that is, A is a 'model' of p, or p 'holds' in A]
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| 1. If p is a sentence symbol S, then A |= p holds if and only if S is in A.
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| 2. If p is q & r, then A |= p if and only if both A |= q and A |= r.
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| 3. If p is ~q, then A |= p iff it is not the case that A |= q.
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| When A |= p, we say that p is 'true' in A, or that p 'holds' in A, or
| that A is a 'model' of p. When it is not the case that A |= p, we say
| that p is 'false' in A, or that p 'fails' in A. The above definition of
| the relation A |= p is an example of a recursive definition based on 1.2.2.
| The proof that the definition is unambiguous for each sentence p is, of course,
| a proof by induction based on 1.2.2.
|
| Chang & Keisler, 'Model Theory', page 7.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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