ONT Re: Model Theory
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Note 15
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| 1. Introduction
|
| 1.2. Model Theory for Sentential Logic (cont.)
|
| The next group of theorems shows connections
| between mathematical operations on models and
| syntactical properties of sentences. The first
| result of this group concerns positive sentences.
| A sentence p is said to be 'positive' iff p is
| built up from sentence symbols using only the
| two connectives & and v. For example,
|
| (S_0 & (S_2 v S_3)) v S_16 is positive,
|
| while ~S_4 and S_3 <=> S_3 are not positive.
|
| A set !S! of sentences is called 'increasing'
| iff A |= !S! and A c B implies B |= !S!.
|
| 1.2.16. Theorem.
|
| 1. A c B
|
| if and only if
|
| every positive sentence which holds in A holds in B.
|
| 2. A consistent theory !C! is increasing
|
| if and only if
|
| !C! has a set of positive axioms.
|
| 3. A sentence p is increasing
|
| if and only if
|
| either p is equivalent to a positive sentence,
| p is valid, or ~p is valid.
|
| Proof. [C&K, pages 13-14].
|
| A completely trivial fact which is analogous to part (1)
| of the above theorem is: A = B if and only if every sentence
| which holds in A holds in B. We shall see later on in this book
| that the situation is very different in predicate logic, where a
| maximal consistent theory ordinarily does not even come close to
| characterizing a single model. This is one thing which makes
| model theory for predicate logic so much more interesting
| and difficult than model theory for sentential logic.
|
| Chang & Keisler, 'Model Theory', pages 13-14.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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