ONT Re: Model Theory
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Note 17
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| 1. Introduction
|
| 1.2. Model Theory for Sentential Logic (cont.)
|
| 1.2.18. Theorem.
|
| 1. A theory !C! is preserved under intersections
| if and only if !C! has a set of conditional axioms.
|
| 2. A sentence p is preserved under intersections
| if and only if p is equivalent to a conditional sentence.
|
| Proof. 1. We leave to the reader the proof that every conditional
| sentence (and hence every set of conditional sentences)
| is preserved under intersections.
|
| Conversely, let !C! be preserved under intersections.
| Consider the set !D! of all conditional consequences
| of !C!. It suffices to show that every model of !D!
| is a model !C!. Let B be an arbitrary model of !D!.
| For each T in $S$ - B, let !S!_T be the set of all
| sentences of the form
|
| S_1 & ... & S_k & ~T
|
| which hold in B. We also let the sentence ~T itself be
| in !S!_T. We first note that the conjunction of finitely
| many sentences in !S!_T is again equivalent to a sentence
| in !S!_T. Consider a sentence p in !S!_T. Then ~p is
| clearly equivalent to a conditional sentence q either
| of the form S or of the form
|
| ~S_1 v ... v ~S_k v T.
|
| But q fails in B, so q does not belong to !D!. This means that q,
| and hence ~p, is not a consequence of !C!, and it follows that
| !C! |_| {p} is satisfiable. Since !S!_T is, up to equivalence,
| closed under finite conjunction, we see that !C! |_| !S!_T is
| finitely satisfiable. Applying the Compactness Theorem, we
| may choose a model A_T of !C! |_| !S!_T .
|
| For each T in $S$ - B, we have T not in A_T and B c A_T.
| Thus, if $S$ - B is not empty, then:
|
| B = |^|_(T not in B) A_T.
|
| Since each A_T is a model of !C! and !C! is
| closed under intersections, we have B |= !C!.
|
| In the remaining case B = $S$, we let !S!
| be the set of all sentences of the form
|
| S_1 & ... & S_k.
|
| Arguing as before, we find that !C! |_| !S!
| is finitely satisfiable and thus has a model.
|
| But B is the only model of !S!, so again B is a model of !C!.
|
| We have now shown that every model of !D! is a model of !C!,
| and it follows that !D! is a set of conditional axioms for !C!.
|
| 2. This follows from (1) by an argument
| similar to the last part of the proof
| of Theorem 1.2.16. -|
|
| Chang & Keisler, 'Model Theory', pages 15-16.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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