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33 (Compactness Theorem) A theory T has a model iff each Finite subtheory of T has a model. f In particular, A is a semantic consequence of the predicate calculus (over L), written \=A, iff A is true in every frame (for L)} | If T \= A, then we also say that A {logically) follows from T, A is entailed by T, or T entails A. 34 Let T consist of the following axioms: Bird(Tweety) Vx. Bird{x) => Flies{x) Vx. Flies{x) zd Ffas- Wings{x) +The notation T \= A was given a somewhat different meaning in the previous section, where T was a propositional theory.

It is easily seen that the restriction to alphabets with computable sets of predicate and function constants guarantees the computability of any first-order language. L al is said to be a first-order language with equality iff ‘ = ’ e AL; otherwise, L al is said to be a first-order language without equality. We use the letters A, B, C and D (resp. a, /? and y), possibly with subscripts and/or primes, as syntactic variables ranging over formulae (resp. terms). Formulae of the form (—|/l)and (A =d B) have the same meaning as in propositional logic.

The former is a connective which may occur in formulae; the latter is a meta-language symbol which is used to speak about formulae. Sec. 2] Step 2. CLASSICAL PROPOSITIONAL LOGIC 31 Move —i inward. In the formula obtained in Step 1, replace —i(B a B by by by — { B v C) C) ^ -iB — i a —iC —B v -nC B until all occurrences of —i immediately precede atomic formulae. Step 3. Distribute possible (B B C) a v (C a v a over D D) v . In the formula constructed in Step 2, replace as long as by by (B (B v D) a (C v v C) a (B v D) D) and denote the resulting formula by A .

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