chapter14(hcompl)

chapter14(hcompl) - 1 Completeness Theorem for Classical...

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Unformatted text preview: 1 Completeness Theorem for Classical Predi- cate Logic The relationship between the first order models defined in terms of structures M = [ M,I ] and valuations s : V AR-→ M and propositional models defined in terms of truth assignments v : P-→ { T,F } is established by the following lemma. Lemma 1.1 (Predicate and Propositional Models) Let M = [ M,I ] be a structure for the language L and let s : V AR-→ M a valuation in M . There is a truth assignments v : P-→ { T,F } such that for all formulas A of L , ( M ,s ) | = A if and only if v * ( A ) = T. In particular, for any set S of sentences of L , if M | = S then S is consistent in sense of propositional logic. Proof For any prime formula A ∈ P we define v ( A ) = ‰ T if ( M ,s ) | = A F otherwise. Since every formula in L is either prime or is built up from prime formulas by means of propositional connectives, the conclusion is obvious. Observe, that the converse of the lemma is far from true. Consider a set S = {∀ x ( A ( x ) ⇒ B ( x )) , ∀ xA ( x ) , ∃ x ¬ B ( x ) } . All formulas of S are different prime formulas, S is hence consistent in the sense of propositional logic and obviously has no (predicate) model. The language L is a predicate language with equality. We adopt a following set of axioms. Equality Axioms For any free variable or constant of L , i.e for any u,w,u i ,w i ∈ ( V AR ∪ C ), E1 u = u , E2 ( u = w ⇒ w = u ) , 1 E3 (( u 1 = u 2 ∩ u 2 = u 3 ) ⇒ u 1 = u 3 ), E4 (( u 1 = w 1 ∩ ... ∩ u n = w n ) ⇒ ( R ( u 1 ,...,u n ) ⇒ R ( w 1 ,...,w n ))), E5 (( u 1 = w 1 ∩ ... ∩ u n = w n ) ⇒ ( t ( u 1 ,...,u n ) ⇒ t ( w 1 ,...,w n ))), where R ∈ bfP and t ∈ T , i.e. R is an arbitrary n-ary relation symbol of L and t is an arbitrary n-ary term of L . Obviously, all equality axioms are first-order tautologies , or are valid formulas of L , i.e. for all M = [ M,I ] and all s : V AR-→ M , and for all A ∈ { E 1 ,E 2 ,E 3 ,E 4 ,E 5 .E 6 } , ( M ,s ) | = A. This is why we still call logic with equality axioms added a logic. Now we are going to define notions that is fundamental to the Henkin’s technique for reducing first-order logic to propositional logic. The first one is that of witnessing expansion of the language L . Witnessing expansion L ( C ) of L We construct an expansion of our lan- guage L by adding a set C of new constants to it, i.e. we define a new language L ( C ) L ( C ) = L ( P , F , ( C ∪ C )) which is usually denoted shortly as L ( C ) = L ∪ C....
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chapter14(hcompl) - 1 Completeness Theorem for Classical...

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