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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 nary relation symbol of L and t is an arbitrary nary term of L . Obviously, all equality axioms are firstorder 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 firstorder 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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 Spring '08
 Bachmair,L
 Computer Science, Logic, Mathematical logic, Model theory, CN, Firstorder logic

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