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Unformatted text preview: 1 Completeness Theorem for First Order Logic There are many proofs of the Completeness Theorem for First Order Logic. We follow here a version of Henkin’s proof, as presented in the Handbook of Mathe matical Logic . It contains a method for reducing certain problems of firstorder logic back to problems about propositional logic. We give independent proof of Compactness Theorem for propositional logic. The Compactness Theorem for firstorder logic and L¨owenheimSkolem Theorems and the G¨odel Completeness Theorem fall out of the Henkin method. 1.1 Compactness Theorem for Propositional Logic Let L = L ( P , F , C ) be a first order language with equality. We assume that the sets P, F, C are infinitely enumerable. We define a propositional logic within it as follows. Prime formulas We consider a subset P of the set F of all formulas of L . Intuitively these are formulas of L which are not direct propositional com bination of simpler formulas, that is, atomic formulas ( A F ) and formulas beginning with quantifiers. Formally, we have that P = { A ∈ F : A ∈ A F or A = ∀ xB, A = ∃ xB for B ∈ F} . Example 1.1 The following are primitive formulas. R ( t 1 ,t 2 ) , ∀ x ( A ( x ) ⇒ ¬ A ( x )) , ( c = c ) , ∃ x ( Q ( x,y ) ∩ ∀ yA ( y )) . The following are not primitive formulas. ( R ( t 1 ,t 2 ) ⇒ ( c = c )) , ( R ( t 1 ,t 2 ) ∪ ∀ x ( A ( x ) ⇒ ¬ A ( x )) . Given a set P of primitive formulas we define in a standard way the set P F of propositional formulas as follows. Propositional formulas The smallest set P F ⊂ F such that 1. P ⊂ P F 2. If A,B ∈ P F , then ( A ⇒ B ) , ( A ∪ B ) , ( A ∩ B ) , and ¬ A ∈ P F is called a set of propositional formulas of the first order language L . We define propositional semantics for propositional formulas in P F as follows. 1 Truth assignment Let P be a set of prime formulas and { T,F } be a two element set, thought as the set of logical values ”true” and ”false”. Any function v : P→ { T,F } is called truth assignment (or variable assignment). Let B = ( { T,F } , ⇒ , ∪ , ∩ , ¬ ) be a twoelement Boolean algebra and PF = ( P F , ⇒ , ∪ , ∩ , ¬ ) a similar algebra of propositional formulas. We extend v to a homomorphism v * : PF→ B in a usual way, i.e. we put v * ( A ) = v ( A ) for A ∈ P , and for any A,B ∈ P F , v * ( A ⇒ B ) = v * ( A ) ⇒ v * ( B ), v * ( A ∪ B ) = v * ( A ) ∪ v * ( B ), v * ( A ∩ B ) = v * ( A ) ∩ v * ( B ), v * ( ¬ A ) = ¬ v * ( A ). Propositional Model A truth assignment v is called a propositional model for a formula A ∈ P F iff v * ( A ) = T ....
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 Spring '08
 Bachmair,L
 Computer Science, Logic, Model theory, Firstorder logic, propositional formulas, ﬁnitely consistent set

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