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Unformatted text preview: CHAPTER 9 Two Proofs of Completeness Theorem There are many proof systems that describe classical propositional logic, i.e. that are complete proof systems with the respect to the classical semantics. We present here a Hilbert proof system for the classical propositional logic and discuss two ways of proving the Completeness Theorem for it. Any proof of the Completeness Theorem consists always of two parts. First we have show that all formulas that have a proof are tautologies . This implication is also called a Soundness Theorem, or soundness part of the Completeness Theorem. The second implication says: if a formula is a tautology then it has a proof . This alone is often called a Completeness Theorem. In our case, we call it a completeness part of the Completeness Theorem. The proof of the soundness part is standard. We concentrate here on the com pleteness part of the Completeness Theorem and present two proofs of it. The first proof is straightforward. It shows how one can use the assumption that a formula A is a tautology in order to construct its formal proof. It is hence called a proof  construction method . The second proof shows how one can deduce that a formula A is not a tautology from the fact that it does not have a proof . It is hence called a countermodel construction method . All these proofs and considerations are relative to a proof system whose com pleteness we discuss and its semantics. The semantics is, of course, that for classical propositional logic, so when we write  = A we mean that A is a classical propositional tautology. As far as the proof system is concerned we define here a certain class S of proof systems, instead of one proof system. We show that the Completeness Theorem holds for any system S from this class S . In particular, our system H 2 from chapter 8 is complete, as it belongs to the class of systems S . 1 1 Classical Propositional System H 2 There are many Hilbert style proof systems for the classical propositional cal culus. We present here one of them as it was called defined in chapter 8, and prove the Completeness theorem for it. H 2 is the following proof system: H 2 = ( L {⇒ , ¬} , A 1 ,A 2 ,A 3 , MP ) (1) where A 1 ,A 2 ,A 3 are axioms of the system defined below, MP is its rule of inference, called Modus Ponens is called a Hilbert proof system for the classical propositional logic. The axioms A 1 A 3 are defined as follows. A1 ( A ⇒ ( B ⇒ A )) , A2 (( A ⇒ ( B ⇒ C )) ⇒ (( A ⇒ B ) ⇒ ( A ⇒ C ))) , A3 (( ¬ B ⇒ ¬ A ) ⇒ (( ¬ B ⇒ A ) ⇒ B ))) MP (Rule of inference) ( MP ) A ; ( A ⇒ B ) B , and A,B,C are any formulas of the propositional language L {⇒ , ¬} . We write, as before ‘ H 2 A to denote that a formula A has a formal proof in H 2 (from the set of logical axioms A 1 ,A 2 ,A 3), and Γ ‘ H 2 A to denote that a formula A has a formal proof in H 2 from a set of formulas Γ (and the set of logical axioms A 1 ,A 2 ,A 3....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Bachmair,L
 Computer Science

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