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Unformatted text preview: Chapter 9 Completeness Theorem: Proof 2 A Counter Model Existence Method We prove now the Completeness Theorem by proving the opposite implication: If 6 A, then 6 = A We will show now how one can define of a countermodel for A from the fact that A is not provable. This means that we deduce that a formula A is not a tautology from the fact that it does not have a proof. We hence call it a a countermodel exis tence method . 1 The construction of a countermodel for any nonprovable A is much more general (and less constructive) then in the case of our first proof. It can be generalized to the case of predi cate logic, and many of nonclassical log ics; propositional and predicate. It is hence a much more general method then the first one and this is the reason we present it here. 2 We remind that 6 = A means that there is a variable assignment v : V AR { T,F } , such that v * ( A ) 6 = T , i.e. in classical se mantics that v * ( A ) = F . a Such v is called a countermodel for A , hence the proof provides a countermodel construction method. Since we assume that A does not have a proof in S ( 6 A ) the method uses this informa tion in order to show that A is not a tautol ogy, i.e. to define v such that v * ( A ) = F . We also have to prove that all steps in that method are correct. This is done in the following steps. 3 Step 1: Definition of * We use the information 6 A to define a spe cial set * , such that A * . Step 2: Counter  model definition We define the variable assignment v : V AR { T,F } as follows: v ( a ) = ( T if * a F if * a. 4 Step 3: Prove that v is a countermodel We first prove a more general property, namely we prove that the set * and v defined in the steps 1 and 2, respectively, are such that for every formula B F , v * ( B ) = ( T if * B F if * B. Then we use the Step 1 to prove that v * ( A ) = F . The definition and the properties of the set * , and hence the Step 1 , are the most essential for the proof. The other steps have only technical character. 5 The main notions involved in this step are: consistent set, complete set and a con sistent complete extension of a set. We are going now to introduce them and to prove some essential facts about them. Consistent and Inconsistent Sets There exist two definitions of consistency; se mantical and syntactical. 6 Semantical definition uses the notion of a model and says: a set is consistent if it has a model . Syntactical definition uses the notion of prov ability and says: a set is consistent if one cant prove a contradiction from it . In our proof of the Completeness Theorem we use assumption that a given formula A does not have a proof to deduce that A is not a tautology....
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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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