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9slides(2)

# 9slides(2) - Chapter 9 Completeness Theorem Proof 2 A...

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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 counter-model 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 counter-model exis- tence method . 1

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The construction of a counter-model for any non-provable 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 non-classical 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 counter-model for A , hence the proof provides a counter-model 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

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Step 1: Definition of Δ * We use the information 6‘ A to define a spe- cial set Δ * , such that ¬ A Δ * . Step 2: Counter - model definition We define the variable assignment v : V AR -→ { T, F } as follows: v ( a ) = ( T if Δ * a F if Δ * ¬ a. 4
Step 3: Prove that v is a counter-model We first prove a more general property, namely we prove that the set Δ * and v defined in the steps 1 and 2, respectively, are such that for every formula B ∈ F , v * ( B ) = ( T if Δ * B F if Δ * ¬ B. Then we use the Step 1 to prove that v * ( A ) = F . The definition and the properties of the set Δ * , and hence the Step 1 , are the most essential for the proof. The other steps have only technical character. 5

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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 can’t 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. We hence use the following syntactical defi- nition of consistency. 7

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Consistent set We say that a set Δ ⊆ F of formulas is con- sistent if and only if there is no a formula A ∈ F such that Δ A and Δ ¬ A.
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9slides(2) - Chapter 9 Completeness Theorem Proof 2 A...

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