Lecture7 - Lecture 7 Completeness of Propositional...

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Lecture 7 Completeness of Propositional Logic—Part I Four general steps in analysis of the notion of logical consequence •Specify logical form by formulating an abstract notion of sentence. •Give a semantics for logic: specify how abstract sentences come to be true and false. •Fix a syntactic notion of proof : specify rules by which abstract sentences can be manipulated in proofs. •Prove completeness .
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Lecture 7 Completeness of Propositional Logic—Part I In symbols, the completeness theorem for propositional logic is Γ | ± ϕ if and only if Γ |= ϕ where Γ is a subset of PROP and ϕ an element of PROP. Today we are going to prove soundness (the left to right direction, the easy direction). We will also define the notion of consistency , and prove some basic facts about it. Consistency is a central notion in the proof of the right to left direction.
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Soundness Lemma 1.5.1 If Γ | ± ϕ then Γ |= ϕ What Γ | ± ϕ means, mathematically: D Γ contains the hypotheses of a derivation ϕ . What Γ |= ϕ means, mathematically: If v is a valuation that is equal to 1 on all elements of Γ , then v ( ϕ ) = [| ϕ |] v = 1. So, to prove Lemma 1.5.1, it suffices to prove: D For any derivation ϕ , if Γ contains all its hypotheses, then for all valuations v equal to 1 on all elements of Γ , v ( ϕ ) = [| ϕ |] v = 1. Since the set of derivations is an inductively defined set, this can be proven by induction on derivations.
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Some terminology Say that satisfies Γ if and only if [| σ |] v = 1 for all σ in Γ . Say that v satisfies ϕ if and only if [| ϕ |] v = 1. Accordingly, the relation Γ |= ϕ is stated as: For all v satisfying Γ , v satisfies ϕ .
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Soundness Proof The base case is a one element derivation ϕ . In the inductive step, there is a case for each of the six rules for building derivations.
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Lecture7 - Lecture 7 Completeness of Propositional...

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