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Unformatted text preview: Lecture 6 Proof in Propositional Logic 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 . Lecture 6 Proof in Propositional Logic The semantics for propositional logic allows us to characterize what it means for a proposition to follow logically from a set of propositions . This characterization is: all the valuations making the propositions in true make true. It is denoted as = What the characterization does not do: connect logical consequence to the notion of argument or proof. Semantic analysis of arguments Example p 1 p 2 p 3 (p 1 p 2 Q Q p p 3 It is not the case that both Dan and Beatrice are going. If Susan is going, Dan is going. Beatrice is going. Therefore, Susan is not going. Analyzing the premises and conclusions as propositional formulas and looking at their truthtables reveals that the conclusion is a logical consequence in the semantic sense. But this analysis seems coarse with respect to logical reasoning . The move from premise to conclusion is made in one giant complicated step. There seem however to be intermediate inferences between the premises and the conclusion. 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 Syntactic analysis of argument The aim of a syntactic analysis of logical consequence is to make this notion of intermediate inference precise. It identifies a fixed number of basic inferences. A conclusion follows logically from a set of premises, accordingly, if it is shown to follow from the premises by a proof consisting of only these basic inferences. More precisely, the basic inferences are analyzed as rules for building configurations made up of elements of PROP . These configurations are termed derivations. Derivations are what correspond to proofs in the analysis. Syntactic analysis of argument Taken together, the rules define a formal system termed Natural Deduction , developed by the logician Gehard Gentzen in 1930s. For every connective, there are two rules:...
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 Winter '09

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