Lecture6 - Lecture 6 Proof in Propositional Logic Four...

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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 truth-tables 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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Lecture6 - Lecture 6 Proof in Propositional Logic Four...

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