Lecture4

# Lecture4 - Lecture 4 Semantics for Propositional Logic Four...

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Lecture 4 Semantics for 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 .

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The inferences modeled by PROP The bike is in the garage or the bike is in the basement. p 1 ˮ p 2 The bike is not in the garage. ¬ p 1 Therefore, the bike is in the basement. p 2 If the plant has been infected, then it has splotches on its leaves. p 1 p 2 The plant does not have splotches on its leaves . ¬ p 2 The plant has not been infected. ¬ p 1 Dan speaks Spanish and Stella speaks German. p 1 ˭ p 2 Dan speaks Spanish.
Just going by the definition of PROP , however, we have no way of distinguishing correct inferences from incorrect ones. By the definition, an element of PROP is just a sequence of symbols. There is nothing in it to rule out the following as incorrect. p 1 p 2 p 1 ˮ p 2 p 2 p 1 p 1 p 2 These inferences, in contrast to the previous three, are not correct logically, given what we intend and ˮ to represent. Why? One answer : it is possible with both inferences for the premises to be true and the conclusion false.

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We thus need a precise way of talking about the elements of PROP as true or false, a way that is compatible with what we intend the connectives ˭ , ˮ , ± ¬ ±  to represent. That is, we need to provide a semantics for PROP . The basic idea : the truth or falsity of a propositional symbol p i is open. Or in other words, the truth values of the propositional symbols can vary arbitrarily.
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Lecture4 - Lecture 4 Semantics for Propositional Logic Four...

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