Lecture2

# Lecture2 - Lecture 2 the sentences of propositional logic...

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Lecture 2: the sentences of propositional logic Four general steps in analysis of the notion of logical consequence •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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Some inferences The bike is in the garage or the bike is in the basement. The bike is not in the garage. Therefore, the bike is in the basement. If the plant has been infected, it has splotches on its leaves. The plant does not have splotches on its leaves. The plant has not been infected. Dan speaks Spanish and Stella speaks German. Dan speaks Spanish.
Some inferences The bike is in the garage or the bike is in the basement. P or Q The bike is not in the garage. not P Therefore, the bike is in the basement. Q If the plant has been infected, then it has splotches on its leaves. If P then Q The plant does not have splotches on its leaves . not Q The plant has not been infected. not P Dan speaks Spanish and Stella speaks German. P & Q Dan speaks Spanish. P

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Propositional Logic Propositional logic provides an analysis of these kinds of inferences. Sentences are either atomic (P, Q, . .) or built from other sentences with words like not, or, and . Accordingly, the language of propositional logic is defined as that consisting of the following symbols (definition 1.1.1 in book): i) proposition symbols: p 1 , p 2 , p 3 ,… ii) connectives: ˭ conjunction (and) ˮ disjunction (or) implication (if. .then) ¬ negation (it is not the case that)  biconditional (if and only if) falsity iii) parentheses: ±² ³
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Lecture2 - Lecture 2 the sentences of propositional logic...

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