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Logic, Proofs - Northwestern University

CHAPTER1 Logic, a declarative sentencethatis eithertrueor false(butnotboth).For instance,thefollowingarepropositions: Parisis in France (true), Londonis in Denmark (false), 2<4 (true), 4= 7 (false) .However thefollowingarenotpropositions: whatisyourname? (thisis a question), doyourhomework (thisis acommand), thissentenceis false (neithertruenorfalse), xis aneven number (itdependsonwhatxrepresents), Socrates (itis notevena sentence).Thetruthor falsehood of a propositionis , Themainonesarethefollowing(pandqrepresen t givenpropositions):NameRepresentedMeanin gNegation p notp Conjunctionp q pandq Disjunctionp q porq(orboth) Exclusive Orp q eitherporq, butnotboth Implicationp q ifpthenq Biconditionalp q pif andonlyifq Thetruthvalueof a compoundpropositiondependsonlyonthevalue of for false andT for true ,wecansummarizethemeaningof theconnectives in pp qp qp qp qp qTTFTTFTTTFFFTTFFFTTFTTTFFFTFFFTTN otethat represents anon-exclusiveor, ,p qis truewhenany ofp,qis represents anexclusiveor, ,p qis trueonlywhenexactlyoneofpandqis , Contradiction, propositionis saidto be atautologyif itstruthvalueis Tforany assignment of truthvaluesto :Thepropositionp pis a A propositionis saidto be acontradictionif itstruthvalueis Fforany assignment of truthvaluesto.

from Chicago” may be false. 1.1. PROPOSITIONS 10 The contrapositive of a conditional proposition p → q is the propo-sition ¬q → ¬p. They are logically equivalent. For instance the con-trapositive of “if John is from Chicago then John is from Illinois” is “if

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