Transcription of Logic, Proofs - Northwestern University
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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.
ditional. It is true precisely when p and q have the same truth value, i.e., they are both true or both false. 1.1.4. Logical Equivalence. Note that the compound proposi-tions p → q and ¬p∨q have the same truth values: p q ¬p ¬p∨q p → q T T F T T T F F F F F T T T T F F T T T When two compound propositions have the same truth values no
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