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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 anexclusi

1.1. PROPOSITIONS 7 p q ¬p p∧q p∨q p⊕q p → q p ↔ q T T F T T F T T T F F F T T F F F T T F T T T F F F T F F F T T Note that ∨ represents a non-exclusive or, i.e., p∨ q is true when any of p, q is true and also when both are true. On the other hand ⊕ represents an exclusive or, i.e., p⊕ q is true only when exactly one of p and q is true. 1.1.2.

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