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Copyright 2008, Scott Gray1 Propositional Logic 3) Truth Tables

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Copyright 2008, Scott Gray 2 Table Definition of NOT ~AA TFTF FTFT

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Copyright 2008, Scott Gray 3 Table Definition of OR A v BA B T T F T T F F F TTTFTTTF

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Copyright 2008, Scott Gray 4 Table Definition of AND A & BA B T T F T T F F F TFFFTFFF

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Copyright 2008, Scott Gray 5 Table Definition of IF A B T T T F F T F F TFTTTFTT This is known as material implication

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Copyright 2008, Scott Gray 6 Table Definition of IF & ONLY IF A B T T F T T F F F TFFTTFFT

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Copyright 2008, Scott Gray 7 Truth Tables A function (mapping out) of all possible combinations of truth and falsity

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Copyright 2008, Scott Gray 8 Truth Table Examples AA TFTFT

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Copyright 2008, Scott Gray 9 Truth Table Examples, cont. A B T T F T T F F F FTTFFTTF Do you recognize this operator? It is the exclusive OR (though that isnt the symbol for it!)

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Copyright 2008, Scott Gray 10 Operator Usage We can use ~ and v to do & A & BA B T T T F F T F F TFFFTFFF ~(~A v ~B) T FT F FT F F T T F T F Prove: A B = ~A v B Prove: A B = ~(~(~A v B) v ~(~B v A))

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Copyright 2008, Scott Gray 11 Further Examples P (~Q v R)P Q R T T T T T F T F T T F F F T T F T F F F T F F F T F T F F F T T T T F T T F F T T T P (~ Q v R) 1 23

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Copyright 2008, Scott Gray 12 Some Additional Definitions Tautology = main column is all true Contingent = main column has at least one true and at least one false Contradiction = main column is all false

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Copyright 2008, Scott Gray 13 Contradiction Example ~(A B) & B A B T T T F F T F F T F T F F F T F 132

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Copyright 2008, Scott Gray 14 A Proof P Q P --------- Q Look of all true premises and a false conclusion, if found, the argument is invalid This argument is valid This argument is known as Modus Ponens P Q T T T F F T F TFTTTFTT P TTFFTTFF Q TFTFTFTF 1 st premise2 nd premiseconclusion

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Copyright 2008, Scott Gray 15 Another Proof P v ~Q ~P --------- ~(P Q) P v ~QP Q T T T F F T F T F T F T ~P FFTTFFTT ~(P Q) F T T F F T Is this valid or invalid? This argument is invalid

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Copyright 2008, Scott Gray 16 Assignments Review this lesson and ask questions if you dont understand Evaluate the following sentences using truth tables: 1.~P ~(P Q) 2.~(P v Q) (~P & ~Q) 3.(P (Q v ~R)) & ~R

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Copyright 2008, Scott Gray 17 Assignments, cont. Prove whether the following are valid or invalid using truth tables: 4.P v Q, ~ Q P 5.~Q v R, ~R Q 6.~(P T), ~R v T ~(P R) 7.~(P v Q), P R ~Q & ~R

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