 # Copyright 2008, Scott Gray1 Propositional Logic 3) Truth Tables.

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Copyright 2008, Scott Gray1 Propositional Logic 3) Truth Tables

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

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

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

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

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

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

Copyright 2008, Scott Gray 8 Truth Table Examples AA TFTFT

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!)

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))

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

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

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

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

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

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

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