Lecture07RAAetc - Lecture 7 - CS 2603 Applied Logic for...

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 7 —CS 2603 Applied Logic for Hardware and Software toBe ( ¬ toBe) and Other Conundrums
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 Rules Galore! —Theorems in Rule Form ¬ (a b) ⎯⎯⎯⎯⎯ {DeM F } ( ¬ a) ( ¬ b) DeMorgan Or Fwd ( ¬ a) ( ¬ b) ⎯⎯⎯⎯⎯ {DeM B } ¬ (a b) DeMorgan Or Bkw a b ⎯⎯⎯ { Comm} b a And Commutes a b, b c ⎯⎯⎯⎯⎯⎯ { Chain} a c Implication Chain Rule a b ⎯⎯⎯ { Comm} b a Or Commutes a ¬ a ⎯⎯⎯⎯ { +&- } False NeverBoth ¬ (a b) ⎯⎯⎯⎯⎯ { ¬ ( )Comm} ¬ (b a) Not Or Commutes a b ¬ b ⎯⎯⎯⎯⎯⎯⎯ {modTol} ¬ a Modus Tollens
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 A Proof Using Contradiction ⎯⎯⎯⎯⎯ ⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { E} ⎯⎯⎯⎯ ± a b, ¬ a |– b (disjunctive syllogism) False ⎯⎯⎯⎯ {CTR} a Plan ² Use E ¬ a {ID} { +&- } False {CTR} remaining assumptions a b a |– c b |– c ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ { E} c discharge what? b b a b a b b conclusion
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 Reductio ad Absurdum ⎯⎯⎯⎯⎯ ± ¬ ( ¬ a) |– a ¬ a |– False ⎯⎯⎯⎯⎯⎯ {RAA} a Plan ² Derive False from ¬ a, given ¬ ( ¬ a) ² Conclude a (citing RAA) { +&- } False ⎯⎯⎯⎯⎯⎯⎯ {RAA} remaining assumption discharge what? ¬ ( ¬ a) ⎯⎯⎯ { ¬¬ F } a Double Negation Fwd What rule for this? a ¬ ( ¬ a) ¬ a conclusion
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 5 Law of the Excluded Middle |– ( ¬ a) a ⎯⎯⎯⎯⎯ { ¬ ( )E R } ¬ a
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Lecture07RAAetc - Lecture 7 - CS 2603 Applied Logic for...

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