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Thursday, July 19, 2007 10:08 AM Inference rules of SL Ampersand In Ampaersand out Wedge In Wedge out Arrow in Arrow out Double arrow in Tilde in Tilde out &I &O vI vO I O I ~I ~O Double arrow out O &I AmpersandIn If you have a formula A and a formula B then you can infer the conjunction of the two formulas. A B ______ A&B 1st conjunction B&A 2nd conjunction &O AmpersandOut If you have a conjunction of A and B then you can infer one or the other of the conjuncts. A&B _____ A B vI WedgeIn If you have a formula A then you can infer the disjunction of A with any other formula. A _____ A v B B v A vO Wedgeout If you have a disjunction of A and B and you have the negation of its first disjunct then you can infer the second. A v B ~A Intro to Logic Page 1 ______ B A v B ~B ______ A I DoubleArrowIn Two conditionals of alternating symbols you can make this into a biconditional AB BA ______ AB O DoubleArrowOut AB _______ AB BA O Arrow Out AB A ______ B AB ~B _____ ~A DN Double Negation ~~A ______ A A _______ ~~A Direct Derivations ~S; RS;(R v ~T); (~P v T) ; (~P~Q)/~Q 1) ~S Pr 2) RS Pr Intro to Logic Page 2 3) 4) 5) 6) 7) 8) 9) 10) R v ~T Pr ~P v T Pr ~P~Q Pr SHOW: ~Q DD Direct derivation ~R 1,2,O ~T 3,7,vO ~P 4,8,vO ~Q 5,9,O 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) (P v Q)~R Pr P&T Pr R v ~S Pr US Pr SHOW: ~U DD P 2,&O P v Q 6,vI ~R 1,7,O ~S 3,8,vO ~U 4,9,O Rule 1) Premise rule: at any point prior to the show line a premise may be written down. The annotation is 'Pr'. Rule 2) InferenceRule: At any point in a derivation, a formula may be written down if it follows from previous available lines by an inference rule. The annotation cites the line numbers and name of the inference rule. Rule 3) Show line rule: At any point in a derivation, on is entitled to write down the expression SHOW: A for any formula A Intro to Logic Page 3 ...
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This note was uploaded on 04/07/2008 for the course PHILOSOPHY 201 taught by Professor Morgan during the Spring '08 term at Rutgers.
- Spring '08