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Natural Deduction-1

# Natural Deduction-1 - Natural Deduction Rules Of...

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Natural Deduction Rules Of Implication 1)Simplification (Conjunction Elimination) Simp (& E) p & q p & q :. p :. q 2)Conjunction (Conjunction Introduction) Conj (& I) p q_______ :. p & q 3)Disjunctive Syllogism (Disjunction Elimination) DS (v E) p v q p v q ~p ____ ~q____ :. q :. p 4)Modus Ponens MP p -> q p_____ : . q (5) Modus Tollens MT p -> q ~q :. ~p 6) Hypothetical Syllogism HS p -> q q -> r :. p -> r 7)Constructive Dillema CD p -> q r -> s p v r___ :. q v s 8)Addition (Disjunction Introduction) ADD (v I) p_____ :. p v q

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Rules of Replacement (Equivalence Rules) (1) Double Negation DN ~ ~ p <=> p (2) De Morgan Rules DM ~ ( p v q) <=> ( ~ p & ~ q ) ~ ( p & q) <=> ( ~ p v ~ q ) (3) Contraposition CONTRA ( p -> q ) <=> ( ~ q -> ~ p ) (4) Implication IMPL ( p -> q ) <=> ( ~ p v q ) (5) Exportation EXP [( p & q ) -> r ] <=> [ p -> ( q -> r )] (6) Equivalence EQUIV ( p <-> q ) <=> [( p -> q) & ( q -> p )] ( p <-> q ) <=> [( p & q ) v ( ~ p & ~ q )] PROOF STRATEGY Natural deduction can ONLY prove VALIDITY. Failure to provide a proof can have two reasons: -the argument is valid but one lacks the practice, imagination, knowledge of how to derive the conclusion (human failure) -the argument is invalid (argument failure). Thus, the failure of deriving the conclusion from the given premises by applying the rules of natural deduction cannot confirm whether the argument is invalid or whether the human fails to see/find the proof.
I. DIRECT PROOF (1) Use the applicable rules to the given premises, derive step-by-step the conclusion. (2) JUSTIFY every step by reference to both: the premises used and the rule applied to them.

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Natural Deduction-1 - Natural Deduction Rules Of...

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