Use truth tables to test the validity of the argument. p ? ~q q ? ~p/
? p ? q
Since a truth table would be so long, and you'd have to have a very wide piece of paper to get it all in, [ I'd just take the four cases and work each one out separately and see if we get T in every case: Case p=T and q=T [(p -> ~q) ^ (q -> ~p)] -> (p V q) [(T -> ~T) ^ (T -> ~T)] -> (T V T) [(T -> F ) ^ (T -> F)] -> T [ F ^ F ] -> T F -> T
T Case p=T and q=F [(p -> ~q) ^ (q -> ~p)] -> (p V q) [(T -> ~F) ^ (F -> ~T)] -> (T V F) [(T -> T ) ^ (F -> F) ] -> T [ T ^ T ] -> T T -> T T Case p=F and q=T [(p -> ~q) ^ (q -> ~p)] -> (p V q) [(F -> ~T) ^ (T -> ~F)] -> (F V T) [(F -> F ) ^ (T -> F )] -> T [ T ^ F ] -> F F -> F T Case p=F and q=F [(p -> ~q) ^ (q -> ~p)] -> (p V q) [(F -> ~F) ^ (F -> ~F)] -> (F V F) [(F -> T ) ^ (F -> T )] -> F [ T ^ F ] -> F F -> F T All four cases come out T, so the argument is valid. ]
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