variable a. To avoid the vacuity and get a falsehood you have to quantify out the free variable, as
((all) a. P(a)) --> ((exists) x. P(x))
here is a proof of this apparent falsehood: (all) a. P(a) --> (all) b. P(b) (exists) b. P(b) --> (exists) x. P(x) -------------------------------- ------------------------------------- (all) a. P(a) --> P(b) P(b) --> (exists) x. P(x) --------------------------------------------------------------------------------------- (all) a. P(a) --> (exists) x. P(x) (it's is just adjunctions in foundations again: we compose the counit of the adjunction of one quantifier with the unit of the other one. note that it is valid constructively. i am not sure, but i think gentzen gave a different proof in an example.) -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]