22 Oct
2011
22 Oct
'11
1:03 p.m.
Dear Vaughan, - On 21 Oct 2011, at 23:06, Vaughan Pratt wrote:
The logical deduction
(all) x. P(x) ------------- P(a) ------------- (exists) x. P(x)
Even simpler: P(a) --> (exists) x. P(x) is a theorem of the pure predicate calculus, which is "clearly" false in the empty universe when P is taken to be the identically true predicate.
No, that's wrong. Your formula P(a) --> (exists) x. P(x) is valid in the empty carrier, because it is (vacuously) true under every interpretation of the free 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)) Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]