The statement is false when its negation is true. What is its negation?
Does the sentence mean "for all x is true?
If so, and we are asked to prove that the sentence is not a tautology, does it mean that all we need to do is come up with some interpretation of what x and P(x) are that would make true and hence false?
Hi SlipEternal, the negation is that there exists some x for which P(x)->Q(x) is false.
I was trying to clarify the how the method of proof where the statement is considered a formula of a first-order language.
Since the statement has no free variables, the proof would only depend on the structure and not the valuation of x.
However, I'm not entirely clear on whether this meant that we could define any structure and interpret/define P(x) and Q(x) in any manner that would suit the proof.
"Every human is a man" is obviously false. That is "for any x, if x is a human, then x is a man" is false. There is an x such that x is a human and x is not a man. Clearly true for almost any x named Sue.