Last section we said that FO validities are the necessary truths of FOL.
FO falsities are just the necessary falsities of FOL.
FO falsities are related to taut-falsities exactly how FO validities are related to tautologies. All taut-falsities are also FO falsities but not vice versa.
Remember that FOL has all of the truth-functional connectives of BOOL and PROP. That means if those connectives make a sentence always false in those systems (taut-falsity), such as P&~P, then it will still be always false in FOL.
But FOL has more than BOOL and PROP. Namely, the quantifiers and identity: ∀, ∃, and =. If those symbols ensure that a sentences is always false, like ~(p=p), then the sentence will be an FO-falsity but not a taut-falsity.
Some FO-falsities are easy to remember, like ~(p=p). Here are two other key ideas to keep in mind:
First, if we negate any FO validity, we get an FO-falsity. For example, here’s an FO validity: ~AxP(x)→Ex~P(x). If we negate that, we get an FO-falsity:
Second, anything that occurs inside the scope of a quantifier, depends on the quantifier.
For example, P(a)&~P(a) is a taut-falsity. It just depends on the truth-functional connectives.
But this sentence:
is an FO-falsity but not a taut-falsity. It depends not just on & and ~, but also on the fact that the contradiction they make still holds inside the scope of the quantifier.