Is this predicate-logic formula a tautology?

Question

$$F(x) \rightarrow \forall x F(x)$$ My teacher says that the formula is a tautology. Because when the antecedent is false, all the formula should always true, and when the antecedent is true, consequent can be infered by using $\forall +$ rule.

Is this correct? I thought some problem in it, but I can't point it out. Sorry for my poor English.

Answer 1

1. The given open formula $$F(x) \rightarrow \forall x F(x)$$ contains a free variable $x,$ so it is merely a propositional function (predicate) rather than a proposition (sentence).
2. It is logically equivalently to $$F(x) \rightarrow \forall y F(y).$$
3. It is invalid (false in some interpretation), and therefore not a tautology (neither an FOL nor a PL tautology):
         domain of discourse: days of $2021$
         $F(x)$ := $\;x$ is Friday
4. But it is satisfiable (true in some interpretation, i.e., not a contradiction):
         domain of discourse: $\mathbb Z$
         $F(x)$ := $\;x+1=x.$