Interpretations of logic formulas

Question

Does

$\forall x (F \land G)$

imply that F and G are both tautologies? So that

$\forall x (F \land G) \iff (\forall xF \land \forall x G)$

means "for all x F(x) and G(x) are true iff F is true for all x and G is true for all x"?

Answer 1

The equivalence you indicate holds, yes, but that does not mean that $F$ and $G$ are both tautologies.

Example:

$$\forall x (Cube(x) \land Large(x)) \Leftrightarrow \forall x \ Cube(x) \land Large(x)$$

holds, since if everything is a large cube, then everything is a cube, and everything is large, and vice versa.

However, neither $Cube(x)$ nor $Large(x)$ are tautologies.