Perhaps a stupid question but important to me:
I know that a second-order logic allows predicates as variables, such as: $$\forall P. P(x) \to P(y).$$
But how about functions? For example, $$f(x) + 1 > 2$$ is a first-order logical formula or a second-order one? And how to evaluate the free variables ($f$ and $x$?) of this formula so that it is true? (I know how to define an evaluation for $x$, like $E(x) = 2$. But how $E(f) = ?$. And in this formula, it seems wrong to evaluate both $f$ and $x$?)
At last, perhaps a more general problem I wish to solve:
Is that possible, or what is a correct way to define the concept of "validation" for all logics?
In other words, is it correct to say "a logical formula is valid if it is true under all evaluations of its free variables" for all kinds of logics?