Is there always true predicate?

Question

Is there such predicate $P(x)$ where the statement is always true, for whatever the $x$ is?

Answer 1

Some (not all) presentations of first-order logic include as primitive symbols "$\top$" and "$\perp$," for "truth" and "falsity" respectively as primitive logical symbols - on the same level as $=$, $\forall,\exists$, and the Boolean connectives and parentheses. (They can be thought of as nullary relation symbols, similarly to how constant symbols can be thought of as nullary function symbols, but since they're included as part of the logical language this isn't necessarily a helpful interpretation.)

The formula "$x=x$" also does the job, although the situation there may be more complicated in certain nonclassical logics. Similarly, formulas like "$P(x)\vee\neg P(x)$" will work in classical logic, and formulas like "$\neg P(x)\vee \neg\neg P(x)$" will work even in intuitionistic logic. Because things do get more complicated when we leave classical logic, I tend to prefer including $\top$ and $\perp$ as logical primitives; but as long as we stay in the context of classical logic, it's unimportant.