Must a predicate $P(x)$ be dependent on its variables?

Question

Must a predicate $P(x)$ be dependent on its variables?

Suppose we have a function $P(x)$ which returns either true or false irrespective of the value of $x$. Is this still a predicate? Is it valid to write $P(x)$ when $x$ is independent of $P$?

I imagine $P$ isn't a predicate when written such; or it's a "type error" to even write $P(x)$ when $x$ is independent of $P$ e.g. a bit like saying:

"$35$ degrees centigrade is tall"

Answer 1

The standard convention is to write $P(x_1, \ldots, x_n)$ as a metanotation for a syntactic predicate whose free variables are contained in $\{x_1, \ldots, x_n\}$. The formulas $x = x$, $x \neq x$, $x \neq 1$ and $1 = 1$ are all perfectly good possibilities for $P(x)$ under this convention even though only one of them has a truth value that depends on the value of $x$. Whether $P(x)$ actually depends on $x$ is a semantic issue and the use of the notation $P(x)$ is neutral about the existence of an actual semantic dependency.

Answer 2

Does $f(x)=3$ not define a function from numbers to numbers? Why would it be any different for a function from numbers, say, to truth values?