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

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 $x = x$, $x \
eq x$, $x \
eq 1$ 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.