In the university course I'm taking, a predicate is defined as a mathematical statement whose truth value depends on the variables involved in the statement. This definition makes me wonder whether a predicate can be a contradiction.
For instance, if we use $P(x)$ to denote the statement "$x$ is an even number", clearly $P(x)$ itself is a predicate.
However, if we consider the statement
$x$ is an even number and $x$ is not an even number,
it can be written as $P(x)\land\lnot P(x)$.
I think based on my knowledge it is a valid statement, but I am not sure whether it is a predicate. The part I am not sure about is:
By the law of excluded middle and well-definedness of mathematical statements, the statement should always be false regardless of the value of $x$. Hence, without knowing the value of $x$, we can say that this statement is false. In this sense, the truth value of this statement does not depend on the value of $x$, so it should not be a predicate.
However, I feel in the above argument (1), I unconsciously use $\forall$ to quantifier the statement, which makes it a proposition (or at least we can directly tell the truth value without considering $x$). If this is true, this statement should be a predicate.
I feel I could not figure out which of my thoughts is true, or both of them are wrong, could someone please help me on this?