If $P$ is a predicate symbol with arity $1$, why is $P(x)$ not a term?

Question

If $P$ is a predicate symbol with arity $1$, why is $P(x)$ not a term?

Is it because terms can only consist of function symbols applied to variables? i.e $F(x)$ where $F$ is a function symbol of arity $1$ and $x$ is a variable.

Answer 1

The basic principle: think of terms as denoting expressions whose role is to pick out objects in the domain (relative to a given assignment of objects as denotations to any lurking free variables). Terms are not sentences, i.e. not expressions which are true or false (relative to an assignment, etc.) but rather the most basic kind of sentence is formed by taking an $n$-place predicate and applying it to $n$ terms.

Thus an expression like '$2$' or '$x$' counts as a simple term, and function expressions like '$2 +3$' (or '$f(2, 3)$') and $f(x, y)$ count as terms too. The term '$2 +3$' is plainly not a sentence (it isn't the sort of thing that can be true or false). But it can feature as part of a sentence if we supply a predicate as in (informally) '$2 + 3$ is odd' or '$2 + 3 = 5$' (or, if it helps, '$=(2 +3, 5)$').

On the other hand, an expression like $P(2)$, with $P$ a one-place predicate, is sentence (a 'closed wff') not a term, and $P(x)$ is is an 'open sentence' in the jargon and again not a term.

This should be all explained in any standard text.