What is a Predicate?

“A predicate is a statement with blanks that become true or false when we fill them in.” More formally, a predicate is a symbol that represents a relation. $P(a)$ is an atomic formula, where $P$ is the predicate, and $a$ is “filled” in $P$ the predicate, hence creating a logical formula.

Example, $P$ is a predicate symbol, $>$ is a predicate symbol, but ‘$x > 0$’ is a predicate.

Predicates are a fundamental notion in first-(and higher)-order-logic, because they let us open up the atoms, peer inside them. This isn’t the case with propositional logic, over there atoms can not be peered inside, ‘$p$’ and ‘$q$’ stand for whole propositions, i.e., the atoms are indivisible. Although compound formulae/propositions such as ‘$p \land q$’ are fine for inspection.

Informally, a predicate is a statement with variables waiting to be filled in.

\[P(x) : x > 10\]

is a predicate expressing a condition on $x$. Once we have $x$, we can evaluate the truth value of $P(15)$, $P(5)$ etc. Although, generally we do need to specify the domain as well.

Also, we don’t always need to specify a value to ‘convert’ a predicate into a statement. We can also do so by quantifying the variable, for example, this is a statement:

\[\exists x \in \mathbb{Z}, x > 10\]

(there exists $x$ in $\mathbb{Z}$, such that $x > 10$)

Also, when thought of as a relation, a predicate has to ‘output’ a boolean true/false. So, while x > 10 is a predicate, x + 10 is not.