Textbook
A proposition is an assertion that is true or false once its meaning is fixed. A predicate is a condition whose truth depends on its arguments. Over the real numbers, the predicate $P(x)$ means $x>3$. Then $P(4)$ is true and $P(2)$ is false.
Before substitution, $x>3$ has no single fixed answer: $x$ remains a free variable. The argument domain is part of the definition. Although $P(x)$ resembles a function call, its result here is a truth value. A predicate may take several arguments, such as the condition $x<y$.
The condition must be defined for every admissible argument. For example, a condition involving division must exclude a zero denominator or explicitly assign a value there. An undefined expression is not automatically false. For a predicate with several arguments, their order matters: “the first number is smaller than the second” can change truth when the numbers are exchanged. A predicate therefore requires a domain, an argument count, and a precise checking rule, not merely a letter.
Definitions and further reading: Mathematics for Computer Science.
In plain language
A predicate resembles a turnstile rule: substituting an admissible visitor or number produces either “passes” or “fails.” The condition “greater than three” has no single answer until a number is supplied. Its domain determines where the rule applies; presenting a sandwich instead of a ticket does not turn an undefined expiry check into falsehood. A mathematical turnstile gives a definite answer without consulting the guard’s mood.
Example
The example applies the predicate $P(x): x>3$ on 2, 3, and 4. The boundary matters: “greater than” does not include equality.
Predicates: Conditions With Arguments
Argument 2
Two is below the boundary, so the result is false.
Step 1 of 3: Argument 2
Read all steps
Argument 2
Two is below the boundary, so the result is false.
- State
- 2currentfalsecurrent
Argument 3
Equality does not satisfy the strict condition.
- State
- 3currentfalsecurrent
Argument 4
Four exceeds three, so the result is true.
- State
- 4currenttruecurrent
Only the argument changes across the three steps; the condition stays fixed. Two and three fail for different reasons: the former is below the boundary, while the latter equals it. Four passes the strict comparison. Including three would silently replace the predicate with “at least three.” A separate boundary check exposes a changed rule even when the remaining answers look plausible.
Related concepts: The universal quantifier.