Definition:Sentence

Definition
A sentence of predicate calculus is a plain WFF with no free variables.

The set of all sentences in the vocabulary $$\mathcal{P}$$ is denoted:
 * $$SENT \left({\mathcal{P}, \varnothing}\right)$$.

Truth Value
A sentence has a truth value as soon as we specify:
 * 1) The meanings of all the symbols in the vocabulary;
 * 2) The range of values which the varables can take.

Example
The WFF:
 * $$\exists x: \forall y: x \le y$$

is true if $$\le$$ has its usual meaning, and the variables range over the set of natural numbers (since $$\forall y \in \N: 0 \le y$$).

However, it is false if the variables range over the set of integers.

Contrast with
The truth value of a WFF with one or more free variables depends on the values of those free variables.

For example, $$x \le y$$ is true if $$x = 2$$ and $$y = 3$$ but not if $$x = 3$$ and $$y = 2$$.

Sentence with Parameters
A WFF of predicate calculus with parameters from $$\mathcal{K}$$ but no free variables is called a sentence with parameters from $$\mathcal{K}$$ and is denoted:
 * $$SENT \left({\mathcal{P}, \mathcal{K}}\right)$$.