Definition:Sentence

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:
 * $$SENT \left({\mathcal{P}, \mathcal{K}}\right)$$.

Truth Value
A sentence with parameters from $$\mathcal{K}$$ 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;
 * 3) The meanings of all the parameter symbols that appear in it.

Example
The WFF:
 * $$\forall y: 0 \le y$$

is true if $$\le$$ and $$0$$ have their usual meanings, and the variable $$y$$ ranges over the set of natural numbers.

Sentence with Parameters from a Model
This is a special case of a sentence with parameters from $$\mathcal{K}$$.

Let $$\mathcal{M}$$ be a model for predicate calculus of type $\mathcal{P}$ whose universe set is $$M$$.

A sentence with parameters from $$M$$ is a sentence whose parameters are taken from $$M$$.

The set of all such sentences is denoted:
 * $$SENT \left({\mathcal{P}, M}\right)$$.

Plain Sentence
A plain sentence (or just sentence) of predicate calculus is a plain WFF with no free variables.

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

Truth Value
A plain 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.

Note
Note that a sentence with parameters from $$\mathcal{K}$$ is, by definition, a sentence whose parameters are all in $$\mathcal{K}$$.

That is, none of its parameters come from outside of $$\mathcal{K}$$.

Hence a plain sentence is a sentence with parameters from $$\mathcal{K}$$ for all $$\mathcal{K}$$.

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 $$y = 2$$.