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 $y = 2$.

Sources

• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 2.3$