# Definition:Classes of WFFs

## Definition

Let $\mathcal L_1$ denote the language of predicate logic.

The set of all WFFs of $\mathcal L_1$ formed with relation symbols from $\mathcal P$ and function symbols from $\mathcal F$ can be denoted $WFF \left({\mathcal P, \mathcal F}\right)$.

If so desired, the parameters can also be emphasized by writing $WFF \left({\mathcal P, \mathcal F, \mathcal K}\right)$ instead.

To specify $\mathcal P$, one speaks of **WFFs with relation symbols from $\mathcal P$**.

To specify $\mathcal F$, one speaks of **WFFs with function symbols from $\mathcal F$**.

To specify $\mathcal K$, one speaks of **WFFs with parameters from $\mathcal K$**.

Of course, combinations of these are possible.

Several classes of WFFs are often considered and have special names.

### Plain WFF

A **plain WFF** of predicate logic is a WFF with no parameters.

Thus $WFF \left({\mathcal P, \mathcal F, \varnothing}\right)$ is the set of all **plain WFFs** with relation symbols from $\mathcal P$ and function symbols from $\mathcal F$.

It is immediate that a **plain WFF** is a WFF with parameters from $\mathcal K$ for *all* choices of $\mathcal K$.

### Sentence

A WFF is said to be a **sentence** if and only if it contains no free variables.

To denote particular classes of **sentences**, $SENT \left({\mathcal P, \mathcal F, \mathcal K}\right)$ and analogues may be used, similar to the notation for classes of WFFs.

### Plain Sentence

A WFF is said to be a **plain sentence** iff it is both plain and a sentence.

That is, if it contains free variables nor parameters.

Thus, **plain sentences** are those WFFs which are in $SENT \left({\mathcal P, \mathcal F, \varnothing}\right)$.

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*: $\S 2.3$