Definition:Classes of WFFs

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Definition

Let $\LL_1$ denote the language of predicate logic.

The set of all WFFs of $\LL_1$ formed with relation symbols from $\PP$ and function symbols from $\FF$ can be denoted $\map {WFF} {\PP, \FF}$.

If so desired, the parameters can also be emphasized by writing $\map {WFF} {\PP, \FF\ ,KK}$ instead.

To specify $\PP$, one speaks of WFFs with relation symbols from $\PP$.

To specify $\FF$, one speaks of WFFs with function symbols from $\FF$.

To specify $\KK$, one speaks of WFFs with parameters from $\KK$.

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 $\map {WFF} {\PP, \FF, \O}$ is the set of all plain WFFs with relation symbols from $\PP$ and function symbols from $\FF$.

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

Sentence

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

To denote particular classes of sentences, $\map {SENT} {\PP, \FF, \KK}$ and analogues may be used, similar to the notation for classes of WFFs.

Plain Sentence

A WFF is said to be a plain sentence if and only if it is both plain and a sentence.

That is, if and only if it contains free variables nor parameters.

Thus, plain sentences are those WFFs which are in $\map {SENT} {\PP, \FF, \O}$.

Sources

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