Definition:Signature (Logic)

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Let $\mathcal L$ be a formal language.

A choice of vocabulary for $\mathcal L$ is called a signature for $\mathcal L$.

Signature for Predicate Logic

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

Then a signature for $\mathcal L_1$ is an explicit choice of the alphabet of $\mathcal L_1$.

That is to say, it amounts to choosing, for each $n \in \N$:

A collection $\mathcal F_n$ of $n$-ary function symbols
A collection $\mathcal P_n$ of $n$-ary relation symbols

It is often conceptually enlightening to explicitly address the $0$-ary function symbols separately, as constant symbols.

Also known as

Some sources refer to a signature as a lexicon.

Others call it a language, particularly in the field of model theory.

However, this is easy to conflate with the generic formal language, and therefore discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see