# Definition:Signature (Logic)/Predicate Logic

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## Definition

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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Sources

- 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\mathrm{II}.5$ First-Order Logic Syntax: Definition $\mathrm{II.5.2}$