Definition:Signature (Logic)/Predicate Logic
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Definition
Let $\LL_1$ be the language of predicate logic.
Then a signature for $\LL_1$ is an explicit choice of the alphabet of $\LL_1$.
That is to say, it amounts to choosing, for each $n \in \N$:
- A collection $\FF_n$ of $n$-ary function symbols
- A collection $\PP_n$ 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): $\text{II}.5$ First-Order Logic Syntax: Definition $\text{II.5.2}$