Structure (First-Order Logic)

first-order-logic

Definition

Structure (First-Order Logic)

A structure is a pair $\mathcal{M}=(\mathcal{D},\mathcal{I})$ where:

• $\mathcal{D}$ is a non-empty set, called the domain.
• $\mathcal{I}$ is an interpretation function that maps:

$$\begin{array}{rl}c/0\in \mathcal{F}& :\mathcal{I}(c)\in \mathcal{D}\\ f/n\in \mathcal{F}& :\mathcal{I}(f):{\mathcal{D}}^n\to \mathcal{D}\\ P/n\in \mathcal{P}& :\mathcal{I}(P)\subseteq {\mathcal{D}}^n\end{array}$$

Notation: $c^{\mathcal{M}},f^{\mathcal{M}},P^{\mathcal{M}}$ may be used instead of $\mathcal{I}(c),\mathcal{I}(f),\mathcal{I}(P)$.