Interpretation (First-Order Logic)

logic

Definition

Interpretation (First-Order Logic)

In first-order logic, an interpretation $I$ (often denoted as a structure $\mathcal{A}$) is a tuple $(\mathcal{U},{\cdot }^I)$ consisting of:

1. Universe ($\mathcal{U}$): A non-empty set called the domain of discourse.
2. Mapping (${\cdot }^I$):
   • For each $n$-ary function symbol $f$, a function $f^I:{\mathcal{U}}^n\to \mathcal{U}$.
   • For each $n$-ary predicate symbol $P$, a relation $P^I\subseteq {\mathcal{U}}^n$.

Interpretation vs. Assignment

Interpretation vs. Assignment

The interpretation $I$ is fixed for a given structure and handles non-logical symbols (constants, functions, predicates). It does not handle logical variables; those are managed by a variable assignment $\sigma$.