Interpretation

logic

Definition

Interpretation

In logic, an interpretation provides the “meaning” of the non-logical symbols in a formal language. Its definition depends on the level of the logic:

Propositional Logic

Definition

Interpretation (Propositional Logic)

In propositional logic, an interpretation is a truth assignment $I:\mathcal{P}\to \{0,1\}$ that maps each proposition symbol to a truth value.

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First-Order 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 $P^I\subseteq {\mathcal{U}}^n$.

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