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 atom to a truth value, where $\mathcal{P}$ is the set of atoms.

The assignment extends to formulas compositionally: the value of a complex formula is determined by the values of its subformulas.

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First-Order Logic

Definition

Interpretation (First-Order Logic)

An interpretation is a pair $(\mathcal{M},\ell )$ where:

• $\mathcal{M}=(\mathcal{D},\mathcal{I})$ is a structure
• $\ell$ is a variable assignment for $\mathcal{M}$

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