Definition:Interpretation

Definition

Let $S$ be the codomain of a mapping $f$ whose domain consists of the set of expressions of a formal language $\FF$.

Then $S$ together with $f$ are known as an interpretation.

Semantic Rule

The mapping $f$ is known as a semantic rule.

Semantic Value

The image $s \in S$ of an expression $e \in \EE$ under $f$ is known as the semantic value of $e$.

Examples

Aristotle

The symbol $\text {Aristotle}$ has no meaning in itself.

However, it acquires meaning when it is assigned an interpretation such that $\text {Aristotle}$ stands for the person Aristotle.

Propositional Calculus

The propositional calculus $\CC$ has a domain $\set {\T, \F}$ representing true and false.

The semantic rules of $\CC$ assign to each WFF of $\CC$ one or other of $\T$ and $\F$.

The connectives of $\CC$ are assumed in this context to have a fixed meaning.

Also see

• Results about interpretations can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): interpretation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): interpretation