An assignment of meanings to the nonlogical symbols (constants, function symbols, predicate symbols) of a formal language together with a domain (universe) of discourse; the assignment determines which formulas of the language are true, false, or indeterminate relative to that assignment.

Principle

Map each nonlogical symbol to a corresponding mathematical object, relation or function on a chosen domain so that atomic formulas receive truth values and complex formulas are evaluated compositionally.

Demonstration

In a first-order language with a constant c, a unary predicate P, and a binary function f, an interpretation can choose the domain N (natural numbers), assign c the value 0, interpret P as “is even,” and interpret f as addition; under this assignment the sentence P(f(c,c)) is true.

Misapplication

Treating a syntactic substitution or a naming convention as an interpretation (for example, renaming symbols without specifying a domain or the meanings of predicates) confuses formal semantics with mere rewriting and can lead to invalid conclusions about truth.

Consequence

Given an interpretation one can evaluate satisfaction, truth in that structure, and thereby separate semantic questions (what is true in that world) from syntactic derivability; interpretations produce models when they make chosen formulas true.

Reversal

Instead of assigning meanings to symbols, one may fix a set of sentences and ask which structures satisfy them; the reversal emphasizes classes of structures (models) rather than a single mapping from symbols to objects.

Boundary

An interpretation concerns only the formal mapping from symbols to domain elements and relations; it does not by itself provide proof procedures, intentions of language users, or empirical justification external to the formal system.

Semantic Tension

Tension arises between an 'intended interpretation' (a natural-language semantics or informal reading) and a 'formal interpretation' (an arbitrary mathematical assignment): both are called interpretations but differ in normative force and intended scope.

Synthesis

An interpretation pairs a domain with a symbol-to-object mapping so that the formal language acquires determinate truth values; it is the semantic construction that links syntax to mathematical structures and enables the notion of satisfaction.