##  [Interpretation](/interpretation-0) 

 Definition

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

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

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

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

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

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

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

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

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.