##  [Signature](/signature-0) 

 Definition

A specification of the nonlogical vocabulary of a formal language: a set of symbol declarations including constant symbols, function symbols with arities, and relation (predicate) symbols with arities that determine the syntactic building blocks available to form terms and atomic formulas.

 

 

 

 

 

 





## Principle

Principle

A signature fixes the language's vocabulary and arities so that structures interpret exactly those symbols; semantic and syntactic notions (structures, homomorphisms, theories, formulas) are defined relative to a chosen signature and change when the signature changes.

 

 

 

 

 





## Demonstration

Demonstration

A typical arithmetic signature Σ = {0 : constant, s : unary function, + : binary function, × : binary function} determines that terms are built from 0 and applications of s, +, ×, and that models for Σ must provide interpretations for these symbols over some domain.

 

 

 

 

## Misapplication

Misapplication

Changing the signature mid-proof or tacitly adding symbols without recording their arities, conflating the signature with a theory by treating axioms as part of the signature, or assuming symbol meanings across different signatures without translation.

 

 

 

 

 





## Consequence

Consequence

Fixing a signature enables precise definitions of structures, morphisms, and satisfaction; it controls what expressions and properties are even expressible and underpins comparisons between models and transfer results (e.g., homomorphisms, embeddings, reducts and expansions).

 

 

 

 

## Reversal

Reversal

A theory is the reversal in that it consists of axioms formulated in a language over a signature; a theory restricts or extends the class of models for a given signature but does not itself change the primitive symbols allowed unless one changes signature.

 

 

 

 

 





## Boundary

Boundary

Signatures concern only nonlogical symbols and arities; they exclude logical connectors, quantifiers and metalogical devices, though many-sorted signatures, typed signatures or signatures with additional syntactic structure are allowed as explicit generalizations.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between 'signature' and 'language' or 'vocabulary' in informal usage: signature is the formal declaration of symbols and arities, whereas language may be used to include syntactic formation rules and logical symbols; another tension is between fixed versus extensible signatures in modular specification.

 

 

 

 

 





## Synthesis

Synthesis

A signature is the formal listing of constant, function and relation symbols with specified arities that defines the allowable primitive expressions of a formal language; it sets the stage for building terms, stating formulas, and interpreting structures and mappings.