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

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

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

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

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

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

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

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

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.