The set of nonlogical symbols (predicate, function, and constant symbols) together with any specified sorts that defines the signature of a formal language.

Principle

A vocabulary (signature) fixes the primitive symbols and their arities and sorts; it determines which formulas and terms are well-formed and constrains the class of interpretations (models) by specifying which symbols must be assigned meanings.

Demonstration

A vocabulary Σ might consist of a binary predicate R, a unary function f, and constants c,d; the language built from Σ permits formation of terms like f(f(c)) and formulas like ∃x R(f(x),d) according to the arities and sorts in Σ.

Misapplication

Using symbols not in the declared vocabulary or changing a symbol's arity mid-proof—e.g., treating R as binary in one step and ternary in the next—violates formation rules and invalidates derivations.

Consequence

A clearly specified vocabulary yields a stable framework for syntax, semantics, and proof theory: it ensures well-formedness, uniform interpretation across models, and precise statements about expressibility and definability.

Reversal

If the vocabulary were allowed to vary arbitrarily within a theory, syntactic coherence and semantic comparisons between models would break down; theories would lack a fixed domain of discourse of primitive symbols.

Boundary

Does not include logical symbols (connectives, quantifiers) which are typically fixed, nor metalinguistic notation; may or may not include equality or type/sort declarations depending on the formalism.

Semantic Tension

Tension appears between using a minimal vocabulary to favor generality and extending the vocabulary to name convenient concepts; adding symbols increases expressive convenience but may obscure definability relations and relative interpretability.

Synthesis

A vocabulary is the language's signature: the finite or infinite collection of predicate, function, and constant symbols (and sorts) whose declared arities and identities determine how terms and formulas are formed and how models must interpret them.