A nonlogical symbol that denotes an n-ary function on the domain and, when applied to terms, yields a new term.

Principle

Function symbols are term-forming operators: an n-ary function symbol applied to n terms produces a compound term whose interpretation is the result of applying the corresponding function to the interpreted arguments.

Demonstration

If f is a binary function symbol and a, b are terms, then f(a,b) is a term; in an interpretation its value is f_M(interp(a), interp(b)), an element of the domain obtained by the model's interpretation of f.

Misapplication

Using function symbols where predicates are required—e.g., treating f(a) as a sentence rather than a term—violates syntax and prevents truth-evaluation since terms are not evaluated as propositions.

Consequence

Proper use of function symbols allows construction of complex terms that can be nested and substituted into predicates, enabling rich term algebra and compositional semantics of expressions.

Reversal

If function symbols were interpreted as relations rather than functions, applying them to terms would not yield a unique term value and would break the syntax-semantics correspondence for term formation.

Boundary

Excludes higher-order functionals unless the language explicitly includes them; does not treat metavariables or semantic functions outside the language's signature; arity is fixed by the signature.

Semantic Tension

Tension exists between viewing function symbols syntactically as term constructors and semantically as operations on the domain; confusion arises when one speaks interchangeably of the symbol and the interpreted mapping.

Synthesis

A function symbol is a signature element that constructs compound terms from argument terms and whose interpretation in a model is an n-ary operation on the domain yielding the term's denotation.