Definition·D077
First-order language
A choice of function symbols and relation symbols, disjoint from each other, together with a rule giving each symbol its arity.
A (first-order) language (or signature) is a triple
where
and
are sets with
(the function symbols and relation symbols) and
assigns each symbol its arity.
In words
A first-order language is a choice of function symbols and relation symbols, with no symbol playing both roles, together with a rule giving each symbol its arity: how many arguments it takes.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Constants are the common special case of a 0-ary function symbol (
): a symbol standing for a fixed element, taking no arguments. Example, a language of arithmetic:
with
,
,
, and
with
; no equality symbol is needed here, since
is a fixed logical symbol available in every language, not a language-specific relation symbol. This is the standard textbook notion of a signature (Enderton, Mendelson). Terms and formulas over a language are built next, as finite sequences of symbols drawn from an alphabet combining
,
, variables, and the fixed logical symbols via disjoint union.