Definition·D086
Structure for a language
A nonempty domain together with an actual function or relation, of matching arity, for every symbol of the language.
For a language
, an
-structure is
where
is a nonempty set (the domain, or universe, of
) and:
In words
An L-structure interprets every function symbol as an actual function on the domain, of matching arity and every relation symbol as an actual subset of tuples on the domain, of matching arity.
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
Nonemptiness of
is a standing convention (needed so quantifiers are never vacuous). A
-ary function symbol
(a constant) gets
(D085), and since
is a single-point set, this amounts to picking out one fixed element of
, the usual reading of a constant symbol. Write
and
for the interpretations. The logical symbols
are not interpreted by
: their meaning is fixed once and for all by the satisfaction relation, defined next by recursion on formula structure.