Editing First-order logic (section)

===First-order structures===
{{Main|Structure (mathematical logic)}}

The most common way of specifying an interpretation (especially in mathematics) is to specify a ''structure'' (also called a ''model''; see below). The structure consists of a domain of discourse ''D'' and an interpretation function {{mvar|I}} mapping non-logical symbols to predicates, functions, and constants.

The domain of discourse ''D'' is a nonempty set of "objects" of some kind. Intuitively, given an interpretation, a first-order formula becomes a statement about these objects; for example, <math>\exists x P(x)</math> states the existence of some object in ''D'' for which the predicate ''P'' is true (or, more precisely, for which the predicate assigned to the predicate symbol ''P'' by the interpretation is true). For example, one can take ''D'' to be the set of [[Integer|integers]].

Non-logical symbols are interpreted as follows:

* The interpretation of an ''n''-ary function symbol is a function from ''D''<sup>''n''</sup> to ''D''. For example, if the domain of discourse is the set of integers, a function symbol ''f'' of arity 2 can be interpreted as the function that gives the sum of its arguments. In other words, the symbol ''f'' is associated with the function {{tmath|I(f)}} which, in this interpretation, is addition.

* The interpretation of a constant symbol (a function symbol of arity 0) is a function from ''D''<sup>0</sup> (a set whose only member is the empty [[tuple]]) to ''D'', which can be simply identified with an object in ''D''. For example, an interpretation may assign the value <math>I(c)=10</math> to the constant symbol <math>c</math>.

* The interpretation of an ''n''-ary predicate symbol is a set of ''n''-tuples of elements of ''D'', giving the arguments for which the predicate is true. For example, an interpretation <math>I(P)</math> of a binary predicate symbol ''P'' may be the set of pairs of integers such that the first one is less than the second. According to this interpretation, the predicate ''P'' would be true if its first argument is less than its second argument. Equivalently, predicate symbols may be assigned [[Boolean-valued function]]s from ''D''<sup>''n''</sup> to <math>\{\mathrm{true, false}\}</math>.