Editing First-order logic (section)

===Additional quantifiers===
Additional quantifiers can be added to first-order logic.
* Sometimes it is useful to say that "{{math|''P''(''x'')}} holds for exactly one ''x''", which can be expressed as {{math|∃!''x'' ''P''(''x'')}}.  This notation, called [[uniqueness quantification]], may be taken to abbreviate a formula such as {{math|1=∃''x'' (''P''(''x'') {{and}} ∀''y'' (''P''(''y'') → (''x'' = ''y'')))}}.
*''First-order logic with extra quantifiers'' has new quantifiers ''Qx'',..., with meanings such as "there are many ''x'' such that ...". Also see [[branching quantifier]]s and the [[plural quantification|plural quantifier]]s of [[George Boolos]] and others.
* ''[[Bounded quantifier]]s'' are often used in the study of set theory or arithmetic.