Editing Necessity and sufficiency (section)

==Relationship between necessity and sufficiency==
[[File:Set intersection.svg|thumb|260 px|Being in the purple region is sufficient for being in A, but not necessary. Being in A is necessary for being in the purple region, but not sufficient. Being in A and being in B is necessary and sufficient for being in the purple region.]]

A condition can be either necessary or sufficient without being the other. For instance, ''being a [[mammal]]'' (''N'') is necessary but not sufficient to ''being human'' (''S''), and that a number <math>x</math> ''is rational'' (''S'') is sufficient but not necessary to <math>x</math> ''being a [[real number]]'' (''N'') (since there are real numbers that are not rational).

A condition can be both necessary and sufficient. For example, at present, "today is the [[Fourth of July]]" is a necessary and sufficient condition for "today is [[Independence Day (United States)|Independence Day]] in the [[United States]]". Similarly, a necessary and sufficient condition for [[Inverse matrix|invertibility]] of a [[matrix (mathematics)|matrix]] ''M'' is that ''M'' has a nonzero [[determinant]].

Mathematically speaking, necessity and sufficiency are [[duality (mathematics)|dual]] to one another. For any statements ''S'' and ''N'', the assertion that "''N'' is necessary for ''S''" is equivalent to the assertion that "''S'' is sufficient for ''N''". Another facet of this duality is that, as illustrated above, conjunctions (using "and") of necessary conditions may achieve sufficiency, while disjunctions (using "or") of sufficient conditions may achieve necessity. For a third facet, identify every mathematical [[predicate (mathematics)|predicate]] ''N'' with the set ''T''(''N'') of objects, events, or statements for which ''N'' holds true; then asserting the necessity of ''N'' for ''S'' is equivalent to claiming that ''T''(''N'') is a [[superset]] of ''T''(''S''), while asserting the sufficiency of ''S'' for ''N'' is equivalent to claiming that ''T''(''S'') is a [[subset]] of ''T''(''N'').

Psychologically speaking, necessity and sufficiency are both key aspects of the classical view of concepts. Under the classical theory of concepts, how human minds represent a category X, gives rise to a set of individually necessary conditions that define X. Together, these individually necessary conditions are sufficient to be X.<ref>{{cite web | url=https://iep.utm.edu/classical-theory-of-concepts/ | title=Classical Theory of Concepts, the &#124; Internet Encyclopedia of Philosophy }}</ref> This contrasts with the probabilistic theory of concepts which states that no defining feature is necessary or sufficient, rather that categories resemble a family tree structure.