Editing Necessity and sufficiency

{{Short description|Terms to describe a conditional relationship between two statements}}
{{about|the formal terminology in logic|causal meanings of the terms|Causality|the concepts in statistics|Sufficient statistic}}
{{redirect|Necessary But Not Sufficient|the novel by Eliyahu Goldratt|Necessary But Not Sufficient (novel)}}
In [[logic]] and [[mathematics]], '''necessity''' and '''sufficiency''' are terms used to describe a [[material conditional|conditional]] or implicational relationship between two [[Statement (logic)|statements]]. For example, in the [[Conditional sentence|conditional statement]]: "If {{mvar|P}} then {{mvar|Q}}", {{mvar|Q}} is '''necessary''' for {{mvar|P}}, because the [[Truth value|truth]] of {{mvar|Q}} is guaranteed by the truth of {{mvar|P}}. (Equivalently, it is impossible to have {{mvar|P}} without {{mvar|Q}}, or the falsity of {{mvar|Q}} ensures the falsity of {{mvar|P}}.)<ref name=":0">{{Cite web|url=https://philosophy.hku.hk/think/meaning/nsc.php|title=[M06] Necessity and sufficiency|website=philosophy.hku.hk|access-date=2019-12-02}}</ref> Similarly, {{mvar|P}} is '''sufficient''' for {{mvar|Q}}, because {{mvar|P}} being true always implies that {{mvar|Q}} is true, but {{mvar|P}} not being true does not always imply that {{mvar|Q}} is not true.<ref>{{Cite book|title=Proofs and Fundamentals: A First Course in Abstract Mathematics|last=Bloch|first=Ethan D.|publisher=Springer|year=2011|isbn=978-1-4419-7126-5|pages=8–9}}</ref>

In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition.<ref>{{Cite web|url=https://www.txstate.edu/philosophy/resources/fallacy-definitions/Confusion-of-Necessary.html|title=Confusion of Necessary with a Sufficient Condition|last=Confusion-of-Necessary|date=2019-05-15|website=www.txstate.edu|language=en|access-date=2019-12-02}}</ref> The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true [[if and only if]] the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false.<ref name=betz>{{cite book|last=Betz|first=Frederick|title=Managing Science: Methodology and Organization of Research|date=2011|publisher=Springer|location=New York|isbn=978-1-4419-7487-7|page=247}}</ref><ref name=Manktelow>{{cite book|last=Manktelow|first=K. I.|title=Reasoning and Thinking|date=1999|publisher=Psychology Press|location=East Sussex, UK|isbn=0-86377-708-2}}</ref><ref name=asnina>{{cite journal|author1=Asnina, Erika |author2=Osis, Janis |author3=Jansone, Asnate |name-list-style=amp |title=Formal Specification of Topological Relations|journal=Databases and Information Systems VII|date=2013|volume=249 |issue=Databases and Information Systems VII |page=175|doi=10.3233/978-1-61499-161-8-175}}</ref>

In [[ordinary English]] (also [[natural language]]) "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. For example, being round is a necessary condition for being a circle, but it is not sufficient since ovals and ellipses are round, but not circles — while being a circle is a necessary and sufficient condition for being round.
Any conditional statement consists of at least one sufficient condition and at least one necessary condition.

In [[Analytics|data analytics]], necessity and sufficiency can refer to different [[Causality|causal]] logics,<ref>{{Cite journal |last=Richter |first=Nicole Franziska |last2=Hauff |first2=Sven |date=2022-08-01 |title=Necessary conditions in international business research–Advancing the field with a new perspective on causality and data analysis |url=https://findresearcher.sdu.dk/ws/files/199180258/1_s2.0_S1090951622000037_main.pdf |journal=Journal of World Business |language=en |volume=57 |issue=5 |pages=101310 |doi=10.1016/j.jwb.2022.101310 |issn=1090-9516|doi-access=free }}</ref> where [[necessary condition analysis]] and [[qualitative comparative analysis]] can be used as analytical techniques for examining necessity and sufficiency of conditions for a particular outcome of interest.

==Definitions==

In the conditional statement, "if ''S'', then ''N''", the expression represented by ''S'' is called the [[Antecedent (logic)|antecedent]], and the expression represented by ''N'' is called the [[consequent]]. This conditional statement may be written in several equivalent ways, such as "''N'' if ''S''", "''S'' only if ''N''", "''S'' implies ''N''", "''N'' is implied by ''S''", {{math|''S'' → ''N''}} , {{math|''S'' ⇒ ''N''}} and "''N'' whenever ''S''".<ref>{{citation|first=Keith|last=Devlin|title=Sets, Functions and Logic / An Introduction to Abstract Mathematics|edition=3rd|publisher=Chapman & Hall|year=2004|isbn=978-1-58488-449-1|pages=22–23}}</ref>

In the above situation of "N whenever S," ''N'' is said to be a '''necessary''' condition for ''S''. In common language, this is equivalent to saying that if the conditional statement is a true statement, then the consequent ''N'' ''must'' be true—if ''S'' is to be true (see third column of "[[truth table]]" immediately below). In other words, the antecedent ''S'' cannot be true without ''N'' being true. For example, in order for someone to be called '''''S'''''ocrates, it is necessary for that someone to be '''''N'''''amed. Similarly, in order for human beings to live, it is necessary that they have air.<ref name=":2">{{Cite web|url=https://www.sfu.ca/~swartz/conditions1.htm#section3|title=The Concept of Necessary Conditions and Sufficient Conditions|website=www.sfu.ca|access-date=2019-12-02}}</ref>

One can also say ''S'' is a '''sufficient''' condition for ''N'' (refer again to the third column of the truth table immediately below). If the conditional statement is true, then if ''S'' is true, ''N'' must be true; whereas if the conditional statement is true and N is true, then S may be true or be false. In common terms, "the truth of ''S'' guarantees the truth of ''N''".<ref name=":2" /> For example, carrying on from the previous example, one can say that knowing that someone is called '''''S'''''ocrates is sufficient to know that someone has a '''''N'''''ame.

A '''''necessary and sufficient''''' condition requires that both of the implications <math>S \Rightarrow N</math> and <math>N \Rightarrow S</math> (the latter of which can also be written as <math>S \Leftarrow N</math>) hold. The first implication suggests that ''S'' is a sufficient condition for ''N'', while the second implication suggests that ''S'' is a necessary condition for ''N''. This is expressed as "''S'' is necessary and sufficient for ''N'' ", "''S'' [[if and only if]] ''N'' ", or <math>S \Leftrightarrow N</math>.

{| class="wikitable" style="margin:1em auto; text-align:center;"
|+ Truth table
|-
! scope="col"  style="width:20%" | {{nobold|{{mvar|S}}}}
! scope="col"  style="width:20%" | {{nobold|{{mvar|N}}}}
! scope="col"  style="width:20%" | <math>S \Rightarrow N</math>
! scope="col"  style="width:20%" | <math>S \Leftarrow N</math>
! scope="col"  style="width:20%" | <math>S \Leftrightarrow N</math>
|-
| T || T || T || T || T
|-
| T || style="background:papayawhip" | F || style="background:papayawhip" | F || T || style="background:papayawhip" | F
|-
| style="background:papayawhip" | F || T || T || style="background:papayawhip" | F || style="background:papayawhip" | F
|-
| style="background:papayawhip" | F || style="background:papayawhip" | F || T || T || T
|}

==Necessity==

[[File:Solar eclipse 1999 4.jpg|thumb|right|200px|The sun being above the horizon is a necessary condition for direct sunlight; but it is not a sufficient condition, as something else may be casting a shadow, e.g., the moon in the case of an [[solar eclipse|eclipse]].]]

The assertion that ''Q'' is necessary for ''P'' is colloquially equivalent to "''P'' cannot be true unless ''Q'' is true" or "if Q is false, then P is false".<ref name=":2" /><ref name=":0" /> By [[contraposition]], this is the same thing as "whenever ''P'' is true, so is ''Q''".

The logical relation between ''P'' and ''Q'' is expressed as "if ''P'', then ''Q''" and denoted "''P'' ⇒ ''Q''" (''P'' [[Logical consequence|implies]] ''Q''). It may also be expressed as any of "''P'' only if ''Q''", "''Q'', if ''P''", "''Q'' whenever ''P''", and "''Q'' when ''P''". One often finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition (i.e., individually necessary and jointly sufficient<ref name=":2" />), as shown in Example 5.

;Example 1: For it to be true that "John is a bachelor", it is necessary that it be also true that he is
:# unmarried,
:# male,
:# adult,
:since to state "John is a bachelor" implies John has each of those three additional [[Predicate (mathematical logic)|predicates]].

;Example 2: For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime.

;Example 3:Consider thunder, the sound caused by lightning. One says that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there is lightning, there is thunder. The thunder ''does not cause'' the lightning (since lightning causes thunder), but because lightning always comes with thunder, we say that thunder is necessary for lightning. (That is, in its formal sense, necessity doesn't imply causality.)

;Example 4:Being at least 30 years old is necessary for serving in the U.S. Senate. If you are under 30 years old, then it is impossible for you to be a senator. That is, if you are a senator, it follows that you must be at least 30 years old.

;Example 5:In [[algebra]], for some [[Set (mathematics)|set]] ''S'' together with an [[Binary operation|operation]] <math>\star</math> to form a [[group (mathematics)|group]], it is necessary that <math>\star</math> be [[associative]]. It is also necessary that ''S'' include a special element ''e'' such that for every ''x'' in ''S'', it is the case that ''e'' <math>\star</math> ''x'' and ''x'' <math>\star</math> ''e'' both equal ''x''. It is also necessary that for every ''x'' in ''S'' there exist a corresponding element ''x&Prime;'', such that both ''x'' <math>\star</math> ''x&Prime;'' and ''x&Prime;'' <math>\star</math> ''x'' equal the special element ''e''. None of these three necessary conditions by itself is sufficient, but the [[conjunction (logic)|conjunction]] of the three is.

==Sufficiency==

[[File:ICE 3 Fahlenbach.jpg|thumb|right|200px|That a train runs on schedule is a sufficient condition for a traveller arriving on time (if one boards the train and it departs on time, then one will arrive on time); but it is not a necessary condition, since there are other ways to travel (if the train does not run to time, one could still arrive on time through other means of transport).]]

If ''P'' is sufficient for ''Q'', then knowing ''P'' to be true is adequate grounds to conclude that ''Q'' is true; however, knowing ''P'' to be false does not meet a minimal need to conclude that ''Q'' is false.

The logical relation is, as before, expressed as "if ''P'', then ''Q''" or "''P'' ⇒ ''Q''". This can also be expressed as "''P'' only if ''Q''", "''P'' implies ''Q''" or several other variants. It may be the case that several sufficient conditions, when taken together, constitute a single necessary condition (i.e., individually sufficient and jointly necessary), as illustrated in example 5.

;Example 1:"John is a king" implies that John is male. So knowing that John is a king is sufficient to knowing that he is a male.

;Example 2:A number's being divisible by 4 is sufficient (but not necessary) for it to be even, but being divisible by 2 is both sufficient and necessary for it to be even.

;Example 3: An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense that hearing thunder, and unambiguously recognizing it as such, justifies concluding that there has been a lightning bolt.

;Example 4:If the U.S. Congress passes a bill, the president's signing of the bill is sufficient to make it law. Note that the case whereby the president did not sign the bill, e.g. through exercising a presidential [[veto#United States|veto]], does not mean that the bill has not become a law (for example, it could still have become a law through a congressional [[veto override|override]]).

;Example 5:That the center of a [[playing card]] should be marked with a single large spade (♠) is sufficient for the card to be an ace. Three other sufficient conditions are that the center of the card be marked with a single diamond (♦), heart (♥), or club (♣). None of these conditions is necessary to the card's being an ace, but their [[disjunction]] is, since no card can be an ace without fulfilling at least (in fact, exactly) one of these conditions.

==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.

==Simultaneous necessity and sufficiency==
{{See also|Material equivalence}}

To say that ''P'' is necessary and sufficient for ''Q'' is to say two things:
# that ''P'' is necessary for ''Q'', <math>P \Leftarrow Q</math>, and that ''P'' is sufficient for ''Q'', <math>P \Rightarrow Q</math>.
# equivalently, it may be understood to say that ''P'' and ''Q'' is necessary for the other, <math>P \Rightarrow Q \land Q \Rightarrow P</math>, which can also be stated as each ''is sufficient for'' or ''implies'' the other.

One may summarize any, and thus all, of these cases by the statement "''P'' [[if and only if]] ''Q''", which is denoted by <math>P \Leftrightarrow Q</math>, whereas cases tell us that <math>P \Leftrightarrow Q</math> is identical to <math>P \Rightarrow Q \land Q \Rightarrow P</math>.

For example, in [[graph theory]] a graph ''G'' is called [[Bipartite graph|bipartite]] if it is possible to assign to each of its vertices the color ''black'' or ''white'' in such a way that every edge of ''G'' has one endpoint of each color. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length [[cycle (graph theory)|cycles]]. Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and conversely. A philosopher<ref name="stan">[http://plato.stanford.edu/entries/logic-intensional/ Stanford University primer, 2006].</ref> might characterize this state of affairs thus: "Although the concepts of bipartiteness and absence of odd cycles differ in [[intension]], they have identical [[extension (semantics)|extension]].<ref>"Meanings, in this sense, are often called ''intensions'', and things designated, ''extensions''. Contexts in which extension is all that matters are, naturally, called ''extensional'', while contexts in which extension is not enough are ''intensional''. Mathematics is typically extensional throughout." [http://plato.stanford.edu/entries/logic-intensional/ Stanford University primer, 2006].</ref>

In mathematics, theorems are often stated in the form "''P'' is true if and only if ''Q'' is true". <!--(The following is irrelevant and not true.)  Their proofs normally first prove sufficiency, e.g. <math>P \Rightarrow Q</math>. Secondly, the opposite is proven, <math>Q \Rightarrow P</math>
# either directly, assuming ''Q'' is true and demonstrating that the Q circle is located within P, or 
# [[Proof by contrapositive|contrapositively]], that is demonstrating that stepping outside circle of P, we fall out the ''Q'': ''assuming not P, not Q results''.

This proves that the circles for Q and P match on the Venn diagrams above.-->

Because, as explained in previous section, necessity of one for the other is equivalent to sufficiency of the other for the first one, e.g. <math>P \Leftarrow Q</math> is [[Logical equivalence|equivalent to]] <math>Q \Rightarrow P</math>, if ''P'' is necessary and sufficient for ''Q'', then ''Q'' is necessary and sufficient for ''P''. We can write <math>P \Leftrightarrow Q \equiv Q \Leftrightarrow P</math> and say that the statements "''P'' is true [[if and only if]] ''Q'', is true" and "''Q'' is true if and only if ''P'' is true" are equivalent.

==See also==
{{cmn|
* [[Affirming the consequent]]
* [[Biological tests of necessity and sufficiency]]
* [[Causality]]
* [[Closed concept]]
* [[Denying the antecedent]]
*[[If and only if]]
* [[Material implication (disambiguation)]]
* [[Principle of sufficient reason]]
* [[Wason selection task]]
* ''[[Modus ponens]]''
* ''[[Modus tollens]]''
}}

==References==
{{reflist}}

==External links==
{{Commons category}}
*Critical thinking web tutorial: [http://philosophy.hku.hk/think/meaning/nsc.php ''Necessary and Sufficient Conditions'']
*Simon Fraser University: [https://www.sfu.ca/~swartz/conditions1.htm Concepts with examples]

{{Logic}}

[[Category:Necessity and sufficiency| ]]
[[Category:Concepts in logic]]
[[Category:Metaphysical properties]]
[[Category:Mathematical terminology]]