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| {{about|the formal terminology in logic|causal meanings of the terms|Causality|the concepts in statistics|Sufficient statistic}}
| | I'm a 41 years old and study at the university (Engineering).<br>In my free time I'm trying to teach myself Korean. I have been twicethere and look forward to returning sometime in the future. I like to read, preferably on my ebook reader. I really love to watch Sons of Anarchy and American Dad as well as docus about nature. I love Sculpting. |
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| In [[logic]], '''necessity''' and '''sufficiency''' are implicational relationships between [[Statement (logic)|statements]].{{Citation needed|date=November 2013}} The assertion that one statement is a ''necessary and sufficient'' [[Condition (philosophy)| 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.
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| In [[Ordinary English | ordinary English]], ‘necessary’ and ‘sufficient’ indicate relations between [[Condition (philosophy)| conditions]] or states of affairs, not statements. Being male sibling is a necessary and sufficient condition for being a brother. Fred’s being a male sibling is necessary and sufficient for the truth of the statement that Fred is a brother.
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| ==Definitions==
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| A true '''''necessary''''' condition in a [[Material conditional|conditional statement]] makes the statement true.{{Citation needed|date=November 2013}} In formal terms, a [[consequent]] ''N'' is a necessary condition for an [[Antecedent (logic)|antecedent]] ''S'', in the conditional statement, "''N'' if ''S''", "''N'' is implied by ''S''", or ''N'' <math>\Leftarrow</math> ''S''. In common words, we would also say "''N'' is weaker than ''S''" or "''S'' cannot occur without ''N''". For example, it is necessary to be Named, to be called "Socrates".
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| A true '''''sufficient''''' condition in a conditional statement ties the statement's truth to its consequent. In formal terms, an antecedent ''S'' is a sufficient condition for a consequent ''N'', in the conditional statement, "if ''S'', then ''N''", "''S'' implies ''N''", or ''S'' <math>\Rightarrow</math> ''N''. In common words, we would also say "''S'' is stronger than ''N''" or "''S'' guarantees ''N''". For example, "Socrates" suffices for a Name.
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| {| class="wikitable" style="margin:1em auto; text-align:center;"
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| |+ '''S implies N'''
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| ! style="width:20%" | ''S''
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| ! style="width:20%" | ''N''
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| ! style="width:20%" | ''S'' <math>\Rightarrow</math> ''N''
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| | T || T || T
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| | T || style="background:papayawhip" | F || style="background:papayawhip" | F
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| | style="background:papayawhip" | F || T || T
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| | style="background:papayawhip" | F || style="background:papayawhip" | F || T
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| ==Necessity==
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| [[File:Solar eclipse 1999 4 NR.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. in the case of an [[solar eclipse|eclipse]].]]
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| 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." By [[contraposition]], this is the same thing as "whenever ''P'' is true, so is ''Q''". The logical relation between them is expressed as "If ''P'' then ''Q''" and denoted "''P'' <math>\Rightarrow</math> ''Q''" (''P'' [[Logical consequence|implies]] ''Q''), and may also be expressed as any of "''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, as shown in Example 5.
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| :'''Example 1''': In order for it to be true that "John is a bachelor," it is necessary that it be also true that he is
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| :# unmarried
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| :# male
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| :# adult
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| :since to state "John is a bachelor" implies John has each of those three additional [[Predicate (mathematical logic)|predicates]].
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| :'''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.
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| :'''Example 3''': Consider thunder, in the technical sense, the acoustic quality demonstrated by the shock wave that inevitably results from any lightning bolt in the atmosphere. It may fairly be said that thunder is necessary for lightning, since lightning cannot occur without thunder, too, occurring. That is, if lightning does occur, then there is thunder.
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| :'''Example 4''': Being at least 30 years old is necessary of 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 are at least 30 years old.
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| :'''Example 5''': In [[algebra]], in order 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''" such that both ''x'' <math>\star</math> ''x''" and ''x''" <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.
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| ==Sufficiency==
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| [[File:ICE 3 Fahlenbach.jpg|thumb|right|200px|That a train runs on schedule can be a sufficient condition for arriving on time (if one boards the train and it departs on time, then one will arrive on time); but it is not always 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).]]
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| To say that ''P'' is sufficient for ''Q'' is to say that, in and of itself, knowing ''P'' to be true is adequate grounds to conclude that ''Q'' is true. (It is to say, at the same time, that knowing ''P'' not to be true does not, in and of itself, provide adequate grounds to conclude that ''Q'' is not true, either.) The logical relation is expressed as "If ''P'' then ''Q''" or "''P'' <math>\Rightarrow</math> ''Q''," and may also be expressed as "''P'' implies ''Q''." Several sufficient conditions may, taken together, constitute a single necessary condition, as illustrated in example 5.
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| :'''Example 1''': Stating that "John is a bachelor" implies that John is male. So knowing that it is true that John is a bachelor is sufficient to know that he is a male.
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| :'''Example 2''': A number's being divisible by 4 is sufficient (but not necessary) for its being even, but being divisible by 2 is both sufficient and necessary.
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| :'''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.
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| :'''Example 4''': A U.S. president's signing a bill that Congress passed is sufficient to make the bill 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 law (it could still have become law through a congressional [[veto override|override]]).
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| :'''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 diamond (♦), heart (♥), or club (♣), respectively. 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 the conditions.
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| ==Relationship between necessity and sufficiency==
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| [[File:Necessary and sufficient venn (set) diagram.svg|thumb|Consider two logical statements. If the sufficient statement (S) is true, the necessary statement (N) must also be true.]]
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| 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>'s ''being a [[real number]]'' (''N'') (since there are real numbers that are not rational).
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| 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]] 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]].
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| 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'').
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| ==Simultaneous necessity and sufficiency==
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| {{See also|Material equivalence}}
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| To say that ''P'' is necessary and sufficient for ''Q'' is to say two things, that ''P'' is necessary for ''Q'' and that ''P'' is sufficient for ''Q''. Of course, it may instead be understood to say ''a different'' two things, namely that each of ''P'' and ''Q'' is necessary for the other. And it may be understood in a third equivalent way: as saying that each is sufficient for 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 ''P'' <math>\Leftrightarrow</math> ''Q''.
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| 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 vice versa. A philosopher<ref name="stan">[http://plato.stanford.edu/entries/logic-intensional/ Stanford University primer, 2006]</ref>
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| 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>
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| ==See also==
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| * [[Causality]]
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| * [[Material implication]]
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| * [[Wason selection task]]
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| * [[Closed concept]]
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| ===Argument forms involving necessary and sufficient conditions===
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| ====Valid forms of argument====
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| * [[Modus ponens]]
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| * [[Modus tollens]]
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| ====Invalid forms of argument (i.e. fallacies)====
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| * [[Affirming the consequent]]
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| * [[Denying the antecedent]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * Critical thinking web tutorial: [http://philosophy.hku.hk/think/meaning/nsc.php ''Necessary and Sufficient Conditions'']
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| * Simon Fraser University: [http://www.sfu.ca/~swartz/conditions1.htm Concepts with examples]
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| {{Logic}}
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| [[Category:Concepts in logic]]
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| [[Category:Concepts in metaphysics]]
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| [[Category:Conditionals]]
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| [[Category:Dichotomies]]
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| [[Category:Mathematical terminology]]
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| [[Category:Necessity and sufficiency| ]]
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I'm a 41 years old and study at the university (Engineering).
In my free time I'm trying to teach myself Korean. I have been twicethere and look forward to returning sometime in the future. I like to read, preferably on my ebook reader. I really love to watch Sons of Anarchy and American Dad as well as docus about nature. I love Sculpting.