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Necessity and sufficiency

This article is about the formal terminology in logic. For causal meanings of the terms, see Causality. For the concepts in statistics, see Sufficient statistic.
"Necessary But Not Sufficient" redirects here. For the novel by Eliyahu Goldratt, see Necessary But Not Sufficient (novel).

In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P. (Equivalently, it is impossible to have P without Q, or the falsity of Q ensures the falsity of P.)[1] Similarly, P is sufficient for Q, because P being true always implies that Q is true, but P not being true does not always imply that Q is not true.[2]

In general, a necessary condition is one (possibly one of multiple conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition.[3] 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.[4][5][6]

In ordinary English (also natural language) "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. For example, being a male is a necessary condition for being a brother, but it is not sufficient—while being a male sibling is a necessary and sufficient condition for being a brother. Any conditional statement consists of at least one sufficient condition and at least one necessary condition.

In data analytics, necessity and sufficiency can refer to different causal logics,[7] 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.

  1. ^ "[M06] Necessity and sufficiency". philosophy.hku.hk. Retrieved 2019-12-02.
  2. ^ Bloch, Ethan D. (2011). Proofs and Fundamentals: A First Course in Abstract Mathematics. Springer. pp. 8–9. ISBN 978-1-4419-7126-5.
  3. ^ Confusion-of-Necessary (2019-05-15). "Confusion of Necessary with a Sufficient Condition". www.txstate.edu. Retrieved 2019-12-02.
  4. ^ Betz, Frederick (2011). Managing Science: Methodology and Organization of Research. New York: Springer. p. 247. ISBN 978-1-4419-7487-7.
  5. ^ Manktelow, K. I. (1999). Reasoning and Thinking. East Sussex, UK: Psychology Press. ISBN 0-86377-708-2.
  6. ^ Asnina, Erika; Osis, Janis & Jansone, Asnate (2013). "Formal Specification of Topological Relations". Databases and Information Systems VII. 249 (Databases and Information Systems VII): 175. doi:10.3233/978-1-61499-161-8-175.
  7. ^ Richter, Nicole Franziska; Hauff, Sven (2022-08-01). "Necessary conditions in international business research–Advancing the field with a new perspective on causality and data analysis" (PDF). Journal of World Business. 57 (5): 101310. doi:10.1016/j.jwb.2022.101310. ISSN 1090-9516.

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