Necessity and sufficiency

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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.

In logic, necessity and sufficiency are implicational relationships between statements. The assertion that one 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.[1][2][3] In ordinary English, 'necessary' and 'sufficient' indicate relations between conditions or states of affairs, not statements. Being a 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.


A true necessary condition in a conditional statement makes the statement true (see "truth table" immediately below). In formal terms, a consequent N is a necessary condition for an antecedent S, in the conditional statement, "N if S ", "N is implied by S ", or N \Leftarrow 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".


A necessary and sufficient condition requires both of these implications (S \Rightarrow N and N \Rightarrow S) to hold. Using the previous statement, this is expressed as "S is necessary and sufficient for N ", "S if and only if N ", or S \Leftrightarrow N.

Truth table
S N S \Rightarrow N S \Leftarrow N S \Leftrightarrow N


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 eclipse.


  1. unmarried
  2. male
  3. adult
Example 3: Consider thunder, the sound caused by lightning. We say that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there's lightning, there's 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 it's 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 are at least 30 years old.


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).



Relationship between necessity and sufficiency[edit]

Consider two logical statements S and N, such that S is sufficient for N (if S is true, then N must also be true). Figuratively, if we are within S then we are within N. It follows that S cannot be true unless N is true – i.e., that N is a necessary condition for S.




Simultaneous necessity and sufficiency[edit]



See also[edit]

Argument forms involving necessary and sufficient conditions[edit]

Valid forms of argument[edit]

Invalid forms of argument (i.e. fallacies)[edit]


  1. ^ Betz, Frederick (2011). Managing Science: Methodology and Organization of Research. New York: Springer. p. 247. ISBN 978-1-4419-7487-7. 
  2. ^ Manktelow, K.I. (1999). Reasoning and Thinking. East Sussex, UK: Psychology Press. ISBN 0-86377-708-2. 
  3. ^ Asnina, Erika; Osis, Janis; and Jansone, Asnate (2013). "Formal Specification of Topological Relations". Databases and Information Systems VII: 175. doi:10.3233/978-1-61499-161-8-175. 
  4. ^ Stanford University primer, 2006

External links[edit]