"Implies" is the connective in propositional calculus which has the meaning "if A is true, then B is also true." In formal terminology, the term conditional is often used to refer to this connective (Mendelson 1997, p. 13). The symbol used to denote "implies" is A=>B, A superset B (Carnap 1958, p. 8; Mendelson 1997, p. 13), or A->B.

The Wolfram Language command Implies[p, q] can be used to represent the logical implication p=>q.

In classical logic, A=>B is an abbreviation for ¬A v B, where ¬A denotes NOT and  v denotes OR (though this is not the case, for example, in intuitionistic logic). => is a binary operator that is implemented in the Wolfram Language as Implies[A, B], and can not be extended to more than two arguments.

A=>B has the following truth table (Carnap 1958, p. 10; Mendelson 1997, p. 13).


If A=>B and B=>A (i.e., A=>B ^ B=>A), then A and B are said to be equivalent, a relationship which is written symbolically as A<=>B, A<->B, or A=B (Carnap 1958, p. 8).

See also

Connective, Equivalent, Exists, For All, Quantifier

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Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, p. 8, 1958.Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.

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Cite this as:

Weisstein, Eric W. "Implies." From MathWorld--A Wolfram Web Resource.

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