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logical implication

Logical implication is a type of relationship between two statements or sentences. The relation translates verbally into "logically implies" or "if/then" and is symbolized by a double-lined arrow pointing toward the right ( ). If A and B represent statements, then A B means "A implies B" or "If A, then B." The word "implies" is used in the strongest possible sense.

As an example of logical implication, suppose the sentences A and B are assigned as follows:

A = The sky is overcast.
B = The sun is not visible.

In this instance, A B is a true statement (assuming we are at the surface of the earth, below the cloud layer.) However, the statement B A is not necessarily true; it might be a clear night. Logical implication does not work both ways. However, the sense of logical implication is reversed if both statements are negated. That is,

(A B) (-B -A)

Using the above sentences as examples, we can say that if the sun is visible, then the sky is not overcast. This is always true. In fact, the two statements A B and -B -A are logically equivalent.

Also see logical equivalence and Mathematical Symbols .

This was last updated in September 2005

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