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.