b (1): a logical relation between two propositions that fails to hold only if the first is true and the second is false — see truth table table(2): a logical relationship between two propositions in which if the first is true the second is true (3): a statement exhibiting a relation of implication
I'm offended by his implication that women can't be good at mathematics.
He condemned the court and, by implication, the entire legal system.
He was shocked by the implication of his partner in the theft.
… but whereas Updike and Roth work to establish connection and coherence in the face of time's chaos, DeLillo is an artist of diffusion and dispersal, of implication and missing information. —A. O. Scott, New York Times Book Review, 21 May 2006
… the power of ideas to transform the world is itself accelerating. Although people readily agree with this observation when simply stated, very few people truly appreciate its profound implications. —Ray Kurzweil, Curious Minds, (2004) 2005
… the astronomer Edwin Hubble found that the universe is expanding, and by implication must have originated a finite time ago in an explosion popularly called the big bang. —Paul Davies, The New Physics, 1989
In logic, a relation that holds between two propositions when they are linked as antecedent and consequent of a true conditional proposition. Logicians distinguish two main types of implication, material and strict. Proposition p materially implies proposition q if and only if the material conditional pq (read if p then q) is true. A proposition of the form pq is false whenever p is true and q is false; it is true in the other three possible cases (i.e., p true and q true; p false and q true; p false and q false). It follows that whenever p is false, pq is automatically true: this is a peculiarity that makes the material conditional inadequate as an interpretation of the meaning of conditional sentences in ordinary English. On the other hand, proposition p strictly implies proposition q if and only if it is impossible for p to be true without q also being true (i.e., if the conjunction of p and not-q is impossible).