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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).
This entry comes from Encyclopædia Britannica Concise. For the full entry on implication, visit Britannica.com.