implication (redirected from logical implication)

implication

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 p ⊃ q (read “if p then q”) is true. A proposition of the form p ⊃ q 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, p ⊃ q 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).

---

implication

Logic

a. the operator that forms a sentence from two given sentences and corresponds to the English if … then …

b. a sentence so formed. Usually written p→q or p⊃q, where p,q are the component sentences, it is true except when p (the antecedent) is true and q (the consequent) is false

c. the relation between such sentences

---

implication [‚im·plə′kā·shən]

(mathematics)

The logical relation between two statementspandq, usually expressed as “ifpthenq.“

A logic operator having the characteristic that ifpandqare statements, the implication ofpandqis false ifpis true andqis false, and is true otherwise. Also known as conditional implication; if-then operation; material implication.