Definition: LOGICAL IMPLICATION from A Dictionary of Philosophical Logic

Logical implication is the relation that holds between A and B if the material conditional: A → B is a logical truth. The formula: A → B is also said to be a logical implication if the relation of logical implication holds between A and B (in that order).

Summary Article: IMPLICATION (LOGICAL)
From The Essentials of Philosophy and Ethics

The relation between two PROPOSITIONS or statements of fact, such that if the first one is true, the second one will be as well. In the early twentieth century Bertrand RUSSELL complicated things by talking of ‘material implication’, where the two propositions have no connection. For example, a false proposition, such as ‘All cats are green’, implies ‘All dogs are blue’ because a false proposition (any false proposition) ‘implies’ every other possible proposition. This strange effect is because, in LOGIC, the only time we can say ‘p does not imply q’ is when the first statement is true and the second one is not, and as the claim about cats here is never going to be true, it is permissible to say that it does imply all the other possible statements.

G.E. MOORE, Russell’s colleague and chum, then suggested a new term, ‘entailment’, which attempts to retain a sensible link between the two propositions. Essentially, this says that if q is deducible from p, then p entails q. Unfortunately, the connection cannot itself be satisfactorily explained.

Truth tables

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pronunciation (15c) 1 a : the act of implicating (see implicate):the state of being implicated b : close connection; esp : an incriminating involve

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Hargrave's Communications Dictionary, Wiley

A Boolean operator relating statements P and Q such that P implies Q is true in all cases except when P is true and Q is false. The written...

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