material implication

English

P Q P → Q
T T T
T F F
F T T
F F T

Figure 1. Truth table for material implication.

Noun

1. (logic) An implication as defined in classical propositional logic, leading to the truth of paradoxes of material implication such as ${\displaystyle Q\vdash P\to Q}$, to be read as "any proposition whatsoever is a sufficient condition for a true proposition".
In the truth table in Figure 1, the first row corresponds to modus ponens, the last row corresponds to modus tollens, the second row could be taken to represent an invalid argument (where P→Q is the argument and P is a premise or conjunction of premises), and the third row helps ensure that an argument of the form ${\displaystyle P\rightarrow Q,\neg P\vdash \neg Q}$ is invalid.
The following paradox (and also axiom) of material implication: ${\displaystyle P\rightarrow (Q\rightarrow P)}$ could be taken to mean the monotonicity of entailment, that is, if 'P' is true then no other or new fact 'Q' should be able to arise which would imply the nullification of P's truth, i.e., it could not be the case, for any 'Q', that Q → ¬P.