What is "material implication"

n. (context logic English) An implication as defined in classical propositional logic, leading to the truth of (w: paradoxes of material implication) such as $Q\; vdash\; P\; to\; Q$, to be read as "any proposition whatsoever is a sufficient condition for a true proposition".

In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction if and only if the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and can replace each other in logical proofs.

$$P \to Q \Leftrightarrow \neg P \or Q$$

Where " ⇔ " is a metalogical symbol representing "can be replaced in a proof with."

Material implication may refer to:

• Material conditional, a logical connective
• Material implication (rule of inference), a rule of replacement for some propositional logic