Rule of Material Implication

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Theorem

The rule of material implication is a valid deduction sequent in propositional logic:

Formulation 1

$p \implies q \dashv \vdash \neg p \lor q$

Formulation 2

$\vdash \left({p \implies q}\right) \iff \left({\neg p \lor q}\right)$


That is:

statement $p$ implies statement $q$

is logically equivalent to:

statement $p$ is true or statement $q$ is false


As a definition

$p \implies q := \neg p \lor q$


Also known as

This rule is sometimes seen referred to as the definition of material implication, as some sources use this rule as a definition of the conditional, so as to justify its semantics.

A material implication is sometimes expressed in the amplified form implication in material meaning.


Also see

The following are related argument forms:


Sources