Material implication (rule of inference)

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Material implication

Type	Rule of replacement
Field	Propositional calculus
Statement	P implies Q is logically equivalent to not-${\displaystyle P}$ or ${\displaystyle Q}$. Either form can replace the other in logical proofs.
Symbolic statement	${\displaystyle P\to Q\Leftrightarrow \neg P\lor Q}$

In propositional logic, material implication^[1][2] is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-${\displaystyle P}$ or ${\displaystyle Q}$ and that either form can replace the other in logical proofs. In other words, if ${\displaystyle P}$ is true, then ${\displaystyle Q}$ must also be true, while if ${\displaystyle Q}$ is not true, then ${\displaystyle P}$ cannot be true either; additionally, when ${\displaystyle P}$ is not true, ${\displaystyle Q}$ may be either true or false.

${\displaystyle P\to Q\Leftrightarrow \neg P\lor Q,}$

where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and ${\displaystyle \neg P\lor Q}$ can be read as "(not P) or Q". To illustrate this, consider the following statements:

• ${\displaystyle P}$: Sam ate an orange for lunch.
• ${\displaystyle Q}$: Sam ate a fruit for lunch.

Then, to say "Sam ate an orange for lunch" implies "Sam ate a fruit for lunch" (${\displaystyle P\to Q}$). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (by contraposition). However, merely saying that Sam did not eat an orange for lunch provides no information on whether or not Sam ate a fruit (of any kind) for lunch.