Material implication (rule of inference)

For other uses, see Material implication.

Not to be confused with material inference.

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 "$\Leftrightarrow$" is a metalogical symbol representing "can be replaced in a proof with."

Formal notation

The material implication rule may be written in sequent notation:

$(P \to Q) \vdash (\neg P \or Q)$

where $\vdash$ is a metalogical symbol meaning that $(\neg P \or Q)$ is a syntactic consequence of $(P \to Q)$ in some logical system;

or in rule form:

$\frac{P \to Q}{\neg P \or Q}$

where the rule is that wherever an instance of "$P \to Q$" appears on a line of a proof, it can be replaced with "$\neg P \or Q$";

or as the statement of a truth-functional tautology or theorem of propositional logic:

$(P \to Q) \to (\neg P \or Q)$

where $P$ and $Q$ are propositions expressed in some formal system.

Example

If it is a bear, then it can swim.

Thus, it is not a bear or it can swim.

where $P$ is the statement "it is a bear" and $Q$ is the statement "it can swim".

If it was found that the bear could not swim, written symbolically as $P \and \neg Q$, then both sentences are false but otherwise they are both true.