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 in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and that either form can replace the other in logical proofs.

$P\to Q\Leftrightarrow \neg P\lor Q$

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:

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

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

or in rule form:

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

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

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

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

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

Proof

Suppose we are given that $P\to Q$. Then, since we have $\neg P\lor P$ by the law of excluded middle, it follows that $\neg P\lor Q$.

Suppose, conversely, we are given $\neg P\lor Q$. Then if P is true that rules out the first disjunct, so we have Q. In short, $P\to Q$.

Example

An example is:

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 $P\land \neg Q$, then both sentences are false but otherwise they are both true.