# Material implication (rule of 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 $Q$ (i.e. either $Q$ must be true, or $P$ must not be true) 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," and P and Q are any given logical statements.

## 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 (i.e. the latter logically follows from the former);

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.

## Partial proof

Suppose we are given that $P\to Q$ . Then, we have $\neg P\lor P$ by the law of excluded middle (i.e. either $P$ must be true, or $P$ must not be true).

Subsequently, since $P\to Q$ , $P$ can be replaced by $Q$ in the statement, and thus it follows that $\neg P\lor Q$ (i.e. either $Q$ must be true, or $P$ must not be true).

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, $P\to Q$ . However if $P$ is false, then this entailment fails, because the first disjunct $\neg P$ is true which puts no constraint on the second disjunct $Q$ . Hence, nothing can be said about $P\to Q$ . In sum, the equivalence in the case of false $P$ is only conventional, and hence the formal proof of equivalence is only partial.

This can also be expressed with a truth table:

P Q ¬P P→Q ¬P ∨ Q
T T F T T
T F F F F
F T T T T
F F T T T

## Example

An example is:

We are given the conditional fact that if it is a bear, then it can swim. Then all 4 possibilities in the truth table are compared to that fact.
1st: If it is a bear, then it can swim — T
2nd: If it is a bear, then it can not swim — F
3rd: If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.
4th: If it is not a bear, then it can not swim — T (as above)

Thus, the conditional fact can be converted to $\neg P\vee Q$ , which is "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".