Material implication (rule of inference)

In propositional logic, material implication[1][2] 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.

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

Not to be confused with Material inference.

Where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with," and P and Q are any given statements.

Formal notation

The material implication rule may be written in sequent notation:

${\displaystyle (P\to Q)\vdash (\neg P\lor Q)}$

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

or in rule form:

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

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

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

${\displaystyle (P\to Q)\to (\neg P\lor Q)}$

where ${\displaystyle P}$ and ${\displaystyle Q}$ are propositions expressed in some formal system.

Partial proof

Suppose we are given that ${\displaystyle P\to Q}$. Then, since we have ${\displaystyle \neg P\lor P}$ by the law of excluded middle, it follows (arguing by cases) that ${\displaystyle \neg P\lor Q}$.

Suppose, conversely, we are given ${\displaystyle \neg P\lor Q}$. Then if ${\displaystyle P}$ is true that rules out the first disjunct, so we have ${\displaystyle Q}$. In short, ${\displaystyle P\to Q}$[3]. However if ${\displaystyle P}$ is false, then this entailment fails, because the first disjunct ${\displaystyle \neg P}$ is true which puts no constraint on the second disjunct ${\displaystyle Q}$. Hence, nothing can be said about ${\displaystyle P\to Q}$. In sum, the equivalence in the case of false ${\displaystyle P}$ 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 ${\displaystyle \neg P\vee Q}$, which is "it is not a bear" or "it can swim", where ${\displaystyle P}$ is the statement "it is a bear" and ${\displaystyle Q}$ is the statement "it can swim".

References

1. Patrick J. Hurley (1 January 2011). A Concise Introduction to Logic. Cengage Learning. ISBN 0-8400-3417-2.
2. Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 371.
3. Math StackExchange: Equivalence of a→b and ¬ a ∨ b