# 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.

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

Where "" 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:

where is a metalogical symbol meaning that is a syntactic consequence of in some logical system;

or in rule form:

where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "";

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

where and are propositions expressed in some formal system.

## Proof

Suppose we are given that . Then, since we have by the law of excluded middle, it follows that .

Suppose, conversely, we are given . Then if P is true that rules out the first disjunct, so we have Q. In short, .[3]

This can also be demonstrated 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:

- If it is a bear, then it can swim.
- Thus, it is not a bear or it can swim.

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

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

## References

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