# Material implication (rule of inference)

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Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

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.

Where "" is a metalogical symbol representing "can be replaced in a proof with."

## Formal notation[edit]

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.

## Example[edit]

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[edit]

**^**Hurley, Patrick (1991).*A Concise Introduction to Logic*(4th ed.). Wadsworth Publishing. pp. 364–5.**^**Copi, Irving M.; Cohen, Carl (2005).*Introduction to Logic*. Prentice Hall. p. 371.