# Mathematical Proof and the Principles of Mathematics/Logic/Rules of inference summary

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This is a list of the rules of inference given in previous sections. In the notation being used, a solid horizontal bar means that the statement below is a valid deduction from the statement(s) above. A vertical bar with a horizontal bar connected to it means that whatever is to the right of the vertical bar is a subproof, and whatever is above the horizontal bar are assumption(s) and whetever is below the horizontal bar is what has been derived. The names given are just placeholders and no guarantees are made that they are standard in any way.

## Propositional logic[edit | edit source]

### Rules not requiring subproofs[edit | edit source]

- Iteration

- Use of contradiction

- Disjunction by first case

or |

- Disjunction by second case

or |

- First use of conjunction

and |

- Second use of conjunction

and |

- Implication from the conclusion

implies |

- Implication from false assumption

not |

implies |

- Double negation

not not |

- Equivalence to implication

iff |

implies |

- Equivalence to converse

iff |

implies |

- Conjunction by components

and |

- Use of disjunction, first alternative false

or |

not |

- Use of disjunction, second alternative false

or |

not |

- Use of implication, from premise (
*modus ponens*)

implies |

- Use of implication, from false conclusion (
*modus tollens*)

implies |

not |

not |

### Rules requiring one subproof[edit | edit source]

- Implication by direct proof

| ||

implies |