# Hypothetical syllogism

In classical logic, **hypothetical syllogism** is a valid argument form which is a syllogism having a conditional statement for one or both of its premises.

Transformation rules |
---|

Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

An example in English:

- If I do not wake up, then I cannot go to work.
- If I cannot go to work, then I will not get paid.
- Therefore, if I do not wake up, then I will not get paid.

The term originated with Theophrastus.[1]

## Propositional logic

In propositional logic, **hypothetical syllogism** is the name of a valid rule of inference (often abbreviated **HS** and sometimes also called the **chain argument**, **chain rule**, or the principle of **transitivity of implication**). Hypothetical syllogism is one of the rules in classical logic that is not always accepted in certain systems of non-classical logic. The rule may be stated:

where the rule is that whenever instances of " ", and " " appear on lines of a proof, " " can be placed on a subsequent line.

Hypothetical syllogism is closely related and similar to disjunctive syllogism, in that it is also type of syllogism, and also the name of a rule of inference.

## Formal notation

The *hypothetical syllogism* inference rule may be written in sequent notation, which amounts to a specialization of the cut rule:

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

and expressed as a truth-functional tautology or theorem of propositional logic:

where , , and are propositions expressed in some formal system.

## Proof

Step |
Proposition |
Derivation |
---|---|---|

1 | Given | |

2 | Material implication | |

3 | Distributivity | |

4 | Conjunction elimination (3) | |

5 | Distributivity | |

6 | Law of noncontradiction | |

7 | Disjunctive syllogism (5,6) | |

8 | Conjunction elimination (7) | |

9 | Material implication |