# 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.^{[1]}^{[2]}

- 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.

In propositional logic, **hypothetical syllogism** is the name of a valid rule of inference^{[3]}^{[4]} (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* rule may be written in sequent notation:

where is a metalogical symbol meaning that is a syntactic consequence of , and 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.