##### Wikipedia

**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.

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:

$$\frac{P \to Q, Q \to R}{\therefore P \to R}$$

where the rule is that whenever instances of "*P* → *Q*", and "*Q* → *R*" appear on lines of a proof, "*P* → *R*" 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.