Hypothetical syllogism

Hypothetical syllogism

Type	Syllogism
Field	Propositional calculus Classical logic Intuitionistic logic Most systems of relevance logic
Statement	Whenever instances of ${\displaystyle P\to Q}$, and ${\displaystyle Q\to R}$ appear on lines of a proof, ${\displaystyle P\to R}$ can be placed on a subsequent line.
Symbolic statement	${\displaystyle {\frac {P\to Q,Q\to R}{\therefore P\to R}}}$

In classical logic, a hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises. Ancient references point to the works of Theophrastus and Eudemus for the first investigation of this kind of syllogism.