Hypothetical syllogism

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 syllogisms.

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