Hypothetical syllogism

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:

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

where the rule is that 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.

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.

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

${\displaystyle {\frac {P\vdash Q\quad Q\vdash R}{P\vdash R}}}$

where ${\displaystyle \vdash }$ is a metalogical symbol and ${\displaystyle A\vdash B}$ meaning that ${\displaystyle B}$ is a syntactic consequence of ${\displaystyle A}$ in some logical system;

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

${\displaystyle ((P\to Q)\land (Q\to R))\to (P\to R)}$

where ${\displaystyle P}$, ${\displaystyle Q}$, and ${\displaystyle R}$ are propositions expressed in some formal system.

• Modus ponens
• Modus tollens
• Affirming the consequent
• Denying the antecedent
• Transitive relation

• Philosophy Index: Hypothetical Syllogism