# 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.

An example in English:

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.

The term originated with Theophrastus.[1]

## Propositional logic

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.

## Formal notation

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.

## Proof

Step Proposition Derivation
1${\displaystyle (P\to Q)\land (Q\to R)}$ Given
2${\displaystyle (\neg P\lor Q)\land (\neg Q\lor R)}$ Material implication
3${\displaystyle ((\neg P\lor Q)\land \neg Q)\lor ((\neg P\lor Q)\land R)}$ Distributivity
4${\displaystyle ((\neg P\lor Q)\land \neg Q)\lor R}$ Conjunction elimination (3)
5${\displaystyle ((\neg P\land \neg Q)\lor (Q\land \neg Q))\lor R}$ Distributivity
6${\displaystyle \neg (Q\land \neg Q)}$ Law of noncontradiction
7${\displaystyle (\neg P\land \neg Q)\lor R}$ Disjunctive syllogism (5,6)
8${\displaystyle \neg P\lor R}$ Conjunction elimination (7)
9${\displaystyle P\to R}$ Material implication