Hypothetical syllogism

Rules of inference
Propositional calculus
Modus ponens Modus tollens Modus ponendo tollens Conjunction introduction Simplification Disjunction introduction Disjunction elimination Disjunctive syllogism Hypothetical syllogism Constructive dilemma Destructive dilemma Biconditional introduction Biconditional elimination
Predicate calculus
Universal generalization Universal instantiation Existential generalization Existential instantiation

In logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a short-hand for the theory of consequence.

Propositional logic

Hypothetical syllogism is one of the proof rules in classical logic that may or may not be available in a non-classical logic. The hypothetical syllogism (abbr. H.S.) is a valid argument of the following form:

If P → Q.

If Q → R.

____________________

Then P → R.

Symbolically, this is expressed:

$P \rightarrow Q, Q \rightarrow R \vdash P \rightarrow R$

Example of use:

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.

See also

• Modus ponens
• Modus tollens
• Modus tollendo ponens
• Affirming the consequent
• Denying the antecedent
• Disjunctive syllogism
• Inference rule
• Transitive relation