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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:

P → Q. Q → R. Therefore, P → R.

Symbolically, this is expressed:

[1]

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

References