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# Conditional syllogism: Wikis

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# Encyclopedia

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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:

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

[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.

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