# Hypothetical syllogism: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

Updated live from Wikipedia, last check: May 17, 2013 10:21 UTC (54 seconds ago)

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.