# Hypothetical syllogism

In logic, a hypothetical syllogism is a valid argument of the following form:

P → Q.
Q → R.
Therefore, P → R.

In logical operator notation

[itex] p \rightarrow q [itex]
[itex] q \rightarrow r, [itex]
[itex] \vdash p \rightarrow r [itex]

In other words, this kind of argument states that if one implies another, and that other implies a third, then the first implies the third. An example hypothetical syllogism:

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.

Hypothetical syllogisms have the advantage that they can be counterfactual: they can be true even if the premises suppose propositions known to be false.

Example counterfactual premises which could be used in a valid hypothetical syllogism:

• If George Washington had a beard, he would look distinguished
• If Yogi Berra had hit 800 home runs, that would be amazing

Other forms of syllogism: categorical syllogism, disjunctive syllogism. he:סילוגיזם_היפותטי

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