Hypothetical syllogism

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

[edit] 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

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.