Hypothetical syllogism

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.

Propositional 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.

Historical role of the term

 * This section needs to be written