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

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:

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

Another example would be...

if M, then ~N. if ~O, then N. if O, then ~P.

seeing that if one is true then the contrapositive must be true, saying if ~N, then O. makes the following make more sense.

if M, then ~N. if ~N, then O. if O, then ~P.

so therefore the conclusion to this argument is

if M, then ~P.

[edit] Historical role of the term

This section needs to be written

[edit] See also

• Modus ponens
• Boethius
• Other syllogistic forms: categorical syllogism, disjunctive syllogism.