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. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as symbolic logic. ... The history of logic documents the development of logic as it occurs in various rival cultures and traditions in history. ...

Propositional logic

The hypothetical syllogism (abbr. H.S.) is a valid argument of the following form: This article discusses validity in logic, for the term in the social sciences see validity (psychometric). ... An argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. ... In logic, the argument form or test form of an argument results from replacing the different words, or sentences, that make up the argument with letters, along the lines of algebra; the letters represent logical variables. ...

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. A counterfactual conditional (sometimes called a subjunctive conditional) is a logical conditional statement whose antecedent is (ordinarily) taken to be contrary to fact by those who utter it. ...

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

See also

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