Hypothetical syllogism

Hypothetical syllogism is closely related and similar to disjunctive syllogism, in that it is also a type of syllogism, and also the name of a rule of inference.

Clearly, (3) does not follow from (1) and (2). (1) is true by default, but fails to hold in the exceptional circumstances of Smith dying. In practice, real-world conditionals always tend to involve default assumptions or contexts, and it may be infeasible or even impossible to specify all the exceptional circumstances in which they might fail to be true. For similar reasons, the rule of hypothetical syllogism does not hold for counterfactual conditionals.

An alternative form of hypothetical syllogism, more useful for classical propositional calculus systems with implication and negation (i.e. without the conjunction symbol), is the following: