Ancient references point to the works of Theophrastus and Eudemus for the first investigation of this kind of syllogisms.

[1][2] Hypothetical syllogisms come in two types: mixed and pure.

For example, In this example, the first premise is a conditional statement in which "P" is the antecedent and "Q" is the consequent.

The conclusion, that the consequent must be true, is deductively valid.

A mixed hypothetical syllogism has four possible forms, two of which are valid, while the other two are invalid.

The antecedent of one premise must match the consequent of the other for the conditional to be valid.

An example in English: In propositional logic, hypothetical syllogism is the name of a valid rule of inference (often abbreviated HS and sometimes also called the chain argument, chain rule, or the principle of transitivity of implication).

The reason for this is that these logics describe defeasible reasoning, and conditionals that appear in real-world contexts typically allow for exceptions, default assumptions, ceteris paribus conditions, or just simple uncertainty.

(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: Yet another form is: An example of the proofs of these theorems in such systems is given below.

We use two of the three axioms used in one of the popular systems described by Jan Łukasiewicz.