In classical logic, hypothetical syllogism is a valid argument form which is a syllogism having a conditional statement for one or both of its premises.
An example in English:
The term originated with Theophrastus.^{[1]}
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 rule may be stated:
where the rule is that whenever instances of "", and "" appear on lines of a proof, "" can be placed on a subsequent line.
Hypothetical syllogism is closely related and similar to disjunctive syllogism, in that it is also type of syllogism, and also the name of a rule of inference.
The rule of hypothetical syllogism holds in classical logic, intuitionistic logic, most systems of relevance logic, and many other systems of logic. However, it does not hold in all logics, including, for example, non-monotonic logic, probabilistic logic and default logic. 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.
An example, derived from Adams, ^{[2]}
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.
The hypothetical syllogism inference rule may be written in sequent notation, which amounts to a specialization of the cut rule:
where is a metalogical symbol and meaning that is a syntactic consequence of in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
where , , and are propositions expressed in some formal system.
Step | Proposition | Derivation |
---|---|---|
1 | Given | |
2 | Material implication | |
3 | Distributivity | |
4 | Conjunction elimination (3) | |
5 | Distributivity | |
6 | Law of noncontradiction | |
7 | Disjunctive syllogism (5,6) | |
8 | Conjunction elimination (7) | |
9 | Material implication |
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 ?ukasiewicz. The proofs relies on two out of the three axioms of this system:
The proof of the (HS1) is as follows:
The proof of the (HS2) is given here.
Whenever we have two theorems of the form and , we can prove by the following steps: