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Concept
Hypothetical syllogism
Summary
In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises.
An example in English:
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.
The term originated with Theophrastus.
A pure hypothetical syllogism is a syllogism in which both premises and conclusions are conditionals. The antecedent of one premise must match the consequent of the other for the conditional to be valid. Consequently, conditionals contain remained antecedent as antecedent and remained consequent as consequent.
If p, then q.
If q, then r.
∴ If p, then r.
A mixed hypothetical syllogism consists of one conditional statement and one statement that expresses either affirmation or denial with either the antecedent or consequence of that conditional. Therefore, such a mixed hypothetical syllogism has four possible forms, of which two are valid, while the other two are invalid(See Table). The first way to get a valid conclusion is to affirm the antecedent. A valid hypothetical syllogism either denies the consequent (modus tollens) or affirms the antecedent (modus ponens).
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 a 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.
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