Editing Hypothetical syllogism (section)

== Types ==
Hypothetical syllogisms come in two types: mixed and pure. A ''mixed'' hypothetical syllogism has two premises: one conditional statement and one statement that either affirms or denies the [[Antecedent (logic)|antecedent]] or [[consequent]] of that conditional statement. For example,

:If P, then Q.
:P.
:∴ Q.

In this example, the first premise is a conditional statement in which "P" is the antecedent and "Q" is the consequent. The second premise "affirms" the antecedent. The conclusion, that the consequent must be true, is [[Deductive reasoning|deductively]] [[Validity (logic)|valid]].

A mixed hypothetical syllogism has four possible forms, two of which are valid, while the other two are invalid. A valid mixed hypothetical syllogism either affirms the antecedent ([[modus ponens]]) or denies the consequent ([[modus tollens]]). An invalid hypothetical syllogism either [[Affirming the consequent|affirms the consequent]] (fallacy of the [[Converse (logic)|converse]]) or [[Denying the antecedent|denies the antecedent]] (fallacy of the [[Inverse (logic)|inverse]]).

A ''pure'' hypothetical syllogism is a syllogism in which both premises and the conclusion are all [[Conditional sentence|conditional statements]]. The antecedent of one premise must match the consequent of the other for the conditional to be valid. Consequently, one of the conditionals contains the remained term as antecedent and the other conditional contains the removed term as consequent.
:If P, then Q.
:If Q, then R.
:∴ If P, then R.
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.