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Syllogism with conditional premise(s)

In classical logic, a hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises. Ancient references point to the works of Theophrastus and Eudemus for the first investigation of this kind of syllogisms.

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 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 deductively 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 affirms the consequent (fallacy of the converse) or denies the antecedent (fallacy of the inverse). [[File:In Quest of Univeral Logic HypoSyll. png|thumbnail|400px|There are four possible forms of mixed hypothetical syllogisms, two of which are valid, while two of which are invalid. Considering a simple example demonstrates why forms are valid or invalid. If "p" represents "Candiru is a fish" and "q" represents "Candiru has gills," try to convince yourself by replacing these statements with p and q in the above table. If Jones wins the election, Smith will retire after the election. If Smith dies before the election, Jones will win the election. If Smith dies before the election, Smith will retire after the election. 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.