(2) A hypothetical syllogism (or, more distinctively, a pure hypothetical syllogism) is a reasoning consisting of two hypothetical premisses and a hypothetical conclusion;[376]

e.g.,— If Q is true, R is true,
If P is true, Q is true,
therefore, If P is true, R is true.

(3) A hypothetico-categorical syllogism (or, as it may also be called, a mixed hypothetical syllogism) is a reasoning consisting of three propositions in which one of the premisses is 349 hypothetical in form, while the other premiss and the conclusion are categorical;[377]

e.g.,— If P is true, Q is true,
P is true,
therefore, Q is true.

[376] To be quite exact, the condition should be added that the premisses and conclusion contain between them three and only three elements (corresponding to the terms of the categorical syllogism).

[377] It seems unnecessary to discuss separately the case in which a conditional premiss and a categorical premiss are combined: e.g., All selfish people are unhappy; If a child is spoilt, he is sure to be selfish; therefore, If a child is spoilt he will be unhappy. Such a syllogism as this is resolvable into an ordinary categorical syllogism by reducing the conditional premiss to the categorical form, “All spoilt children are selfish”; or it may be resolved into a conditional syllogism by transforming the categorical premiss into the corresponding conditional, “If anyone is selfish, he is sure to be unhappy.” The following is another example: If water is salt it will not boil at 212°; Sea water is salt; therefore, Sea water will not boil at 212°. Compare Mr F. B. Tarbell in Mind, 1883, p. 578. The hypothetico-categorical syllogism as above defined cannot be so summarily disposed of.

This nomenclature, so far as concerns the distinction between the hypothetical and the hypothetico-categorical syllogism, is adopted by Spalding and Ueberweg. Sigwart uses the terms “pure hypothetical syllogism,” and “mixed hypothetical syllogism.” Some logicians (e.g., Fowler) give the name “hypothetical syllogism” to all the above forms of reasoning without distinction. Others (e.g., Jevons) define the hypothetical syllogism so as to include the last form only, the others not being recognised as distinct forms of reasoning at all. This view may be to some extent justified by the very close analogy that exists between the syllogism with two conditional or two hypothetical premisses and the categorical syllogism: but the difference in form is worth at least a brief discussion.

302. Distinctions of Mood and Figure in the case of Conditional and Hypothetical Syllogisms.—In the conditional, and in the hypothetical, syllogism, the antecedent of the conclusion is equivalent to the minor term of the categorical syllogism, the consequent of the conclusion to the major term, and the element which does not appear in the conclusion at all to the middle term. Distinctions of mood and figure may be recognised in precisely the same way as in the case of the categorical syllogism. Thus, the conditional syllogism given in the preceding section is in Barbara. The following are examples of other moods: 350

Festino,—Never when E is F, is it the case that C is D,
Sometimes when A is B, C is D,
therefore, Sometimes when A is B, it is not the case that E is F.
Darapti,—Whenever C is B, E is F,
Whenever C is D, A is B,
therefore, Sometimes when A is B, E is F.
Camenes,—Whenever E is F, C is D,
Never when C is D, is it the case that A is B,
therefore, Never when A is B, is it the case that E is F.

In these three examples the form in which the propositions are expressed suggests an assertoric interpretation. On the modal interpretation, either of conditionals or of hypotheticals, the problematic proposition may be regarded as taking the place of the particular, and we shall then again have all the ordinary distinctions of mood and figure. We may illustrate from hypotheticals: