357. Prove that with three given propositions (of the forms A, E, I, O) it is never possible to construct more than one valid syllogism. [K.]
358. On the supposition that no proposition is interpreted as implying the existence either of its subject or of its predicate, find in what cases the reduction of syllogisms to figure 1 is invalid. [K.]
359. Given a valid syllogism, determine the conditions under which the contradictories of the premisses will furnish premisses for another valid syllogism containing the same terms. How will the conclusions of the two syllogisms be related to one another? [K.]
360. Shew that the number of paupers who are blind males is equal to the excess, if any, of the sum of the whole number of blind persons, added to the whole number of male persons, added to the number of those who being paupers are neither blind nor males, above the sum of the whole number of paupers, added to the number of those who not being paupers are blind, and to the number of those who not being paupers are male. [Jevons, Principles of Science.]
361. Shew that, if X and Y are any two propositions containing a common term, then (a) one of the four combinations XY, XYʹ, XʹY, XʹYʹ will always form unstrengthened premisses for a valid syllogism; (b) either only one of the four combinations will do so; or, if two, the syllogisms so formed will be of the same mood. [RR.]
362. Two arguments whose premisses are mutually consistent but which contain sub-contrary conclusions are formed in the same figure with the same middle term.
Find out directly from the general rules of syllogism what can be known with regard to the moods and figure of the two given arguments. [J.]
411 363. Some M is not P, All S is all M. What conclusion follows from the combination of these premisses?
Can you infer anything either about S in terms of P or about P in terms, of S from the knowledge that both the above propositions are false? [K.]
364. (i) Either all M is all P or Some M is not P ; (ii) Some S is not M. What is all that can be inferred (a) about S in terms of P, (b) about P in terms of S, from the knowledge that both the above statements are false? [K.]
365. (a) “A good temper is proof of a good conscience, and the combination of these is proof of a good digestion, which again always produces one or the other.” Shew that this is precisely equivalent to the following: “A good temper is proof of a good digestion, and a good digestion of a good conscience.”
(b) Examine (by diagrams or otherwise) the following argument:—“Patriotism and humanitarianism must be either incompatible or inseparable; and though family-affection and humanitarianism are compatible, yet either may exist without the other; hence, family affection may exist without patriotism.” Reduce the argument, if you can, to ordinary syllogistic form; and determine whether the premisses state anything more than is necessary to prove the conclusion. [J.]