522. If thriftlessness and poverty are inseparable, and virtue and misery are incompatible, and if thrift be a virtue, can any relation be proved to exist between misery and poverty? If moreover all thriftless people are either virtuous or not miserable, what follows? [V.]
523. At a certain examination, all the candidates who were entered for Latin were also entered for either Greek, French, or German, but not for more than one of these languages; all the candidates who were not entered for German were entered for two at least of the other languages; no candidate who was entered for both Greek and French was entered for German, but all candidates who were entered for neither Greek nor French were entered for Latin. Shew that all the candidates were entered for two of the four languages, but none for more than two. [K.]
524. (1) Wherever there is smoke there is also fire or light; (2) Wherever there is light and smoke there is also fire; (3) There is no fire without either smoke or light.
523 Given the truth of the above propositions, what is all that you can infer with regard to (i) circumstances where there is smoke; (ii) circumstances where there is not smoke; (iii) circumstances where there is not light? [W.]
525. In a certain warehouse, when the articles offered are antique, they are costly, and at the same time either beautiful or grotesque, but not both. When they are both modern and grotesque, they are neither beautiful nor costly. Everything which is not beautiful is offered at a low price, and nothing cheap is beautiful. What can we assert (1) about the antique, and (2) about the grotesque articles? [M.]
526. Shew that the following sets of propositions are equivalent to one another:—
(1) All a is b or c ; All b is aCd ; All c is aB ; All D is c.
(2) All A is BC ; All b is aC ; All C is ABd or abd.
(3) All A is B ; All B is A or c ; All c is aB ; All D is c.
(4) All b is aC ; All A is C ; All C is d ; All aC is b.
(5) All c is aB ; All D is aB ; All A is B ; All aB is c.
(6) All A is BC ; All BC is A ; All D is Bc ; All b is C. [K.]
527. Shew that a certain set of four properties must be found somewhere together, if the following facts are known: “Everything that has the first property or is without the last has the two others; and if everything that has both the first and last has one or other but not both of the two others, then something that has the first two must be without the last two.” [J.]
528. Given the propositions: (i) all material goods are external; (ii) no internal (= non-external) goods are dispropriable; (iii) all dispropriable goods are appropriable; (iv) no collective goods are appropriable or immaterial (= non-material); what is all that we can infer about (a) appropriable goods, (b) immaterial goods? [J.]
529. Eliminate X and Y from the following propositions: All aX is BcY or bcy ; No AX is BY ; All AB is Y ; No ABCD is xY. Shew also that it follows from these propositions that All XY is Ab or aBc. [K.]
530. Given (1) All A is Bc or bC, (2) All B is DE or de, (3) All C is De ; shew that (i) All A is BcDE or Bcde or bCDe, (ii) All BcD is E, (iii) All abd is c, (iv) All cd is ab or Be, (v) All bCD is e. [Jevons, Pure Logic, § 160.]