183. Give the contrapositive of the following proposition: If either no P is R or no Q is R, then nothing that is both P and Q is R. [K.]
184. There are three men in a house, Allen, Brown, and Carr, who may go in and out, provided that (1) they never go out all at once, and that (2) Allen never goes out without Brown.
Can Carr ever go out? [LEWIS CARROLL.]
185. There are two propositions, A and B.
Let it be granted that
If A is true, B is true. (i)
Let there be another proposition C, such that
If C is true, then if A is true B is not true. (ii)
274 (ii) amounts to this,—
If C is true, then (i) is not true.
But, ex hypothesi, (i) is true.
Therefore, C cannot be true; for the assumption of C involves an absurdity.
Examine this argument. [LEWIS CARROLL.]
[If the problem in section 184 is regarded as a problem in conditionals, this is the corresponding problem in hypotheticals.]
186. Assuming that rain never falls in Upper Egypt, are the following genuine pairs of contradictories?
(a) The occurrence of rain in Upper Egypt is always succeeded by an earthquake; the occurrence of rain in Upper Egypt is sometimes not succeeded by an earthquake.
(b) If it is true that it rained in Upper Egypt on the 1st of July, it is also true that an earthquake followed on the same day; if it is true that it rained in Upper Egypt on the 1st of July, it is not also true that an earthquake followed on the same day.
If the above are not true contradictories, suggest what should be substituted. [B.]
187. Give the contrapositive and the contradictory of each of the following propositions:
(1) If any nation prospers under a Protective System, its citizens reject all arguments in favour of free-trade;
(2) If any nation prospers under a Protective System, we ought to reject all arguments in favour of free-trade. [J.]
188. Examine the logical relation between the two following propositions; and enquire whether it is logically possible to hold (a) that both are true, (b) that both are false: (i) If volitions are undetermined, then punishments cannot rightly be inflicted; (ii) If punishments can rightly be inflicted, then volitions are undetermined. [J.]