117. If it is false that “Not only the virtuous are happy,” what can we infer (a) with regard to the non-virtuous, (b) with regard to the non-happy? [J.]
118. Write down the contradictory, and also—where possible—the converse, the contrapositive, and the inverse of each of the following propositions:
A bird in the hand is worth two in the bush;
No unjust acts are expedient;
All are not saints that go to church. [K.]
119. Give the contrapositive and the inverse of each of the following propositions,—They never fail who die in a great cause; Whom the Gods love die young.
If A is either B or else both C and D, what do we know about that which is not D? [K.]
120. Take the following propositions in pairs, and in regard to each pair state whether the two propositions are consistent or inconsistent with each other; in the former case, state further whether either proposition can be inferred from the other, and, if it can be, point out the nature of the inference; in the latter case, state whether it is possible for both the propositions to be false:—(a) All S is P ; (b) All not-S is P ; (c) No P is S ; (d) Some not-P is S. [K.]
121. Transform the following propositions in such a way that, without losing any of their force, they may all have the same subject and the same predicate:—No not-P is S ; All P is not-S ; Some P is S ; Some not-P is not not-S. [K.]
122. Describe the logical relations, if any, between each of the following propositions, and each of the others:—
(i) There are no inorganic substances which do not contain carbon; 155
(ii) All organic substances contain carbon;
(iii) Some substances not containing carbon are organic;
(iv) Some inorganic substances do not contain carbon. [C.]
123. “All that love virtue love angling.”
Arrange the following propositions in the three following groups:—(α) those which can be inferred from the above proposition; (β) those which are consistent with it, but which cannot be inferred from it; (γ) those which are inconsistent with it.
(i) None that love not virtue love angling.
(ii) All that love angling love virtue.
(iii) All that love not angling love virtue.
(iv) None that love not angling love virtue.
(v) Some that love not virtue love angling.
(vi) Some that love not virtue love not angling
(vii) Some that love not angling love virtue.
(viii) Some that love not angling love not virtue. [K.]
124. Determine the logical relation between each pair of the following propositions:—
(1) All crystals are solids.
(2) Some solids are not crystals.
(3) Some not crystals are not solids.
(4) No crystals are not solids.
(5) Some solids are crystals.
(6) Some not solids are not crystals.
(7) All solids are crystals. [L.]