The chief difficulty in this problem consists in the accurate statement of the premisses. Call the original class W. We then have
All W is XZ or Yz or YZ or Xy or xZ,
that is, All W is X or Y or Z ; (1)
All xZ is W ; (2)
No xZ is WXZ or WYz or WYZ or WXy,
that is, No xZ is WYZ. (3)
We may now proceed as follows:—By (1), All W is X or Y or Z ; and by (3), All W is X or y or z. Therefore, All W is X or Yz or yZ. (2) affords no information regarding the class W, except that everything that is Z but not X is contained within it.
502. (1) If a nation has natural resources, and a good government, it will be prosperous. (2) If it has natural resources without a good government, or a good government without natural resources, it will be contented, but not prosperous. (3) If it has neither natural resources nor a good government it will be neither contented nor prosperous.
Shew that these statements may be reduced to two propositions of the form of Hamilton’s U. [O’S]
515 Let a nation with natural resources be denoted by R, a nation with a good government by G, a prosperous nation by P, and a contented nation by C. Then the given statements may be expressed as follows:—(1) All RG is P ; (2) All Rg or rG is Cp ; (3) All rg is cp.
By contraposition, (2) may be resolved into the two propositions, All cp is RG or rg, All P is RG or rg. But by (1) No cp is RG ; and by (3) No P is rg. Hence the two propositions into which (2) was resolved may be reduced to the form, All cp is rg, All P is RG.
The three original statements are accordingly equivalent to the two U propositions All RG is all P, All rg is all cp.
503. Let the observation of a class of natural productions be supposed to have led to the following general results.
1st. That in whichsoever of these productions the properties A and C are missing, the property E is found, together with one of the properties B and D, but not with both.
2nd. That wherever the properties A and D are found while E is missing, the properties B and C will either both be found, or both be missing.
3rd. That wherever the property A is found in conjunction with either B or E, or both of them, there either the property C or the property D will be found, but not both of them. And conversely, wherever the property C or D is found singly, there the property A will be found in conjunction with either B or E or both of them.
Shew that it follows that In whatever substances the property A is found, there will also be found either the property C or the property D, but not both, or else the properties B, C, and E will all be wanting. And conversely, Where either the property C or the property D is found singly or the properties B. C, and D are together missing, there the property A will be found. Shew also that If the property A is absent and C present, D is present.
[Boole, Laws of Thought, pp. 146–148. Venn, Symbolic Logic, pp. 280, 281. Johns Hopkins Studies in Logic, pp. 57, 58, 82, 83.]
The premisses are as follows:—
| 1st, | All ac is BdE or bDE ; | (i) |
| 2nd, | All Ade is BC or bc ; | (ii) |
| 3rd, | Whatever is AB or AE is Cd or cD ; | (iii) |
| Whatever is Cd or cD is AB or AE. | (iv) |