| All A is Cd or cD or bcd ; | (α) |
| All Cd is A ; | (β) |
| All cD is A ; | (γ) |
| All bcd is A ; | (δ) |
| All aC is D. | (ε) |
First, By (iii), All A is Cd or cD or bc. But by (ii), All Abe is c or d ; and by (iv), All Abe is CD or cd ; therefore, All Abe is cd. Hence, All A is Cd or cD or bcd. (α)
Secondly, (β) and (γ) follow immediately from (iv).
Thirdly, from (i), we have directly, No ac is bd ; therefore (by conversion), No bcd is a ; therefore, All bcd is A. (δ)
Lastly, by (iv), All Cd is A ; therefore, by contraposition, All aC is D. (ε)
We may obtain a complete solution so far as A is concerned as follows:
By (ii),[517] All A is BC or bc or d or E ;
by (iii), All is be or Cd or cD ;
therefore, All A is Cd or cDE or bcD or bce or bde ;
by (iv). All A is B or E or CD or cd ;
therefore, All A is cDE or bcde or BCd or CdE.
This includes the partial solution with regard to A,—All A is Cd or cD or bcd. Boole contents himself with this because he has started with the intention of eliminating E from his conclusion.
We may now solve for a. (ii) and (iii) give no information with regard to this term. But by (i), All a is BdE or bDE or C ; and by (iv), All a is CD or cd. Therefore, All a is BcdE or CD. And this yields by contraposition, Whatever is bc or Cd or cD or ce is A.
[517] No information whatever with regard to A is given by (i), since a appears as a determinant of the subject. See section [487].
504. Given the same premisses as in the preceding section, shew that,—
1st. If the property B be present in one of the productions, either the properties A, C, and D are all absent, or some one alone of them is absent. And conversely, if they are all absent it may be concluded that the property B is present.
2nd. If A and C are both present or both absent, D will be absent, quite independently of the presence or absence of B. [Boole, Laws of Thought, p. 149.]
We may proceed here by combining all the given premisses in 517 the manner indicated in section [475]. From the result thus obtained the above conclusions as well as those contained in the preceding section will immediately follow.