C = ABC ꖌ AbC ꖌ aBC ꖌ abC.
But substituting for A in the first and for B in the third alternative we get
C = ABCc ꖌ ABbC ꖌ AbC ꖌ aABC ꖌ abC.
The first, second, and fourth alternatives in the above are self-contradictory combinations, and only these; striking them out there remain
C = AbC ꖌ abC = bC,
the required answer. This symbolic reasoning is, I believe, the exact equivalent of Mr. Venn’s reasoning, and I do not believe that the result can be attained in a simpler manner. Mr. Venn adds that he could adduce other similar instances, that is, instances showing the necessity of a better logical method.
Abbreviation of the Process.
Before proceeding to further illustrations of the use of this method, I must point out how much its practical employment can be simplified, and how much more easy it is than would appear from the description. When we want to effect at all a thorough solution of a logical problem it is best to form, in the first place, a complete series of all the combinations of terms involved in it. If there be two terms A and B, the utmost variety of combinations in which they can appear are
| AB | aB |
| Ab | ab. |
The term A appears in the first and second; B in the first and third; a in the third and fourth; and b in the second and fourth. Now if we have any premise, say