[360] See Baldwin’s Dictionary of Philosophy, art. Symbolic Logic. It is shewn in this article that the whole of syllogistic reasoning may be summed up in the following antilogism, the symbolism of section [138] being made use of,—
[(AB = 0)(bC = 0)(AC > 0)] = 0.
The fifteen moods containing neither a strengthened premiss nor a weakened conclusion may, by the aid of conversions and obversions, be obtained from this antilogism according as the contradictory of one or other of the three incompatibles is taken as the conclusion.
266. Equivalence of the Moods of the first three Figures shewn by the Method of Indirect Reduction.—If one of our three equivalent syllogisms is in one of the first three figures, then it can be shewn that the two others will be in the remaining two of these figures.
Thus, let P, Q, ∴ Rʹ be in figure 1, the minor premiss being stated first. It may then be written
| S ⎯ M, M ⎯ P, ∴ (S ⎯ P)ʹ. | (1) |
The second syllogism becomes
| M ⎯ P, S ⎯ P, ∴ (S ⎯ M)ʹ; | (2) |
and the third is
| S ⎯ P, S ⎯ M, ∴ (M ⎯ P)ʹ. | (3) |