222 (2) If SaP is interpreted as implying the existence of S, then it may be expressed existentially S > 0 and SPʹ = 0. These existential forms carry with them the implications P > 0, Either Pʹ = 0 or Sʹ > 0.
The universal negative. Taking the same two suppositions the corresponding existentials will be:—
(1) SP = 0 (carrying with it the implications Either S = 0 or Pʹ > 0, Either P = 0 or Sʹ > 0);
(2) S > 0 and SP = 0 (with the implications Pʹ > 0, Either P = 0 or Sʹ > 0).
These results need no separate discussion.
The particular affirmative. (1) On the supposition that SiP does not carry with it any implication as to the separate existence of its terms, it can be expressed existentially Either S = 0 or SP > 0. It might also be written in the form If S > 0 then SP > 0. Complications resulting from the introduction of considerations of modality will, however, be more easily avoided if the hypothetical form is not made use of.
(2) On the supposition that the existence of S is implied, SiP is reducible to the form SP > 0.
The particular negative. Here the corresponding results are (1) Either S = 0 or SPʹ > 0; (2) SPʹ > 0.
We may sum up our results with reference to the third and fourth of the suppositions formulated in the preceding section.
Let no proposition be interpreted as implying the existence of its separate terms. Then corresponding to the traditional schedule we have the following existential schedule:—
| A,—SPʹ = 0; |
| E,—SP = 0; |
| I,—Either S = 0 or SP > 0; |
| O,—Either S = 0 or SPʹ > 0. |
This represents what may be regarded as the minimum existential import of each of the traditional propositions (interpreted assertorically).