The validity of the above results is at once shewn by reference to the table of equivalences given in section [106]. At least one proposition in which there is no negative term will be found in each line of equivalences except the fourth and the eighth, which are as follows:
| SʹaP | = | SʹePʹ | = | PʹeSʹ | = | PʹaS ; |
| SʹoP | = | SʹiPʹ | = | PʹiSʹ | = | PʹoS. |
In these cases we may indeed get rid of Sʹ (as, for example, from SʹaP), but it is only by introducing Pʹ (thus, SʹaP = PʹaS); there is no getting rid of negative terms altogether. We may here refer back to the results obtained in sections [100] and [106]; with two terms six non-equivalent propositions were obtained, with two terms and their contradictories eight non-equivalent propositions. The ground of this difference is now made clear.
If, however, we are allowed to enlarge our scheme of propositions by recognising certain additional types, and if we work on the assumption that universal propositions are existentially negative while particular propositions are existentially affirmative,[153] then negative terms may always be eliminated.146 Thus, No not-S is not-P is equivalent to the statement Nothing is both not-S and not-P, and this becomes by obversion Everything is either S or P. Again, Some not-S is not-P is equivalent to the statement Something is both not-S and not-P, and this becomes by obversion Something is not either S or P, or, as this proposition may also be written, There is something besides S and P. The elimination of negative terms has now been accomplished in all cases. It will be observed further that we now have eight non-equivalent propositions containing only S and P—namely, All S is P, No S is P, Some S is P, Some S is not P, All P is S, Some P is not S, Everything is either S or P, There is something besides S and P.
[153] It is necessary here to anticipate the results of a discussion that will come at a later stage. See [chapter 8].
Following out this line of treatment, the table of equivalences given in section [106] may be rewritten as follows [columns (ii) and (iii) being omitted, and columns (v) and (vi) taking their places]:
| (i) | (iv) | (v) | (vi) | |||
| SaP | = | PʹaSʹ | = | Nothing is SPʹ | = | Everything is Sʹ or P. |
| SʹaPʹ | = | PaS | = | Nothing is SʹP | = | Everything is S or Pʹ. |
| SaPʹ | = | PaSʹ | = | Nothing is SP | = | Everything is Sʹ or Pʹ. |
| SʹaP | = | PʹaS | = | Nothing is SʹPʹ | = | Everything is S or P. |
| SoP | = | PʹoSʹ | = | Something is SPʹ | = | There is something besides Sʹ and P. |
| SʹoPʹ | = | PoS | = | Something is SʹP | = | There is something besides S and Pʹ. |
| SoPʹ | = | PoSʹ | = | Something is SP | = | There is something besides Sʹ and Pʹ. |
| SʹoP | = | PʹoS | = | Something is SʹPʹ | = | There is something besides S and P. |
Taking the propositions in two divisions of four sets each, the two diagonals from left to right give propositions containing S and P only.[154]
[154] The first four propositions in column (v) may be expressed symbolically SPʹ = 0, &c.; the second four SPʹ > 0, &c.; the first four in column (vi) Sʹ + P = 1, &c.; and the second four Sʹ + P < 1, &c.; where 1 = the universe of discourse, and 0 = nonentity, i.e., the contradictory of the universe of discourse. Compare section [138].
147 The scheme of propositions given in this section may be brought into interesting relation with the three fundamental laws of thought.[155] The scheme is based upon the recognition of the following propositional forms and their contradictories: