150. Sixfold Schedule of Propositions obtained by recognising Y and η, in addition to A, E, I, O.[209]—The schedule of propositions obtained by adding Y and η to the ordinary schedule presents some interesting features, and is worthy of incidental recognition and discussion.[210] It has been shewn in section [100] that in the ordinary scheme there are six and only six independent propositions connecting any two terms, namely, 208 SaP, PaS, SeP (= PeS), SiP (= PiS), PoS, SoP. If we write the second and the last but one of these in forms in which S and P are respectively subject and predicate, we have the schedule which we are now considering, namely,
| SaP | = | All S is P ; |
| SyP | = | Only S is P ; |
| SeP | = | No S is P ; |
| SiP | = | Some S is P ; |
| SηP | = | Not only S is P ; |
| Sop | = | Some S is not P. |
[209] In this schedule some is interpreted throughout in its ordinary logical sense. U is omitted on account of its composite character; its inclusion would also destroy the symmetry of the scheme.
[210] It is not intended that this sixfold schedule should supersede the fourfold schedule in the main body of logical doctrine. It is, however, important to remember that the selection of any one schedule is more or less arbitrary, and that no schedule should be set up as authoritative to the exclusion of all others.
It will be observed that the pair of propositions, SyP and SηP, are contradictories; so that we now have three pairs of contradictories. There are of course other additions to the traditional table of opposition, and some new relations will need to be recognised, e.g., between SaP and SyP. With the help, however, of the discussion contained in section [107], the reader will have no difficulty in working out the required hexagon of opposition for himself.
As regards immediate inferences, we cannot in this scheme obtain any satisfactory obverse of either Y or η, the reason being that they have quantified predicates, and that, therefore, the negation cannot in these propositions be simply attached to the predicate. We have, however, the following interesting table of other immediate inferences:—[211]
| Converse. | Contrapositive. | Inverse. | ||||
| SaP | = | PyS | = | PʹaSʹ | = | SʹyPʹ |
| SyP | = | PaS | = | PʹySʹ | = | SʹaPʹ |
| SeP | = | PeS | = | PʹyS | = | SʹyP |
| SiP | = | PiS | = | PʹηS | = | SʹηP |
| SηP | = | PoS | = | PʹηSʹ | = | SʹoPʹ |
| SoP | = | PηS | = | PʹoSʹ | = | SʹηPʹ |
[211] It will be observed that the impracticability of obverting Y and η leads to a certain want of symmetry in the third and fourth columns.
The main points to notice here are (1) that each proposition now admits of conversion, contraposition, and inversion; and (2) that the inferred proposition is in every case equivalent to the original proposition, so that there is not in any of the 209 inferences any loss of logical force. In other words, we obtain in each case a simple converse, a simple contrapositive, and a simple inverse.