[147] In speaking of the quantity as depressed, it is meant that a universal yields a particular, and a particular yields nothing.
106. Table of Propositions connecting any two terms and their contradictories.—Taking any two terms and their contradictories, S, P, not-S, not-P, and combining them in pairs, we obtain thirty-two propositions of the forms A, E, I, O. The following table, however, shews that only eight of these thirty-two propositions are non-equivalent.
| (i) | (ii) | (iii) | (iv) | ||||
| Universals | |||||||
| A | SaP | = | SePʹ | = | PʹeS | = | PʹaSʹ |
| Aʹ | SʹaPʹ | = | SʹeP | = | PeSʹ | = | PaS |
| E | SaPʹ | = | SeP | = | PeS | = | PaSʹ |
| Eʹ | SʹaP | = | SʹePʹ | = | PʹeSʹ | = | PʹaS |
| Particulars | |||||||
| O | SoP | = | SiPʹ | = | PʹiS | = | PʹoSʹ |
| Oʹ | SʹoPʹ | = | SʹiP | = | PiSʹ | = | PoS |
| I | SoPʹ | = | SiP | = | PiS | = | PoSʹ |
| Iʹ | SʹoP | = | SʹiPʹ | = | PʹiSʹ | = | PʹoS |
In this table, columns (i) and (ii) contain the propositions in which S or Sʹ is subject, and columns (iii) and (iv) the propositions in which P or Pʹ is subject. In columns (i) and (iv) we have the forms which admit of simple contraposition (i.e., A and O), and in columns (ii) and (iii) those which admit of simple conversion (i.e., E and I). Contradictories are shewn by identical places in the universal and particular rows. We pass from column (i) to column (ii) by obversion; from column (ii) to column (iii) by simple conversion; and from column (iii) to column (iv) by obversion.
The forms in black type shew that we may take for our 142 eight non-equivalent propositions the four propositions connecting S and P, and a similar set connecting not-S and not-P.[148] To establish their non-equivalence we may proceed as follows: SaP and SeP are already known to be non-equivalent, and the same is true of SʹaPʹ and SʹePʹ ; but no universal proposition can yield a universal inverse; therefore, no one of these four propositions is equivalent to any other. Again, SiP and SoP are already known to be non-equivalent, and the same is true of SʹiPʹ and SʹoPʹ ; but no particular proposition has any inverse; therefore, no one of these propositions is equivalent to any other. Finally, no universal proposition can be equivalent to a particular proposition.[149]
[148] The former set being denoted by A, E, I, O, the latter set may be denoted by Aʹ, Eʹ, Iʹ, Oʹ.
[149] Mrs Ladd Franklin, in an article on The Proposition in Baldwin’s Dictionary of Philosophy and Psychology, reaches the result arrived at in this section from a different point of view. Mrs Franklin shews that, if we express everything that can be said in the form of existential propositions (that is, propositions affirming or denying existence), it is at once evident that the actual number of different statements possible in terms of X and Y and their contradictories x and y is eight. For the combinations of X and Y and their contradictories are XY, Xy, xY, xy, and we can affirm each of these combinations to exist or to be non-existent. Hence it is clear that eight different statements of fact are possible, and that these eight must remain different, no matter what the form in which they may be expressed.
It may be worth adding that the conditional and disjunctive forms as well as the categorical may here be included on the understanding that all the propositions are interpreted assertorically. Thus, the four following propositions are, on the above understanding, equivalent to one another: All X is Y (categorical); If anything is X, it is Y (conditional); Nothing is Xy (existential); Everything is x or Y (disjunctive).
107. Mutual Relations of the non-equivalent Propositions connecting any two terms and their contradictories.[150]—We may now investigate the mutual relations of our eight non-equivalent propositions. SaP, SeP, SiP, SoP form an ordinary square of opposition; and so do SʹaPʹ, SʹePʹ, SʹiPʹ, SʹoPʹ. Reference to columns (iii) and (iv) in the table will shew further that SaP, SʹePʹ, SʹiPʹ, SoP are equivalent to another square of opposition; and that the same is true of SʹaPʹ, SeP, SiP, SʹoPʹ. This leaves only the following pairs unaccounted for: 143 SaP, SʹaPʹ ; SeP, SʹePʹ ; SoP, SʹoPʹ ; SiP, SʹiPʹ ; SaP, SʹoPʹ ; SʹaPʹ, SoP ; SeP, SʹiPʹ ; SʹePʹ, SiP ; and it will be found that in each of these cases we have an independent pair.
[150] This section may be omitted on a first reading.