[199] The corresponding interpretation when some is used in the sense of some only is given in notes [1] and [2] on page 200, and in note [2] on page 206.
[200] If the Hamiltonian writers had attempted to illustrate their doctrine by means of the Eulerian diagrams, they would I think either have found it to be unworkable, or they would have worked it out to a more distinct and consistent issue.
We have then the following pairs of contradictories—A, O; Y, η; I, E. The contradictory of U is obtained by affirming an alternative between η and O.
Without the use of quantified predicates, the same information may be expressed as follows:—
| U = SaP, PaS ; |
| A = SaP ; |
| Y = PaS ; |
| I = SiP ; |
| E = SeP ; |
| η = PoS ; |
| O = SoP. |
What information, if any, is given by ω will be discussed in section [149].
147. The propositions U and Y.—It must be admitted that these propositions are met with in ordinary discourse. 205 We may not indeed find propositions which are actually written in the form All S is all P ; but we have to all intents and purposes U, whenever there is an unmistakeable affirmation that the subject and the predicate of a proposition are co-extensive. Thus, all definitions are practically U propositions; so are all affirmative propositions of which both the subject and the predicate are singular terms.[201] Take also such propositions as the following: Christianity and civilization are co-extensive; Europe, Asia, Africa, America, and Australia are all the continents;[202] The three whom I have mentioned are all who have ever ascended the mountain by that route; Common salt is the same thing as sodium chloride.[203]
[201] Take the proposition, “Mr Gladstone is the present Prime Minister.” If any one denies that this is U, then he must deny that the proposition “Mr Gladstone is an Englishman” is A. We have at an earlier [stage] discussed the question how far singular propositions may rightly be regarded as constituting a sub-class of universals.
[202] In this and the example that follows the predicate is clearly quantified universally; so that if these are not U propositions, they must be Y propositions. But it is equally clear that the subject denotes the whole of a certain class, however limited that class may be.
[203] These are all examples of what Jevons would call simple identities as distinguished from partial identities. Compare section [138].