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Theoretically, it will be quite as interesting to trace the relation between particular propositions differently interpreted. Take Ix and Ox to represent the usual meaning of "some" in formal logic, and Ia and Oa to represent that of Hamilton and general practice and we shall find a very instructive system of opposition.
In this representation the sub-contraries are the same as in ordinary logic, but the reciprocals imply each other: if one be true, the other is true, and if false, the other is false. But the relation between Ia and Ox, and Ia and Ix is a very interesting and complicated one, which I have denominated "subalterno indeterminate" for the lack of a better term to express it. It can be indicated as follows:
If Ia be true, Ix and Ox are indeterminate.
If Ia be false, Ix and Ox are false.
If Ix be true, Ia and Oa are true.
If Ix be false, Ia and Oa are indeterminate.
No practical interest of any great importance attaches to these facts, except that between the ordinary indefinite "some" and the partitive "some" a contradiction is possible under certain conditions, and they show besides this the complications with which we have to deal in actual discourse and argument, compared with the simple rules of formal logic, which seem to be applicable only to an imaginary world. These may all be summarized in a single figure showing the perplexities due to the ambiguous conception of particular propositions. Moreover the illustration assumes the ordinary interpretation of A and E: otherwise the complications would be still more confusing.
But even all these complications will be greatly increased or modified if we assume A and E to be collective instead of distributive in their meaning, and we need not illustrate it for the
