The Fourth Dimension

by Charles Howard Hinton

Fig. 58.

The propositions I, E, in the various figures are the following, as shown in the accompanying scheme, [fig. 58]:—First figure: some M is P; no S is M. Second figure: some P is M; no S is M. Third figure: some M is P; no M is S. Fourth figure: some P is M; no M is S.

Examining these figures, we see, taking the first, that if some M is P and no S is M, we have no conclusion of the form S is P in the various moods. It is quite indeterminate how the circle representing S lies with regard to the circle representing P. It may lie inside, outside, or partly inside P. The same is true in the other figures 2 and 3. But when we come to the fourth figure, since M and S lie completely outside each other, there cannot lie inside S that part of P which lies inside M. Now we know by the major premiss that some of P does lie in M. Hence S cannot contain the whole of P. In words, some P is M, no M is S, therefore S does not contain the whole of P. If we take P as the subject, this gives us a conclusion in the mood O about P. Some P is not S. But it does not give us conclusion about S in any one of the four forms recognised in the syllogism and called its moods. Hence the breach of the continuity in the poiograph has enabled us to detect a lack of completeness in the relations which are considered in the syllogism.

To take an instance:—Some Americans (P) are of African stock (M); No Aryans (S) are of African stock (M); Aryans (S) do not include all of Americans (P).

In order to draw a conclusion about S we have to admit the statement, “S does not contain the whole of P,” as a valid logical form—it is a statement about S which can be made. The logic which gives us the form, “some P is not S,” and which does not allow us to give the exactly equivalent and equally primary form, “S does not contain the whole of P,” is artificial.

And I wish to point out that this artificiality leads to an error.

If one trusted to the mnemonic lines given above, one would conclude that no logical conclusion about S can be drawn from the statement, “some P are M, no M are S.”

But a conclusion can be drawn: S does not contain the whole of P.

It is not that the result is given expressed in another form. The mnemonic lines deny that any conclusion can be drawn from premisses in the moods I, E, respectively.