Solve this as a Sorites-Problem, taking lines 3 and 4 as the Conclusion to be proved. It is permitted to use, as Premisses, not only all that is here asserted, but also all that we may reasonably understand to be implied.
[pg195]NOTES TO APPENDIX.
[(A)] [See [p. 167, line 6].]
It may, perhaps, occur to the Reader, who has studied Formal Logic that the argument, here applied to the Propositions I and E, will apply equally well to the Propositions I and A (since, in the ordinary text-books, the Propositions “All xy are z” and “Some xy are not z” are regarded as Contradictories). Hence it may appear to him that the argument might have been put as follows:—
“We now have I and A ‘asserting.’ Hence, if the Proposition ‘All xy are z’ be true, some things exist with the Attributes x and y: i.e. ‘Some x are y.’
“Also we know that, if the Proposition ‘Some xy are not-z’ be true the same result follows.
“But these two Propositions are Contradictories, so that one or other of them must be true. Hence this result is always true: i.e. the Proposition ‘Some x are y’ is always true!
“Quod est absurdum. Hence I cannot assert.”
This matter will be discussed in Part II; but I may as well give here what seems to me to be an irresistable proof that this view (that A and I are Contradictories), though adopted in the ordinary text-books, is untenable. The proof is as follows:—
With regard to the relationship existing between the Class ‘xy’ and the two Classes ‘z’ and ‘not-z’, there are four conceivable states of things, viz.