pg195NOTES 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.

Of these four, No. (2) is equivalent to “All xy are z”, No. (3) is equivalent to “All xy are not-z”, and No. (4) is equivalent to “No xy exist.”

Now it is quite undeniable that, of these four states of things, each is, a priori, possible, some one must be true, and the other three must be false.

Hence the Contradictory to (2) is “Either (1) or (3) or (4) is true.” Now the assertion “Either (1) or (3) is true” is equivalent to “Some xy are not-z”; and the assertion “(4) is true” is equivalent to “No xy exist.” Hence the Contradictory to “All xy are z” may be expressed as the Alternative Proposition “Either some xy are not-z, or no xy exist,” but not as the Categorical Proposition “Some y are not-z.”

pg196(B) [See p. 171, at end of Section 2.]

There are yet other views current among “The Logicians”, as to the “Existential Import” of Propositions, which have not been mentioned in this Section.

One is, that the Proposition “some x are y” is to be interpreted, neither as “Some x exist and are y”, nor yet as “If there were any x in existence, some of them would be y”, but merely as “Some x can be y; i.e. the Attributes x and y are compatible”. On this theory, there would be nothing offensive in my telling my friend Jones “Some of your brothers are swindlers”; since, if he indignantly retorted “What do you mean by such insulting language, you scoundrel?”, I should calmly reply “I merely mean that the thing is conceivable——that some of your brothers might possibly be swindlers”. But it may well be doubted whether such an explanation would entirely appease the wrath of Jones!

Another view is, that the Proposition “All x are y” sometimes implies the actual existence of x, and sometimes does not imply it; and that we cannot tell, without having it in concrete form, which interpretation we are to give to it. This view is, I think, strongly supported by common usage; and it will be fully discussed in Part II: but the difficulties, which it introduces, seem to me too formidable to be even alluded to in Part I, which I am trying to make, as far as possible, easily intelligible to mere beginners.

(C) [See p. 173, § 4.]

The three Conclusions are

“No conceited child of mine is greedy”;
“None of my boys could solve this problem”;
“Some unlearned boys are not choristers.”

pg197INDEX.

§ 1.

Tables.

§ 2.

Words &c. explained.