Complete Works of Lewis Carroll

by Lewis Carroll

  44.

  1e1b′0 † 2a′d0 † 3c1h′0 † 4e′a0 † 5d′h0;

  1eb′ † 4e′a † 2a′d † 5d′h † 3ch′ ¶ b′c0 † c1, i.e. ¶ c1b′0

  i.e. Shakespeare was clever.

  45.

  1e′1c′0 † 2hb′0 † 3d1a0 † 4e1a′0 † 5c1b0;

  1e′c′ † 4ea′ † 3da † 5cb † 2hb′ ¶ dh0 † d1, i.e. ¶ d1h0

  i.e. Rainbows are not worth writing odes to.

  46.

  1c′1h′0 † 2e1a0 † 3bd0 † 4a′1h0 † 5d′c0;

  1c′h′ † 4a′h † 2ea † 5d′c † 3bd ¶ eb0 † e1, i.e. ¶ e1b0

  i.e. These Sorites-examples are difficult.

  47.

  1a′1e′0 † 2bk0 † 3c′a0 † 4eh′0 † 5d1b′0 † 6k′h0;

  1a′e′ † 3c′a † 4eh′ † 6k′h † 2bk † 5db′ ¶ c′d0 † d1,

  i.e. ¶ d1c′0

  i.e. All my dreams come true.

  48.

  1a′h0 † 2c′k0 † 3a1d′0 † 4e1h′0 † 5b1k′0 † 6c1e′0;

  1a′h † 3ad′ † 4eh′ † 6ce′ † 2c′k † 5bk′ ¶ d′b0 † b1,

  i.e. ¶ b1d′0

  i.e. All the English pictures here are painted in oils.

  49.

  1k′1e0 † 2c1h0 † 3b1a′0 † 4kd0 † 5h′a0 † 6b′1e′0;

  1k′e † 4kd † 6b′e′ † 3ba′ † 5h′a † 2ch ¶ dc0 † c1,

  i.e. ¶ c1d0

  i.e. Donkeys are not easy to swallow.

  50.

  1ab′0 † 2h′d0 † 3e1c0 † 4b1d′0 † 5a′k0 † 6c′1h0;

  1ab′ † 4bd′ † 2h′d † 6a′k † 5c′h † 3ec ¶ ke0 † e1,

  i.e. ¶ e1k0

  i.e. Opium-eaters never wear white kid gloves.

  51.

  1bc0 † 2k1a′0 † 3eh0 † 4d1b′0 † 5h′c′0 † 6k′1e′0;

  1bc † 4db′ † 5h′c′ † 3eh † 6k′e′ † 2ka′ ¶ da′0 † d1,

  i.e. ¶ d1a′0

  i.e. A good husband always comes home for his tea.

  52.

  1a′1k′0 † 2ch0 † 3h′k0 † 4b1d′0 † 5ea0 † 6d1c′0

  1a′k′ † 3h′k † 2ch † 6dc′ † 4bd′ † 5ea ¶ be0 † b1,

  i.e. ¶ b1e0

  i.e. Bathing-machines are never made of mother-of-pearl.

  53.

  1da′0 † 2k1b′0 † 3c1h0 † 4d′1k′0 † 5e1c′0 † 6a1h′0;

  1da′ † 4d′k′ † 2kb′ † 6ah′ † 5ch † 3ec′

  ¶ b′e0 † e1, i.e. ¶ e1b′0

  i.e. Rainy days are always cloudy.

  54.

  1kb′0 † 1a′1c′0 † 3d′b0 † 4k′1h′0 † 5ea0 † 6d1c0;

  1kb′ † 3d′b † 4k′h′ † 6dc † 2a′c′ † 5ea

  ¶ h′e0

  i.e. No heavy fish is unkind to children.

  55.

  1k′1b′0 † 2eh′0 † 3c′d0 † 4hb0 † 5ac0 † 6kd′0;

  1k′b′ † 4hb † 2eh′ † 6kd′ † 3c′d † 5ac ¶ ea0

  i.e. No engine-driver lives on barley-sugar.

  56.

  1h1b′0 † 2c1d′0 † 3k′a0 † 4e1h′0 † 5b1a′0 † 6k1c′0;

  1hb′ † 4eh′ † 5ba′ † 3k′a † 6kc′ † 2cd′

  ¶ ed′0 † e1, i.e. ¶ e1d′0

  i.e. All the animals in the yard gnaw bones.

  57.

  1h′1d′0 † 2e1c′0 † 3k′a0 † 4cb0 † 5d1l′0 † 6e′h0 † 7kl0;

  1h′d′ † 5dl′ † 7kl † 3k′a † 6e′h † 2ec′ † 4cb ¶ ab0

  i.e. No badger can guess a conundrum.

  58.

  1b′h0 † 2d′1l′0 † 3ca0 † 4d1k′0 † 5h′1e′0 † 6mc′0 † 7a′b0 † 8ek0;

  1b′h † 5h′e′ † 7a′b † 3ca † 6mc′ † 8ek † 4dk′ † 2d′l′ ¶ ml′0

  i.e. No cheque of yours, received by me, is payable to order.

  59.

  1c1l′0 † 2h′e0 † 3kd0 † 4mc′0 † 5b′1e′0 † 6n1a′0 † 7l1d′0 † 8m′b0 † 9ah0;

  1cl′ † 4mc′ † 7ld′ † 3kd † 8m′b † 5b′e′ † 2h′e † 9ah † 6na′

  ¶ kn0

  i.e. I cannot read any of Brown’s letters.

  60.

  1e1c′0 † 2l1n′0 † 3d1a′0 † 4m′b0 † 5ck′0 † 6e′r0 † 7h1n0 † 8b′k0 † 9r′1d′0 † 10m1l′0;

  1ec′ † 5ck′ † 6e′r † 8b′k † 4m′b † 9r′d′ † 3da′ † 10ml′ † 2ln′ † 7hn

  ¶ a′h0 † h1, i.e. ¶ h1a′0

  i.e. I always avoid a kangaroo.

  NOTES.

  (A) .

  One of the favourite objections, brought against the Science of Logic by its detractors, is that a Syllogism has no real validity as an argument, since it involves the Fallacy of Petitio Principii (i.e. “Begging the Question”, the essence of which is that the whole Conclusion is involved in one of the Premisses).

  This formidable objection is refuted, with beautiful clearness and simplicity, by these three Diagrams, which show us that, in each of the three Figures, the Conclusion is really involved in the two Premisses taken together, each contributing its share.

  Thus, in Fig. I., the Premiss xm0 empties the Inner Cell of the N.W. Quarter, while the Premiss ym0 empties its Outer Cell. Hence it needs the two Premisses to empty the whole of the N.W. Quarter, and thus to prove the Conclusion xy0.

  Again, in Fig. II., the Premiss xm0 empties the Inner Cell of the N.W. Quarter. The Premiss ym1 merely tells us that the Inner Portion of the W. Half is occupied, so that we may place a ‘I’ in it, somewhere; but, if this were the whole of our information, we should not know in which Cell to place it, so that it would have to ‘sit on the fence’: it is only when we learn, from the other Premiss, that the upper of these two Cells is empty, that we feel authorised to place the ‘I’ in the lower Cell, and thus to prove the Conclusion x′y1.

  Lastly, in Fig. III., the information, that m exists, merely authorises us to place a ‘I’ somewhere in the Inner Square——but it has large choice of fences to sit upon! It needs the Premiss xm0 to drive it out of the N. Half of that Square; and it needs the Premiss ym0 to drive it out of the W. Half. Hence it needs the two Premisses to drive it into the Inner Portion of the S.E. Quarter, and thus to prove the Conclusion x′y′1.

  APPENDIX,

  ADDRESSED TO TEACHERS.

  § 1.

  Introductory.

  There are several matters, too hard to discuss with Learners, which nevertheless need to be explained to any Teachers, into whose hands this book may fall, in order that they may thoroughly understand what my Symbolic Method is, and in what respects it differs from the many other Methods already published.

  These matters are as follows:—

  The “Existential Import” of Propositions.

  The use of “is-not” (or “are-not”) as a Copula.

  The theory “two Negative Premisses prove nothing.”

  Euler’s Method of Diagrams.

  Venn’s Method of Diagrams.

  My Method of Diagrams.

  The Solution of a Syllogism by various Methods.

  My Method of treating Syllogisms and Sorites.

  Some account of Parts II, III.

  § 2.

  The “Existential Import” of Propositions.

  The writers, and editors, of the Logical text-books which run in the ordinary grooves——to whom I shall hereafter refer by the (I hope inoffensive) title “The Logicians”——take, on this subject, what seems to me to be a more humble position than is at all necessary. They speak of the Copula of a Proposition “with bated breath”, almost as if it were a living, conscious Entity, capable of declaring for itself what it chose to mean, and that we, poor human creatures, had nothing to do but to ascertain what was its sovereign will and pleasure, and submit to it.

  In opposition to this view, I maintain that any writer of a book is fully authorised in attaching any meaning he likes to any word or phrase he intends to use. If I find an
author saying, at the beginning of his book, “Let it be understood that by the word ‘black’ I shall always mean ‘white’, and that by the word ‘white’ I shall always mean ‘black’,” I meekly accept his ruling, however injudicious I may think it.

  And so, with regard to the question whether a Proposition is or is not to be understood as asserting the existence of its Subject, I maintain that every writer may adopt his own rule, provided of course that it is consistent with itself and with the accepted facts of Logic.

  Let us consider certain views that may logically be held, and thus settle which of them may conveniently be held; after which I shall hold myself free to declare which of them I intend to hold.

  The kinds of Propositions, to be considered, are those that begin with “some”, with “no”, and with “all”. These are usually called Propositions “in I”, “in E”, and “in A”.

  First, then, a Proposition in I may be understood as asserting, or else as not asserting, the existence of its Subject. (By “existence” I mean of course whatever kind of existence suits its nature. The two Propositions, “dreams exist” and “drums exist”, denote two totally different kinds of “existence”. A dream is an aggregate of ideas, and exists only in the mind of a dreamer: whereas a drum is an aggregate of wood and parchment, and exists in the hands of a drummer.)

  First, let us suppose that I “asserts” (i.e. “asserts the existence of its Subject”).

  Here, of course, we must regard a Proposition in A as making the same assertion, since it necessarily contains a Proposition in I.

  We now have I and A “asserting”. Does this leave us free to make what supposition we choose as to E? My answer is “No. We are tied down to the supposition that E does not assert.” This can be proved as follows:—

  If possible, let E “assert”. Then (taking x, y, and z to represent Attributes) we see that, if the Proposition “No 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 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. ).

  We see, then, that the supposition “I asserts” necessarily leads to “A asserts, but E does not”. And this is the first of the various views that may conceivably be held.

  Next, let us suppose that I does not “assert.” And, along with this, let us take the supposition that E does “assert.”

  Hence the Proposition “No x are y” means “Some x exist, and none of them are y”: i.e. “all of them are not-y,” which is a Proposition in A. We also know, of course, that the Proposition “All x are not-y” proves “No x are y.” Now two Propositions, each of which proves the other, are equivalent. Hence every Proposition in A is equivalent to one in E, and therefore “asserts”.

  Hence our second conceivable view is “E and A assert, but I does not.”

  This view does not seen to involve any necessary contradiction with itself or with the accepted facts of Logic. But, when we come to test it, as applied to the actual facts of life, we shall find I think, that it fits in with them so badly that its adoption would be, to say the least of it, singularly inconvenient for ordinary folk.

  Let me record a little dialogue I have just held with my friend Jones, who is trying to form a new Club, to be regulated on strictly Logical principles.

  Author. “Well, Jones! Have you got your new Club started yet?”

  Jones (rubbing his hands). “You’ll be glad to hear that some of the Members (mind, I only say ‘some’) are millionaires! Rolling in gold, my boy!”

  Author. “That sounds well. And how many Members have entered?”

  Jones (staring). “None at all. We haven’t got it started yet. What makes you think we have?”

  Author. “Why, I thought you said that some of the Members——”

  Jones (contemptuously). “You don’t seem to be aware that we’re working on strictly Logical principles. A Particular Proposition does not assert the existence of its Subject. I merely meant to say that we’ve made a Rule not to admit any Members till we have at least three Candidates whose incomes are over ten thousand a year!”

  Author. “Oh, that’s what you meant, is it? Let’s hear some more of your Rules.”

  Jones. “Another is, that no one, who has been convicted seven times of forgery, is admissible.”

  Author. “And here, again, I suppose you don’t mean to assert there are any such convicts in existence?”

  Jones. “Why, that’s exactly what I do mean to assert! Don’t you know that a Universal Negative asserts the existence of its Subject? Of course we didn’t make that Rule till we had satisfied ourselves that there are several such convicts now living.”

  The Reader can now decide for himself how far this second conceivable view would fit in with the facts of life. He will, I think, agree with me that Jones’ view, of the ‘Existential Import’ of Propositions, would lead to some inconvenience.

  Thirdly, let us suppose that neither I nor E “asserts”.

  Now the supposition that the two Propositions, “Some x are y” and “No x are not-y”, do not “assert”, necessarily involves the supposition that “All x are y” does not “assert”, since it would be absurd to suppose that they assert, when combined, more than they do when taken separately.

  Hence the third (and last) of the conceivable views is that neither I, nor E, nor A, “asserts”.

  The advocates of this third view would interpret the Proposition “Some x are y” to mean “If there were any x in existence, some of them would be y”; and so with E and A.

  It admits of proof that this view, as regards A, conflicts with the accepted facts of Logic.

  Let us take the Syllogism Darapti, which is universally accepted as valid. Its form is

  “All m are x;

  All m are y.

  ∴ Some y are x”.

  This they would interpret as follows:—

  ”If there were any m in existence, all of them would be x;

  If there were any m in existence, all of them would be y.

  ∴ If there were any y in existence, some of them would be x”.

  That this Conclusion does not follow has been so briefly and clearly explained by Mr. Keynes (in his “Formal Logic”, dated 1894, pp. 356, 357), that I prefer to quote his words:—

  “Let no proposition imply the existence either of its subject or of its predicate.

  “Take, as an example, a syllogism in Darapti:—

  ‘All M is P,

  All M is S,

  ∴ Some S is P.’

  “Taking S, M, P, as the minor, middle, and major terms respectively, the conclusion will imply that, if there is an S, there is some P. Will the premisses also imply this? If so, then the syllogism is valid; but not otherwise.

  “The conclusion implies that if S exists P exists; but, consistently with the premisses, S may be existent while M and P are both non-existent. An implication is, therefore, contained in the conclusion, which is not justified by the premisses.”

  This seems to me entirely clear and convincing. Still, “to make sicker”, I may as well throw the above (soi-disant) Syllogism into a concrete form, which will be within the grasp of even a non-logical Reader.

  Let us suppose that a Boys’ School has been set up, with the following system of Rules:—

  “All boys in the First (the highest) Class are to do French, Greek, and Latin. All in the Second Class are to do Greek only. All in the Third Class are to do Latin only.”

  Suppose also that there are boys in the Third Class, and in the Second; but that no boy has yet risen into the First.

  It is evident that there are no boys in the School doing French: still we know, by the Rules, what would happen if there were any.

  We are authorised, then, by the Dat
a, to assert the following two Propositions:—

  “If there were any boys doing French, all of them would be doing Greek;

  If there were any boys doing French, all of them would be doing Latin.”

  And the Conclusion, according to “The Logicians” would be

  “If there were any boys doing Latin, some of them would be doing Greek.”

  Here, then, we have two true Premisses and a false Conclusion (since we know that there are boys doing Latin, and that none of them are doing Greek). Hence the argument is invalid.

  Similarly it may be shown that this “non-existential” interpretation destroys the validity of Disamis, Datisi, Felapton, and Fresison.

  Some of “The Logicians” will, no doubt, be ready to reply “But we are not Aldrichians! Why should we be responsible for the validity of the Syllogisms of so antiquated an author as Aldrich?”

  Very good. Then, for the special benefit of these “friends” of mine (with what ominous emphasis that name is sometimes used! “I must have a private interview with you, my young friend,” says the bland Dr. Birch, “in my library, at 9 a.m. tomorrow. And you will please to be punctual!”), for their special benefit, I say, I will produce another charge against this “non-existential” interpretation.

  It actually invalidates the ordinary Process of “Conversion”, as applied to Proposition in ‘I’.

  Every logician, Aldrichian or otherwise, accepts it as an established fact that “Some x are y” may be legitimately converted into “Some y are x.”

  But is it equally clear that the Proposition “If there were any x, some of them would be y” may be legitimately converted into “If there were any y, some of them would be x”? I trow not.