As to Syllogisms, I find that their nineteen forms, with about a score of others which they have ignored, can all be arranged under three forms, each with a very simple Rule of its own; and the only question the Reader has to settle, in working any one of the 101 Examples given at p. 101 of this book, is “Does it belong to Fig. I., II., or III.?”
As to Sorites, the only two forms, recognised by the text-books, are the Aristotelian, whose Premisses are a series of Propositions in A, so arranged that the Predicate of each is the Subject of the next, and the Goclenian, whose Premisses are the very same series, written backwards. Goclenius, it seems, was the first who noticed the startling fact that it does not affect the force of a Syllogism to invert the order of its Premisses, and who applied this discovery to a Sorites. If we assume (as surely we may?) that he is the same man as that transcendent genius who first noticed that 4 times 5 is the same thing as 5 times 4, we may apply to him what somebody (Edmund Yates, I think it was) has said of Tupper, viz., “here is a man who, beyond all others of his generation, has been favoured with Glimpses of the Obvious!”
These puerile——not to say infantine——forms of a Sorites I have, in this book, ignored from the very first, and have not only admitted freely Propositions in E, but have purposely stated the Premisses in random order, leaving to the Reader the useful task of arranging them, for himself, in an order which can be worked as a series of regular Syllogisms. In doing this, he can begin with any one of them he likes.
I have tabulated, for curiosity, the various orders in which the Premisses of the Aristotelian Sorites
1. All a are b;
2. All b are c;
3. All c are d;
4. All d are e;
5. All e are h.
∴ All a are h.
may be syllogistically arranged, and I find there are no less than sixteen such orders, viz., 12345, 21345, 23145, 23415, 23451, 32145, 32415, 32451, 34215, 34251, 34521, 43215, 43251, 43521, 45321, 54321. Of these the first and the last have been dignified with names; but the other fourteen——first enumerated by an obscure Writer on Logic, towards the end of the Nineteenth Century——remain without a name!
§ 10.
Some account of Parts II, III.
In Part II. will be found some of the matters mentioned in this Appendix, viz., the “Existential Import” of Propositions, the use of a negative Copula, and the theory that “two negative Premisses prove nothing.” I shall also extend the range of Syllogisms, by introducing Propositions containing alternatives (such as “Not-all x are y”), Propositions containing 3 or more Terms (such as “All ab are c”, which, taken along with “Some bc′ are d”, would prove “Some d are a′”), &c. I shall also discuss Sorites containing Entities, and the very puzzling subjects of Hypotheticals and Dilemmas. I hope, in the course of Part II., to go over all the ground usually traversed in the text-books used in our Schools and Universities, and to enable my Readers to solve Problems of the same kind as, and far harder than, those that are at present set in their Examinations.
In Part III. I hope to deal with many curious and out-of-the-way subjects, some of which are not even alluded to in any of the treatises I have met with. In this Part will be found such matters as the Analysis of Propositions into their Elements (let the Reader, who has never gone into this branch of the subject, try to make out for himself what additional Proposition would be needed to convert “Some a are b” into “Some a are bc”), the treatment of Numerical and Geometrical Problems, the construction of Problems, and the solution of Syllogisms and Sorites containing Propositions more complex than any that I have used in Part II.
I will conclude with eight Problems, as a taste of what is coming in Part II. I shall be very glad to receive, from any Reader, who thinks he has solved any one of them (more especially if he has done so without using any Method of Symbols), what he conceives to be its complete Conclusion.
It may be well to explain what I mean by the complete Conclusion of a Syllogism or a Sorites. I distinguish their Terms as being of two kinds——those which can be eliminated (e.g. the Middle Term of a Syllogism), which I call the “Eliminands,” and those which cannot, which I call the “Retinends”; and I do not call the Conclusion complete, unless it states all the relations among the Retinends only, which can be deduced from the Premisses.
1.
All the boys, in a certain School, sit together in one large room every evening. They are of no less than five nationalities——English, Scotch, Welsh, Irish, and German. One of the Monitors (who is a great reader of Wilkie Collins’ novels) is very observant, and takes MS. notes of almost everything that happens, with the view of being a good sensational witness, in case any conspiracy to commit a murder should be on foot. The following are some of his notes:—
(1) Whenever some of the English boys are singing “Rule Britannia”, and some not, some of the Monitors are wide-awake;
(2) Whenever some of the Scotch are dancing reels, and some of the Irish fighting, some of the Welsh are eating toasted cheese;
(3) Whenever all the Germans are playing chess, some of the Eleven are not oiling their bats;
(4) Whenever some of the Monitors are asleep, and some not, some of the Irish are fighting;
(5) Whenever some of the Germans are playing chess, and none of the Scotch are dancing reels, some of the Welsh are not eating toasted cheese;
(6) Whenever some of the Scotch are not dancing reels, and some of the Irish not fighting, some of the Germans are playing chess;
(7) Whenever some of the Monitors are awake, and some of the Welsh are eating toasted cheese, none of the Scotch are dancing reels;
(8) Whenever some of the Germans are not playing chess, and some of the Welsh are not eating toasted cheese, none of the Irish are fighting;
(9) Whenever all the English are singing “Rule Britannia,” and some of the Scotch are not dancing reels, none of the Germans are playing chess;
(10) Whenever some of the English are singing “Rule Britannia”, and some of the Monitors are asleep, some of the Irish are not fighting;
(11) Whenever some of the Monitors are awake, and some of the Eleven are not oiling their bats, some of the Scotch are dancing reels;
(12) Whenever some of the English are singing “Rule Britannia”, and some of the Scotch are not dancing reels, * * * *
Here the MS. breaks off suddenly. The Problem is to complete the sentence, if possible.
[N.B. In solving this Problem, it is necessary to remember that the Proposition “All x are y” is a Double Proposition, and is equivalent to “Some x are y, and none are y′.” See p. 17.]
2.
(1) A logician, who eats pork-chops for supper, will probably lose money;
(2) A gambler, whose appetite is not ravenous, will probably lose money;
(3) A man who is depressed, having lost money and being likely to lose more, always rises at 5 a.m.;
(4) A man, who neither gambles nor eats pork-chops for supper, is sure to have a ravenous appetite;
(5) A lively man, who goes to bed before 4 a.m., had better take to cab-driving;
(6) A man with a ravenous appetite, who has not lost money and does not rise at 5 a.m., always eats pork-chops for supper;
(7) A logician, who is in danger of losing money, had better take to cab-driving;
(8) An earnest gambler, who is depressed though he has not lost money, is in no danger of losing any;
(9) A man, who does not gamble, and whose appetite is not ravenous, is always lively;
(10) A lively logician, who is really in earnest, is in no danger of losing money;
(11) A man with a ravenous appetite has no need to take to cab-driving, if he is really in earnest;
(12) A gambler, who is depressed though in no danger of losing money, sits up till 4 a.m.
(13) A man, who has lost money and does not eat pork-chops for supper, had better take to cab-driving, unless he gets up at 5 a.m.
(14) A gambler, who goes to b
ed before 4 a.m., need not take to cab-driving, unless he has a ravenous appetite;
(15) A man with a ravenous appetite, who is depressed though in no danger of losing, is a gambler.
Univ. “men”; a = earnest; b = eating pork-chops for supper; c = gamblers; d = getting up at 5; e = having lost money; h = having a ravenous appetite; k = likely to lose money; l = lively; m = logicians; n = men who had better take to cab-driving; r = sitting up till 4.
[N.B. In this Problem, clauses, beginning with “though”, are intended to be treated as essential parts of the Propositions in which they occur, just as if they had begun with “and”.]
3.
(1) When the day is fine, I tell Froggy “You’re quite the dandy, old chap!”;
(2) Whenever I let Froggy forget that £10 he owes me, and he begins to strut about like a peacock, his mother declares “He shall not go out a-wooing!”;
(3) Now that Froggy’s hair is out of curl, he has put away his gorgeous waistcoat;
(4) Whenever I go out on the roof to enjoy a quiet cigar, I’m sure to discover that my purse is empty;
(5) When my tailor calls with his little bill, and I remind Froggy of that £10 he owes me, he does not grin like a hyæna;
(6) When it is very hot, the thermometer is high;
(7) When the day is fine, and I’m not in the humour for a cigar, and Froggy is grinning like a hyæna, I never venture to hint that he’s quite the dandy;
(8) When my tailor calls with his little bill and finds me with an empty purse, I remind Froggy of that £10 he owes me;
(9) My railway-shares are going up like anything!
(10) When my purse is empty, and when, noticing that Froggy has got his gorgeous waistcoat on, I venture to remind him of that £10 he owes me, things are apt to get rather warm;
(11) Now that it looks like rain, and Froggy is grinning like a hyæna, I can do without my cigar;
(12) When the thermometer is high, you need not trouble yourself to take an umbrella;
(13) When Froggy has his gorgeous waistcoat on, but is not strutting about like a peacock, I betake myself to a quiet cigar;
(14) When I tell Froggy that he’s quite the dandy, he grins like a hyæna;
(15) When my purse is tolerably full, and Froggy’s hair is one mass of curls, and when he is not strutting about like a peacock, I go out on the roof;
(16) When my railway-shares are going up, and when it is chilly and looks like rain, I have a quiet cigar;
(17) When Froggy’s mother lets him go a-wooing, he seems nearly mad with joy, and puts on a waistcoat that is gorgeous beyond words;
(18) When it is going to rain, and I am having a quiet cigar, and Froggy is not intending to go a-wooing, you had better take an umbrella;
(19) When my railway-shares are going up, and Froggy seems nearly mad with joy, that is the time my tailor always chooses for calling with his little bill;
(20) When the day is cool and the thermometer low, and I say nothing to Froggy about his being quite the dandy, and there’s not the ghost of a grin on his face, I haven’t the heart for my cigar!
4.
(1) Any one, fit to be an M.P., who is not always speaking, is a public benefactor;
(2) Clear-headed people, who express themselves well, have had a good education;
(3) A woman, who deserves praise, is one who can keep a secret;
(4) People, who benefit the public, but do not use their influence for good purpose, are not fit to go into Parliament;
(5) People, who are worth their weight in gold and who deserve praise, are always unassuming;
(6) Public benefactors, who use their influence for good objects, deserve praise;
(7) People, who are unpopular and not worth their weight in gold, never can keep a secret;
(8) People, who can talk for ever and are fit to be Members of Parliament, deserve praise;
(9) Any one, who can keep a secret and who is unassuming, is a never-to-be-forgotten public benefactor;
(10) A woman, who benefits the public, is always popular;
(11) People, who are worth their weight in gold, who never leave off talking, and whom it is impossible to forget, are just the people whose photographs are in all the shop-windows;
(12) An ill-educated woman, who is not clear-headed, is not fit to go into Parliament;
(13) Any one, who can keep a secret and is not for ever talking, is sure to be unpopular;
(14) A clear-headed person, who has influence and uses it for good objects, is a public benefactor;
(15) A public benefactor, who is unassuming, is not the sort of person whose photograph is in every shop-window;
(16) People, who can keep a secret and who use their influence for good purposes, are worth their weight in gold;
(17) A person, who has no power of expression and who cannot influence others, is certainly not a woman;
(18) People, who are popular and worthy of praise, either are public benefactors or else are unassuming.
Univ. “persons”; a = able to keep a secret; b = clear-headed; c = constantly talking; d = deserving praise; e = exhibited in shop-windows; h = expressing oneself well; k = fit to be an M.P.; l = influential; m = never-to-be-forgotten; n = popular; r = public benefactors; s = unassuming; t = using one’s influence for good objects; v = well-educated; w = women; z = worth one’s weight in gold.
5.
Six friends, and their six wives, are staying in the same hotel; and they all walk out daily, in parties of various size and composition. To ensure variety in these daily walks, they have agree to observe the following Rules:—
(1) If Acres is with (i.e. is in the same party with) his wife, and Barry with his, and Eden with Mrs. Hall, Cole must be with Mrs. Dix;
(2) If Acres is with his wife, and Hall with his, and Barry with Mrs. Cole, Dix must not be with Mrs. Eden;
(3) If Cole and Dix and their wives are all in the same party, and Acres not with Mrs. Barry, Eden must not be with Mrs. Hall;
(4) If Acres is with his wife, and Dix with his, and Barry not with Mrs. Cole, Eden must be with Mrs. Hall;
(5) If Eden is with his wife, and Hall with his, and Cole with Mrs. Dix, Acres must not be with Mrs. Barry;
(6) If Barry and Cole and their wives are all in the same party, and Eden not with Mrs. Hall, Dix must be with Mrs. Eden.
The Problem is to prove that there must be, every day, at least one married couple who are not in the same party.
6.
After the six friends, named in Problem 5, had returned from their tour, three of them, Barry, Cole, and Dix, agreed, with two other friends of theirs, Lang and Mill, that the five should meet, every day, at a certain table d’hôte. Remembering how much amusement they had derived from their code of rules for walking-parties, they devised the following rules to be observed whenever beef appeared on the table:—
(1) If Barry takes salt, then either Cole or Lang takes one only of the two condiments, salt and mustard: if he takes mustard, then either Dix takes neither condiment, or Mill takes both.
(2) If Cole takes salt, then either Barry takes only one condiment, or Mill takes neither: if he takes mustard, then either Dix or Lang takes both.
(3) If Dix takes salt, then either Barry takes neither condiment or Cole take both: if he takes mustard, then either Lang or Mill takes neither.
(4) If Lang takes salt, then Barry or Dix takes only one condiment: if he takes mustard, then either Cole or Mill takes neither.
(5) If Mill takes salt, then either Barry or Lang takes both condiments: if he takes mustard, then either Cole or Dix takes only one.
The Problem is to discover whether these rules are compatible; and, if so, what arrangements are possible.
[N.B. In this Problem, it is assumed that the phrase “if Barry takes salt” allows of two possible cases, viz. (1) “he takes salt only”; (2) “he takes both condiments”. And so with all similar phrases.
It is also assumed that the phrase “either Cole or La
ng takes one only of the two condiments” allows three possible cases, viz. (1) “Cole takes one only, Lang takes both or neither”; (2) “Cole takes both or neither, Lang takes one only”; (3) “Cole takes one only, Lang takes one only”. And so with all similar phrases.
It is also assumed that every rule is to be understood as implying the words “and vice versâ.” Thus the first rule would imply the addition “and, if either Cole or Lang takes only one condiment, then Barry takes salt.”]
7.
(1) Brothers, who are much admired, are apt to be self-conscious;
(2) When two men of the same height are on opposite sides in Politics, if one of them has his admirers, so also has the other;
(3) Brothers, who avoid general Society, look well when walking together;
(4) Whenever you find two men, who differ in Politics and in their views of Society, and who are not both of them ugly, you may be sure that they look well when walking together;
(5) Ugly men, who look well when walking together, are not both of them free from self-consciousness;
(6) Brothers, who differs in Politics, and are not both of them handsome, never give themselves airs;
(7) John declines to go into Society, but never gives himself airs;
(8) Brothers, who are apt to be self-conscious, though not both of them handsome, usually dislike Society;
(9) Men of the same height, who do not give themselves airs, are free from self-consciousness;
(10) Men, who agree on questions of Art, though they differ in Politics, and who are not both of them ugly, are always admired;
(11) Men, who hold opposite views about Art and are not admired, always give themselves airs;
Complete Works of Lewis Carroll Page 116