 |
1 3 5 2 4 cd + c'a' + ed' + he' + h'b > a'b_{0} — = -= -= =
i.e. No pawnbroker is dishonest.
40.
1 2 3 4 5 bd'{0} + c'h{0} + e{1}b'{0} + da{0} + e'c{0};
1 3 4 5 2 bd' + eb' + da + e'c + c'h > ah_{0} — -= = = - =
i.e. No kitten with green eyes will play with a gorilla.
41.
1 2 3 4 5 c{1}a'{0} + h'b{0} + ae{0} + d{1}c'{0} + h{1}e'{0};
1 3 4 5 2 ca' + ae + dc' + he' + h'b > db{0} + d{1}, i.e. > d{1}b{0} — =- = -= =
i.e. All my friends in this College dine at the lower table.
42.
1 2 3 4 5 ca{0} + h{1}d'{0} + c'{1}e'{0} + b'a'{0} + d{1}e{0};
1 3 4 5 2 ca + c'e' + b'a' + de + hd' > b'h{0} + h{1}, — = - = -= =
i.e. > h{1}b'{0}
i.e. My writing-desk is full of live scorpions.
43.
1 2 3 4 5 b'_{1}e_{0} + ah_{0} + dc_{0} + e'_{1}a'_{0} + bc'_{0}
1 4 2 5 3 b'e + e'a' + ah + bc' + dc > hd_{0} - - = - = =- =
i.e. No Mandarin ever reads Hogg's poems. pg162 44.
1 2 3 4 5 e_{1}b'_{0} + a'd_{0} + c_{1}h'_{0} + e'a_{0} + d'h_{0};
1 4 2 5 3 eb' + e'a + a'd + d'h + ch' > b'c{0} + c{1}, - = - = - = - =
i.e. > c{1}b'{0}
i.e. Shakespeare was clever.
45.
1 2 3 4 5 e'_{1}c'_{0} + hb'_{0} + d_{1}a_{0} + e_{1}a'_{0} + c_{1}b_{0};
1 4 3 5 2 e'c' + ea' + da + cb + hb' > dh{0} + d{1}, i.e. > d{1}h{0} - - =- = =- =
i.e. Rainbows are not worth writing odes to.
46.
1 2 3 4 5 c'{1}h'{0} + e{1}a{0} + bd{0} + a'{1}h{0} + d'c{0};
1 4 2 5 3 c'h' + a'h + ea + d'c + bd > eb{0} + e{1}, i.e. > e{1}b{0} - - - = = - = =
i.e. These Sorites-examples are difficult.
47.
1 2 3 4 5 6 a'{1}e'{0} + bk{0} + c'a{0} + eh'{0} + d{1}b'{0} + k'h{0};
1 3 4 6 2 5 a'e' + c'a + eh' + k'h + bk + db' > c'd{0} + d{1}, - - = =- - = -= =
i.e. > d{1}c'{0}
i.e. All my dreams come true.
48.
1 2 3 4 5 6 a'h{0} + c'k{0} + a{1}d'{0} + e{1}h'{0} + b{1}k'{0} + c{1}e'{0};
1 3 4 6 2 5 a'h + ad' + eh' + ce' + c'k + bk' > d'b{0} + b{1}, - - = -= -= = - =
i.e. > b{1}d'{0}
i.e. All the English pictures here are painted in oils.
49.
1 2 3 4 5 6 k'{1}e{0} + c{1}h{0} + b{1}a'{0} + kd{0} + h'a{0} + b'{1}e'{0};
1 4 6 3 5 2 k'e + kd + b'e' + ba' + h'a + ch > dc{0} + c{1}, - - = - = =- - = =
i.e. > c{1}d{0}
i.e. Donkeys are not easy to swallow.
50.
1 2 3 4 5 6 ab'_{0} + h'd_{0} + e_{1}c_{0} + b_{1}d'_{0} + a'k_{0} + c'_{1}h_{0};
1 4 2 5 6 3 ab' + bd' + h'd + a'k + c'h + ec > ke{0} + e{1}, — =- - = = - = =
i.e. > e{1}k{0}
i.e. Opium-eaters never wear white kid gloves.
51.
1 2 3 4 5 6 bc_{0} + k_{1}a'_{0} + eh_{0} + d_{1}b'_{0} + h'c'_{0} + k'_{1}e'_{0};
1 4 5 3 6 2 bc + db' + h'c' + eh + k'e' + ka' > da'{0} + d{1}, — = - = -= - = =
i.e. > d{1}a'{0}
i.e. A good husband always comes home for his tea.
52.
1 2 3 4 5 6 a'_{1}k'_{0} + ch_{0} + h'k_{0} + b_{1}d'_{0} + ea_{0} + d_{1}c'_{0}
1 3 2 6 4 5 a'k' + h'k + ch + dc' + bd' + ea > be{0} + b{1}, - - - = -= -= = =
i.e. > b{1}e{0}
i.e. Bathing-machines are never made of mother-of-pearl. pg163 53.
1 2 3 4 5 da'_{0} + k_{1}b'_{0} + c_{1}h_{0} + d'_{1}k'_{0} + e_{1}c'_{0}
6 + a{1}h'{0};
1 4 2 6 3 5 da' + d'k' + kb' + ah' + ch + ec' — = - = =- -= =
> b'e{0} + e{1}, i.e. > e{1}b'{0}
i.e. Rainy days are always cloudy.
54.
1 2 3 4 5 6 kb'_{0} + a'_{1}c'_{0} + d'b_{0} + k'_{1}h'_{0} + ea_{0} + d_{1}c_{0};
1 3 4 6 2 5 kb' + d'b + k'h' + dc + a'c' + ea — - = = =- - = =
> h'e_{0}
i.e. No heavy fish is unkind to children.
55.
1 2 3 4 5 6 k'_{1}b'_{0} + eh'_{0} + c'd_{0} + hb_{0} + ac_{0} + kd'_{0};
1 4 2 6 3 5 k'b' + hb + eh' + kd' + c'd + ac > ea_{0} - - -= = =- - = =
i.e. No engine-driver lives on barley-sugar.
56.
1 2 3 4 5 h_{1}b'_{0} + c_{1}d'_{0} + k'a_{0} + e_{1}h'_{0} + b_{1}a'_{0}
6 + k{1}c'{0};
1 4 5 3 6 2 hb' + eh' + ba' + k'a + kc' + cd' — = =- - = =- =
> ed'{0} + e{1}, i.e. > e{1}d'{0}
i.e. All the animals in the yard gnaw bones.
57.
1 2 3 4 5 6 h'_{1}d'_{0} + e_{1}c'_{0} + k'a_{0} + cb_{0} + d_{1}l'_{0} + e'h_{0}
7 + kl_{0};
1 5 7 3 6 2 4 h'd' + dl' + kl + k'a + e'h + ec' + cb > ab_{0} - - =- -= = - = =- =
i.e. No badger can guess a conundrum.
58.
1 2 3 4 5 6 b'h_{0} + d'_{1}l'_{0} + ca_{0} + d_{1}k'_{0} + h'_{1}e'_{0} + mc'_{0}
7 8 + a'b{0} + ek{0};
1 5 7 3 6 8 4 2 b'h + h'e' + a'b + ca + mc' + ek + dk' + d'l' > ml'_{0} - - = - - = - = - -= =
i.e. No cheque of yours, received by me, is payable to order.
59.
1 2 3 4 5 6 c_{1}l'_{0} + h'e_{0} + kd_{0} + mc'_{0} + b'_{1}e'_{0} + n_{1}a'_{0}
7 8 9 + l{1}d'{0} + m'b{0} + ah{0};
1 4 7 3 8 5 2 9 6 cl' + mc' + ld' + kd + m'b + b'e' + h'e + ah + na' — -= =- = = - = - - = -= =
> kn_{0}
i.e. I cannot read any of Brown's letters.
60.
1 2 3 4 5 6 e_{1}c'_{0} + l_{1}n'_{0} + d_{1}a'_{0} + m'b_{0} + ck'_{0} + e'r_{0}
7 8 9 10 + h_{1}n_{0} + b'k_{0} + r'_{1}d'_{0} + m_{1}l'_{0};
1 5 6 8 4 9 3 10 2 7 ec' + ck' + e'r + b'k + m'b + r'd' + da' + ml' + ln' + hn — =- = - - = - = = - = =- =- =
> a'h{0} + h{1}, i.e. > h{1}a'{0}
i.e. I always avoid a kangaroo.
pg164
NOTES.
(A) [See p. 80].
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 xm_{0} empties the _Inner_ Cell of the N.W. Quarter, while the Premiss ym_{0} 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 xy_{0}.
Again, in Fig. II., the Premiss xm_{0} empties the Inner Cell of the N.W. Quarter. The Premiss ym_{1} 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'y_{1}.
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 xm_{0} to drive it out of the N. Half of that Square; and it needs the Premiss ym_{0} 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}.
pg165
APPENDIX,
ADDRESSED TO TEACHERS.
Sec. 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.
Sec. 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. pg166 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." pg167 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. (See Note (A), p. 195).
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——" pg168 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". pg169 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. pg170 We are authorised, then, by the Data, 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.
The example I have already used——of a Boys' School with a non-existent First Class——will serve admirably to illustrate this new flaw in the theory of "The Logicians." pg171 Let us suppose that there is yet another Rule in this School, viz. "In each Class, at the end of the Term, the head boy and the second boy shall receive prizes."
This Rule entirely authorises us to assert (in the sense in which "The Logicians" would use the words) "Some boys in the First Class will receive prizes", for this simply means (according to them) "If there were any boys in the First Class, some of them would receive prizes."
Now the Converse of this Proposition is, of course, "Some boys, who will receive prizes, are in the First Class", which means (according to "The Logicians") "If there were any boys about to receive prizes, some of them would be in the First Class" (which Class we know to be empty).
Of this Pair of Converse Propositions, the first is undoubtedly true: the second, as undoubtedly, false.
It is always sad to see a batsman knock down his own wicket: one pities him, as a man and a brother, but, as a cricketer, one can but pronounce him "Out!"
We see, then, that, among all the conceivable views we have here considered, there are only two which can logically be held, viz.
I and A "assert", but E does not. E and A "assert", but I does not.
The second of these I have shown to involve great practical inconvenience.
The first is the one adopted in this book. (See p. 19.)
Some further remarks on this subject will be found in Note (B), at p. 196.
Sec. 3.
The use of "is-not" (or "are-not") as a Copula.
Is it better to say "John is-not in-the-house" or "John is not-in-the-house"? "Some of my acquaintances are-not men-I-should-like-to-be-seen-with" or "Some of my acquaintances are men-I-should-not-like-to-be-seen-with"? That is the sort of question we have now to discuss. pg172 This is no question of Logical Right and Wrong: it is merely a matter of taste, since the two forms mean exactly the same thing. And here, again, "The Logicians" seem to me to take much too humble a position. When they are putting the final touches to the grouping of their Proposition, just before the curtain goes up, and when the Copula——always a rather fussy 'heavy father', asks them "Am I to have the 'not', or will you tack it on to the Predicate?" they are much too ready to answer, like the subtle cab-driver, "Leave it to you, Sir!" The result seems to be, that the grasping Copula constantly gets a "not" that had better have been merged in the Predicate, and that Propositions are differentiated which had better have been recognised as precisely similar. Surely it is simpler to treat "Some men are Jews" and "Some men are Gentiles" as being both of them, affirmative Propositions, instead of translating the latter into "Some men are-not Jews", and regarding it as a negative Propositions?
The fact is, "The Logicians" have somehow acquired a perfectly morbid dread of negative Attributes, which makes them shut their eyes, like frightened children, when they come across such terrible Propositions as "All not-x are y"; and thus they exclude from their system many very useful forms of Syllogisms.
Under the influence of this unreasoning terror, they plead that, in Dichotomy by Contradiction, the negative part is too large to deal with, so that it is better to regard each Thing as either included in, or excluded from, the positive part. I see no force in this plea: and the facts often go the other way. As a personal question, dear Reader, if you were to group your acquaintances into the two Classes, men that you would like to be seen with, and men that you would not like to be seen with, do you think the latter group would be so very much the larger of the two?
For the purposes of Symbolic Logic, it is so much the most convenient plan to regard the two sub-divisions, produced by Dichotomy, on the same footing, and to say, of any Thing, either that it "is" in the one, or that it "is" in the other, that I do not think any Reader of this book is likely to demur to my adopting that course.
pg173 Sec. 4.
The theory that "two Negative Premisses prove nothing".
This I consider to be another craze of "The Logicians", fully as morbid as their dread of a negative Attribute.
It is, perhaps, best refuted by the method of Instantia Contraria.
Take the following Pairs of Premisses:—
"None of my boys are conceited; None of my girls are greedy".
"None of my boys are clever; None but a clever boy could solve this problem".
"None of my boys are learned; Some of my boys are not choristers".
(This last Proposition is, in my system, an affirmative one, since I should read it "are not-choristers"; but, in dealing with "The Logicians," I may fairly treat it as a negative one, since they would read it "are-not choristers".)
If you, dear Reader, declare, after full consideration of these Pairs of Premisses, that you cannot deduce a Conclusion from any of them——why, all I can say is that, like the Duke in Patience, you "will have to be contented with our heart-felt sympathy"! [See Note (C), p. 196.]
Sec. 5.
Euler's Method of Diagrams.
Diagrams seem to have been used, at first, to represent Propositions only. In Euler's well-known Circles, each was supposed to contain a class, and the Diagram consisted of two circles, which exhibited the relations, as to inclusion and exclusion, existing between the two Classes.
/ / / y / x / /
Thus, the Diagram, here given, exhibits the two Classes, whose respective Attributes are x and y, as so related to each other that the following Propositions are all simultaneously true: "All x are y", "No x are not-y", "Some x are y", "Some y are not-x", "Some not-y are not-x", and, of course, the Converses of the last four. pg174 / / / y / x / /
Similarly, with this Diagram, the following Propositions are true:—"All y are x", "No y are not-x", "Some y are x", "Some x are not-y", "Some not-x are not-y", and, of course, the Converses of the last four.
/ / / / x y / / / /
Similarly, with this Diagram, the following are true:—"All x are not-y", "All y are not-x", "No x are y", "Some x are not-y", "Some y are not-x", "Some not-x are not-y", and the Converses of the last four.
/ / / / x y / / / /
Similarly, with this Diagram, the following are true:—"Some x are y", "Some x are not-y", "Some not-x are y", "Some not-x are not-y", and of course, their four Converses.
Note that all Euler's Diagrams assert "Some not-x are not-y." Apparently it never occured to him that it might sometimes fail to be true!
Now, to represent "All x are y", the first of these Diagrams would suffice. Similarly, to represent "No x are y", the third would suffice. But to represent any Particular Proposition, at least three Diagrams would be needed (in order to include all the possible cases), and, for "Some not-x are not-y", all the four.
Sec. 6.
Venn's Method of Diagrams.
Let us represent "not-x" by "x'".
Mr. Venn's Method of Diagrams is a great advance on the above Method.
He uses the last of the above Diagrams to represent any desired relation between x and y, by simply shading a Compartment known to be empty, and placing a + in one known to be occupied.
Thus, he would represent the three Propositions "Some x are y", "No x are y", and "All x are y", as follows:—
/ / / / + / / / /
__ __ _/ \_/ \_ / /# # \_ #/ _/ \__/ \__/
__ __ _/##\_/ \_ /##/ ## + ##\_/ _/ ##/ \__/ pg175 It will be seen that, of the _four_ Classes, whose peculiar Sets of Attributes are xy, xy', x'y, and x'y', only _three_ are here provided with closed Compartments, while the _fourth_ is allowed the rest of the Infinite Plane to range about in!
This arrangement would involve us in very serious trouble, if we ever attempted to represent "No x' are y'." Mr. Venn once (at p. 281) encounters this awful task; but evades it, in a quite masterly fashion, by the simple foot-note "We have not troubled to shade the outside of this diagram"!
To represent two Propositions (containing a common Term) together, a three-letter Diagram is needed. This is the one used by Mr. Venn.
__ _/ \_ _/_ x _\_ _/ \_/ \_ / \_ / _/ _ / \_ m \_/ y _/ \__/ \__/
Here, again, we have only seven closed Compartments, to accommodate the eight Classes whose peculiar Sets of Attributes are xym, xym', &c.
"With four terms in request," Mr. Venn says, "the most simple and symmetrical diagram seems to me that produced by making four ellipses intersect one another in the desired manner". This, however, provides only fifteen closed compartments.
b _ _ c / / a _/_ / _\_ d / / / / / / / / / / / / / / / / / / / / / / / / / // / /\_/_/ / \_\_/_/
For five letters, "the simplest diagram I can suggest," Mr. Venn says, "is one like this (the small ellipse in the centre is to be regarded as a portion of the outside of c; i.e. its four component portions are inside b and d but are no part of c). It must be admitted that such a diagram is not quite so simple to draw as one might wish it to be; but then consider what the alternative is of one undertakes to deal with five terms and all their combinations—nothing short of the disagreeable task of writing out, or in some way putting before us, all the 32 combinations involved."
b c d / / / a / / / / / e / / / / / / / / / / / / / / / / / / / / / / / / / // / / / / / / / / / / / / / / / / / / / /// / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / //// / / / / / pg176 This Diagram gives us 31 closed compartments.
For six letters, Mr. Venn suggests that we might use two Diagrams, like the above, one for the f-part, and the other for the not-f-part, of all the other combinations. "This", he says, "would give the desired 64 subdivisions." This, however, would only give 62 closed Compartments, and one infinite area, which the two Classes, a'b'c'd'e'f and a'b'c'd'e'f', would have to share between them.
Beyond six letters Mr. Venn does not go.
Sec. 7.
My Method of Diagrams.
My Method of Diagrams resembles Mr. Venn's, in having separate Compartments assigned to the various Classes, and in marking these Compartments as occupied or as empty; but it differs from his Method, in assigning a closed area to the Universe of Discourse, so that the Class which, under Mr. Venn's liberal sway, has been ranging at will through Infinite Space, is suddenly dismayed to find itself "cabin'd, cribb'd, confined", in a limited Cell like any other Class! Also I use rectilinear, instead of curvilinear, Figures; and I mark an occupied Cell with a 'I' (meaning that there is at least one Thing in it), and an empty Cell with a 'O' (meaning that there is no Thing in it).
For two letters, I use this Diagram, in which the North Half is assigned to 'x', the South to 'not-x' (or 'x''), the West to y, and the East to y'. Thus the N.W. Cell contains the xy-Class, the N.E. Cell the xy'-Class, and so on.
. -. - - . -.
For three letters, I subdivide these four Cells, by drawing an Inner Square, which I assign to m, the Outer Border being assigned to m'. I thus get eight Cells that are needed to accommodate the eight Classes, whose peculiar Sets of Attributes are xym, xym', &c.
. -. . - -. - - - - . - -. . -.
This last Diagram is the most complex that I use in the Elementary Part of my 'Symbolic Logic.' But I may as well take this opportunity of describing the more complex ones which will appear in Part II. pg177 For four letters (which I call a, b, c, d) I use this Diagram; assigning the North Half to a (and of course the rest of the Diagram to a'), the West Half to b, the Horizontal Oblong to c, and the Upright Oblong to d. We have now got 16 Cells.
. -. . - -. . - - - -. - - - - . - - - -. . - -. . -.
For five letters (adding e) I subdivide the 16 Cells of the previous Diagram by oblique partitions, assigning all the upper portions to e, and all the lower portions to e'. Here, I admit, we lose the advantage of having the e-Class all together, "in a ring-fence", like the other 4 Classes. Still, it is very easy to find; and the operation, of erasing it, is nearly as easy as that of erasing any other Class. We have now got 32 Cells.
. -. / / / . - -. / / / / / . - - - -. / / / / - - - - / / / / . - - - -. / / / / / . - -. / / / . -.
For six letters (adding h, as I avoid tailed letters) I substitute upright crosses for the oblique partitions, assigning the 4 portions, into which each of the 16 Cells is thus divided, to the four Classes eh, eh', e'h, e'h'. We have now got 64 Cells.
============================ H H H H ====H==== H H H H H H H - H H H -H H H H H H H ====H====H====H==== H H H H H H H H H H H H H H H H H H H H H H HH====H====H====H====H H H H H H H H H H H H H H H H H H H H H H H H ====H====H====H==== H H H H H H H - H H H -H H H H H H H ====H==== H H H H ============================ pg178 For seven letters (adding k) I add, to each upright cross, a little inner square. All these 16 little squares are assigned to the k-Class, and all outside them to the k'-Class; so that 8 little Cells (into which each of the 16 Cells is divided) are respectively assigned to the 8 Classes ehk, ehk', &c. We have now got 128 Cells.
#====================================================# H H H H #==========H==========# H H H H H H H . . H . . H . . H . . H H H H H H H - H H H -H H H H H H H . . H . . H . . H . . H H H H H H H #==========H==========H==========H==========# H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H H H H H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H HH========H==========H==========H==========HH H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H H H H H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H #==========H==========H==========H==========# H H H H H H H . . H . . H . . H . . H H H H H H H - H H H -H H H H H H H . . H . . H . . H . . H H H H H H H #==========H==========# H H H H .====================================================#
For eight letters (adding l) I place, in each of the 16 Cells, a lattice, which is a reduced copy of the whole Diagram; and, just as the 16 large Cells of the whole Diagram are assigned to the 16 Classes abcd, abcd', &c., so the 16 little Cells of each lattice are assigned to the 16 Classes ehkl, ehkl', &c. Thus, the lattice in the N.W. corner serves to accommodate the 16 Classes abc'd'ehkl, abc'd'eh'kl', &c. This Octoliteral Diagram (see next page) contains 256 Cells.
For nine letters, I place 2 Octoliteral Diagrams side by side, assigning one of them to m, and the other to m'. We have now got 512 Cells. pg179 ==================================================================== H H H H ==============H============== H H H H H H H . . H . . H . . H . . H H H H H H H . . H . . H . . H . . H H H H H H H -H- -H- -H- H H H H H H H . . H . . H . . H . . H H H H H H H . . H . . H . . H . . H H H H H H H ==============H==============H==============H============== H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H- -H- -H- -H- -H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H HH============H==============H==============H==============HH H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H- -H- -H- -H- -H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H H . . H . . H . . H . . H H H H H H H H H H ==============H==============H==============H============== H H H H H H H . . H . . H . . H . . H H H H H H H . . H . . H . . H . . H H H H H H H -H- -H- -H- H H H H H H H . . H . . H . . H . . H H H H H H H . . H . . H . . H . . H H H H H H H ==============H============== H H H H ====================================================================
Finally, for ten letters, I arrange 4 Octoliteral Diagrams, like the above, in a square, assigning them to the 4 Classes mn, mn', m'n, m'n'. We have now got 1024 Cells.
Sec. 8.
Solution of a Syllogism by various Methods.
The best way, I think, to exhibit the differences between these various Methods of solving Syllogisms, will be to take a concrete example, and solve it by each Method in turn. Let us take, as our example, No. 29 (see p. 102).
"No philosophers are conceited; Some conceited persons are not gamblers. .'. Some persons, who are not gamblers, are not philosophers."
pg180 (1) Solution by ordinary Method.
These Premisses, as they stand, will give no Conclusion, as they are both negative.
If by 'Permutation' or 'Obversion', we write the Minor Premiss thus,
'Some conceited persons are not-gamblers,'
we can get a Conclusion in Fresison, viz.
"No philosophers are conceited; Some conceited persons are not-gamblers. .'. Some not-gamblers are not philosophers"
This can be proved by reduction to Ferio, thus:—
"No conceited persons are philosophers; Some not-gamblers are conceited. .'. Some not-gamblers are not philosophers".
The validity of Ferio follows directly from the Axiom 'De Omni et Nullo'.
(2) Symbolic Representation.
Before proceeding to discuss other Methods of Solution, it is necessary to translate our Syllogism into an abstract form.
Let us take "persons" as our 'Universe of Discourse'; and let x = "philosophers", m = "conceited", and y = "gamblers."
Then the Syllogism may be written thus:—
"No x are m; Some m are y'. .'. Some y' are x'."
(3) Solution by Euler's Method of Diagrams.
The Major Premiss requires only one Diagram, viz.
1 / / / / x m / / / / pg181 The Minor requires three, viz.
2 / / / / y m / / / /
3 / / / / y m / / / /
4 / / / y / m / /
The combination of Major and Minor, in every possible way requires nine, viz.
Figs. 1 and 2 give
5 / / / / / / x y m / / / / / /
6 / / / / / / x y m / / / / / /
7 / / / / xy m / / / /
8 / / / / x / / m y / / / /
9 / / / / y / / m x / / / /
Figs. 1 and 3 give
10 / / / / / / x y m / / / / / /
11 / / / / / / x y m / / / / / /
12 / / / / x / / m y / / / /
Figs. 1 and 4 give
13 / / / / / y x / / m / / /
From this group (Figs. 5 to 13) we have, by disregarding m, to find the relation of x and y. On examination we find that Figs. 5, 10, 13 express the relation of entire mutual exclusion; that Figs. 6, 11 express partial inclusion and partial exclusion; that Fig. 7 expresses coincidence; that Figs. 8, 12 express entire inclusion of x in y; and that Fig. 9 expresses entire inclusion of y in x. pg182 We thus get five Biliteral Diagrams for x and y, viz.
14 / / / / x y / / / /
15 / / / / x y / / / /
16 / / xy / /
17 / / / x / y / /
18 / / / y / x / /
where the only Proposition, represented by them all, is "Some not-y are not-x," i.e. "Some persons, who are not gamblers, are not philosophers"——a result which Euler would hardly have regarded as a valuable one, since he seems to have assumed that a Proposition of this form is always true!
(4) Solution by Venn's Method of Diagrams.
The following Solution has been kindly supplied to me Mr. Venn himself.
"The Minor Premiss declares that some of the constituents in my' must be saved: mark these constituents with a cross.
__ _/ + \_ _/_ _\_ _/ #+#\_/ \_ / #/# _/ + _#_ / \_ m \_/ y _/ \__/ \__/
The Major declares that all xm must be destroyed; erase it.
Then, as some my' is to be saved, it must clearly be my'x'. That is, there must exist my'x'; or eliminating m, y'x'. In common phraseology,
'Some y' are x',' or, 'Some not-gamblers are not-philosophers.'"
pg183 (5) Solution by my Method of Diagrams.
The first Premiss asserts that no xm exist: so we mark the xm-Compartment as empty, by placing a 'O' in each of its Cells.
The second asserts that some my' exist: so we mark the my'-Compartment as occupied, by placing a 'I' in its only available Cell.
. -. . - -. (O) (O) - - - - (I) . - -. . -.
The only information, that this gives us as to x and y, is that the x'y'-Compartment is occupied, i.e. that some x'y' exist.
Hence "Some x' are y'": i.e. "Some persons, who are not philosophers, are not gamblers".
(6) Solution by my Method of Subscripts.
xm_{0} + my'_{1} > x'y'_{1}
i.e. "Some persons, who are not philosophers, are not gamblers."
Sec. 9.
My Method of treating Syllogisms and Sorites.
Of all the strange things, that are to be met with in the ordinary text-books of Formal Logic, perhaps the strangest is the violent contrast one finds to exist between their ways of dealing with these two subjects. While they have elaborately discussed no less than nineteen different forms of Syllogisms——each with its own special and exasperating Rules, while the whole constitute an almost useless machine, for practical purposes, many of the Conclusions being incomplete, and many quite legitimate forms being ignored——they have limited Sorites to two forms only, of childish simplicity; and these they have dignified with special names, apparently under the impression that no other possible forms existed!
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.?" pg184 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!
pg185 Sec. 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.
pg186 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; pg187 (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; pg188 (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 bed 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 L10 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 L10 he owes me, he does not grin like a hyaena; pg189 (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 hyaena, 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 L10 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 L10 he owes me, things are apt to get rather warm;
(11) Now that it looks like rain, and Froggy is grinning like a hyaena, 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 hyaena;
(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!
pg190 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; pg191 (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.
pg192 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'hote. 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 Lang 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 versa." Thus the first rule would imply the addition "and, if either Cole or Lang takes only one condiment, then Barry takes salt."]
pg193 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;
(12) Brothers of the same height always differ in Politics;
(13) Two handsome men, who are neither both of them admired nor both of them self-conscious, are no doubt of different heights;
(14) Brothers, who are self-conscious, and do not both of them like Society, never look well when walking together.
[N.B. See Note at end of Problem 2.]
pg194 8.
(1) A man can always master his father;
(2) An inferior of a man's uncle owes that man money;
(3) The father of an enemy of a friend of a man owes that man nothing;
(4) A man is always persecuted by his son's creditors;
(5) An inferior of the master of a man's son is senior to that man;
(6) A grandson of a man's junior is not his nephew;
(7) A servant of an inferior of a friend of a man's enemy is never persecuted by that man;
(8) A friend of a superior of the master of a man's victim is that man's enemy;
(9) An enemy of a persecutor of a servant of a man's father is that man's friend.
The Problem is to deduce some fact about great-grandsons.
[N.B. In this Problem, it is assumed that all the men, here referred to, live in the same town, and that every pair of them are either "friends" or "enemies," that every pair are related as "senior and junior", "superior and inferior", and that certain pairs are related as "creditor and debtor", "father and son", "master and servant", "persecutor and victim", "uncle and nephew".]
9.
"Jack Sprat could eat no fat: His wife could eat no lean: And so, between them both, They licked the platter clean."
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.
(1) Some xy are z, and some are not-z; (2) " " none " (3) No xy " some " (4) " " none "
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, Sec. 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."
pg197
INDEX.
Sec. 1.
Tables.
I. Biliteral Diagram. Attributes of Classes, and Compartments, or Cells, assigned to them 25
II. do. Representation of Uniliteral Propositions of Existence 34
III. do. Representation of Biliteral Propositions of Existence and of Relation 35
IV. Triliteral Diagram. Attributes of Classes, and Compartments, or Cells, assigned to them 42
V. do. Representation of Particular and Universal Negative Propositions, of Existence and of Relation, in terms of x and m 46
VI. do. do., in terms of y and m 47
VII. do. Representation of Universal Affirmative Propositions of Relation, in terms of x and m 48
VIII. do. do. in terms of y and m 49
IX. Method of Subscripts. Formulae and Rules for Syllogisms 78
Sec. 2.
Words &c. explained.
'Abstract' Proposition 59
'Adjuncts' 1
'Affirmative' Proposition 10
'Attributes' 1
'Biliteral' Diagram 22
" Proposition 27
'Class' 1 1/2
Classes, arbitrary limits of 3 1/2
" , subdivision of 4 pg198 'Classification' 1 1/2
'Codivisional' Classes 3
'Complete' Conclusion of a Sorites 85
'Conclusion' of a Sorites "
" " Syllogism 56
'Concrete' Proposition 59
'Consequent' in a Sorites 85
" " Syllogism 56
'Converse' Propositions 31
'Conversion' of a Proposition "
'Copula' of a Proposition 9
'Definition' 6
'Dichotomy' 3 1/2
'Differentia' 1 1/2
'Division' 3
'Eliminands' of a Sorites 85
" " Syllogism 56
'Entity' 70
'Equivalent' Propositions 17
'Fallacy' 81
'Genus' 1 1/2
'Imaginary' Class "
" Name 4 1/2
'Individual' 2
'Like', and 'Unlike', Signs of Terms 70
'Name' 4
'Negative' Proposition 10
'Normal' form of a Proposition 9
" " of Existence 11
" " of Relation 12
'Nullity' 70
'Partial' Conclusion of a Sorites 85
'Particular' Proposition 9
'Peculiar' Attributes 1 1/2
'Predicate' of a Proposition 9
" of a Proposition of Existence 11
" " Relation 12
'Premisses' of a Sorites 85
" " Syllogism 56 pg199 'Proposition' 8
" 'in I', 'in E', and 'in A' 9
" 'in terms of' certain Letters 27
" of Existence 11
" of Relation 12
'Real' Class 1 1/2
'Retinends' of a Sorites 85
" " Syllogism 56
'Sign of Quantity' in a Proposition 9
'Sitting on the Fence' 26
'Some', technical meaning of 8
'Sorites' 85
'Species' 1 1/2
'Subject' of a Proposition 9
" " of Existence 11
" " of Relation 12
'Subscripts' of Terms 70
'Syllogism' 56
Symbol ".'." "
" "+" and ">" 70
'Terms' of a Proposition 9
'Things' 1
Translation of Proposition from 'concrete' to 'abstract' 59
" " 'abstract' to 'subscript' 72
'Triliteral' Diagram 39
'Underscoring' of letters 91
'Uniliteral' Proposition 27
'Universal' " 10
'Universe of Discourse' (or 'Univ.') 12
'Unreal' Class 1 1/2
'Unreal' Name 4 1/2
pg200
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Transcriber's Note
This book makes extensive use of page references. To assist the reader, page markers in the forms "pg-ix", "pg193" & "px-3" have been included in the right margin at points corresponding closely to the tops of the original pages. These may be searched for to locate the material referred to. In the main section, these page markers are always given with 3 digits including, if necessary, leading zeroes.
This book contains a large number of line drawn illustrations which are un-credited. As these cannot be rendered here in the original manner they have been reproduced as well as possible in the manner known as "ASCII art".
In this ASCII file, the greater-than symbol (>) has bee substituted for the Pilcrow symbol.
A number of transcription errors were found in the original book. As these were clearly not part of the Author's intention they have, as far as possible, been identified and corrected in accordance with the methods given by the Author. These corrections are listed here below with their locations and original text. In these notes the word 'natural' identifies a letter symbol occurring without a prime mark.
Page viii: "189" corrected to "194". Page viii: "188" corrected to "192". Page xv, 'Real' and 'Unreal', or 'Imaginary', Classes: "1 1/2" corrected to "2". Page xix, Propositions beginning with "Some": "18" corrected to "19". Page xix, Rules: "19" corrected to "20". Page xix, The West and East Halves ...: "24" corrected to "23". Page xxi, The Proposition "All x are y" ...: was originally shown as occurring in page 34. Page xxii, And of three other similar arrangements: "37" corrected to "36". Page xxiii, The Proposition 'No xm exist': "43" corrected to "44". Page 060: "contruct" corrected to "construct". Page 094, Paper V: missing word "it" supplied. Page 111, #28: "No h are k-natural" corrected to "No h are k-prime". Page 111, #30: "No a are h-natural" corrected to "No a are h-prime". Page 111, #30: "No c are n-natural" corrected to "No c are n-prime". Page 123, #57 (d): "mortally offended if I fail to notice them" corrected to "mortally offended". Page 128, #34: "Some x-prime are y" corrected to "Some x-natural are y". Page 146, #30: "x{1}m-prime{0}" corrected to "x{1}m-natural{0}". Page 147, #32: "xm-natural{0}" corrected to "xm-prime{0}". Page 156, #18 (4/1): "be" corrected to "be-sub:zero". Page 156, #22 (5/1): "ch" corrected to "ch-sub:zero". Page 157, #23 (5/2): second underline corrected from single to double. Page 157, #23 (2/2): first underline corrected from single to double. Page 157, #23 (>2): "h{1}c" corrected to "h{1}c-sub:zero". Page 157, #26 (5/1): "a{1}c-natural{0}" corrected to "a{1}c-prime{0}". Page 157, #26 (>2): "e{1}c-natural{0}" corrected to "e{1}c-prime{0}". Page 157, #29 (8/2): first underline corrected from single to double. Page 161, #36 (4/2): "e'd-prime" corrected to "e'd-natural". Page 161, #39 (1/2): "c-prime+d" corrected to "c-natural+d". Page 161, #40: "a" and "b" interchanged. Page 161, #43 (1/1): "b-natural{1}e{0}" corrected to "b-prime{1}e{0}". Page 162, #52 (3/1): "h'k-prime{0}" corrected to "h'k-natural{0}". Page 163, #55 (3/2): "c'd-prime" corrected to "c'd-natural". Page 163, #57 (1/2): "h-natural+d'" corrected to "h-prime+d'".
In the original book at the top of page 97, the following text occurred:
[N.B. The numbers at the foot of each page indicate the pages where the corresponding matter may be found.]
In accordance with the un-paged medium here this has been changed to:
[N.B. Reference tags for Examples, Answers & Solutions will be found in the right margin.]
The part of the book to which this relates contains, by sections, "Examples" (Exercises for the student), "Answers" (to the Examples) & "Solutions" (Worked Answers). All Example sections have corresponding Answer sections. For Sections 2 & 3, worked Solutions are not supplied; for Sections 4-7, Solutions are given by 2 different methods. In association with this, the original text contained editorial notes at the foot of each page giving the page numbers for the related Sections. In this version, these notes are replaced with marginal tags such as EX3, AN4, SL5, & SL6-A and SL7-B which are placed at the top of each Section to identify the current location. As with page tags, these may be searched for to locate the material refered to. (A search for "SL4" should find successively both "SL4-A" & "SL4-B".) These tags are unique regardless of case.
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