i.e. My writing-desk is full of live scorpions.
43.
1
1
i.e. No Mandarin ever reads Hogg’s poems.
pg16244.
1
1
i.e. ¶ c1b′0
i.e. Shakespeare was clever.
45.
1
1
i.e. Rainbows are not worth writing odes to.
46.
1
1
i.e. These Sorites-examples are difficult.
47.
1
1
i.e. ¶ d1c′0
i.e. All my dreams come true.
48.
1
1
i.e. ¶ b1d′0
i.e. All the English pictures here are painted in oils.
49.
1
1
i.e. ¶ c1d0
i.e. Donkeys are not easy to swallow.
50.
1
1
i.e. ¶ e1k0
i.e. Opium-eaters never wear white kid gloves.
51.
1
1
i.e. ¶ d1a′0
i.e. A good husband always comes home for his tea.
52.
1
1
i.e. ¶ b1e0
i.e. Bathing-machines are never made of mother-of-pearl.
pg16353.
1
1
¶ b′e0 † e1, i.e. ¶ e1b′0
i.e. Rainy days are always cloudy.
54.
1
1
¶ h′e0
i.e. No heavy fish is unkind to children.
55.
1
1
i.e. No engine-driver lives on barley-sugar.
56.
1
1
¶ ed′0 † e1, i.e. ¶ e1d′0
i.e. All the animals in the yard gnaw bones.
57.
1
1
i.e. No badger can guess a conundrum.
58.
1
1
i.e. No cheque of yours, received by me, is payable to order.
59.
1
1
¶ kn0
i.e. I cannot read any of Brown’s letters.
60.
1
1
¶ a′h0 † h1, i.e. ¶ h1a′0
i.e. I always avoid a kangaroo.
pg164NOTES.
(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 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.
pg165APPENDIX,
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.
pg166In 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.”
pg167Also 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——”
pg168Jones (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”.
pg169This 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.
pg170We 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 pg171with a non-existent First Class——will serve admirably to illustrate this new flaw in the theory of “The Logicians.”
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.
§ 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.
pg172This 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§ 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.]
§ 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.
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.
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.
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.
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.
§ 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:—
pg175It 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.
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.
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.”
pg176This 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.
§ 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.
pg177For 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.
pg178For 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.
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.
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.
§ 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 |
pg181The Minor requires three, viz.
| 2 |
| 3 |
| 4 |
The combination of Major and Minor, in every possible way requires nine, viz.
Figs. 1 and 2 give
| 5 |
| 6 |
| 7 |
| 8 |
| 9 |
Figs. 1 and 3 give
| 10 |
| 11 |
| 12 |
Figs. 1 and 4 give
| 13 |
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.
pg182We thus get five Biliteral Diagrams for x and y, viz.
| 14 |
| 15 |
| 16 |
| 17 |
| 18 |
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.