i.e. My writing-desk is full of live scorpions.
43.1b′1e0†2ah0†3dc0†4e′1a′0†5bc′01b′e†4e′a′†2ah†5bc′†3dc¶hd0
i.e. No Mandarin ever reads Hogg’s poems.
pg16244.1e1b′0†2a′d0†3c1h′0†4e′a0†5d′h0;1eb′†4e′a†2a′d†5d′h†3ch′¶b′c0†c1,i.e. ¶c1b′0
i.e. Shakespeare was clever.
45.1e′1c′0†2hb′0†3d1a0†4e1a′0†5c1b0;1e′c′†4ea′†3da†5cb†2hb′¶dh0†d1, i.e. ¶d1h0
i.e. Rainbows are not worth writing odes to.
46.1c′1h′0†2e1a0†3bd0†4a′1h0†5d′c0;1c′h′†4a′h†2ea†5d′c†3bd¶eb0†e1, i.e. ¶e1b0
i.e. These Sorites-examples are difficult.
47.1a′1e′0†2bk0†3c′a0†4eh′0†5d1b′0†6k′h0;1a′e′†3c′a†4eh′†6k′h†2bk†5db′¶c′d0†d1,i.e. ¶d1c′0
i.e. All my dreams come true.
48.1a′h0†2c′k0†3a1d′0†4e1h′0†5b1k′0†6c1e′0;1a′h†3ad′†4eh′†6ce′†2c′k†5bk′¶d′b0†b1,i.e. ¶b1d′0
i.e. All the English pictures here are painted in oils.
49.1k′1e0†2c1h0†3b1a′0†4kd0†5h′a0†6b′1e′0;1k′e†4kd†6b′e′†3ba′†5h′a†2ch¶dc0†c1,i.e. ¶c1d0
i.e. Donkeys are not easy to swallow.
50.1ab′0†2h′d0†3e1c0†4b1d′0†5a′k0†6c′1h0;1ab′†4bd′†2h′d†6a′k†5c′h†3ec¶ke0†e1,i.e. ¶e1k0
i.e. Opium-eaters never wear white kid gloves.
51.1bc0†2k1a′0†3eh0†4d1b′0†5h′c′0†6k′1e′0;1bc†4db′†5h′c′†3eh†6k′e′†2ka′¶da′0†d1,i.e. ¶d1a′0
i.e. A good husband always comes home for his tea.
52.1a′1k′0†2ch0†3h′k0†4b1d′0†5ea0†6d1c′01a′k′†3h′k†2ch†6dc′†4bd′†5ea¶be0†b1,i.e. ¶b1e0
i.e. Bathing-machines are never made of mother-of-pearl.
pg16353.1da′0†2k1b′0†3c1h0†4d′1k′0†5e1c′0†6a1h′0;1da′†4d′k′†2kb′†6ah′†5ch†3ec′¶b′e0†e1, i.e. ¶e1b′0
i.e. Rainy days are always cloudy.
54.1kb′0†1a′1c′0†3d′b0†4k′1h′0†5ea0†6d1c0;1kb′†3d′b†4k′h′†6dc†2a′c′†5ea¶h′e0
i.e. No heavy fish is unkind to children.
55.1k′1b′0†2eh′0†3c′d0†4hb0†5ac0†6kd′0;1k′b′†4hb†2eh′†6kd′†3c′d†5ac¶ea0
i.e. No engine-driver lives on barley-sugar.
56.1h1b′0†2c1d′0†3k′a0†4e1h′0†5b1a′0†6k1c′0;1hb′†4eh′†5ba′†3k′a†6kc′†2cd′¶ed′0†e1, i.e. ¶e1d′0
i.e. All the animals in the yard gnaw bones.
57.1h′1d′0†2e1c′0†3k′a0†4cb0†5d1l′0†6e′h0†7kl0;1h′d′†5dl′†7kl†3k′a†6e′h†2ec′†4cb¶ab0
i.e. No badger can guess a conundrum.
58.1b′h0†2d′1l′0†3ca0†4d1k′0†5h′1e′0†6mc′0†7a′b0†8ek0;1b′h†5h′e′†7a′b†3ca†6mc′†8ek†4dk′†2d′l′¶ml′0
i.e. No cheque of yours, received by me, is payable to order.
59.1c1l′0†2h′e0†3kd0†4mc′0†5b′1e′0†6n1a′0†7l1d′0†8m′b0†9ah0;1cl′†4mc′†7ld′†3kd†8m′b†5b′e′†2h′e†9ah†6na′¶kn0
i.e. I cannot read any of Brown’s letters.
60.1e1c′0†2l1n′0†3d1a′0†4m′b0†5ck′0†6e′r0†7h1n0†8b′k0†9r′1d′0†10m1l′0;1ec′†5ck′†6e′r†8b′k†4m′b†9r′d′†3da′†10ml′†2ln′†7hn¶a′h0†h1, i.e. ¶h1a′0
i.e. I always avoid a kangaroo.
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 ofPetitio Principii(i.e. “Begging the Question”, the essence of which is that the whole Conclusion is involved inoneof 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 thetwoPremisses taken together, each contributing its share.
Thus, in Fig. I., the Premissxm0empties theInnerCell of the N.W. Quarter, while the Premissym0empties itsOuterCell. Hence it needs thetwoPremisses to empty thewholeof the N.W. Quarter, and thus to prove the Conclusionxy0.
Again, in Fig. II., the Premissxm0empties the Inner Cell of the N.W. Quarter. The Premiss ym1merely tells us that the Inner Portion of the W. Half isoccupied, so that we may place a ‘I’ in it,somewhere; but, if this were thewholeof our information, we should not know inwhichCell to place it, so that it would have to ‘sit on the fence’: it is only when we learn, from the other Premiss, that theupperof these two Cells isempty, that we feel authorised to place the ‘I’ in thelowerCell, and thus to prove the Conclusion x′y1.
Lastly, in Fig. III., the information, thatmexists, merely authorises us to place a ‘I’somewherein the Inner Square——but it has large choice of fences to sit upon! It needs the Premissxm0to drive it out of the N. Half of that Square; and it needs the Premissym0to drive it out of the W. Half. Hence it needs thetwoPremisses to drive it into the Inner Portion of the S.E. Quarter, and thus to prove the Conclusionx′y′1.
There are several matters, too hard to discuss withLearners, which nevertheless need to be explained to anyTeachers, into whose hands this book may fall, in order that they may thoroughly understand what my Symbolic Methodis, 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.
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 ascertainwhatwas 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 maylogicallybe held, and thus settle which of them mayconvenientlybe held; after which I shall hold myself free to declare which of themIintend to hold.
Thekindsof Propositions, to be considered, are those that begin with “some”, with “no”, and with “all”. These are usually called Propositions “inI”, “inE”, and “inA”.
First, then, a Proposition inImay be understood as asserting, or else asnotasserting, the existence of its Subject. (By “existence” I mean of course whatever kind of existence suits its nature. The two Propositions, “dreamsexist” and “drumsexist”, denote two totally different kinds of “existence”. Adreamis an aggregate of ideas, and exists only in themind of a dreamer: whereas adrumis an aggregate of wood and parchment, and exists inthe hands of a drummer.)
First, let us suppose thatI“asserts” (i.e. “asserts the existence of its Subject”).
Here, of course, we must regard a Proposition inAas making thesameassertion, since it necessarilycontainsa Proposition inI.
We now haveIandA“asserting”. Does this leave us free to make what supposition we choose as toE? My answer is “No. We are tied down to the supposition thatEdoesnotassert.” This can be proved as follows:—
If possible, letE“assert”. Then (takingx,y, andzto represent Attributes) we see that, if the Proposition “Noxyarez” be true, some things exist with the Attributesxandy: i.e. “Somexarey.”
pg167Also we know that, if the Proposition “Somexyarez” be true, the same result follows.
But these two Propositions are Contradictories, so that one or other of themmustbe true. Hence this result isalwaystrue: i.e. the Proposition “Somexarey” isalwaystrue!
Quod est absurdum.(SeeNote (A), p. 195).
We see, then, that the supposition “Iasserts” necessarily leads to “Aasserts, butEdoes not”. And this is thefirstof the various views that may conceivably be held.
Next, let us suppose thatIdoesnot“assert.” And, along with this, let us take the supposition thatEdoes“assert.”
Hence the Proposition “Noxarey” means “Somexexist, and none of them arey”: i.e. “allof them arenot-y,” which is a Proposition inA. We also know, of course, that the Proposition “Allxare not-y” proves “Noxarey.” Now two Propositions, each of which proves the other, areequivalent. Hence every Proposition inAis equivalent to one inE, and therefore “asserts”.
Hence oursecondconceivable view is “EandAassert, butIdoes not.”
This view does not seen to involve any necessary contradiction with itself or with the accepted facts of Logic. But, when we come totestit, as applied to the actualfactsof 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 strictlyLogicalprinciples.
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 strictlyLogicalprinciples. AParticularProposition doesnotassert the existence of its Subject. I merely meant to say that we’ve made a Rule not to admitanyMembers till we have at leastthreeCandidates whose incomes are over ten thousand a year!”
Author.“Oh,that’swhat 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 thereareany such convicts in existence?”
Jones.“Why, that’s exactly what Idomean to assert! Don’t you know that a Universal Negativeassertsthe existence of its Subject?Of coursewe 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 thissecondconceivable 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 neitherInorE“asserts”.
Now the supposition that the two Propositions, “Somexarey” and “Noxare not-y”, donot“assert”, necessarily involves the supposition that “Allxarey” doesnot“assert”, since it would be absurd to suppose that they assert, when combined, more than they do when taken separately.
Hence thethird(and last) of the conceivable views is that neitherI, norE, norA, “asserts”.
The advocates of this third view would interpret the Proposition “Somexarey” to mean “If therewereanyxin existence, some of themwouldbey”; and so withEandA.
It admits of proof that this view, as regardsA, conflicts with the accepted facts of Logic.
Let us take the SyllogismDarapti, which is universally accepted as valid. Its form is
“Allmarex;Allmarey.∴ Someyarex”.
pg169This they would interpret as follows:—
”If there were anymin existence, all of them would bex;If there were anymin existence, all of them would bey.∴ If there were anyyin existence, some of them would bex”.
That this Conclusion doesnotfollow 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 inDarapti:—
‘All M is P,All M is S,∴ Some S is P.’
“TakingS,M,P, as the minor, middle, and major terms respectively, the conclusion will imply that, if there is anS, there is someP. Will the premisses also imply this? If so, then the syllogism is valid; but not otherwise.
“The conclusion implies that ifSexistsPexists; but, consistently with the premisses,Smay be existent whileMandPare both non-existent. An implication is, therefore, contained in the conclusion, which is not justified by the premisses.”
This seems tomeentirely 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 anon-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 thereareboys 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 therewereany.
pg170We are authorised, then, by theData, 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 twotruePremisses and afalseConclusion (since we know that thereareboys doing Latin, and thatnoneof them are doing Greek). Hence the argument isinvalid.
Similarly it may be shown that this “non-existential” interpretation destroys the validity ofDisamis,Datisi,Felapton, andFresison.
Some of “The Logicians” will, no doubt, be ready to reply “But we are notAldrichians! Why shouldwebe responsible for the validity of the Syllogisms of so antiquated an author as Aldrich?”
Very good. Then, for thespecialbenefit of these “friends” of mine (with what ominous emphasis that name is sometimes used! “I must have a private interview withyou, my youngfriend,” says the bland Dr. Birch, “in my library, at 9 a.m. tomorrow. And you will please to bepunctual!”), for theirspecialbenefit, I say, I will produceanothercharge against this “non-existential” interpretation.
It actually invalidates the ordinary Process of “Conversion”, as applied to Proposition in ‘I’.
Everylogician, Aldrichian or otherwise, accepts it as an established fact that “Somexarey” may be legitimately converted into “Someyarex.”
But is it equally clear that the Proposition “If therewereanyx, some of themwouldbey” may be legitimately converted into “If therewereanyy, some of them would bex”? I trow not.
The example I have already used——of a Boys’ Schoolpg171with 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 yetanotherRule 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 therewereany boys in the First Class, some of themwouldreceive 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 therewereany boys about to receive prizes, some of themwouldbe in the First Class” (which Class we know to beempty).
Of this Pair of Converse Propositions, the first is undoubtedlytrue: the second,asundoubtedly,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 acricketer, one can but pronounce him “Out!”
We see, then, that, among all the conceivable views we have here considered, there are onlytwowhich canlogicallybe held, viz.
IandA“assert”, butEdoes not.EandA“assert”, butIdoes not.
Thesecondof these I have shown to involve great practical inconvenience.
Thefirstis the one adopted in this book. (Seep. 19.)
Some further remarks on this subject will be found inNote (B), at p. 196.
Is it better to say “Johnis-notin-the-house” or “Johnisnot-in-the-house”? “Some of my acquaintancesare-notmen-I-should-like-to-be-seen-with” or “Some of my acquaintancesaremen-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 oftaste, 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 “AmIto 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 toyou, 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,affirmativePropositions, instead of translating the latter into “Some men are-not Jews”, and regarding it as anegativePropositions?
The fact is, “The Logicians” have somehow acquired a perfectlymorbiddread 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, thenegativepart is too large to deal with, so that it is better to regard each Thing as either included in, or excluded from, thepositivepart. I see no force in this plea: and the facts often go the other way. As a personal question, dear Reader, ifyouwere to group your acquaintances into the two Classes, men that youwouldlike to be seen with, and men that you wouldnotlike to be seen with, do you think the latter group would be soverymuch the larger of the two?
For the purposes of Symbolic Logic, it is somuchthe most convenient plan to regard the two sub-divisions, produced by Dichotomy, on thesamefooting, 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.
This I consider to beanothercraze of “The Logicians”, fully as morbid as their dread of a negative Attribute.
It is, perhaps, best refuted by the method ofInstantia 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, inmysystem, anaffirmativeone, since I should read it “are not-choristers”; but, in dealing with “The Logicians,” I may fairly treat it as anegativeone, sincetheywould read it “are-not choristers”.)
If you, dear Reader, declare, after full consideration of these Pairs of Premisses, that you cannot deduce a Conclusion fromanyof them——why, all I can say is that, like the Duke in Patience, you “will have to be contented with our heart-felt sympathy”! [SeeNote (C), p. 196.]
Diagrams seem to have been used, at first, to representPropositionsonly. 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.
Diagram of circle x inside circle y
Thus, the Diagram, here given, exhibits the two Classes, whose respective Attributes arexandy, as so related to each other that the following Propositions are all simultaneously true:—“Allxarey”, “Noxare not-y”, “Somexarey”, “Someyare not-x”, “Some not-yare not-x”, and, of course, the Converses of the last four.
pg174Diagram of circle y inside circle x
Similarly, with this Diagram, the following Propositions are true:—“Allyarex”, “Noyare not-x”, “Someyarex”, “Somexare not-y”, “Some not-xare not-y”, and, of course, the Converses of the last four.
Diagram of two separate circles x and y
Similarly, with this Diagram, the following are true:—“Allxare not-y”, “Allyare not-x”, “Noxarey”, “Somexare not-y”, “Someyare not-x”, “Some not-xare not-y”, and the Converses of the last four.
Diagram of two intersecting circles x and y
Similarly, with this Diagram, the following are true:—“Somexarey”, “Somexare not-y”, “Some not-xarey”, “Some not-xare not-y”, and of course, their four Converses.
Note thatallEuler’s Diagrams assert “Some not-xare not-y.” Apparently it never occured to him that it mightsometimesfail to be true!
Now, to represent “Allxarey”, thefirstof these Diagrams would suffice. Similarly, to represent “Noxarey”, thethirdwould suffice. But to represent anyParticularProposition, at leastthreeDiagrams would be needed (in order to include all the possible cases), and, for “Some not-xare not-y”, all thefour.
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 representanydesired relation betweenxandy, by simply shading a Compartment known to beempty, and placing a + in one known to beoccupied.
Thus, he would represent the three Propositions “Somexarey”, “Noxarey”, and “Allxarey”, as follows:—
Venn diagram representing x y exists
Venn diagram representing x y does not exist
Venn diagram representing all x are y
pg175It will be seen that, of thefourClasses, whose peculiar Sets of Attributes arexy,xy′,x′y, andx′y′, onlythreeare here provided with closed Compartments, while thefourthis 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 “Nox′arey′.” Mr. Vennonce(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 representtwoPropositions (containing a common Term)together, athree-letter Diagram is needed. This is the one used by Mr. Venn.
Venn diagram of three intersecting circles x y and z
Here, again, we have onlysevenclosed Compartments, to accommodate theeightClasses whose peculiar Sets of Attributes arexym,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 onlyfifteenclosed compartments.
Venn diagram of four intersecting ellipses a b c and d
Forfiveletters, “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 theoutsideofc; i.e. its four component portions are insidebanddbut are no part ofc). 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.”
Venn diagram of five intersecting ellipses a b c d and e and an interior ellipse
pg176This Diagram gives us 31 closed compartments.
Forsixletters, Mr. Venn suggests that we might usetwoDiagrams, like the above, one for thef-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, andoneinfinite area, which the two Classes,a′b′c′d′e′fanda′b′c′d′e′f′, would have to share between them.
Beyondsixletters Mr. Venn does not go.
My Method of DiagramsresemblesMr. Venn’s, in having separate Compartments assigned to the various Classes, and in marking these Compartments asoccupiedor asempty; but itdiffersfrom his Method, in assigning aclosedarea to theUniverse 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 userectilinear, instead ofcurvilinear, Figures; and I mark anoccupiedCell with a ‘I’ (meaning that there is at leastoneThing in it), and anemptyCell with a ‘O’ (meaning that there isnoThing in it).
Fortwoletters, I use this Diagram, in which the North Half is assigned to ‘x’, the South to ‘not-x’ (or ‘x′’), the West toy, and the East toy′. Thus the N.W. Cell contains thexy-Class, the N.E. Cell thexy′-Class, and so on.
Empty biliteral diagram
Forthreeletters, I subdivide these four Cells, by drawing anInnerSquare, which I assign tom, theOuterBorder being assigned tom′. I thus geteightCells that are needed to accommodate the eight Classes, whose peculiar Sets of Attributes arexym,xym′, &c.
Empty triliteral diagram
This last Diagram is the most complex that I use in theElementaryPart of my ‘Symbolic Logic.’ But I may as well take this opportunity of describing the more complex ones which will appear in Part II.
pg177Forfourletters (which I calla,b,c,d) I use this Diagram; assigning the North Half toa(and of course therestof the Diagram toa′), the West Half tob, the Horizontal Oblong toc, and the Upright Oblong tod. We have now got 16 Cells.
Empty quadriliteral diagram
Forfiveletters (adding e) I subdivide the 16 Cells of the previous Diagram byobliquepartitions, assigning all theupperportions toe, and all thelowerportions toe′. Here, I admit, we lose the advantage of having thee-Class alltogether, “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.
Empty pentaliteral diagram
Forsixletters (addingh, as I avoidtailedletters) 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 Classeseh,eh′,e′h,e′h′. We have now got 64 Cells.
Empty hexaliteral diagram
pg178Forsevenletters (addingk) I add, to each upright cross, a little inner square. All these 16 little squares are assigned to thek-Class, and all outside them to thek′-Class; so that 8 little Cells (into which each of the 16 Cells is divided) are respectively assigned to the 8 Classesehk,ehk′, &c. We have now got 128 Cells.
Empty heptaliteral diagram
Foreightletters (addingl) I place, in each of the 16 Cells, alattice, 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 Classesabc′d′ehkl,abc′d′eh′kl′, &c. This Octoliteral Diagram (seenext page) contains 256 Cells.
Fornineletters, I place 2 Octoliteral Diagrams side by side, assigning one of them tom, and the other tom′. We have now got 512 Cells.
pg179Empty octoliteral diagram
Finally, fortenletters, I arrange 4 Octoliteral Diagrams, like the above, in a square, assigning them to the 4 Classesmn,mn′,m′n,m′n′. We have now got 1024 Cells.
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(seep. 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 inFresison, viz.
“No philosophers are conceited;Some conceited persons are not-gamblers.∴ Some not-gamblers are not philosophers”
This can be proved by reduction toFerio, thus:—
“No conceited persons are philosophers;Some not-gamblers are conceited.∴ Some not-gamblers are not philosophers”.
The validity ofFeriofollows 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 anabstractform.
Let us take “persons” as our ‘Universe of Discourse’; and letx= “philosophers”,m= “conceited”, andy= “gamblers.”
Then the Syllogism may be written thus:—
“Noxarem;Somemarey′.∴ Somey′arex′.”
(3)Solution by Euler’s Method of Diagrams.
The Major Premiss requires onlyoneDiagram, viz.
pg181The Minor requiresthree, viz.
The combination of Major and Minor, in every possible way requiresnine, viz.
Figs. 1 and 2 give
Figs. 1 and 3 give
Figs. 1 and 4 give
From this group (Figs. 5 to 13) we have, by disregardingm, to find the relation ofxandy. 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 ofxiny; and that Fig. 9 expresses entire inclusion ofyinx.
pg182We thus get five Biliteral Diagrams forxandy, viz.
where the only Proposition, represented by them all, is “Some not-yare not-x,” i.e. “Some persons, who are not gamblers, are not philosophers”——a result which Euler would hardly have regarded as avaluableone, since he seems to have assumed that a Proposition of this form isalwaystrue!
(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 inmy′must be saved: mark these constituents with a cross.
Venn diagram of three intersecting circles