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.”
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.
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.
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.
§ 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”!
§ 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:—
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 repr
esent 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.”
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.
§ 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.
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.
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.
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.”
(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
The 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.
We 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.
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.’”
(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.
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.
xm0 † my′1 ¶ x′y′1
i.e. “Some persons, who are not philosophers, are not gamblers.”
§ 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!
Complete Works of Lewis Carroll Page 115