Chapter XLII: Appendix: ,

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. 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.

§ 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
§ 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.

_____
_/ ___ \_
/ / 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_.

§ 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.

§ 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
H==H=====H=====H=====H=====H--H
H H | H | H | H | H H
H H--|--H--|--H--|--H--|--H H
H H | H | H | H | H H
H #=====H=====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
H==H===========H===========H===========H===========H==H
H H | H | H | H | H H
H H ·--|--· H ·--|--· H ·--|--· H ·--|--· H H
H H | | | H | | | H | | | H | | | H H
H H--|--|--|--H--|--|--|--H--|--|--|--H--|--|--|--H H
H H | | | H | | | H | | | H | | | H H
H H ·--|--· H ·--|--· H ·--|--· H ·--|--· H H
H H | H | H | H | H H
H #===========H===========H===========H===========# H
H | H | H | H | H
H ·--|--· H ·--|--· H ·--|--· H ·--|--· H
H | | | H | | | H | | | H | | | H
H-----|--|--|--H--|--|--|--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 H==H===============H===============H===============H===============H==H H H | H | H | H | H H H H ·--|--· H ·--|--· H ·--|--· H ·--|--· H H H H | | | H | | | H | | | H | | | H H H H ·--|--|--|--· H ·--|--|--|--· H ·--|--|--|--· H ·--|--|--|--· H H H H | | | | | H | | | | | H | | | | | H | | | | | H H H H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H H H H | | | | | H | | | | | H | | | | | H | | | | | H H H H ·--|--|--|--· H ·--|--|--|--· H ·--|--|--|--· H ·--|--|--|--· H H H H | | | H | | | H | | | H | | | H H H H ·--|--· H ·--|--· H ·--|--· H ·--|--· H H H H | H | H | H | H H H #===============H===============H===============H===============# H H | H | H | H | H H ·--|--· H ·--|--· H ·--|--· H ·--|--· H H | | | H | | | H | | | H | | | H H ·--|--|--|--· H ·--|--|--|--· H ·--|--|--|--· H ·--|--|--|--· H H | | | | | H | | | | | H | | | | | H | | | | | H H----|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|-H-|--|--|--|--|----H H | | | | | H | | | | | H | | | | | H | | | | | H H ·--|--|--|--· H ·--|--|--|--· 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.

§ 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."

§ 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
§ 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