Complete Works of Lewis Carroll
Page 117
(12) Brothers of the same height always differ in Politics;
(13) Two handsome men, who are neither both of them admired nor both of them self-conscious, are no doubt of different heights;
(14) Brothers, who are self-conscious, and do not both of them like Society, never look well when walking together.
[N.B. See Note at end of Problem 2.]
8.
(1) A man can always master his father;
(2) An inferior of a man’s uncle owes that man money;
(3) The father of an enemy of a friend of a man owes that man nothing;
(4) A man is always persecuted by his son’s creditors;
(5) An inferior of the master of a man’s son is senior to that man;
(6) A grandson of a man’s junior is not his nephew;
(7) A servant of an inferior of a friend of a man’s enemy is never persecuted by that man;
(8) A friend of a superior of the master of a man’s victim is that man’s enemy;
(9) An enemy of a persecutor of a servant of a man’s father is that man’s friend.
The Problem is to deduce some fact about great-grandsons.
[N.B. In this Problem, it is assumed that all the men, here referred to, live in the same town, and that every pair of them are either “friends” or “enemies,” that every pair are related as “senior and junior”, “superior and inferior”, and that certain pairs are related as “creditor and debtor”, “father and son”, “master and servant”, “persecutor and victim”, “uncle and nephew”.]
9.
“Jack Sprat could eat no fat:
His wife could eat no lean:
And so, between them both,
They licked the platter clean.”
Solve this as a Sorites-Problem, taking lines 3 and 4 as the Conclusion to be proved. It is permitted to use, as Premisses, not only all that is here asserted, but also all that we may reasonably understand to be implied.
NOTES TO APPENDIX.
(A)
It may, perhaps, occur to the Reader, who has studied Formal Logic that the argument, here applied to the Propositions I and E, will apply equally well to the Propositions I and A (since, in the ordinary text-books, the Propositions “All xy are z” and “Some xy are not z” are regarded as Contradictories). Hence it may appear to him that the argument might have been put as follows:—
“We now have I and A ‘asserting.’ Hence, if the Proposition ‘All xy are z’ be true, some things exist with the Attributes x and y: i.e. ‘Some x are y.’
“Also we know that, if the Proposition ‘Some xy are not-z’ be true the same result follows.
“But these two Propositions are Contradictories, so that one or other of them must be true. Hence this result is always true: i.e. the Proposition ‘Some x are y’ is always true!
“Quod est absurdum. Hence I cannot assert.”
This matter will be discussed in Part II; but I may as well give here what seems to me to be an irresistable proof that this view (that A and I are Contradictories), though adopted in the ordinary text-books, is untenable. The proof is as follows:—
With regard to the relationship existing between the Class ‘xy’ and the two Classes ‘z’ and ‘not-z’, there are four conceivable states of things, viz.
(1)
Some xy
are z,
and
some
are not-z;
(2)
〃
〃
none
〃
(3)
No xy
〃
some
〃
(4)
〃
〃
none
〃
Of these four, No. (2) is equivalent to “All xy are z”, No. (3) is equivalent to “All xy are not-z”, and No. (4) is equivalent to “No xy exist.”
Now it is quite undeniable that, of these four states of things, each is, a priori, possible, some one must be true, and the other three must be false.
Hence the Contradictory to (2) is “Either (1) or (3) or (4) is true.” Now the assertion “Either (1) or (3) is true” is equivalent to “Some xy are not-z”; and the assertion “(4) is true” is equivalent to “No xy exist.” Hence the Contradictory to “All xy are z” may be expressed as the Alternative Proposition “Either some xy are not-z, or no xy exist,” but not as the Categorical Proposition “Some y are not-z.”
(B)
There are yet other views current among “The Logicians”, as to the “Existential Import” of Propositions, which have not been mentioned in this Section.
One is, that the Proposition “some x are y” is to be interpreted, neither as “Some x exist and are y”, nor yet as “If there were any x in existence, some of them would be y”, but merely as “Some x can be y; i.e. the Attributes x and y are compatible”. On this theory, there would be nothing offensive in my telling my friend Jones “Some of your brothers are swindlers”; since, if he indignantly retorted “What do you mean by such insulting language, you scoundrel?”, I should calmly reply “I merely mean that the thing is conceivable——that some of your brothers might possibly be swindlers”. But it may well be doubted whether such an explanation would entirely appease the wrath of Jones!
Another view is, that the Proposition “All x are y” sometimes implies the actual existence of x, and sometimes does not imply it; and that we cannot tell, without having it in concrete form, which interpretation we are to give to it. This view is, I think, strongly supported by common usage; and it will be fully discussed in Part II: but the difficulties, which it introduces, seem to me too formidable to be even alluded to in Part I, which I am trying to make, as far as possible, easily intelligible to mere beginners.
(C)
The three Conclusions are
“No conceited child of mine is greedy”;
“None of my boys could solve this problem”;
“Some unlearned boys are not choristers.”
THE GAME OF LOGIC
To my Child-friend.
I charm in vain; for never again,
All keenly as my glance I bend,
Will Memory, goddess coy,
Embody for my joy
Departed days, nor let me gaze
On thee, my fairy friend!
Yet could thy face, in mystic grace,
A moment smile on me, 'twould send
Far-darting rays of light
From Heaven athwart the night,
By which to read in very deed
Thy spirit, sweetest friend!
So may the stream of Life's long dream
Flow gently onward to its end,
With many a floweret gay,
Adown its willowy way:
May no sigh vex, no care perplex,
My loving little friend!
NOTA BENE.
With each copy of this Book is given an Envelope, containing a Diagram (similar to the frontispiece) on card, and nine Counters, four red and five grey.
The Envelope, &c. can be had separately, at 3d. each.
The Author will be very grateful for suggestions, especially from beginners in Logic, of any alterations, or further explanations, that may seem desirable. Letters should be addressed to him at "29, Bedford Street, Covent Garden, London."
CONTENTS
THE GAME OF LOGIC
PREFACE
CHAPTER I.
NEW LAMPS FOR OLD.
CHAPTER II.
CROSS QUESTIONS.
CHAPTER III.
CROOKED ANSWERS.
CHAPTER IV.
HIT OR MISS.
PREFACE
"There foam'd rebellious Logic, gagg'd and bound."
This Game requires nine Counters—four of one colour and five of another: say four red and five grey.
Besides the nine Counters, it also requires one Player, AT LEAST. I am not aware of any Game that can be played with LESS than this number: while there are seve
ral that require MORE: take Cricket, for instance, which requires twenty-two. How much easier it is, when you want to play a Game, to find ONE Player than twenty-two. At the same time, though one Player is enough, a good deal more amusement may be got by two working at it together, and correcting each other's mistakes.
A second advantage, possessed by this Game, is that, besides being an endless source of amusement (the number of arguments, that may be worked by it, being infinite), it will give the Players a little instruction as well. But is there any great harm in THAT, so long as you get plenty of amusement?
CHAPTER I.
NEW LAMPS FOR OLD.
"Light come, light go." _________
1. Propositions.
"Some new Cakes are nice."
"No new Cakes are nice."
"All new cakes are nice."
There are three 'PROPOSITIONS' for you—the only three kinds we are going to use in this Game: and the first thing to be done is to learn how to express them on the Board.
Let us begin with
"Some new Cakes are nice."
But before doing so, a remark has to be made—one that is rather important, and by no means easy to understand all in a moment: so please to read this VERY carefully.
The world contains many THINGS (such as "Buns", "Babies", "Beetles". "Battledores". &c.); and these Things possess many ATTRIBUTES (such as "baked", "beautiful", "black", "broken", &c.: in fact, whatever can be "attributed to", that is "said to belong to", any Thing, is an Attribute). Whenever we wish to mention a Thing, we use a SUBSTANTIVE: when we wish to mention an Attribute, we use an ADJECTIVE. People have asked the question "Can a Thing exist without any Attributes belonging to it?" It is a very puzzling question, and I'm not going to try to answer it: let us turn up our noses, and treat it with contemptuous silence, as if it really wasn't worth noticing. But, if they put it the other way, and ask "Can an Attribute exist without any Thing for it to belong to?", we may say at once "No: no more than a Baby could go a railway-journey with no one to take care of it!" You never saw "beautiful" floating about in the air, or littered about on the floor, without any Thing to BE beautiful, now did you?
And now what am I driving at, in all this long rigmarole? It is this. You may put "is" or "are" between names of two THINGS (for example, "some Pigs are fat Animals"), or between the names of two ATTRIBUTES (for example, "pink is light-red"), and in each case it will make good sense. But, if you put "is" or "are" between the name of a THING and the name of an ATTRIBUTE (for example, "some Pigs are pink"), you do NOT make good sense (for how can a Thing BE an Attribute?) unless you have an understanding with the person to whom you are speaking. And the simplest understanding would, I think, be this—that the Substantive shall be supposed to be repeated at the end of the sentence, so that the sentence, if written out in full, would be "some Pigs are pink (Pigs)". And now the word "are" makes quite good sense.
Thus, in order to make good sense of the Proposition "some new Cakes are nice", we must suppose it to be written out in full, in the form "some new Cakes are nice (Cakes)". Now this contains two 'TERMS'—"new Cakes" being one of them, and "nice (Cakes)" the other. "New Cakes," being the one we are talking about, is called the 'SUBJECT' of the Proposition, and "nice (Cakes)" the 'PREDICATE'. Also this Proposition is said to be a 'PARTICULAR' one, since it does not speak of the WHOLE of its Subject, but only of a PART of it. The other two kinds are said to be 'UNIVERSAL', because they speak of the WHOLE of their Subjects—the one denying niceness, and the other asserting it, of the WHOLE class of "new Cakes". Lastly, if you would like to have a definition of the word 'PROPOSITION' itself, you may take this:—"a sentence stating that some, or none, or all, of the Things belonging to a certain class, called its 'Subject', are also Things belonging to a certain other class, called its 'Predicate'".
You will find these seven words—PROPOSITION, ATTRIBUTE, TERM, SUBJECT, PREDICATE, PARTICULAR, UNIVERSAL—charmingly useful, if any friend should happen to ask if you have ever studied Logic. Mind you bring all seven words into your answer, and you friend will go away deeply impressed—'a sadder and a wiser man'.
Now please to look at the smaller Diagram on the Board, and suppose it to be a cupboard, intended for all the Cakes in the world (it would have to be a good large one, of course). And let us suppose all the new ones to be put into the upper half (marked 'x'), and all the rest (that is, the NOT-new ones) into the lower half (marked 'x''). Thus the lower half would contain ELDERLY Cakes, AGED Cakes, ANTE-DILUVIAN Cakes—if there are any: I haven't seen many, myself—and so on. Let us also suppose all the nice Cakes to be put into the left-hand half (marked 'y'), and all the rest (that is, the not-nice ones) into the right-hand half (marked 'y''). At present, then, we must understand x to mean "new", x' "not-new", y "nice", and y' "not-nice."
And now what kind of Cakes would you expect to find in compartment
No. 5?
It is part of the upper half, you see; so that, if it has any Cakes in it, they must be NEW: and it is part of the left-hand half; so that they must be NICE. Hence if there are any Cakes in this compartment, they must have the double 'ATTRIBUTE' "new and nice": or, if we use letters, the must be "x y."
Observe that the letters x, y are written on two of the edges of this compartment. This you will find a very convenient rule for knowing what Attributes belong to the Things in any compartment. Take No. 7, for instance. If there are any Cakes there, they must be "x' y", that is, they must be "not-new and nice."
Now let us make another agreement—that a red counter in a compartment shall mean that it is 'OCCUPIED', that is, that there are SOME Cakes in it. (The word 'some,' in Logic, means 'one or more' so that a single Cake in a compartment would be quite enough reason for saying "there are SOME Cakes here"). Also let us agree that a grey counter in a compartment shall mean that it is 'EMPTY', that is that there are NO Cakes in it. In the following Diagrams, I shall put '1' (meaning 'one or more') where you are to put a RED counter, and '0' (meaning 'none') where you are to put a GREY one.
As the Subject of our Proposition is to be "new Cakes", we are only concerned, at present, with the UPPER half of the cupboard, where all the Cakes have the attribute x, that is, "new."
Now, fixing our attention on this upper half, suppose we found it marked like this,
—————-
| | |
| 1 | |
| | |
—————-
that is, with a red counter in No. 5. What would this tell us, with regard to the class of "new Cakes"?
Would it not tell us that there are SOME of them in the x y-compartment? That is, that some of them (besides having the Attribute x, which belongs to both compartments) have the Attribute y (that is, "nice"). This we might express by saying "some x-Cakes are y-(Cakes)", or, putting words instead of letters,
"Some new Cakes are nice (Cakes)",
or, in a shorter form,
"Some new Cakes are nice".
At last we have found out how to represent the first Proposition of this Section. If you have not CLEARLY understood all I have said, go no further, but read it over and over again, till you DO understand it. After that is once mastered, you will find all the rest quite easy.
It will save a little trouble, in doing the other Propositions, if we agree to leave out the word "Cakes" altogether. I find it convenient to call the whole class of Things, for which the cupboard is intended, the 'UNIVERSE.' Thus we might have begun this business by saying "Let us take a Universe of Cakes." (Sounds nice, doesn't it?)
Of course any other Things would have done just as well as Cakes. We might make Propositions about "a Universe of Lizards", or even "a Universe of Hornets". (Wouldn't THAT be a charming Universe to live in?)
So far, then, we have learned that
—————-
| | |
| 1 | |
| | |
—————-
means "some x and y," i.e. "some new are nice."
I think you will see without further explanation, that
—————-
| | |
| | 1 |
| | |
—————-
means "some x are y'," i.e. "some new are not-nice."
Now let us put a GREY counter into No. 5, and ask ourselves the meaning of
—————-
| | |
| 0 | |
| | |
—————-
This tells us that the x y-compartment is EMPTY, which we may express by "no x are y", or, "no new Cakes are nice". This is the second of the three Propositions at the head of this Section.
In the same way,
—————-
| | |
| | 0 |
| | |
—————-
would mean "no x are y'," or, "no new Cakes are not-nice."
What would you make of this, I wonder?
—————-
| | |
| 1 | 1 |
| | |
—————-
I hope you will not have much trouble in making out that this represents a DOUBLE Proposition: namely, "some x are y, AND some are y'," i.e. "some new are nice, and some are not-nice."
The following is a little harder, perhaps:
—————-
| | |
| 0 | 0 |
| | |
—————-
This means "no x are y, AND none are y'," i.e. "no new are nice, AND none are not-nice": which leads to the rather curious result that "no new exist," i.e. "no Cakes are new." This is because "nice" and "not-nice" make what we call an 'EXHAUSTIVE' division of the class "new Cakes": i.e. between them, they EXHAUST the whole class, so that all the new Cakes, that exist, must be found in one or the other of them.
And now suppose you had to represent, with counters the contradictory to "no Cakes are new", which would be "some Cakes are new", or, putting letters for words, "some Cakes are x", how would you do it?