Produced by Gregory D. Weeks, L. Lynn Smith, Reina Hosier,
Brett Fishburne
THE GAME OF LOGIC
By Lewis Carroll
---------------------
|9 | 10|
| | |
| -----x------ |
| |11 | 12| |
| | | | |
|---y-----m------y'---|
| | | | |
| |13 | 14| |
| -----x'----- |
| | |
|15 | 16|
---------------------
COLOURS FOR -------------
COUNTERS |5 | 6|
___ | x |
| | |
See the Sun is overhead, |--y-------y'-|
Shining on us, FULL and | | |
RED! | x' |
|7 | 8|
Now the Sun is gone away, -------------
And the EMPTY sky is
GREY!
___
THE GAME OF LOGIC
By Lewis Carrol
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."
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 several 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?
CONTENTS.
CHAPTER PAGE
I. NEW LAMPS FOR OLD.
1. Propositions . . . . . . . 1
2. Syllogisms . . . . . . . . 20
3. Fallacies . . . . . . . . 32
II. CROSS QUESTIONS.
1. Elementary . . . . . . . . 37
2. Half of Smaller Diagram. Propositions
to be represented . . . . . 40
3. Do. Symbols to be interpreted. . 42
4. Smaller Diagram. Propositions to be
represented . . . . . . . 44
5. Do. Symbols to be interpreted. . 46
6. Larger Diagram. Propositions to be
represented . . . . . . . 48
7. Both Diagrams to be employed . . 51
III. CROOKED ANSWERS.
1. Elementary . . . . . . . . 55
2. Half of Smaller Diagram. Propositions
represented . . . . . . . 59
3. Do. Symbols interpreted . . . 61
4. Smaller Diagram. Propositions represented. 62
5. Do. Symbols interpreted . . . 65
6. Larger Diagram. Propositions represented. 67
7. Both Diagrams employed . . . . 72
IV. HIT OR MISS . . . . . . . . . 85
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?
This will puzzle you a little, I expect. Evidently you must put
a red counter SOMEWHERE in the x-half of the cupboard, since you
know there are SOME new Cakes. But you must not put it into the
LEFT-HAND compartment, since you do not know them to be NICE: nor
may you put it into the RIGHT-HAND one, since you do not know them
to be NOT-NICE.
What, then, are you to do? I think the best way out of the
difficulty is to place the red counter ON THE DIVISION-LINE between
the xy-compartment and the xy'-compartment. This I shall represent
(as I always put '1' where you are to put a red counter) by the
diagram
-----------
| | |
| -1- |
| | |
-----------
Our ingenious American cousins have invented a phrase to express
the position of a man who wants to join one or the other of two
parties--such as their two parties 'Democrats' and 'Republicans'--but
can't make up his mind WHICH. Such a man is said to be "sitting
on the fence." Now that is exactly the position of the red counter
you have just placed on the division-line. He likes the look of
No. 5, and he likes the look of No. 6, and he doesn't know WHICH to
jump down into. So there he sits astride, silly fellow, dangling
his legs, one on each side of the fence!
Now I am going to give you a much harder one to make out. What
does this mean?
-----------
| | |
| 1 | 0 |
| | |
-----------
This is clearly a DOUBLE Proposition. It tells us not only that
"some x are y," but also the "no x are NOT y." Hence the result
is "ALL x are y," i.e. "all new Cakes are nice", which is the last
of the three Propositions at the head of this Section.
We see, then, that the Universal Proposition
"All new Cakes are nice"
consists of TWO Propositions taken together, namely,
"Some new Cakes are nice,"
and "No new Cakes are not-nice."
In the same way
-----------
| | |
| 0 | 1 |
| | |
-----------
would mean "all x are y' ", that is,
"All new Cakes are not-nice."
Now what would you make of such a Proposition as "The Cake you have
given me is nice"? Is it Particular or Universal?
"Particular, of course," you readily reply. "One single Cake is
hardly worth calling 'some,' even."
No, my dear impulsive Reader, it is 'Universal'. Remember that,
few as they are (and I grant you they couldn't well be fewer),
they are (or rather 'it is') ALL that you have given me! Thus, if
(leaving 'red' out of the question) I divide my Universe of Cakes
into two classes--the Cakes you have given me (to which I assign
the upper half of the cupboard), and those you HAVEN'T given me
(which are to go below)--I find the lower half fairly full, and the
upper one as nearly as possible empty. And then, when I am told
to put an upright division into each half, keeping the NICE Cakes
to the left, and the NOT-NICE ones to the right, I begin by carefully
collecting ALL the Cakes you have given me (saying to myself, from
time to time, "Generous creature! How shall I ever repay such
kindness?"), and piling them up in the left-hand compartment. AND
IT DOESN'T TAKE LONG TO DO IT!
Here is another Universal Proposition for you. "Barzillai Beckalegg
is an honest man." That means "ALL the Barzillai Beckaleggs, that
I am now considering, are honest men." (You think I invented that
name, now don't you? But I didn't. It's on a carrier's cart,
somewhere down in Cornwall.)
This kind of Universal Proposition (where the Subject is a single
Thing) is called an 'INDIVIDUAL' Proposition.
Now let us take "NICE Cakes" as the Subject of Proposition: that
is, let us fix our thoughts on the LEFT-HAND half of the cupboard,
where all the Cakes have attribute y, that is, "nice."
-----
Suppose we find it marked like this:-- | |
| 1 |
What would that tell us? | |
-----
| |
| |
| |
-----
I hope that it is not necessary, after explaining the HORIZONTAL
oblong so fully, to spend much time over the UPRIGHT one. I hope
you will see, for yourself, that this means "some y are x", that
is,
"Some nice Cakes are new."
"But," you will say, "we have had this case before. You put a red
counter into No. 5, and you told us it meant 'some new Cakes are
nice'; and NOW you tell us that it means 'some NICE Cakes are NEW'!
Can it mean BOTH?"
The question is a very thoughtful one, and does you GREAT credit,
dear Reader! It DOES mean both. If you choose to take x (that
is, "new Cakes") as your Subject, and to regard No. 5 as part of a
HORIZONTAL oblong, you may read it "some x are y", that is, "some
new Cakes are nice": but, if you choose to take y (that is, "nice
Cake") as your Subject, and to regard No. 5 as part of an UPRIGHT
oblong, THEN you may read it "some y are x", that is, "some nice
Cakes are new". They are merely two different ways of expressing
the very same truth.
Without more words, I will simply set down the other ways in which
this upright oblong might be marked, adding the meaning in each
case. By comparing them with the various cases of the horizontal
oblong, you will, I hope, be able to understand them clearly.
You will find it a good plan to examine yourself on this table,
by covering up first one column and then the other, and 'dodging
about', as the children say.
Also you will do well to write out for yourself two other tables--one
for the LOWER half of the cupboard, and the other for its RIGHT-HAND
half.
And now I think we have said all we need to say about the smaller
Diagram, and may go on to the larger one.
_________________________________________________
|
Symbols. | Meanings.
_______________|_________________________________
----- |
| | |
| | | Some y are x';
| | | i.e. Some nice are not-new.
----- |
| | |
| 1 | |
| | |
----- |
|
----- |
| | | No y are x;
| 0 | | i.e. No nice are new.
| | |
----- | [Observe that this is merely another way of
| | | expressing "No new are nice."]
| | |
| | |
----- |
|
----- |
| | |
| | | No y are x';
| | | i.e. No nice are not-new.
----- |
| | |
| 0 | |
| | |
----- |
|
----- |
| | |
| 1 | | Some y are x, and some are x';
| | | i.e. Some nice are new, and some are
----- | not-new.
| | |
| 1 | |
| | |
----- |
|
----- |
| | |
| 0 | | No y are x, and none are x'; i.e. No y
| | | exist;
----- | i.e. No Cakes are nice.
| | |
| 0 | |
| | |
----- |
|
----- |
| | |
| 1 | | All y are x;
| | | i.e. All nice are new.
----- |
| | |
| 0 | |
| | |
----- |
|
----- |
| | |
| 0 | | All y are x';
| | | i.e. All nice are not-new.
----- |
| | |
| 1 | |
| | |
----- |
_______________|_________________________________
This may be taken to be a cupboard divided in the same way as the
last, but ALSO divided into two portions, for the Attribute m. Let
us give to m the meaning "wholesome": and let us suppose that all
WHOLESOME Cakes are placed INSIDE the central Square, and all the
UNWHOLESOME ones OUTSIDE it, that is, in one or other of the four
queer-shaped OUTER compartments.
We see that, just as, in the smaller Diagram, the Cakes in each
compartment had TWO Attributes, so, here, the Cakes in each compartment
have THREE Attributes: and, just as the letters, representing the
TWO Attributes, were written on the EDGES of the compartment, so,
here, they are written at the CORNERS. (Observe that m' is supposed
to be written at each of the four outer corners.) So that we can
tell in a moment, by looking at a compartment, what three Attributes
belong to the Things in it. For instance, take No. 12. Here we
find x, y', m, at the corners: so we know that the Cakes in it, if
there are any, have the triple Attribute, 'xy'm', that is, "new,
not-nice, and wholesome." Again, take No. 16. Here we find, at
the corners, x', y', m': so the Cakes in it are "not-new, not-nice,
and unwholesome." (Remarkably untempting Cakes!)
It would take far too long to go through all the Propositions,
containing x and y, x and m, and y and m which can be represented
on this diagram (there are ninety-six altogether, so I am sure you
will excuse me!) and I must content myself with doing two or three,
as specimens. You will do well to work out a lot more for yourself.
Taking the upper half by itself, so that our Subject is "new Cakes",
how are we to represent "no new Cakes are wholesome"?
This is, writing letters for words, "no x are m." Now this tells us
that none of the Cakes, belonging to the upper half of the cupboard,
are to be found INSIDE the central Square: that is, the two
compartments, No. 11 and No. 12, are EMPTY. And this, of course,
is represented by
-------------------
| | |
| _____|_____ |
| | | | |
| | 0 | 0 | |
| | | | |
-------------------
And now how are we to represent the contradictory Proposition "SOME x
are m"? This is a difficulty I have already considered. I think
the best way is to place a red counter ON THE DIVISION-LINE between
No. 11 and No. 12, and to understand this to mean that ONE of the
two compartments is 'occupied,' but that we do not at present know WHICH.
This I shall represent thus:--
-------------------
| | |
| _____|_____ |
| | | | |
| | -1- | |
| | | | |
-------------------
Now let us express "all x are m."
This consists, we know, of TWO Propositions,
"Some x are m,"
and "No x are m'."
Let us express the negative part first. This tells us that none
of the Cakes, belonging to the upper half of the cupboard, are to
be found OUTSIDE the central Square: that is, the two compartments,
No. 9 and No. 10, are EMPTY. This, of course, is represented by
-------------------
| 0 | 0 |
| _____|_____ |
| | | | |
| | | | |
| | | | |
-------------------
But we have yet to represent "Some x are m." This tells us that
there are SOME Cakes in the oblong consisting of No. 11 and No.
12: so we place our red counter, as in the previous example, on
the division-line between No. 11 and No. 12, and the result is
-------------------
| 0 | 0 |
| _____|_____ |
| | | | |
| | -1- | |
| | | | |
-------------------
Now let us try one or two interpretations.
What are we to make of this, with regard to x and y?
-------------------
| | 0 |
| _____|_____ |
| | | | |
| | 1 | 0 | |
| | | | |
-------------------
This tells us, with regard to the xy'-Square, that it is wholly
'empty', since BOTH compartments are so marked. With regard to
the xy-Square, it tells us that it is 'occupied'. True, it is only
ONE compartment of it that is so marked; but that is quite enough,
whether the other be 'occupied' or 'empty', to settle the fact that
there is SOMETHING in the Square.
If, then, we transfer our marks to the smaller Diagram, so as to
get rid of the m-subdivisions, we have a right to mark it
-----------
| | |
| 1 | 0 |
| | |
-----------
which means, you know, "all x are y."
The result would have been exactly the same, if the given oblong
had been marked thus:--
-------------------
| 1 | 0 |
| _____|_____ |
| | | | |
| | | 0 | |
| | | | |
-------------------
Once more: how shall we interpret this, with regard to x and y?
-------------------
| 0 | 1 |
| _____|_____ |
| | | | |
| | | | |
| | | | |
-------------------
This tells us, as to the xy-Square, that ONE of its compartments
is 'empty'. But this information is quite useless, as there is no
mark in the OTHER compartment. If the other compartment happened
to be 'empty' too, the Square would be 'empty': and, if it happened
to be 'occupied', the Square would be 'occupied'. So, as we do
not know WHICH is the case, we can say nothing about THIS Square.
The other Square, the xy'-Square, we know (as in the previous
example) to be 'occupied'.
If, then, we transfer our marks to the smaller Diagram, we get
merely this:--
-----------
| | |
| | 1 |
| | |
-----------
which means, you know, "some x are y'."
These principles may be applied to all the other
oblongs. For instance, to represent
"all y' are m'" we should mark the -------
RIGHT-HAND UPRIGHT OBLONG (the one | |
that has the attribute y') thus:-- |--- |
| 0 | |
|---|-1-|
| 0 | |
|--- |
| |
-------
and, if we were told to interpret the lower half of the cupboard,
marked as follows, with regard to x and y,
-------------------
| | | | |
| | | 0 | |
| | | | |
| -----|----- |
| 1 | 0 |
-------------------
we should transfer it to the smaller Diagram thus,
-----------
| | |
| 1 | 0 |
| | |
-----------
and read it "all x' are y."
Two more remarks about Propositions need to be made.
One is that, in every Proposition beginning with "some" or "all",
the ACTUAL EXISTENCE of the 'Subject' is asserted. If, for instance,
I say "all misers are selfish," I mean that misers ACTUALLY EXIST.
If I wished to avoid making this assertion, and merely to state
the LAW that miserliness necessarily involves selfishness, I should
say "no misers are unselfish" which does not assert that any misers
exist at all, but merely that, if any DID exist, they WOULD be
selfish.
The other is that, when a Proposition begins with "some" or "no",
and contains more that two Attributes, these Attributes may be
re-arranged, and shifted from one Term to the other, "ad libitum."
For example, "some abc are def" may be re-arranged as "some bf are
acde," each being equivalent to "some Things are abcdef". Again "No
wise old men are rash and reckless gamblers" may be re-arranged as
"No rash old gamblers are wise and reckless," each being equivalent
to "No men are wise old rash reckless gamblers."
2. Syllogisms
Now suppose we divide our Universe of Things in three ways, with regard
to three different Attributes. Out of these three Attributes, we
may make up three different couples (for instance, if they were a,
b, c, we might make up the three couples ab, ac, bc). Also suppose
we have two Propositions given us, containing two of these three
couples, and that from them we can prove a third Proposition containing
the third couple. (For example, if we divide our Universe for m,
x, and y; and if we have the two Propositions given us, "no m are
x'" and "all m' are y", containing the two couples mx and my, it
might be possible to prove from them a third Proposition, containing
x and y.)
In such a case we call the given Propositions 'THE PREMISSES', the
third one 'THE CONCLUSION' and the whole set 'A SYLLOGISM'.
Evidently, ONE of the Attributes must occur in both Premisses; or
else one must occur in ONE Premiss, and its CONTRADICTORY in the
other.
In the first case (when, for example, the Premisses are "some m
are x" and "no m are y'") the Term, which occurs twice, is called
'THE MIDDLE TERM', because it serves as a sort of link between the
other two Terms.
In the second case (when, for example, the Premisses are "no
m are x'" and "all m' are y") the two Terms, which contain these
contradictory Attributes, may be called 'THE MIDDLE TERMS'.
Thus, in the first case, the class of "m-Things" is the Middle
Term; and, in the second case, the two classes of "m-Things" and
"m'-Things" are the Middle Terms.
The Attribute, which occurs in the Middle Term or Terms, disappears
in the Conclusion, and is said to be "eliminated", which literally
means "turned out of doors".
Now let us try to draw a Conclusion from the two Premisses--
"Some new Cakes are unwholesome;
No nice Cakes are unwholesome."
In order to express them with counters, we need to divide Cakes in
THREE different ways, with regard to newness, to niceness, and to
wholesomeness. For this we must use the larger Diagram, making x
mean "new", y "nice", and m "wholesome". (Everything INSIDE the
central Square is supposed to have the attribute m, and everything
OUTSIDE it the attribute m', i.e. "not-m".)
You had better adopt the rule to make m mean the Attribute which
occurs in the MIDDLE Term or Terms. (I have chosen m as the symbol,
because 'middle' begins with 'm'.)
Now, in representing the two Premisses, I prefer to begin with the
NEGATIVE one (the one beginning with "no"), because GREY counters
can always be placed with CERTAINTY, and will then help to fix the
position of the red counters, which are sometimes a little uncertain
where they will be most welcome.
Let us express, the "no nice Cakes are unwholesome (Cakes)", i.e.
"no y-Cakes are m'-(Cakes)". This tells us that none of the Cakes
belonging to the y-half of the cupboard are in its m'-compartments
(i.e. the ones outside the central Square). Hence the two compartments,
No. 9 and No. 15, are both 'EMPTY'; and we must place a grey counter
in EACH of them, thus:--
-----------
|0 | |
| --|-- |
| | | | |
|--|-----|--|
| | | | |
| --|-- |
|0 | |
-----------
We have now to express the other Premiss, namely, "some new Cakes
are unwholesome (Cakes)", i.e. "some x-Cakes are m'-(Cakes)". This
tells us that some of the Cakes in the x-half of the cupboard are
in its m'-compartments. Hence ONE of the two compartments, No.
9 and No. 10, is 'occupied': and, as we are not told in WHICH of
these two compartments to place the red counter, the usual rule
would be to lay it on the division-line between them: but, in this
case, the other Premiss has settled the matter for us, by declaring
No. 9 to be EMPTY. Hence the red counter has no choice, and MUST
go into No. 10, thus:--
-----------
|0 | 1|
| --|-- |
| | | | |
|--|-----|--|
| | | | |
| --|-- |
|0 | |
-----------
And now what counters will this information enable us to place in
the SMALLER Diagram, so as to get some Proposition involving x and
y only, leaving out m? Let us take its four compartments, one by
one.
First, No. 5. All we know about THIS is that its OUTER portion
is empty: but we know nothing about its inner portion. Thus the
Square MAY be empty, or it MAY have something in it. Who can tell?
So we dare not place ANY counter in this Square.
Secondly, what of No. 6? Here we are a little better off. We
know that there is SOMETHING in it, for there is a red counter in
its outer portion. It is true we do not know whether its inner
portion is empty or occupied: but what does THAT matter? One solitary
Cake, in one corner of the Square, is quite sufficient excuse for
saying "THIS SQUARE IS OCCUPIED", and for marking it with a red
counter.
As to No. 7, we are in the same condition as with No. 5--we find
it PARTLY 'empty', but we do not know whether the other part is
empty or occupied: so we dare not mark this Square.
And as to No. 8, we have simply no information at all.
The result is
-------
| | 1 |
|---|---|
| | |
-------
Our 'Conclusion', then, must be got out of the rather meager piece
of information that there is a red counter in the xy'-Square.
Hence our Conclusion is "some x are y' ", i.e. "some new Cakes are
not-nice (Cakes)": or, if you prefer to take y' as your Subject,
"some not-nice Cakes are new (Cakes)"; but the other looks neatest.
We will now write out the whole Syllogism, putting the symbol
&there4[*] for "therefore", and omitting "Cakes", for the sake of
brevity, at the end of each Proposition.
[*][NOTE from Brett: The use of "&there4" is a rather arbitrary
selection. There is no font available in general practice
which renders the "therefore" symbol correction (three dots in a
triangular formation). This can be done, however, in HTML, so if
this document is read in a browser, then the symbol will be properly
recognized. This is a poor man's excuse.]
"Some new Cakes are unwholesome;
No nice Cakes are unwholesome
&there4 Some new Cakes are not-nice."
And you have now worked out, successfully, your first 'SYLLOGISM'.
Permit me to congratulate you, and to express the hope that it is
but the beginning of a long and glorious series of similar victories!
We will work out one other Syllogism--a rather harder one than the
last--and then, I think, you may be safely left to play the Game
by yourself, or (better) with any friend whom you can find, that
is able and willing to take a share in the sport.
Let us see what we can make of the two Premisses--
"All Dragons are uncanny;
All Scotchmen are canny."
Remember, I don't guarantee the Premisses to be FACTS. In the
first place, I never even saw a Dragon: and, in the second place,
it isn't of the slightest consequence to us, as LOGICIANS, whether
our Premisses are true or false: all WE have to do is to make out
whether they LEAD LOGICALLY TO THE CONCLUSION, so that, if THEY
were true, IT would be true also.
You see, we must give up the "Cakes" now, or our cupboard will
be of no use to us. We must take, as our 'Universe', some class
of things which will include Dragons and Scotchmen: shall we say
'Animals'? And, as "canny" is evidently the Attribute belonging
to the 'Middle Terms', we will let m stand for "canny", x for
"Dragons", and y for "Scotchmen". So that our two Premisses are,
in full,
"All Dragon-Animals are uncanny (Animals);
All Scotchman-Animals are canny (Animals)."
And these may be expressed, using letters for words, thus:--
"All x are m';
All y are m."
The first Premiss consists, as you already know, of two parts:--
"Some x are m',"
and "No x are m."
And the second also consists of two parts:--
"Some y are m,"
and "No y are m'."
Let us take the negative portions first.
We have, then, to mark, on the larger Diagram, first, "no x are
m", and secondly, "no y are m'". I think you will see, without
further explanation, that the two results, separately, are
----------- -----------
| | | |0 | |
| --|-- | | --|-- |
| |0 | 0| | | | | | |
|--|--|--|--| |--|--|--|--|
| | | | | | | | | |
| --|-- | | --|-- |
| | | |0 | |
----------- -----------
and that these two, when combined, give us
-----------
|0 | |
| --|-- |
| |0 | 0| |
|--|--|--|--|
| | | | |
| --|-- |
|0 | |
-----------
We have now to mark the two positive portions, "some x are m'"
and "some y are m".
The only two compartments, available for Things which are xm', are
No. 9 and No. 10. Of these, No. 9 is already marked as 'empty';
so our red counter must go into No. 10.
Similarly, the only two, available for ym, are No. 11 and No. 13.
Of these, No. 11 is already marked as 'empty'; so our red counter
MUST go into No. 13.
The final result is
-----------
|0 | 1|
| --|-- |
| |0 | 0| |
|--|--|--|--|
| |1 | | |
| --|-- |
|0 | |
-----------
And now how much of this information can usefully be transferred
to the smaller Diagram?
Let us take its four compartments, one by one.
As to No. 5? This, we see, is wholly 'empty'. (So mark it with a
grey counter.)
As to No. 6? This, we see, is 'occupied'. (So mark it with a red
counter.)
As to No. 7? Ditto, ditto.
As to No. 8? No information.
The smaller Diagram is now pretty liberally marked:--
-------
| 0 | 1 |
|---|---|
| 1 | |
-------
And now what Conclusion can we read off from this? Well, it is
impossible to pack such abundant information into ONE Proposition:
we shall have to indulge in TWO, this time.
First, by taking x as Subject, we get "all x are y'", that is,
"All Dragons are not-Scotchmen":
secondly, by taking y as Subject, we get "all y are x'", that is,
"All Scotchmen are not-Dragons".
Let us now write out, all together, our two Premisses and our brace
of Conclusions.
"All Dragons are uncanny;
All Scotchmen are canny.
&there4 All Dragons are not-Scotchmen;
All Scotchmen are not-Dragons."
Let me mention, in conclusion, that you may perhaps meet with
logical treatises in which it is not assumed that any Thing EXISTS
at all, by "some x are y" is understood to mean "the Attributes x,
y are COMPATIBLE, so that a Thing can have both at once", and "no
x are y" to mean "the Attributes x, y are INCOMPATIBLE, so that
nothing can have both at once".
In such treatises, Propositions have quite different meanings
from what they have in our 'Game of Logic', and it will be well to
understand exactly what the difference is.
First take "some x are y". Here WE understand "are" to mean "are,
as an actual FACT"--which of course implies that some x-Things EXIST.
But THEY (the writers of these other treatises) only understand
"are" to mean "CAN be", which does not at all imply that any EXIST.
So they mean LESS than we do: our meaning includes theirs (for of
course "some x ARE y" includes "some x CAN BE y"), but theirs does
NOT include ours. For example, "some Welsh hippopotami are heavy"
would be TRUE, according to these writers (since the Attributes
"Welsh" and "heavy" are quite COMPATIBLE in a hippopotamus), but
it would be FALSE in our Game (since there are no Welsh hippopotami
to BE heavy).
Secondly, take "no x are y". Here WE only understand "are" to
mean "are, as an actual FACT"--which does not at all imply that no
x CAN be y. But THEY understand the Proposition to mean, not only
that none ARE y, but that none CAN POSSIBLY be y. So they mean
more than we do: their meaning includes ours (for of course "no x
CAN be y" includes "no x ARE y"), but ours does NOT include theirs.
For example, "no Policemen are eight feet high" would be TRUE
in our Game (since, as an actual fact, no such splendid specimens
are ever found), but it would be FALSE, according to these writers
(since the Attributes "belonging to the Police Force" and "eight
feet high" are quite COMPATIBLE: there is nothing to PREVENT a
Policeman from growing to that height, if sufficiently rubbed with
Rowland's Macassar Oil--which said to make HAIR grow, when rubbed
on hair, and so of course will make a POLICEMAN grow, when rubbed
on a Policeman).
Thirdly, take "all x are y", which consists of the two partial
Propositions "some x are y" and "no x are y'". Here, of course,
the treatises mean LESS than we do in the FIRST part, and more than
we do in the SECOND. But the two operations don't balance each
other--any more than you can console a man, for having knocked down
one of his chimneys, by giving him an extra door-step.
If you meet with Syllogisms of this kind, you may work them, quite
easily, by the system I have given you: you have only to make
'are' mean 'are CAPABLE of being', and all will go smoothly. For
"some x are y" will become "some x are capable of being y", that
is, "the Attributes x, y are COMPATIBLE". And "no x are y" will
become "no x are capable of being y", that is, "the Attributes
x, y are INCOMPATIBLE". And, of course, "all x are y" will become
"some x are capable of being y, and none are capable of being y'",
that is, "the Attributes x, y are COMPATIBLE, and the Attributes
x, y' are INCOMPATIBLE." In using the Diagrams for this system,
you must understand a red counter to mean "there may POSSIBLY be
something in this compartment," and a grey one to mean "there cannot
POSSIBLY be anything in this compartment."
3. Fallacies.
And so you think, do you, that the chief use of Logic, in real life,
is to deduce Conclusions from workable Premisses, and to satisfy
yourself that the Conclusions, deduced by other people, are correct?
I only wish it were! Society would be much less liable to panics
and other delusions, and POLITICAL life, especially, would be a
totally different thing, if even a majority of the arguments, that
scattered broadcast over the world, were correct! But it is all
the other way, I fear. For ONE workable Pair of Premisses (I mean
a Pair that lead to a logical Conclusion) that you meet with in
reading your newspaper or magazine, you will probably find FIVE
that lead to no Conclusion at all: and, even when the Premisses
ARE workable, for ONE instance, where the writer draws a correct
Conclusion, there are probably TEN where he draws an incorrect one.
In the first case, you may say "the PREMISSES are fallacious": in
the second, "the CONCLUSION is fallacious."
The chief use you will find, in such Logical skill as this Game
may teach you, will be in detecting 'FALLACIES' of these two kinds.
The first kind of Fallacy--'Fallacious Premisses'--you will detect
when, after marking them on the larger Diagram, you try to transfer
the marks to the smaller. You will take its four compartments, one
by one, and ask, for each in turn, "What mark can I place HERE?";
and in EVERY one the answer will be "No information!", showing that
there is NO CONCLUSION AT ALL. For instance,
"All soldiers are brave;
Some Englishmen are brave.
&there4 Some Englishmen are soldiers."
looks uncommonly LIKE a Syllogism, and might easily take in a
less experienced Logician. But YOU are not to be caught by such
a trick! You would simply set out the Premisses, and would then
calmly remark "Fallacious PREMISSES!": you wouldn't condescend to
ask what CONCLUSION the writer professed to draw--knowing that,
WHATEVER it is, it MUST be wrong. You would be just as safe as
that wise mother was, who said "Mary, just go up to the nursery,
and see what Baby's doing, AND TELL HIM NOT TO DO IT!"
The other kind of Fallacy--'Fallacious Conclusion'--you will not
detect till you have marked BOTH Diagrams, and have read off the
correct Conclusion, and have compared it with the Conclusion which
the writer has drawn.
But mind, you mustn't say "FALLACIOUS Conclusion," simply because
it is not IDENTICAL with the correct one: it may be a PART of the
correct Conclusion, and so be quite correct, AS FAR AS IT GOES. In
this case you would merely remark, with a pitying smile, "DEFECTIVE
Conclusion!" Suppose, of example, you were to meet with this
Syllogism:--
"All unselfish people are generous;
No misers are generous.
&there4 No misers are unselfish."
the Premisses of which might be thus expressed in letters:--
"All x' are m;
No y are m."
Here the correct Conclusion would be "All x' are y'" (that is,
"All unselfish people are not misers"), while the Conclusion, drawn
by the writer, is "No y are x'," (which is the same as "No x' are
y," and so is PART of "All x' are y'.") Here you would simply say
"DEFECTIVE Conclusion!" The same thing would happen, if you were
in a confectioner's shop, and if a little boy were to come in, put
down twopence, and march off triumphantly with a single penny-bun.
You would shake your head mournfully, and would remark "Defective
Conclusion! Poor little chap!" And perhaps you would ask the
young lady behind the counter whether she would let YOU eat the
bun, which the little boy had paid for and left behind him: and
perhaps SHE would reply "Sha'n't!"
But if, in the above example, the writer had drawn the Conclusion
"All misers are selfish" (that is, "All y are x"), this would
be going BEYOND his legitimate rights (since it would assert the
EXISTENCE of y, which is not contained in the Premisses), and you
would very properly say "Fallacious Conclusion!"
Now, when you read other treatises on Logic, you will meet with
various kinds of (so-called) 'Fallacies' which are by no means
ALWAYS so. For example, if you were to put before one of these
Logicians the Pair of Premisses
"No honest men cheat;
No dishonest men are trustworthy."
and were to ask him what Conclusion followed, he would probably say
"None at all! Your Premisses offend against TWO distinct Rules, and
are as fallacious as they can well be!" Then suppose you were bold
enough to say "The Conclusion is 'No men who cheat are trustworthy',"
I fear your Logical friend would turn away hastily--perhaps angry,
perhaps only scornful: in any case, the result would be unpleasant.
I ADVISE YOU NOT TO TRY THE EXPERIMENT!
"But why is this?" you will say. "Do you mean to tell us that all
these Logicians are wrong?" Far from it, dear Reader! From THEIR
point of view, they are perfectly right. But they do not include,
in their system, anything like ALL the possible forms of Syllogisms.
They have a sort of nervous dread of Attributes beginning with a
negative particle. For example, such Propositions as "All not-x
are y," "No x are not-y," are quite outside their system. And
thus, having (from sheer nervousness) excluded a quantity of very
useful forms, they have made rules which, though quite applicable
to the few forms which they allow of, are no use at all when you
consider all possible forms.
Let us not quarrel with them, dear Reader! There is room enough in
the world for both of us. Let us quietly take our broader system:
and, if they choose to shut their eyes to all these useful forms,
and to say "They are not Syllogisms at all!" we can but stand aside,
and let them Rush upon their Fate! There is scarcely anything of
yours, upon which it is so dangerous to Rush, as your Fate. You
may Rush upon your Potato-beds, or your Strawberry-beds, without
doing much harm: you may even Rush upon your Balcony (unless it
is a new house, built by contract, and with no clerk of the works)
and may survive the foolhardy enterprise: but if you once Rush upon
your FATE--why, you must take the consequences!