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THE GAME OF LOGIC
By Lewis Carroll
--------------------- |9 | 10| | | | | -----x------ | | |11 | 12| | | | | | | |---y-----m------y'---| | | | | | | |13 | 14| | | -----x'----- | | | | |15 | 16| ---------------------
--------------------- |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!
___
COLOURS FOR -------------
COUNTERS |5 | 6|
___ | x |
| | |
Look, the Sun is above us, |--y-------y'-|
Shining down, BRIGHT and | | |
RED! | x' |
|7 | 8|
Now the Sun has slipped away, -------------
And the EMPTY sky is
GRAY!
___THE GAME OF LOGIC
By Lewis Carrol
To my Child-friend.
To my Kid-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!
I charm in vain; for never again,
As sharply as I look, will Memory, the shy goddess,
Create for my delight
The days gone by, nor let me view
You, my enchanting 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!
Yet if your face, in a magical way,
Could smile at me for just a moment, it would send
Bright beams of light
From Heaven across the night,
Through which I could truly see
Your spirit, dearest 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!
So may the stream of life's long dream
Flow gently onward to its end,
With many colorful flowers,
Along its winding path:
May no sigh trouble, no worry confuse,
My dear 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.
With each copy of this book, you'll receive an envelope that contains a diagram (similar to the frontispiece) on cardstock, along with nine counters—four red and five gray.
The Envelope, &c. can be had separately, at 3d. each.
The envelope, etc. can be bought separately for 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."
The author would greatly appreciate suggestions, especially from newcomers to Logic, regarding any changes or additional explanations that might be helpful. Please send letters to him at "29, Bedford Street, Covent Garden, London."
PREFACE
"There foam'd rebellious Logic, gagg'd and bound."
"There foamed rebellious Logic, gagged and bound."
This Game requires nine Counters--four of one colour and five of another: say four red and five grey.
This game needs nine counters—four of one color and five of another: for example, four red and five gray.
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.
Besides the nine Counters, it also needs at least one Player. I don't know of any Game that can be played with fewer than this number, although there are several that need more—take Cricket, for example, which requires twenty-two. It's much easier when you want to play a Game to find one Player than twenty-two. However, while one Player is enough, you can have a lot more fun with two working together and helping each other fix their 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?
A second advantage of this Game is that, in addition to being an endless source of fun (the number of arguments that can come from it is infinite), it also provides players with some learning. But is there really any harm in that, as long as you’re having a good time?
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 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."
"Light comes, light goes."
_________
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.
There are three 'PROPOSITIONS' for you—the only three types we’re going to use in this Game: and the first thing you need to do is learn how to express them on the Board.
Let us begin with
Let's get started with
"Some new Cakes are nice."
"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.
But before proceeding, there's an important point to make—one that's not easy to grasp all at once: so please 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?
The world has lots of THINGS (like "Buns", "Babies", "Beetles", "Battledores", etc.); and these Things have many ATTRIBUTES (like "baked", "beautiful", "black", "broken", etc.: in fact, anything that can be "attributed to", or "said to belong to", a Thing is an Attribute). Whenever we want to mention a Thing, we use a NOUN; when we want to mention an Attribute, we use an ADJECTIVE. People have asked the question, "Can a Thing exist without any Attributes belonging to it?" It's a really tricky question, and I’m not going to try to answer it: let’s just scoff at it and ignore it, as if it isn’t even worth our time. But if they flip the question and ask, "Can an Attribute exist without any Thing for it to belong to?", we can immediately say, "No: just like a Baby couldn’t travel on a train without someone to look after it!" You’ve never seen "beautiful" just hanging around in the air or scattered on the floor, without any Thing to actually BE beautiful, have 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.
And now, what am I getting at with all this long explanation? It's this: You can put "is" or "are" between the names of two THINGS (for example, "some Pigs are fat Animals"), or between two ATTRIBUTES (for example, "pink is light-red"), and it will make sense in both cases. 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"), it doesn't make sense (because how can a Thing BE an Attribute?) unless you have an agreement with the person you're talking to. And the simplest agreement would be this— that you should understand that the noun is implied to be repeated at the end of the sentence, so that if it were written out fully, it would say "some Pigs are pink (Pigs)." Then "are" makes complete 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'".
To understand the statement "some new Cakes are nice," we need to fully express it as "some new Cakes are nice (Cakes)." This includes two 'TERMS': "new Cakes" as one term and "nice (Cakes)" as the other. "New Cakes," which is the term we’re discussing, is called the 'SUBJECT' of the statement, while "nice (Cakes)" is the 'PREDICATE.' This statement is considered 'PARTICULAR' because it doesn’t reference the entire Subject but only a PART of it. The other two types are referred to as 'UNIVERSAL' since they refer to the entire Subject—one denies niceness, and the other asserts it, regarding the whole class of "new Cakes." Finally, if you want a definition of the term 'PROPOSITION,' you can use this: "a sentence that states whether some, none, or all items in a certain class, known as its 'Subject,' are also items in another class, referred to as 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'.
You will find these seven words—PROPOSITION, ATTRIBUTE, TERM, SUBJECT, PREDICATE, PARTICULAR, UNIVERSAL—really helpful if any friend asks whether you've ever studied Logic. Make sure to include all seven words in your response, and your friend will leave feeling both wiser and a bit sadder.
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."
Now, please look at the smaller diagram on the board and imagine it as a cupboard meant for all the cakes in the world (it would definitely need to be a pretty large one). Let's say all the new cakes go into the upper half (marked 'x'), and all the others (the NOT-new ones) go into the lower half (marked 'x''). So, the lower half would have ELDERLY cakes, AGED cakes, ANTE-DILUVIAN cakes—if there are any; I haven't seen many myself—and so on. Also, let’s say all the nice cakes are placed in the left half (marked 'y'), and all the others (the not-nice ones) go into the right half (marked 'y''). For now, we should understand x to mean "new," x' to mean "not-new," y to mean "nice," and y' to mean "not-nice."
And now what kind of Cakes would you expect to find in compartment No. 5?
And now, what kind of cakes do you think you’d 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."
It’s part of the upper half, you see; so if there are any cakes in it, they must be NEW: and it’s part of the left-hand half; so they must be NICE. Therefore, if there are any cakes in this compartment, they must have the double 'ATTRIBUTE' "new and nice": or, if we use letters, they 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."
Observe that the letters x and y are written on two of the edges of this compartment. You'll find this a very handy rule for knowing which Attributes belong to the Things in any compartment. Take No. 7, for example. If there are any Cakes there, they must be "x' y," meaning 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.
Now let's make another agreement—if there's a red counter in a compartment, it means it's 'OCCUPIED,' which means there are SOME Cakes in it. (In Logic, the word 'some' means 'one or more,' so even just one Cake in a compartment is enough reason to say "there are SOME Cakes here.") Also, let's agree that a grey counter in a compartment means it's 'EMPTY,' which means there are NO Cakes in it. In the following Diagrams, I'll place '1' (meaning 'one or more') where you should put a RED counter, and '0' (meaning 'none') where you should 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."
As the topic of our proposal is "new cakes," we are currently focused solely on the upper half of the cupboard, where all the cakes have the characteristic x, meaning "new."
Now, fixing our attention on this upper half, suppose we found it marked like this,
Now, focusing on this upper half, let’s say we found it marked like this,
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-----------that is, with a red counter in No. 5. What would this tell us, with regard to the class of "new Cakes"?
that is, with a red counter in No. 5. What would this tell us about 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,
Wouldn't it indicate that there are some of them in the xy compartment? That is, that some of them (besides having the Attribute x, which belongs to both compartments) have the Attribute y (which means "nice"). We could express this by saying "some x-Cakes are y-Cakes," or, putting words instead of letters,
"Some new Cakes are nice (Cakes)",
"Some new cakes are nice (cakes),"
or, in a shorter form,
or, in shorter form,
"Some new Cakes are nice".
"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.
At last, we have figured out how to explain the first Proposition of this Section. If you haven't clearly understood everything I've said, stop here and read it repeatedly until you do understand it. Once you've got that down, you'll find the rest fairly simple.
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?)
It will make things a bit easier for the other proposals if we agree to drop the word "Cakes" completely. I think it's helpful to refer to the entire category of items that the cupboard is meant for as the 'UNIVERSE.' So, we could have started this by saying "Let’s take a Universe of Cakes." (Sounds good, right?)
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?)
Of course, any other things would have worked just as well as cakes. We could propose "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
So far, we have learned that
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-----------means "some x and y," i.e. "some new are nice."
means "some x and y," i.e. "some new things are nice."
I think you will see without further explanation, that
I think you’ll understand without needing more explanation, that
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-----------means "some x are y'," i.e. "some new are not-nice."
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
Now let's put a GREY counter into No. 5 and ask ourselves the meaning of
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-----------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.
This tells us that the x y-compartment is EMPTY, which we can say as "no x are y", or, "no new Cakes are nice". This is the second of the three Propositions at the start of this Section.
In the same way,
Similarly,
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-----------would mean "no x are y'," or, "no new Cakes are not-nice."
would mean "no x are y," or, "no new Cakes are not nice."
What would you make of this, I wonder?
What do you think about this, I wonder?
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-----------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."
I hope you won’t have too much trouble understanding 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:
The following might be a bit more challenging, perhaps:
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-----------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.
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" create what we call an 'EXHAUSTIVE' division of the class "new Cakes": i.e. together, they EXHAUST the entire class, so that all the new Cakes that exist must fall into one category or the other.
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?
And now imagine you had to use counters to show the opposite of "no Cakes are new," which would be "some Cakes are new," or, using letters for words, "some Cakes are x." How would you do that?
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.
This might confuse you a bit, I imagine. Clearly, you need to place a red counter SOMEWHERE in the x-half of the cupboard since you know there are SOME new Cakes. However, you can't put it in the LEFT-HAND compartment because you don’t know if they’re NICE; nor can you put it in the RIGHT-HAND compartment because you don’t know if they’re 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
What should you do then? I believe the best solution is to put the red counter ON THE DIVISION-LINE between the xy-compartment and the xy'-compartment. I'll show this (as I always indicate '1' where you should place a red counter) with the diagram.
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-----------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!
Our clever American cousins have come up with a phrase to describe a person who wants to pick between two groups—like their two parties, 'Democrats' and 'Republicans'—but can't decide WHICH one to choose. This person is said to be "sitting on the fence." That’s exactly the situation of the red counter you just placed on the dividing line. He likes the look of No. 5 and he likes the look of No. 6, but he doesn’t know WHICH one to jump into. So there he is, sitting straddled, 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?
Now I'm going to give you a much tougher one to figure out. What does this mean?
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-----------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.
This is clearly a DOUBLE Proposition. It tells us not only that "some x are y," but also that "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
We see, then, that the Universal Proposition
"All new Cakes are nice"
"All new cakes are nice."
consists of TWO Propositions taken together, namely,
consists of TWO Propositions combined, specifically,
"Some new Cakes are nice,"
and "No new Cakes are not-nice."
"Some new cakes are nice,"
and "No new cakes are not nice."
In the same way
Similarly
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-----------would mean "all x are y' ", that is,
would mean "all x are y," that is,
"All new Cakes are not-nice."
"All new cakes are bad."
Now what would you make of such a Proposition as "The Cake you have given me is nice"? Is it Particular or Universal?
Now, what do you think about the statement "The cake you gave me is nice"? Is it specific or general?
"Particular, of course," you readily reply. "One single Cake is hardly worth calling 'some,' even."
"Particular, of course," you quickly respond. "One single cake is barely enough to be called '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!
No, my dear impulsive Reader, it is 'Universal'. Remember that, as few as they are (and I admit they couldn't be fewer), they are (or rather 'it is') ALL that you've given me! So, if I leave 'red' out of the equation and divide my Universe of Cakes into two categories— the Cakes you've given me (which I’ll place in the upper half of the cupboard) and those you HAVEN'T given me (which will go below)—I see that the lower half is pretty full, and the upper half is almost completely empty. Then, when I'm told to add an upright divider to each half, keeping the NICE Cakes to the left and the NOT-NICE ones to the right, I start by carefully gathering ALL the Cakes you've given me (thinking to myself, every now and then, "Generous creature! How will I ever repay such kindness?"), and stacking them in the left 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.)
Here’s another universal statement for you. "Barzillai Beckalegg is an honest man." That means "ALL the Barzillai Beckaleggs that I'm currently thinking about are honest men." (You think I made up that name, don’t you? But I didn’t. It's on a delivery cart somewhere down in Cornwall.)
This kind of Universal Proposition (where the Subject is a single Thing) is called an 'INDIVIDUAL' Proposition.
This type of Universal Proposition (where the Subject is one specific 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."
Now let’s take "NICE Cakes" as the topic of discussion: that is, let’s focus on the LEFT-HAND half of the cupboard, where all the Cakes have the quality y, which is "nice."
-----
Suppose we find it marked like this:-- | |
| 1 |
What would that tell us? | |
-----
| |
| |
| |
-----
-----
Suppose we find it marked like this:-- | |
| 1 |
What does 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,
I hope that, after explaining the HORIZONTAL oblong in detail, we won’t need to spend much time on the UPRIGHT one. I trust you will understand that this means "some y are x", that is,
"Some nice Cakes are new."
"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?"
"But," you might say, "we've been through this before. You placed a red counter in No. 5 and told us it meant 'some new Cakes are nice'; and NOW you say 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.
The question is really insightful, and it says a lot about you, dear Reader! It does mean both. If you choose to take x (that is, "new Cakes") as your subject and see No. 5 as part of a horizontal rectangle, you can read it as "some x are y," which means "some new Cakes are nice." But, if you decide to take y (that is, "nice Cake") as your subject and view No. 5 as part of a vertical rectangle, then you can read it as "some y are x," meaning "some nice Cakes are new." They are just two different ways of expressing the 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.
Without further ado, I will list the other ways this upright rectangle could be marked, including the meaning for each. By comparing them with the different cases of the horizontal rectangle, I hope you'll 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.
You’ll find it helpful to reflect on this table by first covering one column and then the other, and “dodging around,” as the kids 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.
Also, it would be a good idea to create two more tables for yourself—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.
And now I think we've covered everything we need to about the smaller diagram, so we can move 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 | |
| | |
----- |
_______________|_________________________________
_________________________________________________
|
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.
| | |
----- | [Note that this is just another way of
| | | saying "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.
This can be seen as a cupboard set up similarly to the last one, but also divided into two sections for the Attribute m. Let's define m as meaning "wholesome": and let's imagine that all WHOLESOME Cakes are positioned INSIDE the central square, while all the UNWHOLESOME ones are placed OUTSIDE it, specifically in one of the four odd-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!)
We see that, just like in the smaller diagram where the cakes in each section had two attributes, here the cakes in each section have three attributes. Just as the letters representing the two attributes were written on the edges of the section, here they are written at the corners. (Note that m' is meant to be written at each of the four outer corners.) This lets us quickly tell what three attributes belong to the items in that section just by looking at it. For example, take No. 12. Here we see x, y', m at the corners: so we know that the cakes in it, if there are any, have the triple attribute 'xy'm', which means "new, not-nice, and wholesome." Again, take No. 16. Here we find x', y', m' at the corners: 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.
It would take way too long to go through all the Propositions involving x and y, x and m, and y and m that can be shown on this diagram (there are ninety-six total, so I’m sure you’ll understand!) and I’ll have to settle for doing just two or three as examples. It would be a good idea for you to work out a lot more on your own.
Taking the upper half by itself, so that our Subject is "new Cakes", how are we to represent "no new Cakes are wholesome"?
Taking the upper half on its own, so that our subject is "new Cakes," how can we show that "no new Cakes are healthy"?
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
This is, writing letters for words, "no x are m." Now this tells us that none of the cakes from the upper half of the cupboard are located inside the central square: in other words, compartments No. 11 and No. 12 are empty. And this, of course, is represented by
-------------------
| | |
| _____|_____ |
| | | | |
| | 0 | 0 | |
| | | | |
-------------------
-------------------
| | |
| _____|_____ |
| | | | |
| | 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:--
And now how should we represent the contradictory statement "SOME x are m"? This is a challenge I've already thought about. I believe the best approach is to place a red counter ON THE DIVISION-LINE between No. 11 and No. 12, indicating that ONE of the two sections is 'occupied,' but we don't currently know WHICH one. I will represent this as follows:--
-------------------
| | |
| _____|_____ |
| | | | |
| | -1- | |
| | | | |
-------------------
-------------------
| | |
| _____|_____ |
| | | | |
| | -1- | |
| | | | |
-------------------Now let us express "all x are m."
Now let's express "all x are m."
This consists, we know, of TWO Propositions,
This consists, as we know, of TWO Propositions,
"Some x are m,"
and "No x are m'."
"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
Let’s talk about the negative part first. This means that none of the Cakes from the upper half of the cupboard are outside the central Square: in other words, the two compartments, No. 9 and No. 10, are EMPTY. This, of course, is represented by
-------------------
| 0 | 0 |
| _____|_____ |
| | | | |
| | | | |
| | | | |
-------------------
-------------------
| 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
But we still need to show "Some x are m." This tells us that there are SOME Cakes in the shape that includes No. 11 and No. 12: so we put our red counter, like in the earlier example, on the boundary between No. 11 and No. 12, and the result is
-------------------
| 0 | 0 |
| _____|_____ |
| | | | |
| | -1- | |
| | | | |
-------------------
-------------------
| 0 | 0 |
| _____|_____ |
| | | | |
| | -1- | |
| | | | |
-------------------Now let us try one or two interpretations.
Now let's explore a couple of interpretations.
What are we to make of this, with regard to x and y?
What should we think about this, in relation to x and y?
-------------------
| | 0 |
| _____|_____ |
| | | | |
| | 1 | 0 | |
| | | | |
-------------------
-------------------
| | 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.
This tells us about the xy'-Square that it is completely 'empty' since BOTH compartments are marked that way. As for the xy-Square, it indicates that it is 'occupied'. It's true that only ONE of its compartments is marked like that, but that's enough, regardless of whether the other compartment is 'occupied' or 'empty', to confirm 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
If we move our marks to the smaller Diagram to eliminate the m-subdivisions, we can mark it.
-----------
| | |
| 1 | 0 |
| | |
-----------
-----------
| | |
| 1 | 0 |
| | |
-----------which means, you know, "all x are y."
which means, you know, "all x are y."
The result would have been exactly the same, if the given oblong had been marked thus:--
The result would have been the same, even if the given rectangle had been marked like this:--
-------------------
| 1 | 0 |
| _____|_____ |
| | | | |
| | | 0 | |
| | | | |
-------------------
-------------------
| 1 | 0 |
| _____|_____ |
| | | | |
| | | 0 | |
| | | | |
-------------------Once more: how shall we interpret this, with regard to x and y?
Once again: how should we understand this in relation to x and y?
-------------------
| 0 | 1 |
| _____|_____ |
| | | | |
| | | | |
| | | | |
-------------------
-------------------
| 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.
This tells us, regarding the xy-Square, that one of its sections is 'empty.' However, this information is pretty useless because there’s no indication in the other section. If the other section is also 'empty,' then the Square would be 'empty'; and if it’s 'occupied,' then the Square would be 'occupied.' Since we don’t know which situation applies, we can’t make any conclusions about this Square.
The other Square, the xy'-Square, we know (as in the previous example) to be 'occupied'.
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:--
If we move our marks to the smaller Diagram, we just get this:—
-----------
| | |
| | 1 |
| | |
-----------
-----------
| | |
| | 1 |
| | |
-----------which means, you know, "some x are y'."
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 | |
|--- |
| |
-------
These principles can be applied to all the other rectangles. For example, to represent "all y' are m'" we should mark the ------- RIGHT-HAND UPRIGHT RECTANGLE (the one that has the attribute y') like this:-- | | |--- | | 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,
and, if we were asked to interpret the lower part of the cupboard, marked as follows, concerning x and y,
-------------------
| | | | |
| | | 0 | |
| | | | |
| -----|----- |
| 1 | 0 |
-------------------
-------------------
| | | | |
| | | 0 | |
| | | | |
| -----|----- |
| 1 | 0 |
-------------------we should transfer it to the smaller Diagram thus,
we should move it to the smaller Diagram like this,
-----------
| | |
| 1 | 0 |
| | |
-----------
-----------
| | |
| 1 | 0 |
| | |
-----------and read it "all x' are y."
and read it "all x's are y."
Two more remarks about Propositions need to be made.
Two more points 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.
One point is that in every statement starting with "some" or "all," the REALITY of the 'Subject' is claimed. For example, when I say "all misers are selfish," I mean that misers REALLY EXIST. If I wanted to avoid making this claim and just talk about the RULE that being miserly necessarily includes being selfish, I would say "no misers are unselfish." This doesn't claim that any misers exist at all; it just means 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."
The other point is that when a proposition starts with "some" or "no," and includes more than two attributes, these attributes can be rearranged and moved between terms freely. For example, "some abc are def" can be rearranged as "some bf are acde," both meaning "some things are abcdef." Similarly, "No wise old men are rash and reckless gamblers" can be rearranged as "No rash old gamblers are wise and reckless," which can also be interpreted as "No men are wise old rash reckless gamblers."
2. Syllogisms
2. Logical arguments
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.)
Now let’s say we divide our Universe of Things in three ways based on three different Attributes. From these three Attributes, we can create three different pairs (for example, if they are a, b, and c, we could form the pairs ab, ac, and bc). Also, suppose we have two Propositions provided to us that include two of these three pairs, and we can use them to prove a third Proposition that includes the third pair. (For instance, if we divide our Universe into m, x, and y; and if we have the two Propositions "no m are x" and "all m are y," which include the pairs mx and my, it could be possible to prove a third Proposition that involves 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'.
In this case, we refer to the given statements as 'THE PREMISES', the third one as 'THE CONCLUSION', and the entire set as '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.
Clearly, one of the attributes must be present in both premises; otherwise, one must be in one premise and its opposite must be 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 first case (when, for example, the Premises are "some m are x" and "no m are y") the Term that appears twice is called 'THE MIDDLE TERM' because it acts as a 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'.
In the second case (when, for example, the premises are "no m are x" and "all m' are y"), the two terms that contain these contradictory attributes can 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.
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".
The Attribute, which appears in the Middle Term or Terms, vanishes in the Conclusion and is described as being "eliminated," which literally means "kicked out."
Now let us try to draw a Conclusion from the two Premisses--
Now let's try to draw a conclusion from the two premises--
"Some new Cakes are unwholesome;
No nice Cakes are unwholesome."
"Some new cakes are unhealthy;
No good cakes are unhealthy."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".)
To express them with counters, we need to categorize Cakes in three different ways: based on newness, niceness, and wholesomeness. For this, we will use the larger Diagram, where x represents "new," y stands for "nice," and m means "wholesome." (Everything within the central Square is considered to have the attribute m, while everything outside has the attribute m', meaning "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'.)
You should adopt the rule that makes m represent the Attribute that appears in the MIDDLE Term or Terms. (I chose m as the symbol because 'middle' starts 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.
Now, when it comes to representing the two premises, I like to start with the negative one (the one that begins with "no"), because grey counters can always be placed with certainty, and they will then help determine the position of the red counters, which can sometimes be a bit uncertain about where they'll be most effective.
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:--
Let’s say, “no nice cakes are unhealthy (cakes),” meaning “no y-cakes are m’-(cakes).” This means that none of the cakes in the y-half of the cupboard are in its m’-compartments (i.e., the ones outside the central square). Therefore, both compartments, No. 9 and No. 15, are ‘EMPTY’; and we need to put a grey counter in EACH of them, like this:--
-----------
|0 | |
| --|-- |
| | | | |
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-----------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:--
We now need to state the other premise, which is "some new cakes are unhealthy (cakes)," or "some x-cakes are m'-(cakes)." This means that some of the cakes in the x-half of the cupboard are located in its m'-compartments. Therefore, one of the two compartments, No. 9 or No. 10, is 'occupied'; and since we’re not told which of these two compartments should hold the red counter, the usual rule would be to place it on the dividing line between them. However, in this case, the other premise has resolved the issue for us by stating that No. 9 is EMPTY. Thus, the red counter has no option but to go into No. 10, like this:--
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-----------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.
And now, what counters can we use this information for in the SMALLER Diagram to create a Proposition involving only x and y, excluding m? Let’s examine its four sections, one at a time.
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.
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.
Secondly, what about No. 6? We’re a bit better off here. We know that there’s SOMETHING in it because there’s a red counter on the outside. It’s true we don’t know if the inside is empty or filled, but does that really matter? One single Cake, in one corner of the Square, is more than enough reason to say "THIS SQUARE IS OCCUPIED" and to mark 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.
As for No. 7, we're in the same situation as with No. 5—we find it PARTLY 'empty', but we don't know if the other part is empty or occupied: so we can't mark this Square.
And as to No. 8, we have simply no information at all.
And regarding No. 8, we really don’t have any information whatsoever.
The result is
The outcome is
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-------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.
Our 'Conclusion', then, must come from the rather limited information that there is a red counter in the xy'-Square. So our Conclusion is "some x are y'", which means "some new Cakes are not-nice (Cakes)"; or, if you want to take y' as your Subject, "some not-nice Cakes are new (Cakes)"; but the first option looks the best.
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.
We will now write out the entire Syllogism, using the symbol &there4[*] for "therefore," and leaving out "Cakes" at the end of each Proposition for the sake of brevity.
[*][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.]
[*][NOTE from Brett: The use of "&there4" is a somewhat random choice. There isn’t a commonly used font that displays the "therefore" symbol correctly (three dots in a triangular formation). However, this can be achieved in HTML, so if this document is viewed in a browser, the symbol will be displayed properly. This is a lackluster justification.]
"Some new Cakes are unwholesome;
No nice Cakes are unwholesome
&there4 Some new Cakes are not-nice."
"Some new cakes are unhealthy;
No nice cakes are unhealthy
& therefore 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!
And you have now successfully figured out your first 'SYLLOGISM.' Let me congratulate you and express the hope that this is just the start of a long and impressive series of similar achievements!
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.
We will figure out one more syllogism—a bit tougher than the last one—and then, I think, you'll be good to play the game on your own, or (even better) with any friend you can find who is able and willing to join in the fun.
Let us see what we can make of the two Premisses--
Let’s see what we can make of the two premises—
"All Dragons are uncanny;
All Scotchmen are canny."
"All dragons are strange;
All Scots are shrewd."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.
Remember, I don’t guarantee that the premises are facts. First, I’ve never even seen a dragon; and second, it doesn’t matter to us as logicians whether our premises are true or false. All we need to do is determine whether they logically lead to the conclusion, so that if they were true, the conclusion would be true as well.
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,
You see, we need to give up the "Cakes" now, or our cupboard won't be useful to us. We have to choose a category for our 'Universe' that includes Dragons and Scotsmen: should we say 'Animals'? And, since "canny" clearly belongs to the 'Middle Terms', let's use m to represent "canny", x for "Dragons", and y for "Scotsmen". Therefore, our two premises are, in full,
"All Dragon-Animals are uncanny (Animals); All Scotchman-Animals are canny (Animals)."
"All dragon animals are strange (animals); All Scottish animals are clever (animals)."
And these may be expressed, using letters for words, thus:--
And these can be represented by using letters for words, like this:--
"All x are m';
All y are m."
"All x are m';
All y are m."The first Premiss consists, as you already know, of two parts:--
The first premise consists, as you already know, of two parts:--
"Some x are m',"
and "No x are m."
"Some x are m," and "No x are m."
And the second also consists of two parts:--
And the second also has two parts:--
"Some y are m,"
and "No y are m'."
"Some y are m," and "No y are m."
Let us take the negative portions first.
Let’s start with the negative parts 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
We need to indicate, on the larger Diagram, first, "no x are m," and secondly, "no y are m'." I believe you will see, without any further explanation, that the two results, individually, are
----------- -----------
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|--|--|--|--| |--|--|--|--|
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----------- -----------
----------- -----------
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----------- -----------and that these two, when combined, give us
and that these two, when combined, give us
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-----------
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-----------We have now to mark the two positive portions, "some x are m'" and "some y are m".
We now need to highlight the two positive parts, "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.
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.
Similarly, the only two available for me 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
The end result is
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|--|--|--|--|
| |1 | | |
| --|-- |
|0 | |
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| --|-- |
| |0 | 0| |
|--|--|--|--|
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|0 | |
-----------And now how much of this information can usefully be transferred to the smaller Diagram?
And now, how much of this information can be effectively transferred to the smaller Diagram?
Let us take its four compartments, one by one.
Let's examine its four sections, one by one.
As to No. 5? This, we see, is wholly 'empty'. (So mark it with a grey counter.)
As for No. 5? This, we see, is completely 'empty'. (So mark it with a grey token.)
As to No. 6? This, we see, is 'occupied'. (So mark it with a red counter.)
As for No. 6? This one is 'occupied.' (So mark it with a red counter.)
As to No. 7? Ditto, ditto.
As for No. 7? Same here, same here.
As to No. 8? No information.
As for No. 8? No info.
The smaller Diagram is now pretty liberally marked:--
The smaller diagram is now quite clearly marked:--
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-------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.
And now, what conclusion can we draw from this? Well, it’s impossible to fit such abundant information into ONE statement; we’ll have to rely on TWO this time.
First, by taking x as Subject, we get "all x are y'", that is,
First, by taking x as the Subject, we get "all x are y," that is,
"All Dragons are not-Scotchmen":
"All Dragons are not Scotsmen"
secondly, by taking y as Subject, we get "all y are x'", that is,
secondly, by taking y as the Subject, we get "all y are x," that is,
"All Scotchmen are not-Dragons".
"Not all Scots are Dragons."
Let us now write out, all together, our two Premisses and our brace of Conclusions.
Let's now write out, all together, our two premises and our pair of conclusions.
"All Dragons are uncanny;
All Scotchmen are canny.
&there4 All Dragons are not-Scotchmen;
All Scotchmen are not-Dragons."
"All dragons are unusual;
All Scots are clever.
&there4 All dragons are not Scots;
All Scots 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".
Let me conclude by saying that you might come across logical discussions where it’s not assumed that anything actually EXISTS. In these discussions, "some x are y" means "the attributes x and y are COMPATIBLE, allowing for a thing to have both at the same time," while "no x are y" indicates "the attributes x and y are INCOMPATIBLE, meaning nothing can have both at the same time."
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.
In these discussions, Propositions mean something completely different than they do in our 'Game of Logic', and it's important to understand exactly what that 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).
First, consider "some x are y." Here, we understand "are" to mean "exist, in reality," which, of course, implies that some x-things actually exist. But they (the authors of those other writings) only interpret "are" as "can exist," which doesn’t imply that any actually exist. So they mean less than we do: our meaning includes theirs (since "some x ARE y" includes "some x CAN BE y"), but theirs does not include ours. For example, "some Welsh hippopotamuses are heavy" would be considered true by these writers (because the attributes "Welsh" and "heavy" can apply to a hippopotamus), but it would be false in our understanding (since there are no Welsh hippopotamuses that can 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).
Secondly, consider "no x are y." Here, we only understand "are" to mean "are, as an actual FACT," which doesn’t imply that no x CAN be y. But they interpret the statement to mean not just that none ARE y, but that none CAN POSSIBLY be y. So their meaning is broader than ours: their interpretation includes ours (because "no x CAN be y" includes "no x ARE y"), but ours doesn’t include theirs. For example, "no Policemen are eight feet high" would be TRUE in our context (since, as a matter of fact, there are no such impressive specimens), but it would be FALSE according to these writers (since the qualities "belonging to the Police Force" and "eight feet high" are completely COMPATIBLE: there’s nothing preventing a Policeman from reaching that height, if sufficiently treated with Rowland's Macassar Oil—which is said to promote HAIR growth when applied to hair, and will of course make a POLICEMAN grow when applied to 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.
Thirdly, consider "all x are y," which includes the two partial statements "some x are y" and "no x are y." Here, the discussions convey LESS than we do in the FIRST part and more than we do in the SECOND. However, the two actions don't offset each other—just like you can't comfort someone for knocking down one of their chimneys by adding an extra doorstep.
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."
If you come across Syllogisms like this, you can handle them quite easily using the system I've given you: just make 'are' mean 'are CAPABLE of being,' and everything will go smoothly. For "some x are y" will turn into "some x are capable of being y," which means "the Attributes x, y are COMPATIBLE." And "no x are y" will change to "no x are capable of being y," meaning "the Attributes x, y are INCOMPATIBLE." Also, "all x are y" will become "some x are capable of being y, and none are capable of being y'," which means "the Attributes x, y are COMPATIBLE, and the Attributes x, y' are INCOMPATIBLE." While using the Diagrams for this system, you should understand that a red counter means "there may POSSIBLY be something in this compartment," and a grey one means "there cannot POSSIBLY be anything in this compartment."
3. Fallacies.
3. Logical 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.
So you think that the main purpose of Logic in real life is to draw Conclusions from valid Premises and to convince yourself that the Conclusions drawn by others are right? I wish that were the case! Society would be much less prone to panics and other misconceptions, and POLITICAL life, in particular, would be completely different if even a majority of the arguments that are spread around the world were correct! But sadly, it’s the opposite. For every ONE reliable Pair of Premises (I mean a Pair that leads to a logical Conclusion) you come across in a newspaper or magazine, you'll probably find FIVE that lead to no Conclusion at all; and even when the Premises are valid, for every ONE instance where the writer reaches a correct Conclusion, there are likely TEN where they reach an incorrect one.
In the first case, you may say "the PREMISSES are fallacious": in the second, "the CONCLUSION is fallacious."
In the first case, you might say "the premises are flawed": in the second, "the conclusion is flawed."
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 main benefit you’ll get from the logical skills this game can teach you is in spotting 'FALLACIES' of these two types.
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,
The first type of Fallacy—'Fallacious Premises'—you'll notice when, after marking them on the larger Diagram, you attempt to transfer the markings to the smaller one. You'll take its four sections, one at a time, and for each one you'll ask, "What mark can I put HERE?"; and in EVERY case, the answer will be "No information!", indicating that there is NO CONCLUSION AT ALL. For example,
"All soldiers are brave;
Some Englishmen are brave.
&there4 Some Englishmen are soldiers."
"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!"
looks strangely like a syllogism, and might easily confuse a less experienced logician. But you won't fall for such a trick! You would simply lay out the premises and then calmly say, "Fallacious premises!": you wouldn’t waste your time asking what conclusion the writer claims to draw—knowing that whatever it is, it has to be wrong. You would be just as safe as that wise mother who said, "Mary, just go up to the nursery, see what the 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.
The other type of Fallacy—'Fallacious Conclusion'—you won't notice until you have examined BOTH Diagrams, read the correct Conclusion, and compared it with the Conclusion that the writer has made.
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:--
But keep in mind, you shouldn’t call it a "FALLACIOUS Conclusion" just because it doesn’t match the correct one exactly: it might be a part of the correct Conclusion, and therefore quite accurate, AS FAR AS IT GOES. In this case, you would simply say, with a sympathetic smile, "DEFECTIVE Conclusion!" Let’s say, for example, you encountered this Syllogism:--
"All unselfish people are generous;
No misers are generous.
&there4 No misers are unselfish."
"Everyone who is selfless is generous;
No stingy people are generous.
&there4 No stingy people are selfless."the Premisses of which might be thus expressed in letters:--
the premises of which could be expressed in letters like this:--
"All x' are m;
No y are m."
"All x's are m;
No y's 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!"
Here, the correct conclusion would be "All x' are y'" (which means "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 is thus PART of "All x' are y'.") Here, you would simply say "DEFECTIVE conclusion!" The same thing would happen if you were in a candy shop, and a little boy came in, dropped down two pence, and proudly walked away with a single penny bun. You would shake your head sadly and say, "Defective conclusion! Poor little kid!" And maybe you would ask the young woman behind the counter if she would let YOU eat the bun that the little boy paid for and left behind: and maybe SHE would respond, "No way!"
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!"
But if, in the example above, the writer had concluded "All misers are selfish" (meaning "All y are x"), this would be going BEYOND his legitimate rights (since it would claim the EXISTENCE of y, which isn’t included in the Premises), and you would rightly 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
Now, when you read other writings on Logic, you'll come across different types of (so-called) 'Fallacies' that are not ALWAYS the case. For example, if you were to present one of these Logicians with the Pair of Premises
"No honest men cheat;
No dishonest men are trustworthy."
"Honest people don’t cheat;
Dishonest people aren’t 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!
and if you were to ask him what conclusion followed, he would probably say, "None at all! Your premises violate two distinct rules and are as fallacious as they could possibly 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 quickly—maybe angry, maybe just scornful: either way, the outcome 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.
"But why is this?" you might ask. "Are you really saying that all these Logicians are wrong?" Not at all, dear Reader! From THEIR perspective, they're completely correct. However, they don't account for all the possible forms of Syllogisms in their system.
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.
They have a kind of anxious fear of attributes that start with a negative term. For example, propositions like "All not-x are y" and "No x are not-y" don’t fit into their system at all. As a result, they’ve excluded a lot of really useful forms due to their anxiety, and they’ve created rules that, while they work for the few forms they accept, are completely unhelpful when you think about 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!
Let's not argue with them, dear Reader! There's enough space in the world for both of us. Let’s calmly embrace our broader perspective; if they decide to ignore all these useful forms and insist, "They aren't Syllogisms at all!" we can only step aside and watch them rush toward their destiny! There’s hardly anything more dangerous to rush into than your fate. You can rush into your potato patch or strawberry patch without causing much trouble; you might even leap onto your balcony (unless it's a new house built by contract without a clerk overseeing the work) and survive the reckless adventure. But once you rush toward your FATE—well, you’ll have to deal with the consequences!
CHAPTER II.
CROSS QUESTIONS.
"The Man in the Wilderness asked of me
'How many strawberries grow in the sea?'"
__________
"The Man in the Wilderness asked me,
'How many strawberries grow in the sea?'"
__________1. Elementary.
Basic.
1. What is an 'Attribute'? Give examples.
1. What is an 'Attribute'? Give examples.
2. When is it good sense to put "is" or "are" between two names? Give examples.
2. When does it make sense to use "is" or "are" between two names? Give examples.
3. When is it NOT good sense? Give examples.
3. When is it NOT common sense? Give examples.
4. When it is NOT good sense, what is the simplest agreement to make, in order to make good sense?
4. When it doesn’t make sense, what’s the easiest agreement to reach in order to make sense?
5. Explain 'Proposition', 'Term', 'Subject', and 'Predicate'. Give examples.
5. Explain 'Proposition', 'Term', 'Subject', and 'Predicate'. Give examples.
6. What are 'Particular' and 'Universal' Propositions? Give examples.
6. What are 'Particular' and 'Universal' Propositions? Give examples.
7. Give a rule for knowing, when we look at the smaller Diagram, what Attributes belong to the things in each compartment.
7. Provide a guideline for understanding, when we look at the smaller Diagram, which Attributes belong to the items in each section.
8. What does "some" mean in Logic? [See pp. 55, 6]
8. What does "some" mean in Logic? [See pp. 55, 6]
9. In what sense do we use the word 'Universe' in this Game?
9. How are we using the term 'Universe' in this Game?
10. What is a 'Double' Proposition? Give examples.
10. What is a 'Double' Proposition? Give examples.
11. When is a class of Things said to be 'exhaustively' divided? Give examples.
11. When is a category of things considered to be ‘exhaustively’ divided? Give examples.
12. Explain the phrase "sitting on the fence."
12. Explain the phrase "sitting on the fence."
13. What two partial Propositions make up, when taken together, "all x are y"?
13. What two partial statements combine to form the idea that "all x are y"?
14. What are 'Individual' Propositions? Give examples.
14. What are 'Individual' Propositions? Give examples.
15. What kinds of Propositions imply, in this Game, the EXISTENCE of their Subjects?
15. What types of statements imply, in this game, the EXISTENCE of their subjects?
16. When a Proposition contains more than two Attributes, these Attributes may in some cases be re-arranged, and shifted from one Term to the other. In what cases may this be done? Give examples.
16. When a Proposition has more than two Attributes, these Attributes can sometimes be rearranged and moved from one Term to another. In what situations can this be done? Provide examples.
__________
__________
Break up each of the following into two partial Propositions:
Break each of the following into two separate propositions:
17. All tigers are fierce.
All tigers are fierce.
18. All hard-boiled eggs are unwholesome.
18. All hard-boiled eggs are unhealthy.
19. I am happy.
I'm happy.
20. John is not at home.
John's not home.
__________
__________
[See pp. 56, 7]
[See pp. 56, 7]
21. Give a rule for knowing, when we look at the larger Diagram, what Attributes belong to the Things contained in each compartment.
21. Provide a guideline for identifying, when we examine the larger Diagram, which Attributes are associated with the Things found in each section.
22. Explain 'Premisses', 'Conclusion', and 'Syllogism'. Give examples.
22. Explain 'Premises', 'Conclusion', and 'Syllogism'. Give examples.
23. Explain the phrases 'Middle Term' and 'Middle Terms'.
23. Explain the terms 'Middle Term' and 'Middle Terms'.
24. In marking a pair of Premisses on the larger Diagram, why is it best to mark NEGATIVE Propositions before AFFIRMATIVE ones?
24. In marking a pair of premises on the larger diagram, why is it better to mark negative propositions before affirmative ones?
25. Why is it of no consequence to us, as Logicians, whether the Premisses are true or false?
25. Why does it not matter to us, as Logicians, if the Premises are true or false?
26. How can we work Syllogisms in which we are told that "some x are y" is to be understood to mean "the Attribute x, y are COMPATIBLE", and "no x are y" to mean "the Attributes x, y are INCOMPATIBLE"?
26. How can we use syllogisms when we understand that "some x are y" means "the attributes x and y are compatible," and "no x are y" means "the attributes x and y are incompatible"?
27. What are the two kinds of 'Fallacies'?
27. What are the two types of 'Fallacies'?
28. How may we detect 'Fallacious Premisses'?
28. How can we identify 'Fallacious Premises'?
29. How may we detect a 'Fallacious Conclusion'?
29. How can we spot a 'Fallacious Conclusion'?
30. Sometimes the Conclusion, offered to us, is not identical with the correct Conclusion, and yet cannot be fairly called 'Fallacious'. When does this happen? And what name may we give to such a Conclusion?
30. Sometimes the conclusion presented to us is not the same as the correct conclusion, and yet it can't be fairly labeled as 'fallacious'. When does this occur? And what term can we use for such a conclusion?
[See pp. 57-59]
[See pages 57-59]
2. Half of Smaller Diagram.
Half of the Smaller Diagram.
Propositions to be represented.
Proposals to be represented.
-----------
| | |
| x |
| | |
--y-----y'-
Here is the paragraph:
-----------
| | |
| x |
| | |
--y-----y'-__________
__________
1. Some x are not-y.
Some x aren't y.
2. All x are not-y.
Not all x are y.
3. Some x are y, and some are not-y.
3. Some x are y, and some are not y.
4. No x exist.
4. No x exists.
5. Some x exist.
Some x are real.
6. No x are not-y.
6. No x are not-y.
7. Some x are not-y, and some x exist.
7. Some x aren’t y, and some x exist.
__________
__________
Taking x="judges"; y="just";
Taking x="judges"; y="fair";
8. No judges are just.
8. No judges are fair.
9. Some judges are unjust.
Some judges are unfair.
10. All judges are just.
All judges are fair.
__________
__________
Taking x="plums"; y="wholesome";
Taking x="plums"; y="wholesome";
11. Some plums are wholesome.
Some plums are nutritious.
12. There are no wholesome plums.
12. There are no good plums.
13. Plums are some of them wholesome, and some not.
13. Some plums are healthy, and some are not.
14. All plums are unwholesome.
All plums are unhealthy.
[See pp. 59, 60]
[See pp. 59, 60]
-----
| |
| x
| |
|--y--|
| |
| x'
| |
-----
-----
| |
| x
| |
|--y--|
| |
| x'
| |
-----__________
__________
Taking y="diligent students"; x="successful";
Diligent students are successful.
15. No diligent students are unsuccessful.
No dedicated students fail.
16. All diligent students are successful.
16. All hardworking students are successful.
17. No students are diligent.
No students are hardworking.
18. There are some diligent, but unsuccessful, students.
18. There are some hardworking, but unsuccessful, students.
19. Some students are diligent.
Some students are hard-working.
[See pp. 60, 1]
[See pp. 60, 1]
3. Half of Smaller Diagram.
Half of Small Diagram.
Symbols to be interpreted.
Symbols to interpret.
__________
__________
-----------
| | |
| x |
| | |
--y-----y'-
-----------
| | |
| x |
| | |
--y-----y'-
__________
__________
------- -------
| | | | | |
1. | | 0 | 2. | 0 | 0 |
| | | | | |
------- -------
------- -------
| | | | | |
1. | | 0 | 2. | 0 | 0 |
| | | | | |
------- ------- ------- -------
| | | | | |
3. | - | 4. | 0 | 1 |
| | | | | |
------- -------
------- -------
| | | | | |
3. | - | 4. | 0 | 1 |
| | | | | |
------- -------__________
__________
Taking x="good riddles"; y="hard";
good riddles
------- -------
| | | | | |
5. | 1 | | 6. | 1 | 0 |
| | | | | |
------- -------
------- -------
| | | | | |
5. | 1 | | 6. | 1 | 0 |
| | | | | |
------- ------- ------- -------
| | | | | |
7. | 0 | 0 | 8. | 0 | |
| | | | | |
------- -------
------- -------
| | | | | |
7. | 0 | 0 | 8. | 0 | |
| | | | | |
------- -------__________
__________
[See pp. 61, 2]
[See pp. 61, 2]
Taking x="lobster"; y="selfish";
Taking x="lobster"; y="selfish";
------- -------
| | | | | |
9. | | 1 | 10. | 0 | |
| | | | | |
------- -------
------- -------
| | | | | |
9. | | 1 | 10. | 0 | |
| | | | | |
------- ------- ------- -------
| | | | | |
11. | 0 | 1 | 12. | 1 | 1 |
| | | | | |
------- -------
------- -------
| | | | | |
11. | 0 | 1 | 12. | 1 | 1 |
| | | | | |
------- -------__________
__________
-----
| |
x |
| |
|--y'-|
| |
x' |
| |
-----
-----
| |
x |
| |
|--y'-|
| |
x' |
| |
-----Taking y="healthy people"; x="happy";
Taking y="healthy people"; x="happy";
--- --- --- ---
| 0 | | | | 1 | | 0 |
13. |---| 14. |-1-| 15. |---| 16. |---|
| 1 | | | | 1 | | |
--- --- --- ---
--- --- --- ---
| 0 | | | | 1 | | 0 |
13. |---| 14. |-1-| 15. |---| 16. |---|
| 1 | | | | 1 | | |
--- --- --- ---[See p. 62]
[See p. 62]
4. Smaller Diagram.
4. Small Diagram.
Propositions to be represented.
Proposals to be represented.
-----------
| | |
| x |
|--y--|--y'-|
| x' |
| | |
-----------
-----------
| | |
| x |
|--y--|--y'-|
| x' |
| | |
-----------__________
__________
1. All y are x.
All of you are x.
2. Some y are not-x.
Some y are not-x.
3. No not-x are not-y.
No not-x are not-y.
4. Some x are not-y.
4. Some x aren't y.
5. Some not-y are x.
Some are not y.
6. No not-x are y.
No not-x are y.
7. Some not-x are not-y.
Some non-x are non-y.
8. All not-x are not-y.
All not-x aren't y.
9. Some not-y exist.
Some non-binary people exist.
10. No not-x exist.
No non-x exist.
11. Some y are x, and some are not-x.
11. Some y are x, and some are not x.
12. All x are y, and all not-y are not-x.
12. All x are y, and everything that isn't y isn't x.
[See pp. 62, 3]
[See pp. 62, 3]
Taking "nations" as Universe; x="civilised"; y="warlike";
Taking "nations" as the Universe; x="civilized"; y="militant";
13. No uncivilised nation is warlike.
13. No uncivilized nation is aggressive.
14. All unwarlike nations are uncivilised.
14. All peaceful nations are uncivilized.
15. Some nations are unwarlike.
Some countries are peaceful.
16. All warlike nations are civilised, and all civilised nations are warlike.
16. All militaristic nations are civilized, and all civilized nations are militaristic.
17. No nation is uncivilised.
No nation is uncivilized.
__________
__________
Taking "crocodiles" as Universe; x="hungry"; and y="amiable";
Taking "crocodiles" as the Universe; x="hungry"; and y="friendly";
18. All hungry crocodiles are unamiable.
18. All hungry crocodiles are unfriendly.
19. No crocodiles are amiable when hungry.
19. No crocodiles are friendly when they're hungry.
20. Some crocodiles, when not hungry, are amiable; but some are not.
20. Some crocodiles are friendly when they're not hungry, but some aren't.
21. No crocodiles are amiable, and some are hungry.
21. No crocodiles are friendly, and some are aggressive.
22. All crocodiles, when not hungry, are amiable; and all unamiable crocodiles are hungry.
22. All crocodiles are friendly when they're not hungry, and any crocodile that's unfriendly is hungry.
23. Some hungry crocodiles are amiable, and some that are not hungry are unamiable.
23. Some hungry crocodiles are friendly, while some that aren't hungry are unfriendly.
[See pp. 63, 4]
[See pp. 63, 4]
5. Smaller Diagram.
Smaller Diagram.
Symbols to be interpreted.
Symbols to interpret.
__________
__________
-----------
| | |
| x |
|--y--|--y'-|
| x' |
| | |
-----------
-----------
| | |
| x |
|--y--|--y'-|
| x' |
| | |
-----------
__________
__________
------- -------
| | | | | |
1. |---|---| 2. |---|---|
| 1 | | | | 0 |
------- -------
------- -------
| | | | | |
1. |---|---| 2. |---|---|
| 1 | | | | 0 |
------- ------- ------- -------
| | 1 | | | |
3. |---|---| 4. |---|---|
| | 0 | | 0 | 0 |
------- -------
------- -------
| | 1 | | | |
3. |---|---| 4. |---|---|
| | 0 | | 0 | 0 |
------- -------__________
__________
Taking "houses" as Universe; x="built of brick"; and y="two-storied"; interpret
Taking "houses" as the universe; x="made of brick"; and y="two stories"; interpret.
------- -------
| 0 | | | | |
5. |---|---| 6. |---|---|
| 0 | | | - |
------- ---|---
------- -------
| 0 | | | | |
5. |---|---| 6. |---|---|
| 0 | | | - |
------- ---|--- ------- -------
| | 0 | | | |
7. |---|---| 8. |---|---|
| | | | 0 | 1 |
------- -------
------- -------
| | 0 | | | |
7. |---|---| 8. |---|---|
| | | | 0 | 1 |
------- -------[See p. 65]
[See p. 65]
Taking "boys" as Universe; x="fat"; and y="active"; interpret
Taking "boys" as the Universe; x="overweight"; and y="energetic"; interpret
------- -------
| 1 | 1 | | | 0 |
9. |---|---| 10. |---|---|
| | | | | 1 |
------- -------
------- -------
| 1 | 1 | | | 0 |
9. |---|---| 10. |---|---|
| | | | | 1 |
------- ------- ------- -------
| 0 | 1 | | 1 | |
11. |---|---| 12. |---|---|
| | 0 | | 0 | 1 |
------- -------
------- -------
| 0 | 1 | | 1 | |
11. |---|---| 12. |---|---|
| | 0 | | 0 | 1 |
------- -------__________
__________
Taking "cats" as Universe; x="green-eyed"; and y="good-tempered"; interpret
Taking "cats" as the Universe; x="green-eyed"; and y="good-natured"; interpret
------- -------
| 0 | 0 | | | 1 |
13. |---|---| 14. |---|---|
| | 0 | | 1 | |
------- -------
------- -------
| 0 | 0 | | | 1 |
13. |---|---| 14. |---|---|
| | 0 | | 1 | |
------- ------- ------- -------
| 1 | | | 0 | 1 |
15. |---|---| 16. |---|---|
| | 0 | | 1 | 0 |
------- -------
------- -------
| 1 | | | 0 | 1 |
15. |---|---| 16. |---|---|
| | 0 | | 1 | 0 |
------- -------[See pp. 65, 6]
[See pp. 65, 6]
6. Larger Diagram.
6. Bigger Diagram.
Propositions to be represented.
Proposals to be represented.
__________
__________
-----------
| | |
| --x-- |
| | | | |
|--y--m--y'-|
| | | | |
| --x'- |
| | |
-----------
-----------
| | |
| --x-- |
| | | | |
|--y--m--y'-|
| | | | |
| --x'- |
| | |
-----------__________
__________
1. No x are m.
No x are m.
2. Some y are m'.
Some y are m'.
3. All m are x'.
All m are x's.
4. No m' are y'.
No more, are you?
5. No m are x; All y are m.
5. No m are x; All y are m.
6. Some x are m; No y are m.
6. Some x are m; No y are m.
7. All m are x'; No m are y.
7. All m are x; No m are y.
8. No x' are m; No y' are m'.
8. No x are m; No y are m'.
[See pp. 67,8]
[See pp. 67, 68]
Taking "rabbits" as Universe; m="greedy"; x="old"; and y="black"; represent
Taking "rabbits" as the Universe; m="greedy"; x="old"; and y="black"; represent
9. No old rabbits are greedy.
9. No old rabbits are greedy.
10. Some not-greedy rabbits are black.
10. Some non-greedy rabbits are black.
11. All white rabbits are free from greediness.
11. All white rabbits are not greedy.
12. All greedy rabbits are young.
12. All greedy rabbits are young.
13. No old rabbits are greedy; All black rabbits are greedy.
13. No old rabbits are greedy; all black rabbits are greedy.
14. All rabbits, that are not greedy, are black; No old rabbits are free from greediness.
14. All rabbits that aren't greedy are black; no old rabbits are free from greed.
__________
__________
Taking "birds" as Universe; m="that sing loud"; x="well-fed"; and y="happy"; represent
Taking "birds" as the Universe; m="that sing loudly"; x="well-fed"; and y="happy"; represent
15. All well-fed birds sing loud; No birds, that sing loud, are unhappy.
15. All well-fed birds sing loudly; no birds that sing loudly are unhappy.
16. All birds, that do not sing loud, are unhappy; No well-fed birds fail to sing loud.
16. All birds that don't sing loudly are unhappy; no well-fed birds fail to sing loudly.
__________
__________
Taking "persons" as Universe; m="in the house"; x="John"; and y="having a tooth-ache"; represent
Taking "people" as Universe; m="in the house"; x="John"; and y="having a toothache"; represent
17. John is in the house; Everybody in the house is suffering from tooth-ache.
17. John is in the house; everyone in the house is suffering from a toothache.
18. There is no one in the house but John; Nobody, out of the house, has a tooth-ache.
18. There's no one in the house except John; Nobody outside the house has a toothache.
__________
__________
[See pp. 68-70]
[See pages 68-70]
Taking "persons" as Universe; m="I"; x="that has taken a walk"; y="that feels better"; represent
Taking "people" as Universe; m="I"; x="that has taken a walk"; y="that feels better"; represent
19. I have been out for a walk; I feel much better.
19. I went out for a walk; I feel way better.
__________
__________
Choosing your own 'Universe' &c., represent
Choosing your own 'Universe' &c., represent
20. I sent him to bring me a kitten; He brought me a kettle by mistake.
20. I sent him to get me a kitten; he accidentally brought me a kettle.
[See pp. 70, 1]
[See pp. 70, 1]
7. Both Diagrams to be employed.
Use both charts.
__________
__________
-----------
| | | -----------
| --x-- | | | |
| | | | | | x |
|--y--m--y'-| |--y--|--y'-|
| | | | | | x' |
| --x'- | | | |
| | | -----------
-----------
-----------
| | | -----------
| --x-- | | | |
| | | | | | x |
|--y--m--y'-| |--y--|--y'-|
| | | | | | x' |
| --x'- | | | |
| | | -----------
-----------__________
__________
N.B. In each Question, a small Diagram should be drawn, for x and y only, and marked in accordance with the given large Diagram: and then as many Propositions as possible, for x and y, should be read off from this small Diagram.
N.B. In each Question, a small Diagram should be drawn for x and y only, and labeled according to the provided large Diagram; then as many Propositions as possible for x and y should be derived from this small Diagram.
----------- -----------
|0 | | | | |
| --|-- | | --|-- |
| |0 | 0| | | |0 | 1| |
1. |--|--|--|--| 2. |--|--|--|--|
| |1 | | | | |0 | | |
| --|-- | | --|-- |
|0 | | | | |
----------- -----------
----------- -----------
|0 | | | | |
| --|-- | | --|-- |
| |0 | 0| | | |0 | 1| |
1. |--|--|--|--| 2. |--|--|--|--|
| |1 | | | | |0 | | |
| --|-- | | --|-- |
|0 | | | | |
----------- -----------[See p. 72]
[See p. 72]
----------- -----------
| | | | | 0|
| --|-- | | --|-- |
| |0 | 0| | | | | | |
3. |--|--|--|--| 4. |--|--|--|--|
| |1 | 0| | | |0 | | |
| --|-- | | --|-- |
| | | | | 0|
----------- -----------
----------- -----------
| | | | | 0|
| --|-- | | --|-- |
| |0 | 0| | | | | | |
3. |--|--|--|--| 4. |--|--|--|--|
| |1 | 0| | | |0 | | |
| --|-- | | --|-- |
| | | | | 0|
----------- -----------__________
__________
Mark, in a large Diagram, the following pairs of Propositions from the preceding Section: then mark a small Diagram in accordance with it, &c.
Mark, in a large diagram, the following pairs of propositions from the previous section; then mark a small diagram based on it, etc.
5. No. 13. [see p. 49] 9. No. 17.
6. No. 14. 10. No. 18.
7. No. 15. 11. No. 19. [see p. 50]
8. No. 16. 12. No. 20.
5. No. 13. [see p. 49] 9. No. 17.
6. No. 14. 10. No. 18.
7. No. 15. 11. No. 19. [see p. 50]
8. No. 16. 12. No. 20.
__________
__________
Mark, on a large Diagram, the following Pairs of Propositions: then mark a small Diagram, &c. These are, in fact, Pairs of PREMISSES for Syllogisms: and the results, read off from the small Diagram, are the CONCLUSIONS.
Mark, on a large diagram, the following pairs of propositions; then mark a small diagram, etc. These are, in fact, pairs of premises for syllogisms, and the results read from the small diagram are the conclusions.
13. No exciting books suit feverish patients; Unexciting books make one drowsy.
13. No thrilling books are suitable for patients with fevers; boring books just make you sleepy.
14. Some, who deserve the fair, get their deserts; None but the brave deserve the fair.
14. Some people who deserve a good outcome get what they deserve; only the brave truly deserve the good things.
15. No children are patient; No impatient person can sit still.
15. No kids are patient; no impatient person can sit still.
[See pp. 72-5]
[See pp. 72-75]
16. All pigs are fat; No skeletons are fat.
16. All pigs are overweight; No skeletons are overweight.
17. No monkeys are soldiers; All monkeys are mischievous.
17. No monkeys are soldiers; all monkeys are troublemakers.
18. None of my cousins are just; No judges are unjust.
18. None of my cousins are fair; no judges are unfair.
19. Some days are rainy; Rainy days are tiresome.
19. Some days are rainy; rainy days are exhausting.
20. All medicine is nasty; Senna is a medicine.
20. All medicine is unpleasant; Senna is a medicine.
21. Some Jews are rich; All Patagonians are Gentiles.
21. Some Jews are wealthy; all Patagonians are non-Jews.
22. All teetotalers like sugar; No nightingale drinks wine.
22. All non-drinkers like sugar; No nightingale drinks wine.
23. No muffins are wholesome; All buns are unwholesome.
23. No muffins are healthy; all buns are unhealthy.
24. No fat creatures run well; Some greyhounds run well.
24. No overweight animals run well; some greyhounds run great.
25. All soldiers march; Some youths are not soldiers.
25. All soldiers march; Some young people are not soldiers.
26. Sugar is sweet; Salt is not sweet.
26. Sugar is sweet; salt isn’t sweet.
27. Some eggs are hard-boiled; No eggs are uncrackable.
27. Some eggs are hard-boiled; no eggs are uncrackable.
28. There are no Jews in the house; There are no Gentiles in the garden.
28. There are no Jews in the house; there are no non-Jews in the garden.
[See pp. 75-82]
[See pages 75-82]
29. All battles are noisy; What makes no noise may escape notice.
29. All battles are loud; what doesn't make a sound might go unnoticed.
30. No Jews are mad; All Rabbis are Jews.
30. No Jews are crazy; All Rabbis are Jews.
31. There are no fish that cannot swim; Some skates are fish.
31. There are no fish that can’t swim; some skates are fish.
32. All passionate people are unreasonable; Some orators are passionate.
32. All passionate people are unreasonable; some speakers are passionate.
[See pp. 82-84]
[See pp. 82-84]
CHAPTER III.
CROOKED ANSWERS.
"I answered him, as I thought good,
'As many as red-herrings grow in the wood'."
"I responded to him, as I thought best,
'As many as red herrings grow in the woods'."
__________
__________
1. Elementary.
Basic.
1. Whatever can be "attributed to", that is "said to belong to", a Thing, is called an 'Attribute'. For example, "baked", which can (frequently) be attributed to "Buns", and "beautiful", which can (seldom) be attributed to "Babies".
1. Anything that can be "attributed to," or "said to belong to," a thing is called an 'Attribute.' For example, "baked," which can often be attributed to "Buns," and "beautiful," which can rarely be attributed to "Babies."
2. When they are the Names of two Things (for example, "these Pigs are fat Animals"), or of two Attributes (for example, "pink is light red").
2. When they refer to two things (for example, "these pigs are fat animals"), or two attributes (for example, "pink is light red").
3. When one is the Name of a Thing, and the other the Name of an Attribute (for example, "these Pigs are pink"), since a Thing cannot actually BE an Attribute.
3. When one is the Name of a Thing and the other is the Name of an Attribute (for example, "these pigs are pink"), since a Thing cannot actually BE an Attribute.
4. That the Substantive shall be supposed to be repeated at the end of the sentence (for example, "these Pigs are pink (Pigs)").
4. That the noun should be assumed to be repeated at the end of the sentence (for example, "these pigs are pink (pigs)").
5. A 'Proposition' is a sentence stating that some, or none, or all, of the Things belonging to a certain class, called the 'Subject', are also Things belonging to a certain other class, called the 'Predicate'. For example, "some new Cakes are not nice", that is (written in full) "some new Cakes are not nice Cakes"; where the class "new Cakes" is the Subject, and the class "not-nice Cakes" is the Predicate.
5. A 'Proposition' is a statement that indicates whether some, none, or all of the items in a specific group, known as the 'Subject', also belong to another group, referred to as the 'Predicate'. For example, "some new Cakes are not nice," which can be expressed fully as "some new Cakes are not nice Cakes"; here, the group "new Cakes" is the Subject, and the group "not-nice Cakes" is the Predicate.
6. A Proposition, stating that SOME of the Things belonging to its Subject are so-and-so, is called 'Particular'. For example, "some new Cakes are nice", "some new Cakes are not nice."
6. A proposition that says SOME of the things related to its subject are a certain way is called 'Particular'. For instance, "some new cakes are nice," "some new cakes are not nice."
A Proposition, stating that NONE of the Things belonging to its Subject, or that ALL of them, are so-and-so, is called 'Universal'. For example, "no new Cakes are nice", "all new Cakes are not nice".
A proposition that claims NONE of the things related to its subject, or that ALL of them, are this way or that, is called 'Universal.' For example, "no new cakes are nice," "all new cakes are not nice."
7. The Things in each compartment possess TWO Attributes, whose symbols will be found written on two of the EDGES of that compartment.
7. The items in each compartment have TWO Attributes, and their symbols can be found written on two of the EDGES of that compartment.
8. "One or more."
"One or more."
9. As a name of the class of Things to which the whole Diagram is assigned.
9. As the name for the category of things that the entire diagram is associated with.
10. A Proposition containing two statements. For example, "some new Cakes are nice and some are not-nice."
10. A proposition that has two statements. For example, "some new cakes are nice and some are not nice."
11. When the whole class, thus divided, is "exhausted" among the sets into which it is divided, there being no member of it which does not belong to some one of them. For example, the class "new Cakes" is "exhaustively" divided into "nice" and "not-nice" since EVERY new Cake must be one or the other.
11. When the entire class is divided up like this, it's "exhausted" among the subsets that it's split into, with no member left out of one of them. For instance, the class "new Cakes" is "exhaustively" divided into "nice" and "not-nice" since EVERY new Cake has to fall into one of those categories.
12. When a man cannot make up his mind which of two parties he will join, he is said to be "sitting on the fence"--not being able to decide on which side he will jump down.
12. When a guy can't decide which of two groups to join, he's said to be "sitting on the fence"—unable to choose which side he'll jump down on.
13. "Some x are y" and "no x are y'".
13. "Some x are y" and "no x are y."
14. A Proposition, whose Subject is a single Thing, is called 'Individual'. For example, "I am happy", "John is not at home". These are Universal Propositions, being the same as "all the I's that exist are happy", "ALL the Johns, that I am now considering, are not at home".
14. A proposition that focuses on a single thing is called 'Individual.' For example, "I am happy," "John is not at home." These are Universal Propositions, equivalent to "all the I's that exist are happy," "ALL the Johns that I'm considering right now are not at home."
15. Propositions beginning with "some" or "all".
15. Propositions that start with "some" or "all".
16. When they begin with "some" or "no". For example, "some abc are def" may be re-arranged as "some bf are acde", each being equivalent to "some abcdef exist".
16. When they start with "some" or "no". For example, "some abc are def" can be re-arranged as "some bf are acde", both meaning "some abcdef exist".
17. Some tigers are fierce, No tigers are not-fierce.
17. Some tigers are fierce, and no tigers are not fierce.
18. Some hard-boiled eggs are unwholesome, No hard-boiled eggs are wholesome.
18. Some hard-boiled eggs are unhealthy, No hard-boiled eggs are healthy.
19. Some I's are happy, No I's are unhappy.
19. Some I's are happy; no I's are unhappy.
20. Some Johns are not at home, No Johns are at home.
20. Some Johns aren't home, No Johns are home.
21. The Things, in each compartment of the larger Diagram, possess THREE Attributes, whose symbols will be found written at three of the CORNERS of the compartment (except in the case of m', which is not actually inserted in the Diagram, but is SUPPOSED to stand at each of its four outer corners).
21. The Things in each section of the larger Diagram have THREE Attributes, represented by symbols located at three of the CORNERS of the section (except for m', which isn't actually included in the Diagram but is SUPPOSED to be at each of its four outer corners).
22. If the Universe of Things be divided with regard to three different Attributes; and if two Propositions be given, containing two different couples of these Attributes; and if from these we can prove a third Proposition, containing the two Attributes that have not yet occurred together; the given Propositions are called 'the Premisses', the third one 'the Conclusion', and the whole set 'a Syllogism'. For example, the Premisses might be "no m are x'" and "all m' are y"; and it might be possible to prove from them a Conclusion containing x and y.
22. If we divide the Universe of Things based on three different Attributes, and if we have two Propositions that include two different pairs of these Attributes, and if we can prove a third Proposition that contains the two Attributes that haven't been paired up yet, the given Propositions are referred to as 'the Premises', the third one as 'the Conclusion', and the entire set as 'a Syllogism'. For example, the Premises could be "no m are x" and "all m are y"; from these, it might be possible to prove a Conclusion that includes x and y.
23. If an Attribute occurs in both Premisses, the Term containing it is called 'the Middle Term'. For example, if the Premisses are "some m are x" and "no m are y'", the class of "m-Things" is 'the Middle Term.'
23. If an attribute appears in both premises, the term that includes it is referred to as 'the Middle Term.' For instance, if the premises are "some m are x" and "no m are y," the category of "m-Things" is 'the Middle Term.'
If an Attribute occurs in one Premiss, and its contradictory in the other, the Terms containing them may be called 'the Middle Terms'. For example, if the Premisses are "no m are x'" and "all m' are y", the two classes of "m-Things" and "m'-Things" may be called 'the Middle Terms'.
If an attribute appears in one premise and its opposite appears in the other, the terms containing them can be referred to as 'the Middle Terms'. For example, if the premises are "no m are x" and "all m' are y", the two classes of "m-Things" and "m'-Things" can be called 'the Middle Terms'.
24. Because they can be marked with CERTAINTY: whereas AFFIRMATIVE Propositions (that is, those that begin with "some" or "all") sometimes require us to place a red counter 'sitting on a fence'.
24. Because they can be marked with CERTAINTY: while AFFIRMATIVE Propositions (those that start with "some" or "all") sometimes need us to place a red counter 'sitting on a fence'.
25. Because the only question we are concerned with is whether the Conclusion FOLLOWS LOGICALLY from the Premisses, so that, if THEY were true, IT also would be true.
25. Because the only question we care about is whether the Conclusion logically follows from the Premises, so that if they were true, it would also be true.
26. By understanding a red counter to mean "this compartment CAN be occupied", and a grey one to mean "this compartment CANNOT be occupied" or "this compartment MUST be empty".
26. By understanding a red counter to mean "this compartment CAN be occupied," and a grey one to mean "this compartment CANNOT be occupied" or "this compartment MUST be empty."
27. 'Fallacious Premisses' and 'Fallacious Conclusion'.
27. 'False Premises' and 'False Conclusion'.
28. By finding, when we try to transfer marks from the larger Diagram to the smaller, that there is 'no information' for any of its four compartments.
28. By discovering, when we attempt to move marks from the larger diagram to the smaller one, that there is 'no information' for any of its four sections.
29. By finding the correct Conclusion, and then observing that the Conclusion, offered to us, is neither identical with it nor a part of it.
29. By identifying the right Conclusion, and then noting that the Conclusion presented to us is neither the same as it nor a part of it.
30. When the offered Conclusion is PART of the correct Conclusion. In this case, we may call it a 'Defective Conclusion'.
30. When the offered Conclusion is a PART of the correct Conclusion. In this case, we can call it a 'Defective Conclusion'.
2. Half of Smaller Diagram.
Half of Small Diagram.
Propositions represented.
Propositions displayed.
__________
__________
------- -------
| | | | | |
1. | | 1 | 2. | 0 | 1 |
| | | | | |
------- -------
Here is the paragraph:
------- -------
| | | | | |
1. | | 1 | 2. | 0 | 1 |
| | | | | |
------- ------- ------- -------
| | | | | |
3. | 1 | 1 | 4. | 0 | 0 |
| | | | | |
------- -------
------- -------
| | | | | |
3. | 1 | 1 | 4. | 0 | 0 |
| | | | | |
------- ------- ------- -------
| | | | | |
5. | 1 | 6. | | 0 |
| | | | | |
------- -------
------- -------
| | | | | |
5. | 1 | 6. | | 0 |
| | | | | |
------- ------- -------
| | |
7. | 1 | 1 | It might be thought that the proper
| | |
------- -------
| | |
Diagram would be | 1 1 |, in order to express "some
| | |
-------
x exist": but this is really contained in "some x are y'."
To put a red counter on the division-line would only tell
us "ONE OF THE compartments is occupied", which we
know already, in knowing that ONE is occupied.
-------
| | |
8. No x are y. i.e. | 0 | |
| | |
-------
-------
| | |
7. | 1 | 1 | It might be thought that the correct
| | |
------- -------
| | |
The diagram should be | 1 1 |, to indicate "some
| | |
-------
x exists": but this is actually included in "some x are y."
Putting a red counter on the division line would only show
us "ONE OF THE compartments is filled," which we
already know by recognizing that ONE is filled.
-------
| | |
8. No x are y. i.e. | 0 | |
| | |
------- -------
| | |
9. Some x are y'. i.e. | | 1 |
| | |
-------
-------
| | |
9. Some x are y'. i.e. | | 1 |
| | |
------- -------
| | |
10. All x are y. i.e. | 1 | 0 |
| | |
-------
-------
| | |
10. Every x is a y. i.e. | 1 | 0 |
| | |
------- -------
| | |
11. Some x are y. i.e. | 1 | |
| | |
-------
-------
| | |
11. Some x are y. i.e. | 1 | |
| | |
------- -------
| | |
12. No x are y. i.e. | 0 | |
| | |
-------
-------
| | |
12. No x are y. i.e. | 0 | |
| | |
------- -------
| | |
13. Some x are y, and some are y'. i.e. | 1 | 1 |
| | |
-------
-------
| | |
13. Some x are y, and some are y'. i.e. | 1 | 1 |
| | |
------- -------
| | |
14. All x are y'. i.e. | 0 | 1 |
| | |
-------
-------
| | |
14. All x are y'. i.e. | 0 | 1 |
| | |
------- ---
| |
15. No y are x'. i.e. |---|
| 0 |
---
15. No you're x'. i.e. |---| 0 | ---
---
| 1 |
16. All y are x. i.e. |---|
| 0 |
---
---
| 1 |
16. All of you are x. i.e. |---|
| 0 |
--- ---
| 0 |
17. No y exist. i.e. |---|
| 0 |
---
No y exists. i.e. |---|
---
| |
18. Some y are x'. i.e. |---|
| 1 |
---
---
| |
18. Some y are x's. i.e. |---|
| 1 |
--- ---
| |
15. Some y exist. i.e. |-1-|
| |
---
---
| |
15. Some y exists. i.e. |-1-|
| |
---3. Half of Smaller Diagram.
Half of Smaller Diagram.
Symbols interpreted.
Symbols decoded.
__________
__________
1. No x are y'.
No x are y.
2. No x exist.
No x exists.
3. Some x exist.
Some x are real.
4. All x are y'.
All x are y.
5. Some x are y. i.e. Some good riddles are hard.
5. Some x are y. i.e. Some good riddles are difficult.
6. All x are y. i.e. All good riddles are hard.
6. All x are y. i.e. All good riddles are tough.
7. No x exist. i.e. No riddles are good.
7. No x exists. In other words, no riddles are good.
8. No x are y. i.e. No good riddles are hard.
8. No x are y. i.e. No good riddles are difficult.
9. Some x are y'. i.e. Some lobsters are unselfish.
9. Some x are y'. i.e. Some lobsters are unselfish.
10. No x are y. i.e. No lobsters are selfish.
10. No x are y. i.e. No lobsters are selfish.
11. All x are y'. i.e. All lobsters are unselfish.
11. All x are y. i.e. All lobsters are generous.
12. Some x are y, and some are y'. i.e. Some lobsters are selfish, and some are unselfish.
12. Some x are y, and some are y'. For example, some lobsters are selfish, and some are unselfish.
13. All y' are x'. i.e. All invalids are unhappy.
13. All of you are x'. i.e. All invalids are unhappy.
14. Some y' exist. i.e. Some people are unhealthy.
14. Some people exist. i.e. Some people are unhealthy.
15. Some y' are x, and some are x'. i.e. Some invalids are happy, and some are unhappy.
15. Some of you are x, and some are x'. i.e. Some people with disabilities are happy, and some are unhappy.
16. No y' exist. i.e. Nobody is unhealthy.
Everyone is healthy.
4. Smaller Diagram.
4. Mini Diagram.
Propositions represented.
Propositions expressed.
__________
__________
------- -------
| 1 | | | | |
1. |---|---| 2. |---|---|
| 0 | | | 1 | |
------- -------
------- -------
| 1 | | | | |
1. |---|---| 2. |---|---|
| 0 | | | 1 | |
------- ------- ------- -------
| | | | | 1 |
3. |---|---| 4. |---|---|
| | 0 | | | |
------- -------
------- -------
| | | | | 1 |
3. |---|---| 4. |---|---|
| | 0 | | | |
------- ------- ------- -------
| | 1 | | | |
5. |---|---| 6. |---|---|
| | | | 0 | |
------- -------
------- -------
| | 1 | | | |
5. |---|---| 6. |---|---|
| | | | 0 | |
------- ------- ------- -------
| | | | | |
7. |---|---| 8. |---|---|
| | 1 | | 0 | 1 |
------- -------
------- -------
| | | | | |
7. |---|---| 8. |---|---|
| | 1 | | 0 | 1 |
------- ------- ------- -------
| | | | | |
9. |---|-1-| 10. |---|---|
| | | | 0 | 0 |
------- -------
------- -------
| | | | | |
9. |---|-1-| 10. |---|---|
| | | | 0 | 0 |
------- ------- ------- -------
| 1 | | | 1 | 0 |
11. |---|---| 12. |---|---|
| 1 | | | | 1 |
------- -------
------- -------
| 1 | | | 1 | 0 |
11. |---|---| 12. |---|---|
| 1 | | | | 1 |
------- ------- -------
| | |
13. No x' are y. i.e. |---|---|
| 0 | |
-------
-------
| | |
13. No x's are y's. i.e. |---|---|
| 0 | |
------- -------
| | 0 |
14. All y' are x'. i.e. |---|---|
| | 1 |
-------
-------
| | 0 |
14. All of you are x'. i.e. |---|---|
| | 1 |
------- -------
| | |
15. Some y' exist. i.e. |---|-1-|
| | |
-------
-------
| | |
15. Some of you exist. i.e. |---|-1-|
| | |
------- -------
| 1 | 0 |
16. All y are x, and all x are y. i.e. |---|---|
| 0 | |
-------
-------
| 1 | 0 |
16. All y are x, and all x are y. i.e. |---|---|
| 0 | |
------- -------
| | |
17. No x' exist. i.e. |---|---|
| 0 | 0 |
-------
-------
| | |
17. No x' exist. i.e. |---|---|
| 0 | 0 |
------- -------
| 0 | 1 |
18. All x are y'. i.e. |---|---|
| | |
-------
-------
| 0 | 1 |
18. All x are y'. i.e. |---|---|
| | |
------- -------
| 0 | |
19. No x are y. i.e. |---|---|
| | |
-------
-------
| 0 | |
19. No x are y. That is, |---|---|
| | |
------- -------
| | |
20. Some x' are y, and some are y'. i.e. |---|---|
| 1 | 1 |
-------
-------
| | |
20. Some x are y, and some are y'. i.e. |---|---|
| 1 | 1 |
------- -------
| 0 | 1 |
21. No y exist, and some x exist. i.e. |---|---|
| 0 | |
-------
-------
| 0 | 1 |
21. No y exist, and some x exist. i.e. |---|---|
| 0 | |
------- -------
| | 1 |
22. All x' are y, and all y' are x. i.e. |---|---|
| 1 | 0 |
-------
-------
| | 1 |
22. All x' are y, and all y' are x. i.e. |---|---|
| 1 | 0 |
------- -------
| 1 | |
17. Some x are y, and some x' are y'. i.e. |---|---|
| | 1 |
-------
Here is the paragraph:
-------
| 1 | |
17. Some x are y, and some x' are y'. i.e. |---|---|
| | 1 |
-------5. Smaller Diagram.
Small Diagram.
Symbols interpreted.
Symbols decoded.
__________
__________
1. Some y are not-x, or, Some not-x are y.
1. Some y are not-x, or, Some not-x are y.
2. No not-x are not-y, or, No not-y are not-x.
2. No non-X are non-Y, or, No non-Y are non-X.
3. No not-y are x.
No, you are not.
4. No not-x exist. i.e. No Things are not-x.
4. No non-x exists. i.e. No Things are not-x.
5. No y exist. i.e. No houses are two-storied.
5. No y exist. i.e. No houses are two stories.
6. Some x' exist. i.e. Some houses are not built of brick.
6. Some x' exist. i.e. Some houses aren't made of brick.
7. No x are y'. Or, no y' are x. i.e. No houses, built of brick, are other than two-storied. Or, no houses, that are not two-storied, are built of brick.
7. No x are y'. Or, no y' are x. i.e. No houses built of brick are anything other than two stories. Or, no houses that aren't two stories are built of brick.
8. All x' are y'. i.e. All houses, that are not built of brick, are not two-storied.
8. All x' are y'. i.e. All houses that aren't made of brick are not two stories.
9. Some x are y, and some are y'. i.e. Some fat boys are active, and some are not.
9. Some x are y, and some are y'. i.e. Some heavy boys are active, and some are not.
10. All y' are x'. i.e. All lazy boys are thin.
10. All of you are thin. i.e. All lazy boys are thin.
11. All x are y', and all y' are x. i.e. All fat boys are lazy, and all lazy ones are fat.
11. All x are y, and all y are x. i.e. All fat boys are lazy, and all lazy ones are fat.
12. All y are x, and all x' are y. i.e. All active boys are fat, and all thin ones are lazy.
12. All y are x, and all x' are y. i.e. All active boys are overweight, and all thin ones are lazy.
13. No x exist, and no y' exist. i.e. No cats have green eyes, and none have bad tempers.
13. No x exists, and no y' exists. i.e. No cats have green eyes, and none have bad tempers.
14. Some x are y', and some x' are y. Or some y are x', and some y' are x. i.e. Some green-eyed cats are bad-tempered, and some, that have not green eyes, are good-tempered. Or, some good-tempered cats have not green eyes, and some bad-tempered ones have green eyes.
14. Some x are y, and some x' are y. Or some y are x', and some y' are x. That is, some green-eyed cats are bad-tempered, and some that don’t have green eyes are good-tempered. Or, some good-tempered cats don’t have green eyes, and some bad-tempered ones do have green eyes.
15. Some x are y, and no x' are y'. Or, some y are x, and no y' are x'. i.e. Some green-eyed cats are good-tempered, and none, that are not green-eyed, are bad-tempered. Or, some good-tempered cats have green eyes, and none, that are bad-tempered, have not green eyes.
15. Some x are y, and none of the x' are y'. Or, some y are x, and none of the y' are x'. For example, some green-eyed cats have good temperaments, and none that are not green-eyed have bad tempers. Or, some good-tempered cats have green eyes, and none that are bad-tempered do not have green eyes.
16. All x are y', and all x' are y. Or, all y are x', and all y' are x. i.e. All green-eyed cats are bad-tempered and all, that have not green eyes, are good-tempered. Or, all good-tempered ones have eyes that are not green, and all bad-tempered ones have green eyes.
16. All x are y', and all x' are y. Or, all y are x', and all y' are x. That is, all green-eyed cats are bad-tempered, and all cats that don't have green eyes are good-tempered. Or, all good-tempered cats have eyes that aren't green, and all bad-tempered ones have green eyes.
6. Larger Diagram.
6. Bigger Diagram.
Propositions represented.
Propositions displayed.
__________
__________
--------------- ---------------
| | | | | |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | | | |
1. |---|---|---|---| 2. |-1-|---|---|---|
| | | | | | | | | |
| ---|--- | | ---|--- |
| | | | | |
--------------- ---------------
--------------- ---------------
| | | | | |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | | | |
1. |---|---|---|---| 2. |-1-|---|---|---|
| | | | | | | | | |
| ---|--- | | ---|--- |
| | | | | |
--------------- --------------- --------------- ---------------
| | | | | 0 |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | | | |
3. |---|---|---|---| 4. |---|---|---|---|
| | - | | | | | | |
| ---|--- | | ---|--- |
| | | | | 0 |
--------------- ---------------
--------------- ---------------
| | | | | 0 |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | | | |
3. |---|---|---|---| 4. |---|---|---|---|
| | - | | | | | | |
| ---|--- | | ---|--- |
| | | | | 0 |
--------------- --------------- --------------- ---------------
| 0 | | | | |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | 0 | 1 | |
5. |---|---|---|---| 6. |---|---|---|---|
| | 1 | | | | | 0 | | |
| ---|--- | | ---|--- |
| 0 | | | | |
--------------- ---------------
--------------- ---------------
| 0 | | | | |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | 0 | 1 | |
5. |---|---|---|---| 6. |---|---|---|---|
| | 1 | | | | | 0 | | |
| ---|--- | | ---|--- |
| 0 | | | | |
--------------- --------------- --------------- ---------------
| | | | | 0 |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | | | |
7. |---|---|---|---| 8. |---|---|---|---|
| | 0 | 1 | | | | 0 | 0 | |
| ---|--- | | ---|--- |
| | | | | 0 |
--------------- ---------------
--------------- ---------------
| | | | | 0 |
| ---|--- | | ---|--- |
| | 0 | 0 | | | | | | |
7. |---|---|---|---| 8. |---|---|---|---|
| | 0 | 1 | | | | 0 | 0 | |
| ---|--- | | ---|--- |
| | | | | 0 |
--------------- --------------- ---------------
| | |
| ---|--- |
| | 0 | 0 | |
9. No x are m. i.e. |---|---|---|---|
| | 0 | | |
| ---|--- |
| | |
---------------
---------------
| | |
| ---|--- |
| | 0 | 0 | |
9. No x are m. i.e. |---|---|---|---|
| | 0 | | |
| ---|--- |
| | |
--------------- ---------------
| | |
| ---|--- |
| | | | |
10. Some m' are y. i.e. |-1-|---|---|---|
| | | | |
| ---|--- |
| | |
---------------
10. Some m' are y. i.e. |-1-|---|---|---|
---------------
| | |
| ---|--- |
| | | 0 | |
11. All y' are m'. i.e. |---|---|---|-1-|
| | | 0 | |
| ---|--- |
| | |
---------------
---------------
| | |
| ---|--- |
| | | 0 | |
11. All of you are m'. i.e. |---|---|---|-1-|
| | | 0 | |
| ---|--- |
| | |
--------------- ---------------
| | |
| ---|--- |
| | 0 | 0 | |
12. All m are x'. i.e. |---|---|---|---|
| | 1 | |
| ---|--- |
| | |
---------------
---------------
| | |
| ---|--- |
| | 0 | 0 | |
12. All m are x's. i.e. |---|---|---|---|
| | 1 | |
| ---|--- |
| | |
--------------- ---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
13. No x are m; i.e. |---|---|---|---|
All y are m. | | 1 | | |
| ---|--- |
| 0 | |
---------------
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
13. No x are m; i.e. |---|---|---|---|
All y are m. | | 1 | | |
| ---|--- |
| 0 | |
--------------- ---------------
| 0 | 0 |
| ---|--- |
| | | | |
14. All m' are y; i.e. |---|---|---|---|
No x are m'. | | | | |
| ---|--- |
| 1 | 0 |
---------------
---------------
| 0 | 0 |
| ---|--- |
| | | | |
14. All m' are yours; i.e. |---|---|---|---|
No x are m'. | | | | |
| ---|--- |
| 1 | 0 |
--------------- ---------------
| 0 | 0 |
| ---|--- |
| | 1 | 0 | |
15. All x are m; i.e. |---|---|---|---|
No m are y'. | | | 0 | |
| ---|--- |
| | |
---------------
---------------
| 0 | 0 |
| ---|--- |
| | 1 | 0 | |
15. All x are m; i.e. |---|---|---|---|
No m are y'. | | | 0 | |
| ---|--- |
| | |
--------------- ---------------
| 0 | 0 |
| ---|--- |
| | | | |
16. All m' are y'; i.e. |---|---|---|---|
No x are m'. | | | | |
| ---|--- |
| 0 | 1 |
---------------
---------------
| 0 | 0 |
| ---|--- |
| | | | |
16. All my things are yours; i.e. |---|---|---|---|
No x are my things. | | | | |
| ---|--- |
| 0 | 1 |
--------------- ---------------
| 0 | 0 |
| ---|--- |
| | 1 | 0 | |
17. All x are m; i.e. |---|---|---|---|
All m are y. | | | 0 | |
| ---|--- |
[See remarks on No. 7, p. 60.] | | |
---------------
---------------
| 0 | 0 |
| ---|--- |
| | 1 | 0 | |
17. Every x is an m; i.e. |---|---|---|---|
Every m is a y. | | | 0 | |
| ---|--- |
[See remarks on No. 7, p. 60.] | | |
--------------- ---------------
| 0 | |
| ---|--- |
| | | | |
18. No x' are m; i.e. |---|---|---|---|
No m' are y. | | 0 | 0 | |
| ---|--- |
| 0 | |
---------------
---------------
| 0 | |
| ---|--- |
| | | | |
18. No x' are m; i.e. |---|---|---|---|
No m' are y. | | 0 | 0 | |
| ---|--- |
| 0 | |
--------------- ---------------
| | |
| ---|--- |
| | 1 | 0 | |
19. All m are x; i.e. |---|---|---|---|
All m are y. | | 0 | 0 | |
| ---|--- |
| | |
---------------
19. All m are x; that is, all m are y.
20. We had better take "persons" as Universe. We
may choose "myself" as 'Middle Term', in which case
the Premisses will take the form
20. We should consider "persons" as the Universe. We
can use "myself" as the 'Middle Term', which means the Premises will look like
I am a-person-who-sent-him-to-bring-a-kitten; I am a-person-to-whom-he-brought-a-kettle-by-mistake.
I’m the person he was sent to bring a kitten to; I’m the person he mistakenly brought a kettle to.
Or we may choose "he" as 'Middle Term', in which case the Premisses will take the form
Or we may choose "he" as the 'Middle Term,' in which case the premises will take the form
He is a-person-whom-I-sent-to-bring-me-a-kitten; He is a-person-who-brought-me-a-kettle-by-mistake.
He is someone I sent to get me a kitten; He is someone who accidentally brought me a kettle.
The latter form seems best, as the interest of the anecdote clearly depends on HIS stupidity--not on what happened to ME. Let us then make m = "he"; x = "persons whom I sent, &c."; and y = "persons who brought, &c."
The latter form seems best, as the interest of the anecdote clearly depends on HIS stupidity—not on what happened to ME. Let’s then make m = "he"; x = "the people I sent, etc."; and y = "the people who brought, etc."
Hence, All m are x;
All m are y. and the required Diagram is
Hence, All m are x;
All m are y. and the required Diagram is ---------------
| | |
| ---|--- |
| | 1 | 0 | |
|---|---|---|---|
| | 0 | 0 | |
| ---|--- |
| | |
---------------
---------------
| | |
| ---|--- |
| | 1 | 0 | |
|---|---|---|---|
| | 0 | 0 | |
| ---|--- |
| | |
---------------7. Both Diagrams employed.
Both diagrams used.
-------
| 0 | |
1. |---|---| i.e. All y are x'.
| 1 | |
-------
-------
| 0 | |
1. |---|---| i.e. All y are x'.
| 1 | |
------- -------
| | 1 |
2. |---|---| i.e. Some x are y'; or, Some y' are x.
| | |
-------
-------
| | 1 |
2. |---|---| i.e. Some x are y'; or, Some y' are x.
| | |
------- -------
| | |
3. |---|---| i.e. Some y are x'; or, Some x' are y.
| 1 | |
-------
-------
| | |
3. |---|---| i.e. Some y are x'; or, Some x' are y.
| 1 | |
------- -------
| | |
4. |---|---| i.e. No x' are y'; or, No y' are x'.
| | 0 |
-------
-------
| | |
4. |---|---| i.e. No x's are y's; or, No y's are x's.
| | 0 |
------- -------
| 0 | |
5. |---|---| i.e. All y are x'. i.e. All black rabbits
| 1 | | are young.
-------
-------
| 0 | |
5. |---|---| i.e. All y are x'. i.e. All black rabbits
| 1 | | are young.
------- -------
| | |
6. |---|---| i.e. Some y are x'. i.e. Some black
| 1 | | rabbits are young.
-------
-------
| | |
6. |---|---| i.e. Some y are x'. i.e. Some black
| 1 | | rabbits are young.
------- -------
| 1 | 0 |
7. |---|---| i.e. All x are y. i.e. All well-fed birds
| | | are happy.
-------
-------
| 1 | 0 |
7. |---|---| i.e. All x are y. i.e. All well-fed birds
| | | are happy.
------- -------
| | | i.e. Some x' are y'. i.e. Some birds,
8. |---|---| that are not well-fed, are unhappy;
| | 1 | or, Some unhappy birds are not
------- well-fed.
-------
| | | i.e. Some x' are y'. i.e. Some birds,
8. |---|---| that aren’t well-fed, are unhappy;
| | 1 | or, Some unhappy birds aren’t
------- well-fed.
-------
| 1 | 0 |
9. |---|---| i.e. All x are y. i.e. John has got a
| | | tooth-ache.
-------
-------
| 1 | 0 |
9. |---|---| i.e. All x are y. i.e. John has a
| | | toothache.
------- -------
| | |
10. |---|---| i.e. No x' are y. i.e. No one, but John,
| 0 | | has got a tooth-ache.
-------
-------
| | |
10. |---|---| i.e. No x' are y. i.e. No one except John,
| 0 | | has a toothache.
------- -------
| 1 | |
11. |---|---| i.e. Some x are y. i.e. Some one, who
| | | has taken a walk, feels better.
-------
-------
| 1 | |
11. |---|---| i.e. Some x are y. i.e. Some person who
| | | has taken a walk feels better.
------- -------
| 1 | | i.e. Some x are y. i.e. Some one,
12. |---|---| whom I sent to bring me a kitten,
| | | brought me a kettle by mistake.
-------
-------
| 1 | | i.e. Some x are y. i.e. Someone,
12. |---|---| whom I asked to get me a kitten,
| | | accidentally brought me a kettle.
------- ---------------
| | 0 |
| ---|--- |
| | 0 | 0 | |
13. |-1-|---|---|---| -------
| | | | | | | 0 |
| ---|--- | |---|---|
| | 0 | | | |
--------------- -------
---------------
| | 0 |
| ---|--- |
| | 0 | 0 | |
13. |-1-|---|---|---| -------
| | | | | | | 0 |
| ---|--- | |---|---|
| | 0 | | | |
--------------- -------Let "books" be Universe; m="exciting",
x="that suit feverish patients"; y="that make
one drowsy".
Let "books" be Universe; m="exciting",
x="that suit restless readers"; y="that make
one sleepy".
No m are x; &there4 No y' are x.
All m' are y.
No m are x; &there4 No y' are x.
All m' are y.
i.e. No books suit feverish patients, except such as make
one drowsy.
i.e. No books are good for restless patients, except those that make
you sleepy.
---------------
| | |
| ---|--- |
| | 1 | 0 | |
14. |---|---|---|---| -------
| | | 0 | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| | |
| ---|--- |
| | 1 | 0 | |
14. |---|---|---|---| -------
| | | 0 | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "persons" be Universe; m="that deserve the fair";
x="that get their deserts"; y="brave".
Let "people" be Universe; m="that deserve what's fair";
x="that get what they deserve"; y="brave".
Some m are x; &there4 Some y are x.
No y' are m.
Some m are x; and therefore some y are x.
No y are m.
i.e. Some brave persons get their deserts.
i.e. Some brave people get what they deserve.
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
15. |---|---|---|---| -------
| | | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | | |
--------------- -------
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
15. |---|---|---|---| -------
| | | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | | |
--------------- -------Let "persons" be Universe; m="patient";
x="children"; y="that can sit still".
Let "people" be Universe; m="patient";
x="kids"; y="who can sit still".
No x are m; &there4 No x are y.
No m' are y.
No x are m; and therefore, no x are y.
No m' are y.
i.e. No children can sit still.
i.e. No kids can sit still.
---------------
| 0 | 0 |
| ---|--- |
| | 0 | 1 | |
16. |---|---|---|---| -------
| | 0 | | | | 0 | 1 |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| 0 | 0 |
| ---|--- |
| | 0 | 1 | |
16. |---|---|---|---| -------
| | 0 | | | | 0 | 1 |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "things" be Universe; m="fat"; x="pigs";
y="skeletons".
Let "things" be Universe; m="fat"; x="pigs";
y="skeletons".
All x are m; &there4 All x are y'.
No y are m.
All x are m; therefore, all x are y.
No y are m.
i.e. All pigs are not-skeletons.
All pigs aren't skeletons.
---------------
| | |
| ---|--- |
| | 0 | 0 | |
17. |---|---|---|---| -------
| | 1 | 0 | | | | |
| ---|--- | |---|---|
| | | | 1 | |
--------------- -------
---------------
| | |
| ---|--- |
| | 0 | 0 | |
17. |---|---|---|---| -------
| | 1 | 0 | | | | |
| ---|--- | |---|---|
| | | | 1 | |
--------------- -------Let "creatures" be Universe; m="monkeys";
x="soldiers"; y="mischievous".
Let "creatures" mean Universe; m="monkeys";
x="soldiers"; y="mischievous".
No m are x; &there4 Some y are x'.
All m are y.
No m are x; &there4 Some y are x'.
All m are y.
i.e. Some mischievous creatures are not soldiers.
i.e. Some naughty creatures aren't soldiers.
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
18. |---|---|---|---| -------
| | | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | | |
--------------- -------
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
18. |---|---|---|---| -------
| | | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | | |
--------------- -------Let "persons" be Universe; m="just";
x="my cousins"; y="judges".
Let "people" be Universe; m="just";
x="my cousins"; y="judges".
No x are m; &there4 No x are y.
No y are m'.
No x are m; &there4 No x are y.
No y are m'.
i.e. None of my cousins are judges.
i.e. None of my cousins are judges.
---------------
| | |
| ---|--- |
| | 1 | 0 | |
19. |---|---|---|---| -------
| | | | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| | |
| ---|--- |
| | 1 | 0 | |
19. |---|---|---|---| -------
| | | | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "periods" be Universe; m="days";
x="rainy"; y="tiresome".
Let "periods" be Universe; m="days";
x="rainy"; y="tiresome".
Some m are x; &there4 Some x are y.
All xm are y.
Some m are x; and therefore some x are y.
All xm are y.
i.e. Some rainy periods are tiresome.
i.e. Some rainy days can be annoying.
N.B. These are not legitimate Premisses, since the Conclusion is really part of the second Premiss, so that the first Premiss is superfluous. This may be shown, in letters, thus:--
N.B. These are not valid premises, since the conclusion is actually part of the second premise, making the first premise unnecessary. This can be shown in letters, like this:--
"All xm are y" contains "Some xm are y", which contains "Some x are y". Or, in words, "All rainy days are tiresome" contains "Some rainy days are tiresome", which contains "Some rainy periods are tiresome".
"All xm are y" includes "Some xm are y", which includes "Some x are y". In other words, "All rainy days are tiresome" includes "Some rainy days are tiresome", which includes "Some rainy periods are tiresome".
Moreover, the first Premiss, besides being superfluous, is actually contained in the second; since it is equivalent to "Some rainy days exist", which, as we know, is implied in the Proposition "All rainy days are tiresome".
Moreover, the first premise, besides being unnecessary, is actually included in the second; since it is equivalent to "Some rainy days exist," which, as we know, is implied in the statement "All rainy days are tiresome."
Altogether, a most unsatisfactory Pair of Premisses!
Altogether, a very unsatisfactory set of premises!
---------------
| 0 | |
| ---|--- |
| | 1 | | |
20. |---|---|---|---| -------
| | 0 | 0 | | | 1 | |
| ---|--- | |---|---|
| 0 | | | 0 | |
--------------- -------
---------------
| 0 | |
| ---|--- |
| | 1 | | |
20. |---|---|---|---| -------
| | 0 | 0 | | | 1 | |
| ---|--- | |---|---|
| 0 | | | 0 | |
--------------- -------Let "things" be Universe; m="medicine";
x="nasty"; y="senna".
Let "things" be Universe; m="medicine";
x="nasty"; y="senna".
All m are x; &there4 All y are x.
All y are m.
All m are x; and therefore, all y are x.
All y are m.
i.e. Senna is nasty.
Senna is mean.
[See remarks on No. 7, p 60.]
[See remarks on No. 7, p 60.]
---------------
| | |
| ---|--- |
| | 0 | 1 | |
21. |-1-|---|---|---| -------
| | 0 | | | | | 1 |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| | |
| ---|--- |
| | 0 | 1 | |
21. |-1-|---|---|---| -------
| | 0 | | | | | 1 |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "persons" be Universe; m="Jews";
x="rich"; y="Patagonians".
Let "people" be Universe; m="Jews";
x="wealthy"; y="Patagonians".
Some m are x; &there4 Some x are y'.
All y are m'.
Some m are x; and therefore some x are y.
All y are m.
i.e. Some rich persons are not Patagonians.
i.e. Some wealthy people are not Patagonians.
---------------
| 0 | |
| ---|--- |
| | - | |
22. |---|---|---|---| -------
| | 0 | 0 | | | | |
| ---|--- | |---|---|
| 0 | | | 0 | |
--------------- -------
---------------
| 0 | |
| ---|--- |
| | - | |
22. |---|---|---|---| -------
| | 0 | 0 | | | | |
| ---|--- | |---|---|
| 0 | | | 0 | |
--------------- -------Let "creatures" be Universe; m="teetotalers";
x="that like sugar"; y="nightingales".
Let "creatures" represent the Universe; m="teetotalers";
x="that enjoy sugar"; y="nightingales".
All m are x; &there4 No y are x'.
No y are m'.
All m are x; &there4 No y are x'.
No y are m'.
i.e. No nightingales dislike sugar.
No nightingales hate sugar.
---------------
| | |
| ---|--- |
| | 0 | 0 | |
23. |-1-|---|---|---| -------
| | 0 | | | | | |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| | |
| ---|--- |
| | 0 | 0 | |
23. |-1-|---|---|---| -------
| | 0 | | | | | |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "food" be Universe; m="wholesome";
x="muffins"; y="buns".
Let "food" be Universe; m="wholesome";
x="muffins"; y="buns".
No x are m;
All y are m.
No x are m;
All y are m.
There is 'no information' for the smaller Diagram; so no Conclusion can be drawn.
There is 'no information' for the smaller Diagram, so no Conclusion can be drawn.
---------------
| | |
| ---|--- |
| | 0 | 0 | |
24. |---|---|---|---| -------
| | 1 | | | | | |
| ---|--- | |---|---|
| | | | 1 | |
--------------- -------
---------------
| | |
| ---|--- |
| | 0 | 0 | |
24. |---|---|---|---| -------
| | 1 | | | | | |
| ---|--- | |---|---|
| | | | 1 | |
--------------- -------Let "creatures" be Universe; m="that run well";
x="fat"; y="greyhounds".
Let "creatures" be Universe; m="that run well";
x="fat"; y="greyhounds".
No x are m; &there4 Some y are x'.
Some y are m.
No x are m; &there4 Some y are x'.
Some y are m.
i.e. Some greyhounds are not fat.
Some greyhounds aren't fat.
---------------
| | |
| ---|--- |
| | - | |
25. |-1-|---|---|---| -------
| | 0 | 0 | | | | |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| | |
| ---|--- |
| | - | |
25. |-1-|---|---|---| -------
| | 0 | 0 | | | | |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "persons" be Universe; m="soldiers";
x="that march"; y="youths".
Let "people" be Universe; m="soldiers";
x="who march"; y="young people".
All m are x;
Some y are m'.
All m are x;
Some y are m'.
There is 'no information' for the smaller Diagram; so no Conclusion can be drawn.
There is 'no information' for the smaller diagram; so no conclusion can be drawn.
---------------
| 0 | 0 |
| ---|--- |
| | 0 | 1 | |
26. |---|---|---|---| -------
| | 0 | | | | 0 | 1 |
| ---|--- | |---|---|
| 1 | | | 1 | |
--------------- -------
---------------
| 0 | 0 |
| ---|--- |
| | 0 | 1 | |
26. |---|---|---|---| -------
| | 0 | | | | 0 | 1 |
| ---|--- | |---|---|
| 1 | | | 1 | |
--------------- -------Let "food" be Universe; m="sweet";
x="sugar"; y="salt".
Let "food" be Universe; m="sweet";
x="sugar"; y="salt".
All x are m; &there4 All x are y'.
All y are m'. All y are x'.
All x are m; &there4 All x are y'.
All y are m'. All y are x'.
i.e. Sugar is not salt.
Salt is not sugar.
i.e. Sugar isn't salt.
Salt isn't sugar.
---------------
| | |
| ---|--- |
| | 1 | 0 | |
27. |---|---|---|---| -------
| | | 0 | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| | |
| ---|--- |
| | 1 | 0 | |
27. |---|---|---|---| -------
| | | 0 | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "Things" be Universe; m="eggs";
x="hard-boiled"; y="crackable".
Let "Things" be Universe; m="eggs";
x="hard-boiled"; y="crackable".
Some m are x; &there4 Some x are y.
No m are y'.
Some m are x; and therefore some x are y.
No m are y.
i.e. Some hard-boiled things can be cracked.
i.e. Some tough things can be broken.
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
28. |---|---|---|---| -------
| | | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | | |
--------------- -------
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
28. |---|---|---|---| -------
| | | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | | |
--------------- -------Let "persons" be Universe; m="Jews"; x="that are in the house"; y="that are in the garden".
Let "people" be Universe; m="Jews"; x="who are in the house"; y="who are in the garden".
No m are x; &there4 No x are y.
No m' are y.
No m are x; and therefore no x are y.
No m' are y.
i.e. No persons, that are in the house, are also in
the garden.
i.e. No one who is in the house is also in the garden.
---------------
| 0 | 0 |
| ---|--- |
| | - | |
29. |---|---|---|---| -------
| | | | | | | |
| ---|--- | |---|---|
| 1 | 0 | | 1 | |
--------------- -------
---------------
| 0 | 0 |
| ---|--- |
| | - | |
29. |---|---|---|---| -------
| | | | | | | |
| ---|--- | |---|---|
| 1 | 0 | | 1 | |
--------------- -------Let "Things" be Universe; m="noisy";
x="battles"; y="that may escape notice".
Let "Things" be Universe; m="noisy";
x="battles"; y="that might go unnoticed".
All x are m; &there4 Some x' are y.
All m' are y.
All x are m; and therefore some x' are y.
All m' are y.
i.e. Some things, that are not battles, may escape notice.
i.e. Some things that aren't battles might go unnoticed.
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
30. |---|---|---|---| -------
| | 1 | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | 1 | |
--------------- -------
---------------
| 0 | |
| ---|--- |
| | 0 | 0 | |
30. |---|---|---|---| -------
| | 1 | | | | 0 | |
| ---|--- | |---|---|
| 0 | | | 1 | |
--------------- -------Let "persons" be Universe; m="Jews";
x="mad"; y="Rabbis".
Let "people" be Universe; m="Jews";
x="crazy"; y="Rabbis".
No m are x; &there4 All y are x'.
All y are m.
No m are x; &there4 All y are x'.
All y are m.
i.e. All Rabbis are sane.
All Rabbis are sane.
---------------
| | |
| ---|--- |
| | 1 | | |
31. |---|---|---|---| -------
| | 0 | 0 | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------
---------------
| | |
| ---|--- |
| | 1 | | |
31. |---|---|---|---| -------
| | 0 | 0 | | | 1 | |
| ---|--- | |---|---|
| | | | | |
--------------- -------Let "Things" be Universe; m="fish";
x="that can swim"; y="skates".
Let "Things" be Universe; m="fish";
x="that can swim"; y="skates".
No m are x'; &there4 Some y are x.
Some y are m.
No m are x'; &there4 Some y are x.
Some y are m.
i.e. Some skates can swim.
Some skates can swim.
---------------
| | |
| ---|--- |
| | 0 | 0 | |
32. |---|---|---|---| -------
| | 1 | | | | | |
| ---|--- | |---|---|
| | | | 1 | |
--------------- -------
---------------
| | |
| ---|--- |
| | 0 | 0 | |
32. |---|---|---|---| -------
| | 1 | | | | | |
| ---|--- | |---|---|
| | | | 1 | |
--------------- -------Let "people" be Universe; m="passionate";
x="reasonable"; y="orators".
Let "people" be Universe; m="passionate";
x="reasonable"; y="speakers".
All m are x'; &there4 Some y are x'.
Some y are m.
All m are x'; &there4 Some y are x'.
Some y are m.
i.e. Some orators are unreasonable.
Some speakers are unreasonable.
[See remarks on No. 7, p. 60.]
[See remarks on No. 7, p. 60.]
CHAPTER IV.
HIT OR MISS.
"Thou canst not hit it, hit it, hit it,
Thou canst not hit it, my good man."
"You can't hit it, hit it, hit it,
You can't hit it, my good man."__________
__________
1. Pain is wearisome; No pain is eagerly wished for.
1. Pain is exhausting; No one looks forward to experiencing pain.
2. No bald person needs a hair-brush; No lizards have hair.
2. No bald person needs a hairbrush; no lizards have hair.
3. All thoughtless people do mischief; No thoughtful person forgets a promise.
3. All careless people cause trouble; No considerate person forgets a promise.
4. I do not like John; Some of my friends like John.
4. I don't like John; some of my friends like him.
5. No potatoes are pine-apples; All pine-apples are nice.
5. No potatoes are pineapples; all pineapples are nice.
6. No pins are ambitious; No needles are pins.
6. No pins are ambitious; no needles are pins.
7. All my friends have colds; No one can sing who has a cold.
7. All my friends have colds; no one can sing when they have a cold.
8. All these dishes are well-cooked; Some dishes are unwholesome if not well-cooked.
8. All these dishes are well-cooked; some dishes can be unhealthy if they're not cooked properly.
9. No medicine is nice; Senna is a medicine.
9. No medicine is pleasant; Senna is a medicine.
10. Some oysters are silent; No silent creatures are amusing.
10. Some oysters are quiet; no quiet creatures are entertaining.
11. All wise men walk on their feet; All unwise men walk on their hands.
11. All wise people walk on their feet; all unwise people walk on their hands.
12. "Mind your own business; This quarrel is no business of yours."
12. "Mind your own business; this argument has nothing to do with you."
13. No bridges are made of sugar; Some bridges are picturesque.
13. No bridges are made of sugar; some bridges are pretty to look at.
14. No riddles interest me that can be solved; All these riddles are insoluble.
14. I'm not interested in any riddles that can be solved; all these riddles are unsolvable.
15. John is industrious; All industrious people are happy.
15. John works hard; all hardworking people are happy.
16. No frogs write books; Some people use ink in writing books.
16. No frogs write books; some people use ink to write books.
17. No pokers are soft; All pillows are soft.
17. No pokers are soft; all pillows are soft.
18. No antelope is ungraceful; Graceful animals delight the eye.
18. No antelope is clumsy; graceful animals please the eye.
19. Some uncles are ungenerous; All merchants are generous.
19. Some uncles are not generous; all merchants are generous.
20. No unhappy people chuckle; No happy people groan.
20. Unhappy people don’t laugh; happy people don’t complain.
21. Audible music causes vibration in the air; Inaudible music is not worth paying for.
21. Audible music creates vibrations in the air; inaudible music isn't worth the money.
22. He gave me five pounds; I was delighted.
22. He gave me five pounds; I was thrilled.
23. No old Jews are fat millers; All my friends are old millers.
23. No elderly Jews are heavyset millers; all my friends are older millers.
24. Flour is good for food; Oatmeal is a kind of flour.
24. Flour is great for food; oatmeal is a type of flour.
25. Some dreams are terrible; No lambs are terrible.
25. Some dreams are awful; No lambs are awful.
26. No rich man begs in the street; All who are not rich should keep accounts.
26. No wealthy person begs on the street; everyone who isn't wealthy should manage their finances.
27. No thieves are honest; Some dishonest people are found out.
27. No thieves are honest; some dishonest people get caught.
28. All wasps are unfriendly; All puppies are friendly.
28. All wasps are unfriendly; all puppies are friendly.
29. All improbable stories are doubted; None of these stories are probable.
29. All unlikely stories are questioned; none of these stories are believable.
30. "He told me you had gone away." "He never says one word of truth."
30. "He said you left." "He never speaks a word of truth."
31. His songs never last an hour; A song, that lasts an hour, is tedious.
31. His songs never go on for an hour; a song that lasts an hour is boring.
32. No bride-cakes are wholesome; Unwholesome food should be avoided.
32. No bride cakes are healthy; unhealthy food should be avoided.
33. No old misers are cheerful; Some old misers are thin.
33. No old cheapskates are happy; Some old cheapskates are skinny.
34. All ducks waddle; Nothing that waddles is graceful.
34. All ducks waddle; nothing that waddles is graceful.
35. No Professors are ignorant; Some ignorant people are conceited.
35. No professors are clueless; Some clueless people are arrogant.
36. Toothache is never pleasant; Warmth is never unpleasant.
36. A toothache is never fun; warmth is always nice.
37. Bores are terrible; You are a bore.
37. Bores are awful; You’re a bore.
38. Some mountains are insurmountable; All stiles can be surmounted.
38. Some mountains can't be climbed; all fences can be crossed.
39. No Frenchmen like plumpudding; All Englishmen like plumpudding.
39. No French people like plum pudding; All English people like plum pudding.
40. No idlers win fame; Some painters are not idle.
40. No slackers gain fame; Some artists are not lazy.
41. No lobsters are unreasonable; No reasonable creatures expect impossibilities.
41. No lobsters are unreasonable; no sensible beings expect the impossible.
42. No kind deed is unlawful; What is lawful may be done without fear.
42. No good deed is illegal; What is legal can be done without worry.
43. No fossils can be crossed in love; Any oyster may be crossed in love.
43. No fossils can fall in love; any oyster can fall in love.
44. "This is beyond endurance!" "Well, nothing beyond endurance has ever happened to me."
44. "This is too much to handle!" "Well, I've never experienced anything I couldn't handle."
45. All uneducated men are shallow; All these students are educated.
45. All uneducated people are superficial; All these students are educated.
46. All my cousins are unjust; No judges are unjust.
46. All my cousins are unfair; No judges are unfair.
47. No country, that has been explored, is infested by dragons; Unexplored countries are fascinating.
47. No country that has been explored is filled with dragons; unexplored countries are intriguing.
48. No misers are generous; Some old men are not generous.
48. No stingy people are generous; some older men are not generous.
49. A prudent man shuns hyaenas; No banker is imprudent.
49. A wise person avoids hyenas; no banker is careless.
50. Some poetry is original; No original work is producible at will.
50. Some poetry is original; no original work can be created on demand.
51. No misers are unselfish; None but misers save egg-shells.
51. No cheapskates are generous; only cheapskates save egg shells.
52. All pale people are phlegmatic; No one, who is not pale, looks poetical.
52. All pale people are calm; No one who isn't pale looks poetic.
53. All spiders spin webs; Some creatures, that do not spin webs, are savage.
53. All spiders make webs; some creatures that don't make webs are fierce.
54. None of my cousins are just; All judges are just.
54. None of my cousins are fair; all judges are fair.
55. John is industrious; No industrious people are unhappy.
55. John works hard; no hardworking people are unhappy.
56. Umbrellas are useful on a journey; What is useless on a journey should be left behind.
56. Umbrellas are handy on a trip; what isn’t useful on a trip should be left behind.
57. Some pillows are soft; No pokers are soft.
57. Some pillows are soft; no pokers are soft.
58. I am old and lame; No old merchant is a lame gambler.
58. I'm old and weak; No old merchant is a broken gambler.
59. No eventful journey is ever forgotten; Uneventful journeys are not worth writing a book about.
59. No exciting journey is ever forgotten; boring journeys aren't worth writing a book about.
60. Sugar is sweet; Some sweet things are liked by children.
60. Sugar is sweet; some sweet things are liked by kids.
61. Richard is out of temper; No one but Richard can ride that horse.
61. Richard is in a bad mood; no one but Richard can ride that horse.
62. All jokes are meant to amuse; No Act of Parliament is a joke.
62. All jokes are meant to be funny; no law made by Parliament is a joke.
63. "I saw it in a newspaper." "All newspapers tell lies."
63. "I read it in a newspaper." "All newspapers lie."
64. No nightmare is pleasant; Unpleasant experiences are not anxiously desired.
64. No nightmare is enjoyable; Unpleasant experiences are not eagerly wanted.
65. Prudent travellers carry plenty of small change; Imprudent travellers lose their luggage.
65. Smart travelers carry a lot of spare change; careless travelers lose their bags.
66. All wasps are unfriendly; No puppies are unfriendly.
66. All wasps are aggressive; No puppies are aggressive.
67. He called here yesterday; He is no friend of mine.
67. He called here yesterday; he's not my friend.
68. No quadrupeds can whistle; Some cats are quadrupeds.
68. No four-legged animals can whistle; Some cats are four-legged animals.
69. No cooked meat is sold by butchers; No uncooked meat is served at dinner.
69. No cooked meat is sold by butchers; no raw meat is served at dinner.
70. Gold is heavy; Nothing but gold will silence him.
70. Gold is heavy; only gold can silence him.
71. Some pigs are wild; There are no pigs that are not fat.
71. Some pigs are wild; there are no pigs that aren't fat.
72. No emperors are dentists; All dentists are dreaded by children.
72. No emperors are dentists; all dentists are feared by kids.
73. All, who are not old, like walking; Neither you nor I are old.
73. Everyone who isn't old likes to walk; neither you nor I are old.
74. All blades are sharp; Some grasses are blades.
74. All blades are sharp; some grasses are blades.
75. No dictatorial person is popular; She is dictatorial.
75. No one likes a controlling person; she is controlling.
76. Some sweet things are unwholesome; No muffins are sweet.
76. Some sweet things aren't good for you; No muffins are sweet.
77. No military men write poetry; No generals are civilians.
77. No soldiers write poetry; No generals are regular people.
78. Bores are dreaded; A bore is never begged to prolong his visit.
78. Bores are feared; a bore is never asked to stay longer.
79. All owls are satisfactory; Some excuses are unsatisfactory.
79. All owls are fine; some excuses are not good enough.
80. All my cousins are unjust; All judges are just.
80. All my cousins are unfair; all judges are fair.
81. Some buns are rich; All buns are nice.
81. Some buns are rich; all buns are nice.
82. No medicine is nice; No pills are unmedicinal.
82. No medicine is pleasant; no pills are without medicinal properties.
83. Some lessons are difficult; What is difficult needs attention.
83. Some lessons are tough; what’s tough needs focus.
84. No unexpected pleasure annoys me; Your visit is an unexpected pleasure.
84. Nothing unexpected bothers me; your visit is a nice surprise.
85. Caterpillars are not eloquent; Jones is eloquent.
85. Caterpillars don't speak well; Jones speaks very well.
86. Some bald people wear wigs; All your children have hair.
86. Some bald people wear wigs; all your kids have hair.
87. All wasps are unfriendly; Unfriendly creatures are always unwelcome.
87. All wasps are hostile; Hostile creatures are always unwelcome.
88. No bankrupts are rich; Some merchants are not bankrupts.
88. No bankrupts are wealthy; some merchants aren't bankrupts.
89. Weasels sometimes sleep; All animals sometimes sleep.
89. Weasels sometimes sleep; all animals sometimes sleep.
90. Ill-managed concerns are unprofitable; Railways are never ill-managed.
90. Poorly managed operations are not profitable; Railways are never poorly managed.
91. Everybody has seen a pig; Nobody admires a pig.
91. Everyone has seen a pig; no one admires a pig.
______________
______________
Extract a Pair of Premisses out of each of the following: and deduce the Conclusion, if there is one:--
Extract a pair of premises from each of the following: and deduce the conclusion, if there is one:--
92. "The Lion, as any one can tell you who has been chased by them as often as I have, is a very savage animal: and there are certain individuals among them, though I will not guarantee it as a general law, who do not drink coffee."
92. "The Lion, as anyone can tell you who has been chased by them as often as I have, is a very fierce animal: and there are certain individuals among them, though I won't say this applies to all, who don’t drink coffee."
93. "It was most absurd of you to offer it! You might have known, if you had had any sense, that no old sailors ever like gruel!"
93. "It was really ridiculous of you to offer that! You should have realized, if you had any common sense, that no old sailors ever like gruel!"
"But I thought, as he was an uncle of yours--"
"But I thought, since he was your uncle--"
"An uncle of mine, indeed! Stuff!"
"An uncle of mine, really! What nonsense!"
"You may call it stuff, if you like. All I know is, MY uncles are all old men: and they like gruel like anything!"
"You can call it stuff if you want. All I know is, my uncles are all old men, and they really like gruel!"
"Well, then YOUR uncles are--"
"Well, then YOUR uncles are—"
94. "Do come away! I can't stand this squeezing any more. No crowded shops are comfortable, you know very well."
94. "Please come away! I can't take this crowding anymore. You know crowded shops are never comfortable."
"Well, who expects to be comfortable, out shopping?"
"Well, who expects to be comfortable while out shopping?"
"Why, I do, of course! And I'm sure there are some shops, further down the street, that are not crowded. So--"
"Of course I do! And I’m sure there are some stores further down the street that aren't crowded. So—"
95. "They say no doctors are metaphysical organists: and that lets me into a little fact about YOU, you know."
95. "People say that no doctors are metaphysical organists, and that reveals a little something about YOU, you know."
"Why, how do you make THAT out? You never heard me play the organ."
"How do you come to that conclusion? You’ve never heard me play the organ."
"No, doctor, but I've heard you talk about Browning's poetry: and that showed me that you're METAPHYSICAL, at any rate. So--"
"No, doctor, but I've heard you talk about Browning's poetry: and that showed me that you're METAPHYSICAL, at least. So--"
___________________
___________________
Extract a Syllogism out of each of the following: and test its correctness:--
Extract a syllogism from each of the following and check its accuracy:--
96. "Don't talk to me! I've known more rich merchants than you have: and I can tell you not ONE of them was ever an old miser since the world began!"
96. "Don't talk to me! I've known more wealthy merchants than you have, and I can tell you that not a single one of them has ever been an old miser since the world began!"
"And what has that got to do with old Mr. Brown?"
"And what does that have to do with old Mr. Brown?"
"Why, isn't he very rich?"
"Isn't he super rich?"
"Yes, of course he is. And what then?"
"Yeah, of course he is. So what now?"
"Why, don't you see that it's absurd to call him a miserly merchant? Either he's not a merchant, or he's not a miser!"
"Why don’t you see that it's ridiculous to call him a stingy merchant? Either he’s not a merchant, or he’s not stingy!"
97. "It IS so kind of you to enquire! I'm really feeling a great deal better to-day."
97. "That's really sweet of you to ask! I'm feeling a lot better today."
"And is it Nature, or Art, that is to have the credit of this happy change?"
"And is it Nature or Art that should get the credit for this wonderful change?"
"Art, I think. The Doctor has given me some of that patent medicine of his."
"Art, I believe. The doctor has given me some of his patent medicine."
"Well, I'll never call him a humbug again. There's SOMEBODY, at any rate, that feels better after taking his medicine!"
"Well, I’ll never call him a fraud again. There’s definitely someone who feels better after taking his medicine!"
98. "No, I don't like you one bit. And I'll go and play with my doll. DOLLS are never unkind."
98. "No, I don't like you at all. And I'm going to play with my doll. DOLLS are never mean."
"So you like a doll better than a cousin? Oh you little silly!"
"So you like a doll more than a cousin? Oh, you little silly!"
"Of course I do! COUSINS are never kind--at least no cousins I've ever seen."
"Of course I do! COUSINS are never nice—at least none I've ever met."
"Well, and what does THAT prove, I'd like to know! If you mean that cousins aren't dolls, who ever said they were?"
"Well, what does THAT even prove? If you're saying that cousins aren’t dolls, who ever claimed they were?"
99. "What are you talking about geraniums for? You can't tell one flower from another, at this distance! I grant you they're all RED flowers: it doesn't need a telescope to know THAT."
99. "What are you even talking about geraniums for? You can't tell one flower from another at this distance! I get that they're all RED flowers: you don't need a telescope to see THAT."
"Well, some geraniums are red, aren't they?"
"Well, some geraniums are red, right?"
"I don't deny it. And what then? I suppose you'll be telling me some of those flowers are geraniums!"
"I won't deny it. So what? I guess you'll be telling me some of those flowers are geraniums!"
"Of course that's what I should tell you, if you'd the sense to follow an argument! But what's the good of proving anything to YOU, I should like to know?"
"Of course that's what I should say to you, if you had the sense to follow an argument! But what's the point of proving anything to YOU, if I may ask?"
100. "Boys, you've passed a fairly good examination, all things considered. Now let me give you a word of advice before I go. Remember that all, who are really anxious to learn, work HARD."
100. "Guys, you've done pretty well on the exam, all things considered. Now let me give you a piece of advice before I go. Remember that anyone who genuinely wants to learn puts in the HARD work."
"I thank you, Sir, in the name of my scholars! And proud am I to think there are SOME of them, at least, that are really ANXIOUS to learn."
"I thank you, Sir, on behalf of my students! And I am proud to think that there are some of them, at least, who are truly eager to learn."
"Very glad to hear it: and how do you make it out to be so?"
"That's great to hear! How do you figure that out?"
"Why, Sir, I know how hard they work--some of them, that is. Who should know better?"
"Well, sir, I know how hard they work—some of them, at least. Who would know better?"
___________________
___________________
Extract from the following speech a series of Syllogisms, or arguments having the form of Syllogisms: and test their correctness.
Extract from the following speech a series of syllogisms, or arguments that follow the form of syllogisms, and check if they are correct.
It is supposed to be spoken by a fond mother, in answer to a friend's cautious suggestion that she is perhaps a LITTLE overdoing it, in the way of lessons, with her children.
It’s meant to be spoken by a loving mother, in response to a friend’s careful suggestion that she might be overdoing it just a bit when it comes to lessons for her children.
101. "Well, they've got their own way to make in the world. WE can't leave them a fortune apiece. And money's not to be had, as YOU know, without money's worth: they must WORK if they want to live. And how are they to work, if they don't know anything? Take my word for it, there's no place for ignorance in THESE times! And all authorities agree that the time to learn is when you're young. One's got no memory afterwards, worth speaking of. A child will learn more in an hour than a grown man in five. So those, that have to learn, must learn when they're young, if ever they're to learn at all. Of course that doesn't do unless children are HEALTHY: I quite allow THAT. Well, the doctor tells me no children are healthy unless they've got a good colour in their cheeks. And only just look at my darlings! Why, their cheeks bloom like peonies! Well, now, they tell me that, to keep children in health, you should never give them more than six hours altogether at lessons in the day, and at least two half-holidays in the week. And that's EXACTLY our plan I can assure you! We never go beyond six hours, and every Wednesday and Saturday, as ever is, not one syllable of lessons do they do after their one o'clock dinner! So how you can imagine I'm running any risk in the education of my precious pets is more than I can understand, I promise you!"
101. "Well, they’ve got their own way to make it in the world. We can't leave them a fortune each. And money doesn't come easy, as you know, without putting in the effort: they have to WORK if they want to live. And how can they work if they don’t know anything? Trust me, there’s no place for ignorance in these times! Plus, all experts agree that the best time to learn is when you’re young. You don’t really have a good memory afterwards. A child can learn more in an hour than an adult can in five. So those who need to learn must do so when they’re young, if they’re ever going to learn at all. Of course, that only works if children are HEALTHY: I completely agree with that. Well, the doctor tells me no child is healthy unless they have a good color in their cheeks. And just look at my darlings! Their cheeks are blooming like peonies! Now, they say that to keep children healthy, you should never give them more than six hours of lessons in a day, and at least two half-holidays a week. And that’s EXACTLY our plan, I assure you! We never go beyond six hours, and every Wednesday and Saturday, without fail, they don’t do a single lesson after their one o'clock lunch! So how you can think I’m risking my precious pets’ education is beyond me, I promise you!"
THE END.
THE END.
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