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diff --git a/old/thgmf10.txt b/old/thgmf10.txt new file mode 100644 index 0000000..39eb5d1 --- /dev/null +++ b/old/thgmf10.txt @@ -0,0 +1,3998 @@ +The Project Gutenberg EBook of The Game of Logic, by Lewis Carroll +(#6 in our series by Lewis Carroll) + +Copyright laws are changing all over the world. Be sure to check the +copyright laws for your country before downloading or redistributing +this or any other Project Gutenberg eBook. + +This header should be the first thing seen when viewing this Project +Gutenberg file. Please do not remove it. Do not change or edit the +header without written permission. + +Please read the "legal small print," and other information about the +eBook and Project Gutenberg at the bottom of this file. Included is +important information about your specific rights and restrictions in +how the file may be used. You can also find out about how to make a +donation to Project Gutenberg, and how to get involved. + + +**Welcome To The World of Free Plain Vanilla Electronic Texts** + +**eBooks Readable By Both Humans and By Computers, Since 1971** + +*****These eBooks Were Prepared By Thousands of Volunteers!***** + + +Title: The Game of Logic + +Author: Lewis Carroll + +Release Date: December, 2003 [EBook #4763] +[Yes, we are more than one year ahead of schedule] +[This file was first posted on March 13, 2002] +[Date last updated: May 10, 2004] + +Edition: 10 + +Language: English + +Character set encoding: ASCII + +*** START OF THE PROJECT GUTENBERG EBOOK, THE GAME OF LOGIC *** + + + + +Scanned by Gregory D. Weeks +Transcribed by L. Lynn Smith +Proofed by Reina Hosier and Brett Fishburne + + + +THE GAME OF LOGIC + +By Lewis Carroll + + + --------------------- +|9 | 10| +| | | +| -----x------ | +| |11 | 12| | +| | | | | +|---y-----m------y'---| +| | | | | +| |13 | 14| | +| -----x'----- | +| | | +|15 | 16| + --------------------- + + COLOURS FOR ------------- + COUNTERS |5 | 6| + ___ | x | + | | | +See the Sun is overhead, |--y-------y'-| +Shining on us, FULL and | | | + RED! | x' | + |7 | 8| +Now the Sun is gone away, ------------- +And the EMPTY sky is + GREY! + ___ + + + +THE GAME OF LOGIC + +By Lewis Carrol + + + +To my Child-friend. + +I charm in vain; for never again, +All keenly as my glance I bend, + Will Memory, goddess coy, + Embody for my joy +Departed days, nor let me gaze + On thee, my fairy friend! + +Yet could thy face, in mystic grace, +A moment smile on me, 'twould send + Far-darting rays of light + From Heaven athwart the night, +By which to read in very deed + Thy spirit, sweetest friend! + +So may the stream of Life's long dream +Flow gently onward to its end, + With many a floweret gay, + Adown its willowy way: +May no sigh vex, no care perplex, + My loving little friend! + + + +NOTA BENE. + +With each copy of this Book is given an Envelope, containing a +Diagram (similar to the frontispiece) on card, and nine Counters, +four red and five grey. + +The Envelope, &c. can be had separately, at 3d. each. + +The Author will be very grateful for suggestions, especially from +beginners in Logic, of any alterations, or further explanations, +that may seem desirable. Letters should be addressed to him at +"29, Bedford Street, Covent Garden, London." + + + +PREFACE + + + +"There foam'd rebellious Logic, gagg'd and bound." + + +This Game requires nine Counters--four of one colour and five of +another: say four red and five grey. + +Besides the nine Counters, it also requires one Player, AT LEAST. +I am not aware of any Game that can be played with LESS than this +number: while there are several that require MORE: take Cricket, +for instance, which requires twenty-two. How much easier it is, +when you want to play a Game, to find ONE Player than twenty-two. +At the same time, though one Player is enough, a good deal more +amusement may be got by two working at it together, and correcting +each other's mistakes. + +A second advantage, possessed by this Game, is that, besides being +an endless source of amusement (the number of arguments, that may +be worked by it, being infinite), it will give the Players a little +instruction as well. But is there any great harm in THAT, so long +as you get plenty of amusement? + + + +CONTENTS. + + +CHAPTER PAGE + + I. NEW LAMPS FOR OLD. + 1. Propositions . . . . . . . 1 + 2. Syllogisms . . . . . . . . 20 + 3. Fallacies . . . . . . . . 32 + + II. CROSS QUESTIONS. + 1. Elementary . . . . . . . . 37 + 2. Half of Smaller Diagram. Propositions + to be represented . . . . . 40 + 3. Do. Symbols to be interpreted. . 42 + 4. Smaller Diagram. Propositions to be + represented . . . . . . . 44 + 5. Do. Symbols to be interpreted. . 46 + 6. Larger Diagram. Propositions to be + represented . . . . . . . 48 + 7. Both Diagrams to be employed . . 51 + + III. CROOKED ANSWERS. + 1. Elementary . . . . . . . . 55 + 2. Half of Smaller Diagram. Propositions + represented . . . . . . . 59 + 3. Do. Symbols interpreted . . . 61 + 4. Smaller Diagram. Propositions represented. 62 + 5. Do. Symbols interpreted . . . 65 + 6. Larger Diagram. Propositions represented. 67 + 7. Both Diagrams employed . . . . 72 + + IV. HIT OR MISS . . . . . . . . . 85 + + + + +CHAPTER I. + +NEW LAMPS FOR OLD. + + + + + + "Light come, light go." + _________ + + 1. Propositions. + + "Some new Cakes are nice." + "No new Cakes are nice." + "All new cakes are nice." + +There are three 'PROPOSITIONS' for you--the only three kinds we +are going to use in this Game: and the first thing to be done is +to learn how to express them on the Board. + +Let us begin with + +"Some new Cakes are nice." + +But before doing so, a remark has to be made--one that is rather +important, and by no means easy to understand all in a moment: so +please to read this VERY carefully. + +The world contains many THINGS (such as "Buns", "Babies", "Beetles". +"Battledores". &c.); and these Things possess many ATTRIBUTES +(such as "baked", "beautiful", "black", "broken", &c.: in fact, +whatever can be "attributed to", that is "said to belong to", any +Thing, is an Attribute). Whenever we wish to mention a Thing, we +use a SUBSTANTIVE: when we wish to mention an Attribute, we use +an ADJECTIVE. People have asked the question "Can a Thing exist +without any Attributes belonging to it?" It is a very puzzling +question, and I'm not going to try to answer it: let us turn up +our noses, and treat it with contemptuous silence, as if it really +wasn't worth noticing. But, if they put it the other way, and ask +"Can an Attribute exist without any Thing for it to belong to?", we +may say at once "No: no more than a Baby could go a railway-journey +with no one to take care of it!" You never saw "beautiful" floating +about in the air, or littered about on the floor, without any Thing +to BE beautiful, now did you? + +And now what am I driving at, in all this long rigmarole? It is +this. You may put "is" or "are" between names of two THINGS (for +example, "some Pigs are fat Animals"), or between the names of two +ATTRIBUTES (for example, "pink is light-red"), and in each case it +will make good sense. But, if you put "is" or "are" between the +name of a THING and the name of an ATTRIBUTE (for example, "some +Pigs are pink"), you do NOT make good sense (for how can a Thing +BE an Attribute?) unless you have an understanding with the person +to whom you are speaking. And the simplest understanding would, I +think, be this--that the Substantive shall be supposed to be repeated +at the end of the sentence, so that the sentence, if written out +in full, would be "some Pigs are pink (Pigs)". And now the word +"are" makes quite good sense. + +Thus, in order to make good sense of the Proposition "some new Cakes +are nice", we must suppose it to be written out in full, in the +form "some new Cakes are nice (Cakes)". Now this contains two +'TERMS'--"new Cakes" being one of them, and "nice (Cakes)" the +other. "New Cakes," being the one we are talking about, is called +the 'SUBJECT' of the Proposition, and "nice (Cakes)" the 'PREDICATE'. +Also this Proposition is said to be a 'PARTICULAR' one, since it +does not speak of the WHOLE of its Subject, but only of a PART of +it. The other two kinds are said to be 'UNIVERSAL', because they +speak of the WHOLE of their Subjects--the one denying niceness, and +the other asserting it, of the WHOLE class of "new Cakes". Lastly, +if you would like to have a definition of the word 'PROPOSITION' +itself, you may take this:--"a sentence stating that some, or +none, or all, of the Things belonging to a certain class, called +its 'Subject', are also Things belonging to a certain other class, +called its 'Predicate'". + +You will find these seven words--PROPOSITION, ATTRIBUTE, TERM, +SUBJECT, PREDICATE, PARTICULAR, UNIVERSAL--charmingly useful, if +any friend should happen to ask if you have ever studied Logic. +Mind you bring all seven words into your answer, and you friend +will go away deeply impressed--'a sadder and a wiser man'. + +Now please to look at the smaller Diagram on the Board, and suppose +it to be a cupboard, intended for all the Cakes in the world (it +would have to be a good large one, of course). And let us suppose +all the new ones to be put into the upper half (marked 'x'), and all +the rest (that is, the NOT-new ones) into the lower half (marked +'x''). Thus the lower half would contain ELDERLY Cakes, AGED +Cakes, ANTE-DILUVIAN Cakes--if there are any: I haven't seen many, +myself--and so on. Let us also suppose all the nice Cakes to be +put into the left-hand half (marked 'y'), and all the rest (that +is, the not-nice ones) into the right-hand half (marked 'y''). At +present, then, we must understand x to mean "new", x' "not-new", +y "nice", and y' "not-nice." + +And now what kind of Cakes would you expect to find in compartment +No. 5? + +It is part of the upper half, you see; so that, if it has any Cakes +in it, they must be NEW: and it is part of the left-hand half; +so that they must be NICE. Hence if there are any Cakes in this +compartment, they must have the double 'ATTRIBUTE' "new and nice": +or, if we use letters, the must be "x y." + +Observe that the letters x, y are written on two of the edges of +this compartment. This you will find a very convenient rule for +knowing what Attributes belong to the Things in any compartment. +Take No. 7, for instance. If there are any Cakes there, they must +be "x' y", that is, they must be "not-new and nice." + +Now let us make another agreement--that a red counter in a +compartment shall mean that it is 'OCCUPIED', that is, that there +are SOME Cakes in it. (The word 'some,' in Logic, means 'one or +more' so that a single Cake in a compartment would be quite enough +reason for saying "there are SOME Cakes here"). Also let us agree +that a grey counter in a compartment shall mean that it is 'EMPTY', +that is that there are NO Cakes in it. In the following Diagrams, +I shall put '1' (meaning 'one or more') where you are to put a RED +counter, and '0' (meaning 'none') where you are to put a GREY one. + +As the Subject of our Proposition is to be "new Cakes", we are only +concerned, at present, with the UPPER half of the cupboard, where +all the Cakes have the attribute x, that is, "new." + +Now, fixing our attention on this upper half, suppose we found it +marked like this, + + + ----------- + | | | + | 1 | | + | | | + ----------- + + +that is, with a red counter in No. 5. What would this tell us, +with regard to the class of "new Cakes"? + +Would it not tell us that there are SOME of them in the x y-compartment? +That is, that some of them (besides having the Attribute x, which +belongs to both compartments) have the Attribute y (that is, "nice"). +This we might express by saying "some x-Cakes are y-(Cakes)", or, +putting words instead of letters, + + +"Some new Cakes are nice (Cakes)", + + +or, in a shorter form, + + +"Some new Cakes are nice". + + +At last we have found out how to represent the first Proposition +of this Section. If you have not CLEARLY understood all I have +said, go no further, but read it over and over again, till you DO +understand it. After that is once mastered, you will find all the +rest quite easy. + +It will save a little trouble, in doing the other Propositions, +if we agree to leave out the word "Cakes" altogether. I find it +convenient to call the whole class of Things, for which the cupboard +is intended, the 'UNIVERSE.' Thus we might have begun this business +by saying "Let us take a Universe of Cakes." (Sounds nice, doesn't +it?) + +Of course any other Things would have done just as well as Cakes. +We might make Propositions about "a Universe of Lizards", or even +"a Universe of Hornets". (Wouldn't THAT be a charming Universe to +live in?) + +So far, then, we have learned that + + + ----------- + | | | + | 1 | | + | | | + ----------- + + +means "some x and y," i.e. "some new are nice." + +I think you will see without further explanation, that + + + ----------- + | | | + | | 1 | + | | | + ----------- + + +means "some x are y'," i.e. "some new are not-nice." + +Now let us put a GREY counter into No. 5, and ask ourselves the +meaning of + + + ----------- + | | | + | 0 | | + | | | + ----------- + + +This tells us that the x y-compartment is EMPTY, which we may express +by "no x are y", or, "no new Cakes are nice". This is the second +of the three Propositions at the head of this Section. + +In the same way, + + + ----------- + | | | + | | 0 | + | | | + ----------- + + +would mean "no x are y'," or, "no new Cakes are not-nice." + +What would you make of this, I wonder? + + + ----------- + | | | + | 1 | 1 | + | | | + ----------- + + +I hope you will not have much trouble in making out that this +represents a DOUBLE Proposition: namely, "some x are y, AND some +are y'," i.e. "some new are nice, and some are not-nice." + +The following is a little harder, perhaps: + + + ----------- + | | | + | 0 | 0 | + | | | + ----------- + + +This means "no x are y, AND none are y'," i.e. "no new are nice, +AND none are not-nice": which leads to the rather curious result +that "no new exist," i.e. "no Cakes are new." This is because +"nice" and "not-nice" make what we call an 'EXHAUSTIVE' division +of the class "new Cakes": i.e. between them, they EXHAUST the whole +class, so that all the new Cakes, that exist, must be found in one +or the other of them. + +And now suppose you had to represent, with counters the contradictory +to "no Cakes are new", which would be "some Cakes are new", or, +putting letters for words, "some Cakes are x", how would you do +it? + +This will puzzle you a little, I expect. Evidently you must put +a red counter SOMEWHERE in the x-half of the cupboard, since you +know there are SOME new Cakes. But you must not put it into the +LEFT-HAND compartment, since you do not know them to be NICE: nor +may you put it into the RIGHT-HAND one, since you do not know them +to be NOT-NICE. + +What, then, are you to do? I think the best way out of the +difficulty is to place the red counter ON THE DIVISION-LINE between +the xy-compartment and the xy'-compartment. This I shall represent +(as I always put '1' where you are to put a red counter) by the +diagram + + + ----------- + | | | + | -1- | + | | | + ----------- + + +Our ingenious American cousins have invented a phrase to express +the position of a man who wants to join one or the other of two +parties--such as their two parties 'Democrats' and 'Republicans'--but +can't make up his mind WHICH. Such a man is said to be "sitting +on the fence." Now that is exactly the position of the red counter +you have just placed on the division-line. He likes the look of +No. 5, and he likes the look of No. 6, and he doesn't know WHICH to +jump down into. So there he sits astride, silly fellow, dangling +his legs, one on each side of the fence! + +Now I am going to give you a much harder one to make out. What +does this mean? + + + ----------- + | | | + | 1 | 0 | + | | | + ----------- + + +This is clearly a DOUBLE Proposition. It tells us not only that +"some x are y," but also the "no x are NOT y." Hence the result +is "ALL x are y," i.e. "all new Cakes are nice", which is the last +of the three Propositions at the head of this Section. + +We see, then, that the Universal Proposition + + +"All new Cakes are nice" + + +consists of TWO Propositions taken together, namely, + + + "Some new Cakes are nice," + and "No new Cakes are not-nice." + + +In the same way + + + ----------- + | | | + | 0 | 1 | + | | | + ----------- + + +would mean "all x are y' ", that is, + + +"All new Cakes are not-nice." + + +Now what would you make of such a Proposition as "The Cake you have +given me is nice"? Is it Particular or Universal? + + +"Particular, of course," you readily reply. "One single Cake is +hardly worth calling 'some,' even." + + +No, my dear impulsive Reader, it is 'Universal'. Remember that, +few as they are (and I grant you they couldn't well be fewer), +they are (or rather 'it is') ALL that you have given me! Thus, if +(leaving 'red' out of the question) I divide my Universe of Cakes +into two classes--the Cakes you have given me (to which I assign +the upper half of the cupboard), and those you HAVEN'T given me +(which are to go below)--I find the lower half fairly full, and the +upper one as nearly as possible empty. And then, when I am told +to put an upright division into each half, keeping the NICE Cakes +to the left, and the NOT-NICE ones to the right, I begin by carefully +collecting ALL the Cakes you have given me (saying to myself, from +time to time, "Generous creature! How shall I ever repay such +kindness?"), and piling them up in the left-hand compartment. AND +IT DOESN'T TAKE LONG TO DO IT! + +Here is another Universal Proposition for you. "Barzillai Beckalegg +is an honest man." That means "ALL the Barzillai Beckaleggs, that +I am now considering, are honest men." (You think I invented that +name, now don't you? But I didn't. It's on a carrier's cart, +somewhere down in Cornwall.) + +This kind of Universal Proposition (where the Subject is a single +Thing) is called an 'INDIVIDUAL' Proposition. + +Now let us take "NICE Cakes" as the Subject of Proposition: that +is, let us fix our thoughts on the LEFT-HAND half of the cupboard, +where all the Cakes have attribute y, that is, "nice." + + + ----- + Suppose we find it marked like this:-- | | + | 1 | + What would that tell us? | | + ----- + | | + | | + | | + ----- + + +I hope that it is not necessary, after explaining the HORIZONTAL +oblong so fully, to spend much time over the UPRIGHT one. I hope +you will see, for yourself, that this means "some y are x", that +is, + + +"Some nice Cakes are new." + + +"But," you will say, "we have had this case before. You put a red +counter into No. 5, and you told us it meant 'some new Cakes are +nice'; and NOW you tell us that it means 'some NICE Cakes are NEW'! +Can it mean BOTH?" + +The question is a very thoughtful one, and does you GREAT credit, +dear Reader! It DOES mean both. If you choose to take x (that +is, "new Cakes") as your Subject, and to regard No. 5 as part of a +HORIZONTAL oblong, you may read it "some x are y", that is, "some +new Cakes are nice": but, if you choose to take y (that is, "nice +Cake") as your Subject, and to regard No. 5 as part of an UPRIGHT +oblong, THEN you may read it "some y are x", that is, "some nice +Cakes are new". They are merely two different ways of expressing +the very same truth. + +Without more words, I will simply set down the other ways in which +this upright oblong might be marked, adding the meaning in each +case. By comparing them with the various cases of the horizontal +oblong, you will, I hope, be able to understand them clearly. + +You will find it a good plan to examine yourself on this table, +by covering up first one column and then the other, and 'dodging +about', as the children say. + +Also you will do well to write out for yourself two other tables--one +for the LOWER half of the cupboard, and the other for its RIGHT-HAND +half. + +And now I think we have said all we need to say about the smaller +Diagram, and may go on to the larger one. + +_________________________________________________ + | + Symbols. | Meanings. +_______________|_________________________________ + ----- | + | | | + | | | Some y are x'; + | | | i.e. Some nice are not-new. + ----- | + | | | + | 1 | | + | | | + ----- | + | + ----- | + | | | No y are x; + | 0 | | i.e. No nice are new. + | | | + ----- | [Observe that this is merely another way of + | | | expressing "No new are nice."] + | | | + | | | + ----- | + | + ----- | + | | | + | | | No y are x'; + | | | i.e. No nice are not-new. + ----- | + | | | + | 0 | | + | | | + ----- | + | + ----- | + | | | + | 1 | | Some y are x, and some are x'; + | | | i.e. Some nice are new, and some are + ----- | not-new. + | | | + | 1 | | + | | | + ----- | + | + ----- | + | | | + | 0 | | No y are x, and none are x'; i.e. No y + | | | exist; + ----- | i.e. No Cakes are nice. + | | | + | 0 | | + | | | + ----- | + | + ----- | + | | | + | 1 | | All y are x; + | | | i.e. All nice are new. + ----- | + | | | + | 0 | | + | | | + ----- | + | + ----- | + | | | + | 0 | | All y are x'; + | | | i.e. All nice are not-new. + ----- | + | | | + | 1 | | + | | | + ----- | +_______________|_________________________________ + +This may be taken to be a cupboard divided in the same way as the +last, but ALSO divided into two portions, for the Attribute m. Let +us give to m the meaning "wholesome": and let us suppose that all +WHOLESOME Cakes are placed INSIDE the central Square, and all the +UNWHOLESOME ones OUTSIDE it, that is, in one or other of the four +queer-shaped OUTER compartments. + +We see that, just as, in the smaller Diagram, the Cakes in each +compartment had TWO Attributes, so, here, the Cakes in each compartment +have THREE Attributes: and, just as the letters, representing the +TWO Attributes, were written on the EDGES of the compartment, so, +here, they are written at the CORNERS. (Observe that m' is supposed +to be written at each of the four outer corners.) So that we can +tell in a moment, by looking at a compartment, what three Attributes +belong to the Things in it. For instance, take No. 12. Here we +find x, y', m, at the corners: so we know that the Cakes in it, if +there are any, have the triple Attribute, 'xy'm', that is, "new, +not-nice, and wholesome." Again, take No. 16. Here we find, at +the corners, x', y', m': so the Cakes in it are "not-new, not-nice, +and unwholesome." (Remarkably untempting Cakes!) + +It would take far too long to go through all the Propositions, +containing x and y, x and m, and y and m which can be represented +on this diagram (there are ninety-six altogether, so I am sure you +will excuse me!) and I must content myself with doing two or three, +as specimens. You will do well to work out a lot more for yourself. + +Taking the upper half by itself, so that our Subject is "new Cakes", +how are we to represent "no new Cakes are wholesome"? + +This is, writing letters for words, "no x are m." Now this tells us +that none of the Cakes, belonging to the upper half of the cupboard, +are to be found INSIDE the central Square: that is, the two +compartments, No. 11 and No. 12, are EMPTY. And this, of course, +is represented by + + + ------------------- + | | | + | _____|_____ | + | | | | | + | | 0 | 0 | | + | | | | | + ------------------- + + +And now how are we to represent the contradictory Proposition "SOME x +are m"? This is a difficulty I have already considered. I think +the best way is to place a red counter ON THE DIVISION-LINE between +No. 11 and No. 12, and to understand this to mean that ONE of the +two compartments is 'occupied,' but that we do not at present know WHICH. +This I shall represent thus:-- + + + ------------------- + | | | + | _____|_____ | + | | | | | + | | -1- | | + | | | | | + ------------------- + +Now let us express "all x are m." + +This consists, we know, of TWO Propositions, + + + "Some x are m," + and "No x are m'." + +Let us express the negative part first. This tells us that none +of the Cakes, belonging to the upper half of the cupboard, are to +be found OUTSIDE the central Square: that is, the two compartments, +No. 9 and No. 10, are EMPTY. This, of course, is represented by + + + ------------------- + | 0 | 0 | + | _____|_____ | + | | | | | + | | | | | + | | | | | + ------------------- + +But we have yet to represent "Some x are m." This tells us that +there are SOME Cakes in the oblong consisting of No. 11 and No. +12: so we place our red counter, as in the previous example, on +the division-line between No. 11 and No. 12, and the result is + + + ------------------- + | 0 | 0 | + | _____|_____ | + | | | | | + | | -1- | | + | | | | | + ------------------- + + +Now let us try one or two interpretations. + +What are we to make of this, with regard to x and y? + + + ------------------- + | | 0 | + | _____|_____ | + | | | | | + | | 1 | 0 | | + | | | | | + ------------------- + + +This tells us, with regard to the xy'-Square, that it is wholly +'empty', since BOTH compartments are so marked. With regard to +the xy-Square, it tells us that it is 'occupied'. True, it is only +ONE compartment of it that is so marked; but that is quite enough, +whether the other be 'occupied' or 'empty', to settle the fact that +there is SOMETHING in the Square. + +If, then, we transfer our marks to the smaller Diagram, so as to +get rid of the m-subdivisions, we have a right to mark it + + + ----------- + | | | + | 1 | 0 | + | | | + ----------- + + +which means, you know, "all x are y." + +The result would have been exactly the same, if the given oblong +had been marked thus:-- + + + ------------------- + | 1 | 0 | + | _____|_____ | + | | | | | + | | | 0 | | + | | | | | + ------------------- + + +Once more: how shall we interpret this, with regard to x and y? + + + ------------------- + | 0 | 1 | + | _____|_____ | + | | | | | + | | | | | + | | | | | + ------------------- + + +This tells us, as to the xy-Square, that ONE of its compartments +is 'empty'. But this information is quite useless, as there is no +mark in the OTHER compartment. If the other compartment happened +to be 'empty' too, the Square would be 'empty': and, if it happened +to be 'occupied', the Square would be 'occupied'. So, as we do +not know WHICH is the case, we can say nothing about THIS Square. + +The other Square, the xy'-Square, we know (as in the previous +example) to be 'occupied'. + +If, then, we transfer our marks to the smaller Diagram, we get +merely this:-- + + + ----------- + | | | + | | 1 | + | | | + ----------- + + +which means, you know, "some x are y'." + +These principles may be applied to all the other +oblongs. For instance, to represent +"all y' are m'" we should mark the ------- +RIGHT-HAND UPRIGHT OBLONG (the one | | +that has the attribute y') thus:-- |--- | + | 0 | | + |---|-1-| + | 0 | | + |--- | + | | + ------- + +and, if we were told to interpret the lower half of the cupboard, +marked as follows, with regard to x and y, + + + ------------------- + | | | | | + | | | 0 | | + | | | | | + | -----|----- | + | 1 | 0 | + ------------------- + + +we should transfer it to the smaller Diagram thus, + + + ----------- + | | | + | 1 | 0 | + | | | + ----------- + + +and read it "all x' are y." + +Two more remarks about Propositions need to be made. + +One is that, in every Proposition beginning with "some" or "all", +the ACTUAL EXISTENCE of the 'Subject' is asserted. If, for instance, +I say "all misers are selfish," I mean that misers ACTUALLY EXIST. +If I wished to avoid making this assertion, and merely to state +the LAW that miserliness necessarily involves selfishness, I should +say "no misers are unselfish" which does not assert that any misers +exist at all, but merely that, if any DID exist, they WOULD be +selfish. + +The other is that, when a Proposition begins with "some" or "no", +and contains more that two Attributes, these Attributes may be +re-arranged, and shifted from one Term to the other, "ad libitum." +For example, "some abc are def" may be re-arranged as "some bf are +acde," each being equivalent to "some Things are abcdef". Again "No +wise old men are rash and reckless gamblers" may be re-arranged as +"No rash old gamblers are wise and reckless," each being equivalent +to "No men are wise old rash reckless gamblers." + + + +2. Syllogisms + + +Now suppose we divide our Universe of Things in three ways, with regard +to three different Attributes. Out of these three Attributes, we +may make up three different couples (for instance, if they were a, +b, c, we might make up the three couples ab, ac, bc). Also suppose +we have two Propositions given us, containing two of these three +couples, and that from them we can prove a third Proposition containing +the third couple. (For example, if we divide our Universe for m, +x, and y; and if we have the two Propositions given us, "no m are +x'" and "all m' are y", containing the two couples mx and my, it +might be possible to prove from them a third Proposition, containing +x and y.) + +In such a case we call the given Propositions 'THE PREMISSES', the +third one 'THE CONCLUSION' and the whole set 'A SYLLOGISM'. + +Evidently, ONE of the Attributes must occur in both Premisses; or +else one must occur in ONE Premiss, and its CONTRADICTORY in the +other. + +In the first case (when, for example, the Premisses are "some m +are x" and "no m are y'") the Term, which occurs twice, is called +'THE MIDDLE TERM', because it serves as a sort of link between the +other two Terms. + +In the second case (when, for example, the Premisses are "no +m are x'" and "all m' are y") the two Terms, which contain these +contradictory Attributes, may be called 'THE MIDDLE TERMS'. + +Thus, in the first case, the class of "m-Things" is the Middle +Term; and, in the second case, the two classes of "m-Things" and +"m'-Things" are the Middle Terms. + +The Attribute, which occurs in the Middle Term or Terms, disappears +in the Conclusion, and is said to be "eliminated", which literally +means "turned out of doors". + +Now let us try to draw a Conclusion from the two Premisses-- + + "Some new Cakes are unwholesome; + No nice Cakes are unwholesome." + +In order to express them with counters, we need to divide Cakes in +THREE different ways, with regard to newness, to niceness, and to +wholesomeness. For this we must use the larger Diagram, making x +mean "new", y "nice", and m "wholesome". (Everything INSIDE the +central Square is supposed to have the attribute m, and everything +OUTSIDE it the attribute m', i.e. "not-m".) + +You had better adopt the rule to make m mean the Attribute which +occurs in the MIDDLE Term or Terms. (I have chosen m as the symbol, +because 'middle' begins with 'm'.) + +Now, in representing the two Premisses, I prefer to begin with the +NEGATIVE one (the one beginning with "no"), because GREY counters +can always be placed with CERTAINTY, and will then help to fix the +position of the red counters, which are sometimes a little uncertain +where they will be most welcome. + +Let us express, the "no nice Cakes are unwholesome (Cakes)", i.e. +"no y-Cakes are m'-(Cakes)". This tells us that none of the Cakes +belonging to the y-half of the cupboard are in its m'-compartments +(i.e. the ones outside the central Square). Hence the two compartments, +No. 9 and No. 15, are both 'EMPTY'; and we must place a grey counter +in EACH of them, thus:-- + + + ----------- + |0 | | + | --|-- | + | | | | | + |--|-----|--| + | | | | | + | --|-- | + |0 | | + ----------- + + +We have now to express the other Premiss, namely, "some new Cakes +are unwholesome (Cakes)", i.e. "some x-Cakes are m'-(Cakes)". This +tells us that some of the Cakes in the x-half of the cupboard are +in its m'-compartments. Hence ONE of the two compartments, No. +9 and No. 10, is 'occupied': and, as we are not told in WHICH of +these two compartments to place the red counter, the usual rule +would be to lay it on the division-line between them: but, in this +case, the other Premiss has settled the matter for us, by declaring +No. 9 to be EMPTY. Hence the red counter has no choice, and MUST +go into No. 10, thus:-- + + + ----------- + |0 | 1| + | --|-- | + | | | | | + |--|-----|--| + | | | | | + | --|-- | + |0 | | + ----------- + + +And now what counters will this information enable us to place in +the SMALLER Diagram, so as to get some Proposition involving x and +y only, leaving out m? Let us take its four compartments, one by +one. + +First, No. 5. All we know about THIS is that its OUTER portion +is empty: but we know nothing about its inner portion. Thus the +Square MAY be empty, or it MAY have something in it. Who can tell? +So we dare not place ANY counter in this Square. + +Secondly, what of No. 6? Here we are a little better off. We +know that there is SOMETHING in it, for there is a red counter in +its outer portion. It is true we do not know whether its inner +portion is empty or occupied: but what does THAT matter? One solitary +Cake, in one corner of the Square, is quite sufficient excuse for +saying "THIS SQUARE IS OCCUPIED", and for marking it with a red +counter. + +As to No. 7, we are in the same condition as with No. 5--we find +it PARTLY 'empty', but we do not know whether the other part is +empty or occupied: so we dare not mark this Square. + +And as to No. 8, we have simply no information at all. + +The result is + + ------- + | | 1 | + |---|---| + | | | + ------- + + +Our 'Conclusion', then, must be got out of the rather meager piece +of information that there is a red counter in the xy'-Square. +Hence our Conclusion is "some x are y' ", i.e. "some new Cakes are +not-nice (Cakes)": or, if you prefer to take y' as your Subject, +"some not-nice Cakes are new (Cakes)"; but the other looks neatest. + +We will now write out the whole Syllogism, putting the symbol +&there4[*] for "therefore", and omitting "Cakes", for the sake of +brevity, at the end of each Proposition. + +[*][NOTE from Brett: The use of "&there4" is a rather arbitrary +selection. There is no font available in general practice +which renders the "therefore" symbol correction (three dots in a +triangular formation). This can be done, however, in HTML, so if +this document is read in a browser, then the symbol will be properly +recognized. This is a poor man's excuse.] + + "Some new Cakes are unwholesome; + No nice Cakes are unwholesome + &there4 Some new Cakes are not-nice." + +And you have now worked out, successfully, your first 'SYLLOGISM'. +Permit me to congratulate you, and to express the hope that it is +but the beginning of a long and glorious series of similar victories! + +We will work out one other Syllogism--a rather harder one than the +last--and then, I think, you may be safely left to play the Game +by yourself, or (better) with any friend whom you can find, that +is able and willing to take a share in the sport. + +Let us see what we can make of the two Premisses-- + + + "All Dragons are uncanny; + All Scotchmen are canny." + + +Remember, I don't guarantee the Premisses to be FACTS. In the +first place, I never even saw a Dragon: and, in the second place, +it isn't of the slightest consequence to us, as LOGICIANS, whether +our Premisses are true or false: all WE have to do is to make out +whether they LEAD LOGICALLY TO THE CONCLUSION, so that, if THEY +were true, IT would be true also. + +You see, we must give up the "Cakes" now, or our cupboard will +be of no use to us. We must take, as our 'Universe', some class +of things which will include Dragons and Scotchmen: shall we say +'Animals'? And, as "canny" is evidently the Attribute belonging +to the 'Middle Terms', we will let m stand for "canny", x for +"Dragons", and y for "Scotchmen". So that our two Premisses are, +in full, + + + "All Dragon-Animals are uncanny (Animals); + All Scotchman-Animals are canny (Animals)." + + +And these may be expressed, using letters for words, thus:-- + + + "All x are m'; + All y are m." + + +The first Premiss consists, as you already know, of two parts:-- + + + "Some x are m'," + and "No x are m." + + +And the second also consists of two parts:-- + + + "Some y are m," + and "No y are m'." + + +Let us take the negative portions first. + +We have, then, to mark, on the larger Diagram, first, "no x are +m", and secondly, "no y are m'". I think you will see, without +further explanation, that the two results, separately, are + + + ----------- ----------- + | | | |0 | | + | --|-- | | --|-- | + | |0 | 0| | | | | | | + |--|--|--|--| |--|--|--|--| + | | | | | | | | | | + | --|-- | | --|-- | + | | | |0 | | + ----------- ----------- + + + +and that these two, when combined, give us + + + ----------- + |0 | | + | --|-- | + | |0 | 0| | + |--|--|--|--| + | | | | | + | --|-- | + |0 | | + ----------- + + +We have now to mark the two positive portions, "some x are m'" +and "some y are m". + +The only two compartments, available for Things which are xm', are +No. 9 and No. 10. Of these, No. 9 is already marked as 'empty'; +so our red counter must go into No. 10. + +Similarly, the only two, available for ym, are No. 11 and No. 13. +Of these, No. 11 is already marked as 'empty'; so our red counter +MUST go into No. 13. + +The final result is + + + ----------- + |0 | 1| + | --|-- | + | |0 | 0| | + |--|--|--|--| + | |1 | | | + | --|-- | + |0 | | + ----------- + + + +And now how much of this information can usefully be transferred +to the smaller Diagram? + +Let us take its four compartments, one by one. + +As to No. 5? This, we see, is wholly 'empty'. (So mark it with a +grey counter.) + +As to No. 6? This, we see, is 'occupied'. (So mark it with a red +counter.) + +As to No. 7? Ditto, ditto. + +As to No. 8? No information. + +The smaller Diagram is now pretty liberally marked:-- + + + ------- + | 0 | 1 | + |---|---| + | 1 | | + ------- + + +And now what Conclusion can we read off from this? Well, it is +impossible to pack such abundant information into ONE Proposition: +we shall have to indulge in TWO, this time. + +First, by taking x as Subject, we get "all x are y'", that is, + + +"All Dragons are not-Scotchmen": + + +secondly, by taking y as Subject, we get "all y are x'", that is, + + +"All Scotchmen are not-Dragons". + + +Let us now write out, all together, our two Premisses and our brace +of Conclusions. + + + "All Dragons are uncanny; + All Scotchmen are canny. + &there4 All Dragons are not-Scotchmen; + All Scotchmen are not-Dragons." + + +Let me mention, in conclusion, that you may perhaps meet with +logical treatises in which it is not assumed that any Thing EXISTS +at all, by "some x are y" is understood to mean "the Attributes x, +y are COMPATIBLE, so that a Thing can have both at once", and "no +x are y" to mean "the Attributes x, y are INCOMPATIBLE, so that +nothing can have both at once". + +In such treatises, Propositions have quite different meanings +from what they have in our 'Game of Logic', and it will be well to +understand exactly what the difference is. + +First take "some x are y". Here WE understand "are" to mean "are, +as an actual FACT"--which of course implies that some x-Things EXIST. +But THEY (the writers of these other treatises) only understand +"are" to mean "CAN be", which does not at all imply that any EXIST. +So they mean LESS than we do: our meaning includes theirs (for of +course "some x ARE y" includes "some x CAN BE y"), but theirs does +NOT include ours. For example, "some Welsh hippopotami are heavy" +would be TRUE, according to these writers (since the Attributes +"Welsh" and "heavy" are quite COMPATIBLE in a hippopotamus), but +it would be FALSE in our Game (since there are no Welsh hippopotami +to BE heavy). + +Secondly, take "no x are y". Here WE only understand "are" to +mean "are, as an actual FACT"--which does not at all imply that no +x CAN be y. But THEY understand the Proposition to mean, not only +that none ARE y, but that none CAN POSSIBLY be y. So they mean +more than we do: their meaning includes ours (for of course "no x +CAN be y" includes "no x ARE y"), but ours does NOT include theirs. +For example, "no Policemen are eight feet high" would be TRUE +in our Game (since, as an actual fact, no such splendid specimens +are ever found), but it would be FALSE, according to these writers +(since the Attributes "belonging to the Police Force" and "eight +feet high" are quite COMPATIBLE: there is nothing to PREVENT a +Policeman from growing to that height, if sufficiently rubbed with +Rowland's Macassar Oil--which said to make HAIR grow, when rubbed +on hair, and so of course will make a POLICEMAN grow, when rubbed +on a Policeman). + +Thirdly, take "all x are y", which consists of the two partial +Propositions "some x are y" and "no x are y'". Here, of course, +the treatises mean LESS than we do in the FIRST part, and more than +we do in the SECOND. But the two operations don't balance each +other--any more than you can console a man, for having knocked down +one of his chimneys, by giving him an extra door-step. + +If you meet with Syllogisms of this kind, you may work them, quite +easily, by the system I have given you: you have only to make +'are' mean 'are CAPABLE of being', and all will go smoothly. For +"some x are y" will become "some x are capable of being y", that +is, "the Attributes x, y are COMPATIBLE". And "no x are y" will +become "no x are capable of being y", that is, "the Attributes +x, y are INCOMPATIBLE". And, of course, "all x are y" will become +"some x are capable of being y, and none are capable of being y'", +that is, "the Attributes x, y are COMPATIBLE, and the Attributes +x, y' are INCOMPATIBLE." In using the Diagrams for this system, +you must understand a red counter to mean "there may POSSIBLY be +something in this compartment," and a grey one to mean "there cannot +POSSIBLY be anything in this compartment." + + + +3. Fallacies. + + +And so you think, do you, that the chief use of Logic, in real life, +is to deduce Conclusions from workable Premisses, and to satisfy +yourself that the Conclusions, deduced by other people, are correct? +I only wish it were! Society would be much less liable to panics +and other delusions, and POLITICAL life, especially, would be a +totally different thing, if even a majority of the arguments, that +scattered broadcast over the world, were correct! But it is all +the other way, I fear. For ONE workable Pair of Premisses (I mean +a Pair that lead to a logical Conclusion) that you meet with in +reading your newspaper or magazine, you will probably find FIVE +that lead to no Conclusion at all: and, even when the Premisses +ARE workable, for ONE instance, where the writer draws a correct +Conclusion, there are probably TEN where he draws an incorrect one. + +In the first case, you may say "the PREMISSES are fallacious": in +the second, "the CONCLUSION is fallacious." + +The chief use you will find, in such Logical skill as this Game +may teach you, will be in detecting 'FALLACIES' of these two kinds. + +The first kind of Fallacy--'Fallacious Premisses'--you will detect +when, after marking them on the larger Diagram, you try to transfer +the marks to the smaller. You will take its four compartments, one +by one, and ask, for each in turn, "What mark can I place HERE?"; +and in EVERY one the answer will be "No information!", showing that +there is NO CONCLUSION AT ALL. For instance, + + + "All soldiers are brave; + Some Englishmen are brave. + &there4 Some Englishmen are soldiers." + + +looks uncommonly LIKE a Syllogism, and might easily take in a +less experienced Logician. But YOU are not to be caught by such +a trick! You would simply set out the Premisses, and would then +calmly remark "Fallacious PREMISSES!": you wouldn't condescend to +ask what CONCLUSION the writer professed to draw--knowing that, +WHATEVER it is, it MUST be wrong. You would be just as safe as +that wise mother was, who said "Mary, just go up to the nursery, +and see what Baby's doing, AND TELL HIM NOT TO DO IT!" + +The other kind of Fallacy--'Fallacious Conclusion'--you will not +detect till you have marked BOTH Diagrams, and have read off the +correct Conclusion, and have compared it with the Conclusion which +the writer has drawn. + +But mind, you mustn't say "FALLACIOUS Conclusion," simply because +it is not IDENTICAL with the correct one: it may be a PART of the +correct Conclusion, and so be quite correct, AS FAR AS IT GOES. In +this case you would merely remark, with a pitying smile, "DEFECTIVE +Conclusion!" Suppose, of example, you were to meet with this +Syllogism:-- + + + "All unselfish people are generous; + No misers are generous. + &there4 No misers are unselfish." + + +the Premisses of which might be thus expressed in letters:-- + + + "All x' are m; + No y are m." + + +Here the correct Conclusion would be "All x' are y'" (that is, +"All unselfish people are not misers"), while the Conclusion, drawn +by the writer, is "No y are x'," (which is the same as "No x' are +y," and so is PART of "All x' are y'.") Here you would simply say +"DEFECTIVE Conclusion!" The same thing would happen, if you were +in a confectioner's shop, and if a little boy were to come in, put +down twopence, and march off triumphantly with a single penny-bun. +You would shake your head mournfully, and would remark "Defective +Conclusion! Poor little chap!" And perhaps you would ask the +young lady behind the counter whether she would let YOU eat the +bun, which the little boy had paid for and left behind him: and +perhaps SHE would reply "Sha'n't!" + +But if, in the above example, the writer had drawn the Conclusion +"All misers are selfish" (that is, "All y are x"), this would +be going BEYOND his legitimate rights (since it would assert the +EXISTENCE of y, which is not contained in the Premisses), and you +would very properly say "Fallacious Conclusion!" + +Now, when you read other treatises on Logic, you will meet with +various kinds of (so-called) 'Fallacies' which are by no means +ALWAYS so. For example, if you were to put before one of these +Logicians the Pair of Premisses + + + "No honest men cheat; + No dishonest men are trustworthy." + + +and were to ask him what Conclusion followed, he would probably say +"None at all! Your Premisses offend against TWO distinct Rules, and +are as fallacious as they can well be!" Then suppose you were bold +enough to say "The Conclusion is 'No men who cheat are trustworthy'," +I fear your Logical friend would turn away hastily--perhaps angry, +perhaps only scornful: in any case, the result would be unpleasant. +I ADVISE YOU NOT TO TRY THE EXPERIMENT! + +"But why is this?" you will say. "Do you mean to tell us that all +these Logicians are wrong?" Far from it, dear Reader! From THEIR +point of view, they are perfectly right. But they do not include, +in their system, anything like ALL the possible forms of Syllogisms. + +They have a sort of nervous dread of Attributes beginning with a +negative particle. For example, such Propositions as "All not-x +are y," "No x are not-y," are quite outside their system. And +thus, having (from sheer nervousness) excluded a quantity of very +useful forms, they have made rules which, though quite applicable +to the few forms which they allow of, are no use at all when you +consider all possible forms. + +Let us not quarrel with them, dear Reader! There is room enough in +the world for both of us. Let us quietly take our broader system: +and, if they choose to shut their eyes to all these useful forms, +and to say "They are not Syllogisms at all!" we can but stand aside, +and let them Rush upon their Fate! There is scarcely anything of +yours, upon which it is so dangerous to Rush, as your Fate. You +may Rush upon your Potato-beds, or your Strawberry-beds, without +doing much harm: you may even Rush upon your Balcony (unless it +is a new house, built by contract, and with no clerk of the works) +and may survive the foolhardy enterprise: but if you once Rush upon +your FATE--why, you must take the consequences! + + + + + +CHAPTER II. + +CROSS QUESTIONS. + + "The Man in the Wilderness asked of me + 'How many strawberries grow in the sea?'" + __________ + + + + + +1. Elementary. + +1. What is an 'Attribute'? Give examples. + +2. When is it good sense to put "is" or "are" between two names? +Give examples. + +3. When is it NOT good sense? Give examples. + +4. When it is NOT good sense, what is the simplest agreement to +make, in order to make good sense? + +5. Explain 'Proposition', 'Term', 'Subject', and 'Predicate'. +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. + +8. What does "some" mean in Logic? [See pp. 55, 6] + +9. In what sense do we use the word 'Universe' in this Game? + +10. What is a 'Double' Proposition? Give examples. + +11. When is a class of Things said to be 'exhaustively' divided? +Give examples. + +12. Explain the phrase "sitting on the fence." + +13. What two partial Propositions make up, when taken together, +"all x are y"? + +14. What are 'Individual' Propositions? Give examples. + +15. What kinds of Propositions 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. + +__________ + + +Break up each of the following into two partial +Propositions: + +17. All tigers are fierce. + +18. All hard-boiled eggs are unwholesome. + +19. I am happy. + +20. John is not at home. + +__________ + +[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. + +22. Explain 'Premisses', 'Conclusion', and 'Syllogism'. Give +examples. + +23. Explain the phrases '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? + +25. Why is it of no consequence to us, as Logicians, whether the +Premisses 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"? + +27. What are the two kinds of 'Fallacies'? + +28. How may we detect 'Fallacious Premisses'? + +29. How may we detect 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? + +[See pp. 57-59] + + + + +2. Half of Smaller Diagram. + + +Propositions to be represented. + + + ----------- + | | | + | x | + | | | + --y-----y'- + +__________ + +1. Some x are not-y. + +2. All x are not-y. + +3. Some x are y, and some are not-y. + +4. No x exist. + +5. Some x exist. + +6. No x are not-y. + +7. Some x are not-y, and some x exist. + +__________ + +Taking x="judges"; y="just"; + +8. No judges are just. + +9. Some judges are unjust. + +10. All judges are just. + +__________ + +Taking x="plums"; y="wholesome"; + +11. Some plums are wholesome. + +12. There are no wholesome plums. + +13. Plums are some of them wholesome, and some not. + +14. All plums are unwholesome. + +[See pp. 59, 60] + + ----- + | | + | x + | | + |--y--| + | | + | x' + | | + ----- + +__________ + +Taking y="diligent students"; x="successful"; + +15. No diligent students are unsuccessful. + +16. All diligent students are successful. + +17. No students are diligent. + +18. There are some diligent, but unsuccessful, students. + +19. Some students are diligent. + +[See pp. 60, 1] + + + + +3. Half of Smaller Diagram. + + + +Symbols to be interpreted. + +__________ + + + ----------- + | | | + | x | + | | | + --y-----y'- + +__________ + + + ------- ------- + | | | | | | + 1. | | 0 | 2. | 0 | 0 | + | | | | | | + ------- ------- + + + ------- ------- + | | | | | | + 3. | - | 4. | 0 | 1 | + | | | | | | + ------- ------- + +__________ + +Taking x="good riddles"; y="hard"; + + + ------- ------- + | | | | | | + 5. | 1 | | 6. | 1 | 0 | + | | | | | | + ------- ------- + + ------- ------- + | | | | | | + 7. | 0 | 0 | 8. | 0 | | + | | | | | | + ------- ------- + +__________ + +[See pp. 61, 2] + +Taking x="lobster"; y="selfish"; + + ------- ------- + | | | | | | + 9. | | 1 | 10. | 0 | | + | | | | | | + ------- ------- + + ------- ------- + | | | | | | + 11. | 0 | 1 | 12. | 1 | 1 | + | | | | | | + ------- ------- + +__________ + + + ----- + | | + x | + | | + |--y'-| + | | + x' | + | | + ----- + +Taking y="healthy people"; x="happy"; + + --- --- --- --- + | 0 | | | | 1 | | 0 | + 13. |---| 14. |-1-| 15. |---| 16. |---| + | 1 | | | | 1 | | | + --- --- --- --- + +[See p. 62] + + + + +4. Smaller Diagram. + + + +Propositions to be represented. + + ----------- + | | | + | x | + |--y--|--y'-| + | x' | + | | | + ----------- + +__________ + + +1. All y are x. + +2. Some y are not-x. + +3. No not-x are not-y. + +4. Some x are not-y. + +5. Some not-y are x. + +6. No not-x are y. + +7. Some not-x are not-y. + +8. All not-x are not-y. + +9. Some not-y exist. + +10. No not-x exist. + +11. Some y are x, and some are not-x. + +12. All x are y, and all not-y are not-x. + + +[See pp. 62, 3] + +Taking "nations" as Universe; x="civilised"; +y="warlike"; + +13. No uncivilised nation is warlike. + +14. All unwarlike nations are uncivilised. + +15. Some nations are unwarlike. + +16. All warlike nations are civilised, and all civilised nations +are warlike. + +17. No nation is uncivilised. + +__________ + +Taking "crocodiles" as Universe; x="hungry"; and +y="amiable"; + +18. All hungry crocodiles are unamiable. + +19. No crocodiles are amiable when hungry. + +20. Some crocodiles, when not hungry, are amiable; but some +are not. + +21. No crocodiles are amiable, and some are hungry. + +22. All crocodiles, when not hungry, are amiable; and all +unamiable crocodiles are hungry. + +23. Some hungry crocodiles are amiable, and some that are +not hungry are unamiable. + +[See pp. 63, 4] + + + + +5. Smaller Diagram. + + + +Symbols to be interpreted. + +__________ + + + ----------- + | | | + | x | + |--y--|--y'-| + | x' | + | | | + ----------- + + __________ + + + ------- ------- + | | | | | | + 1. |---|---| 2. |---|---| + | 1 | | | | 0 | + ------- ------- + + ------- ------- + | | 1 | | | | + 3. |---|---| 4. |---|---| + | | 0 | | 0 | 0 | + ------- ------- + +__________ + +Taking "houses" as Universe; x="built of brick"; and +y="two-storied"; interpret + + ------- ------- + | 0 | | | | | + 5. |---|---| 6. |---|---| + | 0 | | | - | + ------- ---|--- + + ------- ------- + | | 0 | | | | + 7. |---|---| 8. |---|---| + | | | | 0 | 1 | + ------- ------- + +[See p. 65] + +Taking "boys" as Universe; x="fat"; and y="active"; +interpret + + ------- ------- + | 1 | 1 | | | 0 | + 9. |---|---| 10. |---|---| + | | | | | 1 | + ------- ------- + + ------- ------- + | 0 | 1 | | 1 | | + 11. |---|---| 12. |---|---| + | | 0 | | 0 | 1 | + ------- ------- + +__________ + +Taking "cats" as Universe; x="green-eyed"; and +y="good-tempered"; interpret + + ------- ------- + | 0 | 0 | | | 1 | + 13. |---|---| 14. |---|---| + | | 0 | | 1 | | + ------- ------- + + ------- ------- + | 1 | | | 0 | 1 | + 15. |---|---| 16. |---|---| + | | 0 | | 1 | 0 | + ------- ------- + +[See pp. 65, 6] + + + + +6. Larger Diagram. + + + +Propositions to be represented. + +__________ + + + ----------- + | | | + | --x-- | + | | | | | + |--y--m--y'-| + | | | | | + | --x'- | + | | | + ----------- + +__________ + + +1. No x are m. + +2. Some y are m'. + +3. All m are x'. + +4. No m' are y'. + +5. No m are x; All y are m. + +6. Some x are m; No y are m. + +7. All m are x'; No m are y. + +8. No x' are m; No y' are m'. + +[See pp. 67,8] + +Taking "rabbits" as Universe; m="greedy"; x="old"; and +y="black"; represent + +9. No old rabbits are greedy. + +10. Some not-greedy rabbits are black. + +11. All white rabbits are free from greediness. + +12. All greedy rabbits are young. + +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. + +__________ + +Taking "birds" as Universe; m="that sing loud"; x="well-fed"; +and y="happy"; represent + +15. All well-fed birds sing loud; No birds, that sing loud, +are unhappy. + +16. All birds, that do not sing loud, are unhappy; No well-fed +birds fail to sing loud. + +__________ + +Taking "persons" as Universe; m="in the house"; x="John"; +and y="having a tooth-ache"; represent + +17. John is in the house; Everybody in the house is suffering +from tooth-ache. + +18. There is no one in the house but John; Nobody, out of +the house, has a tooth-ache. + +__________ + +[See pp. 68-70] + +Taking "persons" 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. + +__________ + +Choosing your own 'Universe' &c., represent + +20. I sent him to bring me a kitten; He brought me a kettle +by mistake. + + +[See pp. 70, 1] + + + + +7. Both Diagrams to be employed. + +__________ + + + ----------- + | | | ----------- + | --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. + + + ----------- ----------- + |0 | | | | | + | --|-- | | --|-- | + | |0 | 0| | | |0 | 1| | + 1. |--|--|--|--| 2. |--|--|--|--| + | |1 | | | | |0 | | | + | --|-- | | --|-- | + |0 | | | | | + ----------- ----------- + +[See p. 72] + + ----------- ----------- + | | | | | 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. + + 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. + +13. No exciting books suit feverish patients; Unexciting +books make one drowsy. + +14. Some, who deserve the fair, get their deserts; None +but the brave deserve the fair. + +15. No children are patient; No impatient person can sit +still. + +[See pp. 72-5] + +16. All pigs are fat; No skeletons are fat. + +17. No monkeys are soldiers; All monkeys are mischievous. + +18. None of my cousins are just; No judges are unjust. + +19. Some days are rainy; Rainy days are tiresome. + +20. All medicine is nasty; Senna is a medicine. + +21. Some Jews are rich; All Patagonians are Gentiles. + +22. All teetotalers like sugar; No nightingale drinks wine. + +23. No muffins are wholesome; All buns are unwholesome. + +24. No fat creatures run well; Some greyhounds run well. + +25. All soldiers march; Some youths are not soldiers. + +26. Sugar is sweet; Salt is not sweet. + +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. + +[See pp. 75-82] + +29. All battles are noisy; What makes no noise may escape +notice. + +30. No Jews are mad; All Rabbis are Jews. + +31. There are no fish that cannot swim; Some skates are +fish. + +32. All passionate people are unreasonable; Some orators +are passionate. + + + + +[See pp. 82-84] + + + + + +CHAPTER III. + +CROOKED ANSWERS. + + "I answered him, as I thought good, + 'As many as red-herrings grow in the wood'." + +__________ + + + + + + +1. Elementary. + + + +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". + +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"). + +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. + +4. That the Substantive shall be supposed 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. + +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." + +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". + +7. The Things in each compartment possess TWO Attributes, whose +symbols will be found written on two of the EDGES of that compartment. + +8. "One or more." + +9. As a name of the class of Things to which the whole Diagram is +assigned. + +10. A Proposition containing 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. + +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. + +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". + +15. Propositions beginning 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". + +17. Some tigers are fierce, No tigers are not-fierce. + +18. Some hard-boiled eggs are unwholesome, No hard-boiled +eggs are wholesome. + +19. Some I's are happy, No I's are unhappy. + +20. Some Johns are not at home, No Johns are at 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). + +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. + +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.' + +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'. + +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'. + +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. + +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'. + +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. + +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. + +30. When the offered Conclusion is PART of the correct Conclusion. +In this case, we may call it a 'Defective Conclusion'. + + + + +2. Half of Smaller Diagram. + + + +Propositions represented. + +__________ + + + ------- ------- + | | | | | | + 1. | | 1 | 2. | 0 | 1 | + | | | | | | + ------- ------- + + + ------- ------- + | | | | | | + 3. | 1 | 1 | 4. | 0 | 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 | | + | | | + ------- + + ------- + | | | + 9. Some x are y'. i.e. | | 1 | + | | | + ------- + + ------- + | | | + 10. All x are y. i.e. | 1 | 0 | + | | | + ------- + + ------- + | | | + 11. Some x are y. i.e. | 1 | | + | | | + ------- + + ------- + | | | + 12. No x are y. i.e. | 0 | | + | | | + ------- + + ------- + | | | + 13. Some x are y, and some are y'. i.e. | 1 | 1 | + | | | + ------- + + ------- + | | | + 14. All x are y'. i.e. | 0 | 1 | + | | | + ------- + --- + | | + 15. No y are x'. i.e. |---| + | 0 | + --- + + --- + | 1 | + 16. All y are x. i.e. |---| + | 0 | + --- + + --- + | 0 | + 17. No y exist. i.e. |---| + | 0 | + --- + + --- + | | + 18. Some y are x'. i.e. |---| + | 1 | + --- + + --- + | | + 15. Some y exist. i.e. |-1-| + | | + --- + + + + +3. Half of Smaller Diagram. + + + +Symbols interpreted. + +__________ + +1. No x are y'. + +2. No x exist. + +3. Some x exist. + +4. All x are y'. + +5. Some x are y. i.e. Some good riddles are hard. + +6. All x are y. i.e. All good riddles are hard. + +7. No x exist. i.e. No riddles are good. + +8. No x are y. i.e. No good riddles are hard. + +9. Some x are y'. i.e. Some lobsters are unselfish. + +10. No x are y. i.e. No lobsters are selfish. + +11. All x are y'. i.e. All lobsters are unselfish. + +12. Some x are y, and some are y'. i.e. Some lobsters are +selfish, and some are unselfish. + +13. All y' are x'. i.e. All invalids are unhappy. + +14. Some y' 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. + +16. No y' exist. i.e. Nobody is unhealthy. + + + + + +4. Smaller Diagram. + + + +Propositions represented. + +__________ + + ------- ------- + | 1 | | | | | + 1. |---|---| 2. |---|---| + | 0 | | | 1 | | + ------- ------- + + ------- ------- + | | | | | 1 | + 3. |---|---| 4. |---|---| + | | 0 | | | | + ------- ------- + + ------- ------- + | | 1 | | | | + 5. |---|---| 6. |---|---| + | | | | 0 | | + ------- ------- + + ------- ------- + | | | | | | + 7. |---|---| 8. |---|---| + | | 1 | | 0 | 1 | + ------- ------- + + ------- ------- + | | | | | | + 9. |---|-1-| 10. |---|---| + | | | | 0 | 0 | + ------- ------- + + ------- ------- + | 1 | | | 1 | 0 | + 11. |---|---| 12. |---|---| + | 1 | | | | 1 | + ------- ------- + + + ------- + | | | + 13. No x' are y. i.e. |---|---| + | 0 | | + ------- + + ------- + | | 0 | + 14. All y' are x'. i.e. |---|---| + | | 1 | + ------- + + + ------- + | | | + 15. Some y' exist. i.e. |---|-1-| + | | | + ------- + + ------- + | 1 | 0 | + 16. All y are x, and all x are y. i.e. |---|---| + | 0 | | + ------- + ------- + | | | + 17. No x' exist. i.e. |---|---| + | 0 | 0 | + ------- + + ------- + | 0 | 1 | + 18. All x are y'. i.e. |---|---| + | | | + ------- + + ------- + | 0 | | + 19. No x are y. i.e. |---|---| + | | | + ------- + + + ------- + | | | + 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 | | + ------- + + ------- + | | 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 | + ------- + + + + + +5. Smaller Diagram. + + + +Symbols interpreted. + +__________ + + +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. + +3. No not-y are x. + +4. No not-x exist. i.e. No Things are not-x. + +5. No y exist. i.e. No houses are two-storied. + +6. Some x' exist. i.e. Some houses are not built 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. + +8. All x' are y'. i.e. All houses, that are not built of +brick, are not two-storied. + +9. Some x are y, and some are y'. i.e. Some fat boys are +active, and some are not. + +10. All y' are x'. 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. + +12. All y are x, and all x' are y. i.e. All active boys +are fat, 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. + +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. + +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. + +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. + + + + +6. Larger Diagram. + + + +Propositions represented. + +__________ + + --------------- --------------- + | | | | | | + | ---|--- | | ---|--- | + | | 0 | 0 | | | | | | | + 1. |---|---|---|---| 2. |-1-|---|---|---| + | | | | | | | | | | + | ---|--- | | ---|--- | + | | | | | | + --------------- --------------- + + + --------------- --------------- + | | | | | 0 | + | ---|--- | | ---|--- | + | | 0 | 0 | | | | | | | + 3. |---|---|---|---| 4. |---|---|---|---| + | | - | | | | | | | + | ---|--- | | ---|--- | + | | | | | 0 | + --------------- --------------- + + + --------------- --------------- + | 0 | | | | | + | ---|--- | | ---|--- | + | | 0 | 0 | | | | 0 | 1 | | + 5. |---|---|---|---| 6. |---|---|---|---| + | | 1 | | | | | 0 | | | + | ---|--- | | ---|--- | + | 0 | | | | | + --------------- --------------- + + + --------------- --------------- + | | | | | 0 | + | ---|--- | | ---|--- | + | | 0 | 0 | | | | | | | + 7. |---|---|---|---| 8. |---|---|---|---| + | | 0 | 1 | | | | 0 | 0 | | + | ---|--- | | ---|--- | + | | | | | 0 | + --------------- --------------- + + + --------------- + | | | + | ---|--- | + | | 0 | 0 | | + 9. No x are m. i.e. |---|---|---|---| + | | 0 | | | + | ---|--- | + | | | + --------------- + + + --------------- + | | | + | ---|--- | + | | | | | + 10. Some m' are y. i.e. |-1-|---|---|---| + | | | | | + | ---|--- | + | | | + --------------- + + + --------------- + | | | + | ---|--- | + | | | 0 | | + 11. All y' are m'. i.e. |---|---|---|-1-| + | | | 0 | | + | ---|--- | + | | | + --------------- + + --------------- + | | | + | ---|--- | + | | 0 | 0 | | + 12. All m are x'. i.e. |---|---|---|---| + | | 1 | | + | ---|--- | + | | | + --------------- + + --------------- + | 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 | + | ---|--- | + | | 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 | + | ---|--- | + | | 1 | 0 | | + 17. All x are m; i.e. |---|---|---|---| + All m are y. | | | 0 | | + | ---|--- | +[See remarks on No. 7, p. 60.] | | | + --------------- + + --------------- + | 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 | | + | ---|--- | + | | | + --------------- + + 20. We had better take "persons" as Universe. We +may choose "myself" as 'Middle Term', in which case +the Premisses will take the form + + I am a-person-who-sent-him-to-bring-a-kitten; + I am a-person-to-whom-he-brought-a-kettle-by-mistake. + +Or we may choose "he" as 'Middle Term', in which case the Premisses +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. + +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." + + Hence, All m are x; + All m are y. and the required Diagram is + + + --------------- + | | | + | ---|--- | + | | 1 | 0 | | + |---|---|---|---| + | | 0 | 0 | | + | ---|--- | + | | | + --------------- + + + + +7. Both Diagrams employed. + + ------- + | 0 | | + 1. |---|---| i.e. All y are x'. + | 1 | | + ------- + + ------- + | | 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 | | + ------- + + ------- + | | | + 4. |---|---| i.e. No x' are y'; or, No y' are x'. + | | 0 | + ------- + + ------- + | 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. + ------- + + ------- + | 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. + + + ------- + | 1 | 0 | + 9. |---|---| i.e. All x are y. i.e. John has got a + | | | tooth-ache. + ------- + + + ------- + | | | + 10. |---|---| i.e. No x' are y. i.e. No one, but John, + | 0 | | has got a tooth-ache. + ------- + + + ------- + | 1 | | + 11. |---|---| i.e. Some x are y. i.e. Some one, 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. + ------- + + --------------- + | | 0 | + | ---|--- | + | | 0 | 0 | | +13. |-1-|---|---|---| ------- + | | | | | | | 0 | + | ---|--- | |---|---| + | | 0 | | | | + --------------- ------- + + + Let "books" be Universe; m="exciting", + x="that suit feverish patients"; y="that make + one drowsy". + + 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. + + --------------- + | | | + | ---|--- | + | | 1 | 0 | | +14. |---|---|---|---| ------- + | | | 0 | | | 1 | | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + +Let "persons" be Universe; m="that deserve the fair"; + x="that get their deserts"; y="brave". + + Some m are x; &there4 Some y are x. + No y' are m. + + i.e. Some brave persons get their deserts. + + --------------- + | 0 | | + | ---|--- | + | | 0 | 0 | | +15. |---|---|---|---| ------- + | | | | | | 0 | | + | ---|--- | |---|---| + | 0 | | | | | + --------------- ------- + + Let "persons" be Universe; m="patient"; + x="children"; y="that can sit still". + + No x are m; &there4 No x are y. + No m' are y. + + i.e. No children can sit still. + + + --------------- + | 0 | 0 | + | ---|--- | + | | 0 | 1 | | +16. |---|---|---|---| ------- + | | 0 | | | | 0 | 1 | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + + Let "things" be Universe; m="fat"; x="pigs"; + y="skeletons". + + All x are m; &there4 All x are y'. + No y are m. + + i.e. All pigs are not-skeletons. + + --------------- + | | | + | ---|--- | + | | 0 | 0 | | +17. |---|---|---|---| ------- + | | 1 | 0 | | | | | + | ---|--- | |---|---| + | | | | 1 | | + --------------- ------- + + Let "creatures" be Universe; m="monkeys"; + x="soldiers"; y="mischievous". + + No m are x; &there4 Some y are x'. + All m are y. + + i.e. Some mischievous creatures are not soldiers. + + + --------------- + | 0 | | + | ---|--- | + | | 0 | 0 | | +18. |---|---|---|---| ------- + | | | | | | 0 | | + | ---|--- | |---|---| + | 0 | | | | | + --------------- ------- + + Let "persons" be Universe; m="just"; + x="my cousins"; y="judges". + + No x are m; &there4 No x are y. + No y are m'. + + i.e. None of my cousins are judges. + + --------------- + | | | + | ---|--- | + | | 1 | 0 | | +19. |---|---|---|---| ------- + | | | | | | 1 | | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + Let "periods" be Universe; m="days"; + x="rainy"; y="tiresome". + + Some m are x; &there4 Some x are y. + All xm are y. + + i.e. Some rainy periods are tiresome. + +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:-- + +"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". + +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". + +Altogether, a most unsatisfactory Pair of Premisses! + + --------------- + | 0 | | + | ---|--- | + | | 1 | | | +20. |---|---|---|---| ------- + | | 0 | 0 | | | 1 | | + | ---|--- | |---|---| + | 0 | | | 0 | | + --------------- ------- + + Let "things" be Universe; m="medicine"; + x="nasty"; y="senna". + + All m are x; &there4 All y are x. + All y are m. + + i.e. Senna is nasty. + + [See remarks on No. 7, p 60.] + + --------------- + | | | + | ---|--- | + | | 0 | 1 | | +21. |-1-|---|---|---| ------- + | | 0 | | | | | 1 | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + Let "persons" be Universe; m="Jews"; + x="rich"; y="Patagonians". + + Some m are x; &there4 Some x are y'. + All y are m'. + + i.e. Some rich persons are not Patagonians. + + --------------- + | 0 | | + | ---|--- | + | | - | | +22. |---|---|---|---| ------- + | | 0 | 0 | | | | | + | ---|--- | |---|---| + | 0 | | | 0 | | + --------------- ------- + + Let "creatures" be Universe; m="teetotalers"; + x="that like sugar"; y="nightingales". + + All m are x; &there4 No y are x'. + No y are m'. + + i.e. No nightingales dislike sugar. + + --------------- + | | | + | ---|--- | + | | 0 | 0 | | +23. |-1-|---|---|---| ------- + | | 0 | | | | | | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + Let "food" be Universe; m="wholesome"; + x="muffins"; y="buns". + + No x are m; + All y are m. + + There is 'no information' for the smaller Diagram; so +no Conclusion can be drawn. + + --------------- + | | | + | ---|--- | + | | 0 | 0 | | +24. |---|---|---|---| ------- + | | 1 | | | | | | + | ---|--- | |---|---| + | | | | 1 | | + --------------- ------- + + 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. + + i.e. Some greyhounds are not fat. + + + --------------- + | | | + | ---|--- | + | | - | | +25. |-1-|---|---|---| ------- + | | 0 | 0 | | | | | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + Let "persons" be Universe; m="soldiers"; + x="that march"; y="youths". + + All m are x; + Some y are m'. + + There is 'no information' for the smaller Diagram; so +no Conclusion can be drawn. + + --------------- + | 0 | 0 | + | ---|--- | + | | 0 | 1 | | +26. |---|---|---|---| ------- + | | 0 | | | | 0 | 1 | + | ---|--- | |---|---| + | 1 | | | 1 | | + --------------- ------- + + + 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'. + + + i.e. Sugar is not salt. + Salt is not sugar. + + --------------- + | | | + | ---|--- | + | | 1 | 0 | | +27. |---|---|---|---| ------- + | | | 0 | | | 1 | | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + Let "Things" be Universe; m="eggs"; + x="hard-boiled"; y="crackable". + + Some m are x; &there4 Some x are y. + No m are y'. + + i.e. Some hard-boiled things can be cracked. + + --------------- + | 0 | | + | ---|--- | + | | 0 | 0 | | +28. |---|---|---|---| ------- + | | | | | | 0 | | + | ---|--- | |---|---| + | 0 | | | | | + --------------- ------- + +Let "persons" be Universe; m="Jews"; x="that + are in the house"; y="that are in the garden". + + No m are x; &there4 No x are y. + No m' are y. + + i.e. No persons, that are in the house, are also in + the garden. + + + --------------- + | 0 | 0 | + | ---|--- | + | | - | | +29. |---|---|---|---| ------- + | | | | | | | | + | ---|--- | |---|---| + | 1 | 0 | | 1 | | + --------------- ------- + + + Let "Things" be Universe; m="noisy"; + x="battles"; y="that may escape notice". + + All x are m; &there4 Some x' are y. + All m' are y. + + i.e. Some things, that are not battles, may escape notice. + + --------------- + | 0 | | + | ---|--- | + | | 0 | 0 | | +30. |---|---|---|---| ------- + | | 1 | | | | 0 | | + | ---|--- | |---|---| + | 0 | | | 1 | | + --------------- ------- + + Let "persons" be Universe; m="Jews"; + x="mad"; y="Rabbis". + + No m are x; &there4 All y are x'. + All y are m. + + i.e. All Rabbis are sane. + + --------------- + | | | + | ---|--- | + | | 1 | | | +31. |---|---|---|---| ------- + | | 0 | 0 | | | 1 | | + | ---|--- | |---|---| + | | | | | | + --------------- ------- + + 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. + + i.e. Some skates can swim. + + --------------- + | | | + | ---|--- | + | | 0 | 0 | | +32. |---|---|---|---| ------- + | | 1 | | | | | | + | ---|--- | |---|---| + | | | | 1 | | + --------------- ------- + + Let "people" be Universe; m="passionate"; + x="reasonable"; y="orators". + + All m are x'; &there4 Some y are x'. + Some y are m. + + i.e. Some orators are unreasonable. + + [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." + +__________ + + +1. Pain is wearisome; No pain is eagerly wished for. + +2. No bald person needs a hair-brush; No lizards have hair. + +3. All thoughtless people do mischief; No thoughtful person +forgets a promise. + +4. I do not like John; Some of my friends like John. + +5. No potatoes are pine-apples; All pine-apples are nice. + +6. No pins are ambitious; No needles are pins. + +7. All my friends have colds; No one can sing who has a cold. + +8. All these dishes are well-cooked; Some dishes are unwholesome +if not well-cooked. + +9. No medicine is nice; Senna is a medicine. + +10. Some oysters are silent; No silent creatures are amusing. + +11. All wise men walk on their feet; All unwise men walk on +their hands. + +12. "Mind your own business; This quarrel is no business of +yours." + +13. No bridges are made of sugar; Some bridges are picturesque. + +14. No riddles interest me that can be solved; All these +riddles are insoluble. + +15. John is industrious; All industrious people are happy. + +16. No frogs write books; Some people use ink in writing +books. + +17. No pokers are soft; All pillows are soft. + +18. No antelope is ungraceful; Graceful animals delight the +eye. + +19. Some uncles are ungenerous; All merchants are generous. + +20. No unhappy people chuckle; No happy people groan. + +21. Audible music causes vibration in the air; Inaudible +music is not worth paying for. + +22. He gave me five pounds; I was delighted. + +23. No old Jews are fat millers; All my friends are old +millers. + +24. Flour is good for food; Oatmeal is a kind of flour. + +25. Some dreams are terrible; No lambs are terrible. + +26. No rich man begs in the street; All who are not rich +should keep accounts. + +27. No thieves are honest; Some dishonest people are found +out. + +28. All wasps are unfriendly; All puppies are friendly. + +29. All improbable stories are doubted; None of these +stories are probable. + +30. "He told me you had gone away." "He never says one word +of truth." + +31. His songs never last an hour; A song, that lasts an +hour, is tedious. + +32. No bride-cakes are wholesome; Unwholesome food should +be avoided. + +33. No old misers are cheerful; Some old misers are thin. + +34. All ducks waddle; Nothing that waddles is graceful. + +35. No Professors are ignorant; Some ignorant people are +conceited. + +36. Toothache is never pleasant; Warmth is never unpleasant. + +37. Bores are terrible; You are a bore. + +38. Some mountains are insurmountable; All stiles can be +surmounted. + +39. No Frenchmen like plumpudding; All Englishmen like +plumpudding. + +40. No idlers win fame; Some painters are not idle. + +41. No lobsters are unreasonable; No reasonable creatures +expect impossibilities. + +42. No kind deed is unlawful; What is lawful may be done +without fear. + +43. No fossils can be crossed in love; Any oyster may be +crossed in love. + +44. "This is beyond endurance!" "Well, nothing beyond +endurance has ever happened to me." + +45. All uneducated men are shallow; All these students are +educated. + +46. All my cousins are unjust; No judges are unjust. + +47. No country, that has been explored, is infested +by dragons; Unexplored countries are fascinating. + +48. No misers are generous; Some old men are not generous. + +49. A prudent man shuns hyaenas; No banker is imprudent. + +50. Some poetry is original; No original work is producible +at will. + +51. No misers are unselfish; None but misers save egg-shells. + +52. All pale people are phlegmatic; No one, who is not +pale, looks poetical. + +53. All spiders spin webs; Some creatures, that do not spin +webs, are savage. + +54. None of my cousins are just; All judges are just. + +55. John is industrious; No industrious people are unhappy. + +56. Umbrellas are useful on a journey; What is useless on +a journey should be left behind. + +57. Some pillows are soft; No pokers are soft. + +58. I am old and lame; No old merchant is a lame gambler. + +59. No eventful journey is ever forgotten; Uneventful +journeys are not worth writing a book about. + +60. Sugar is sweet; Some sweet things are liked by children. + +61. Richard is out of temper; No one but Richard can ride +that horse. + +62. All jokes are meant to amuse; No Act of Parliament is +a joke. + +63. "I saw it in a newspaper." "All newspapers tell lies." + +64. No nightmare is pleasant; Unpleasant experiences are +not anxiously desired. + +65. Prudent travellers carry plenty of small change; Imprudent +travellers lose their luggage. + +66. All wasps are unfriendly; No puppies are unfriendly. + +67. He called here yesterday; He is no friend of mine. + +68. No quadrupeds can whistle; Some cats are quadrupeds. + +69. No cooked meat is sold by butchers; No uncooked meat +is served at dinner. + +70. Gold is heavy; Nothing but gold will silence him. + +71. Some pigs are wild; There are no pigs that are not fat. + +72. No emperors are dentists; All dentists are dreaded by +children. + +73. All, who are not old, like walking; Neither you nor I +are old. + +74. All blades are sharp; Some grasses are blades. + +75. No dictatorial person is popular; She is dictatorial. + +76. Some sweet things are unwholesome; No muffins are sweet. + +77. No military men write poetry; No generals are civilians. + +78. Bores are dreaded; A bore is never begged to prolong +his visit. + +79. All owls are satisfactory; Some excuses are unsatisfactory. + +80. All my cousins are unjust; All judges are just. + +81. Some buns are rich; All buns are nice. + +82. No medicine is nice; No pills are unmedicinal. + +83. Some lessons are difficult; What is difficult needs +attention. + +84. No unexpected pleasure annoys me; Your visit is an +unexpected pleasure. + +85. Caterpillars are not eloquent; Jones is eloquent. + +86. Some bald people wear wigs; All your children have +hair. + +87. All wasps are unfriendly; Unfriendly creatures are +always unwelcome. + +88. No bankrupts are rich; Some merchants are not bankrupts. + +89. Weasels sometimes sleep; All animals sometimes sleep. + +90. Ill-managed concerns are unprofitable; Railways are +never ill-managed. + +91. Everybody has seen a pig; Nobody admires a pig. + +______________ + + +Extract a Pair of Premisses out of 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." + +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!" + +"But I thought, as he was an uncle of yours--" + +"An uncle of mine, indeed! Stuff!" + +"You may call it stuff, if you like. All I know is, MY uncles are +all old men: and they like gruel like anything!" + +"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." + +"Well, who expects to be comfortable, out shopping?" + +"Why, I do, of course! And I'm sure there are some shops, further +down the street, that are not crowded. So--" + +95. "They say no doctors are metaphysical organists: and that lets +me into a little fact about YOU, you know." + +"Why, how do you make THAT out? You 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--" + +___________________ + +Extract a Syllogism out of each of the following: and +test its correctness:-- + +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!" + +"And what has that got to do with old Mr. Brown?" + +"Why, isn't he very rich?" + +"Yes, of course he is. And what then?" + +"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!" + +97. "It IS so kind of you to enquire! I'm really feeling a great +deal better to-day." + +"And is it Nature, or Art, that is to have the credit of this happy +change?" + +"Art, I think. The Doctor has given me some of that patent medicine +of his." + +"Well, I'll never call him a humbug again. There's SOMEBODY, at +any rate, that 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." + +"So you like a doll better than a cousin? Oh you little silly!" + +"Of course I do! COUSINS are never kind--at least no cousins I've +ever seen." + +"Well, and what does THAT prove, I'd like to know! If you mean +that cousins aren't dolls, who ever said 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." + +"Well, some geraniums are red, aren't they?" + +"I don't deny it. And what then? I suppose 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?" + +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." + +"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." + +"Very glad to hear it: and how do you make it out to be so?" + +"Why, Sir, I know how hard they work--some of them, that is. Who +should know better?" + +___________________ + +Extract from the following speech a series of Syllogisms, or +arguments having the form of Syllogisms: and test their correctness. + +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. + +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!" + +THE END. + + + + +*** END OF THE PROJECT GUTENBERG EBOOK, THE GAME OF LOGIC *** + +This file should be named thgmf10.txt or thgmf10.zip +Corrected EDITIONS of our eBooks get a new NUMBER, thgmf11.txt +VERSIONS based on separate sources get new LETTER, thgmf10a.txt + +Project Gutenberg eBooks are often created from several printed +editions, all of which are confirmed as Public Domain in the US +unless a copyright notice is included. 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