mind volume xv issue 60 1906 [doi 10.1093%2fmind%2fxv.60.504] maccoll, hugh -- iv.—symbolic...

8/13/2019 Mind Volume xv issue 60 1906 [doi 10.1093%2Fmind%2Fxv.60.504] MACCOLL, HUGH -- IV.SYMBOLIC REASONIN

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IV.SYMBOLIC REASONING (VIII.) .1B Y H U G H M A C C O L L .

1. T HE mnin subject of this article will be paradoxes. W em eet w ith them everywherein logic, in m athe m atics , andin science generally. T hey nearly always spring from theambiguities and obscurities more or less inherent in alllanguagesthe symbolic languages of logic and mathematicsno t excepted. T he same words or symbols suggest differentconcep ts to different minds, and even to the same mind atdifferent times. T ake the word infinite or infinity, whichmathematicians usually represent by the symboloo . Workson modern geometry often speak of a series of straight linesmeeting at "the point at infinity," when, as a matter of fact,their points of intersection at infinity, instead of being one,m ay be m any, or may even be non-existent. And the y alsospeak of a series of points being all in " the line at infinity,"when ther e may be no real line containing all the said poin ts,eithe r at infinity or elsewhere. Similarly algebraists some-times speak of the infinityaoas if it were one definite hugenu m ber or ratio which differed from a million or a billion inonly one respect, that of being much larger; whereas thereare numberless infinities, each of which differs from a millionor a billion not only in being much larger, but also in anotherim portant quality of which I shall speak presently. T hetru th is tha t the real ' infinity' of m athem atics denotes no ta single individual ratio but a whole class, and that thesymbol oo, when it represents a reality (which it does notalways), sometimes stands for an infinite ratiolf at anothertime-or in another place for a different infinite ra tioao atano ther tim e or place "for an infinite ratio oo,, and so on.T hu s, when we meet such a statement asoo - 2oo- 4oo,which seems to assert the absurdity that infinity is equal toits double and also to itshalf,we m ust understand it to meanoo, - 2oo, - loo a perfectly self-consistent s tatem en t w hichonly asserts that the infinity oo, is double the infinity GO , and

1 For VII. see MISD , July, 1906.

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SYMBOLIC REASONING. 5 0 5half the infinityoot. Similarly an infinityoo" may be eitherinfinite or infinitesimal in comparison with another in-finityoo".2. Bat, it may be asked, what is it exactly that separatesthe infinite from the finite ? W here is the exact line ofdem arcation? L et F denote the class of finite positivenum bers or ratios, mad e u p of th e individual ratios F j, F 2 , F , ,etc.; and let H deno te the class of positivein finities,H l tHj, H ,,etc. Logical consistency requir.es that these tw o classesshall be considered mutually exclusive; for it is clear that,speaking of any num ber or ratio A, the two state m ents A rand AH are mutua lly inconsistent. If any real ratio A is-finite, it can not be inf init e; and if it is infinite it c anno t b efinite. Neither can it, however large, be on the borderlandbetween the two classes. T hese stateme nts may be ex-pressed by a single formula, (AFAH)1'.3. For example, let M denote a million. T h e n u m b e rMM (which means M x M x M x . . . etc., up to a millionfactors) is inconceivably large so large th a t th e volum e ofthe earth (or even of the sun , or of a sphere enclosing ou rwhole solar system) divided by that of the smallest drop ofwater, would be an exceedingly small number in comparison;yet the number MMbelongs to the finite class F and not t othe infinite class H . So does the num ber M 1"1, which isinconceivably large even in comparison with the inconceiv-ably large number MM; and BO does MMM M, which is incon-ceivably large even in comparison with M MM. And we m ightcarry the ascending comparison further, till the hand gotweary of the repetition of exponents, without finding anynumber or ratio that belongs to the infinite class H, or thatdoes not belong to the finite class F . Are the finite F a ndth e infinite H then both indefinable? Is there no qualityQ which we can assert of every F and d eny of every H ?T he re i s , and i t i s th i s : Every finite num ber or ratio F , howeverlarge, is expressible, either exactly or (like IT) approxima tely, inthe decimal or some other conventional notation, in terms of somefinitenumber or ratio(such as 10 or 100 or 1,000,000, etc.) ;whereas no infinite number or ratio is, either exactly or approxi-mately, expressiblesolely in terms of any finite number or numbers.T h u s , M , MMM , M * " , etc., though inconceivably large, areall finite, because they are all expressible in terms of thefinite and known n um ber M ; whereas H j, 2H l f J^Hj, andgenerally F H (whatever be the finite num ber F and theinfinite number H) are all infinities, because they are toolarge to be expressible, either exactly or approxim ately, solelyin terms of any known numbers or ratios however large. 3

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6 0 6 HUGH MACCOLL :4. Sim ilarly we may define the class of infinitesimal ratios,a class which we will here denote by h. Ju st as every

infinite number or ratio H is too large to be expressible,either exactly or approximately, solely in terms of any finitenumber F, or finite numbers F,, F 2 , etc., so every infinites-imal ratio h is too small to be expressible, either exactly orapproximately, solely in terms of any finite number or ratioF, or finite numbers Fj, F s , etc. T hus , just as M M, thoughinconceivably large, is still not infinite, so its reciprocal1 -5-MM, though inconceivably small, is still not infinitesimal.5 . Using the symbol A* to assert th at the nu m ber or ratioA belongs to the classx, wherex may stand for F or H or h,thes e conventions or definitions give us several evident for-mulae, of which the following are a few:

(1) (FH)H; (2)(F*); (3)( J Y ; (4)( 1 Y ; (5)( ? ) ;(6) (p-)*; (7) (F h)*; (8) (H F )H ; (9) H , + F - H

What leads to much confusion is the fact that mathe-m atician s also use the wordinfinityand th e symboloo to denote1 2such expressions as -, -, etc.,which represent noreal ratios atall but pure non-existences, such a3 in my two preceding ar-ticles I have denoted by the symbol 0. B ut as these pseudo-1 2ratios -, -, etc., form a different class of non-existences fromthe pseudo-ratios -, -, etc., it would be convenient in mathe-

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matical reasoning to restrict the symbol 0 to the latter class,and the symbol oo to the former. T hu s, the symbol 0 repre-sents, as it were, the death of a real infinitesimal ratio h inpassing from the positive to the negative state, orviceversa;while the other non-existence symbolGO (which may be calledpseudo-infinity similarly represents, as it were, the death of areal infinite ratio H,.in mak ing the same transition. W eshall then get the following self-evident formulae, in which(to prevent ambiguity) the symbol :: will be used (insteadof =) to assert equivalence of propositions, not equivalence ofratios.


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SYMBOLIC SEASONING. 5 0 7If we denote negative infinity by K, negative infinitesimalby k, and use the symbol tan1A. as an abbreviation for th e

itaiemmt (tanA)*, and similarly for other trigonometricalratios, we get( 2 2 ) t a n H (? - f c) ; (23) ta n K ( ? +fc (24) t a n " (? ) ;(25) co t ( | ) ; (26) cot*(^ - hj ; (27) cot * ^ +h) ;(28) aec-(j) ; (29) cos( | ) ; (30) Be"with numberless others on the same principle of notation.6. T he following is a geometrical illustration bearing bothon the ambiguity (as commonly employed) of the wordinfinity,and on the (to my m ind inadmissible) paradox ofthe non-E uclidean geometry, tha t a point moving always inthe same straight line and in the same direction may never-theless finally find itself at the point of starting.1 Le t a

P \straight line of unlimited length, such as AB produced bothways indefinitely, revolve uniformly in the unscrewing direc-tion round a fixed point C, and cut afixed straight line, alsoof unlimited length, at t he variable point P . L et MC, theperpendicular from C upon the fixed straight line, be ourlinear unit . As the moving line revolves uniformly roundC, the point P moves farther and farther, and with fastincreasing velocity, to the right of M, while the angle PCM


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5 0 8 HU GH MACCOLL:numberless succession of increasing real positive infinitevalues, H l f H j, H ,, etc., th at is to say, positive values alltoo large to be expressible solely in terms of any finite values(see 3). W hen the angle BCM becomes exactly a rig htangle, the point P, the infinite straight line MP, and theM Pinfinite ratio ^ j ^ which represents tan PC M , all van ish,MOand the revolving line becomes parallel to the fixed line.T hen, the revolution still continuing , ano ther anddifferentpoin t Q , representing the intersection of the other branch (thebranch CA produced) of the revolving line with the fixed line,springs into existence at an infinite distance to the leftof M,and the straight line QM , after passing throu gh a num ber-less series of really infinite but diminishing values, eventuallybecomes finite and continues to diminish till it finally van-ishes, or becomes zero, just as the variable point Q coincideswith M.7. T he mistake made by non-E uclideans, when they appeaLto geometrical examples like the preceding, is that theywrongly identify the variable point P which moves always.through contiguous positions to the right of M, and far ther andfarther away from M till it finally vanishes, with the pointQ which immediately after springs into existence at aninfinite distance to the left of M, and then moves throughcontiguous positions nearer and nearer to M till it finallycoincides w ith it. W hen this coincidence takes place, theinfinite branch containing both the fixed point C and th erevolving point B will be pointing perpendicularly upw ards ,with B above C, and will contain neither the point P,whichit lost when parallel to the fixed line and neverrecovered after,nor the po in t Q , wh ich had never belonged to it but to the o therinfinite branch con taining the revolving point A. T h e fa llacy ofth e non-E uclideans is analogous to t h at of the lawyers who-assert that " th e king never dies," because the instant th eking P dies, his successor Q becomes,ipso facto,king in his.place. T o both we may return analogous answ ers: to thelawyers we reply that the dead king P is nevertheless not t h eliving king Q* and to the non-E uclideans we reply th at thevanished point P is neverthelessnotthe new point Q. W h enthe angle B CM is a right angle ther e is no point P , and the reis no po int Q, so th at in this position of the revolving lin e(a position parallel to the fixed line) the distances M P an dQM arepseudo-infinities which have only symbolic existence.Suppose the revolving line to remain for a moment stationaryin th is parallel position. If it then revolves thro ug h aninfinitesimal angle in the screwing direction, we get the

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SYMBOLIC RE ASONING. 5 0 9

K int P to the right of M, and a real infinite distance M P .on the contrary, it revolves from its parallel positionthrough an infinitesimal angle in the unscrewing direction,we get the *point Q to the left of M, and a real infinitenegative distance MQ.8. It would conduce to logical accuracy in dealing withthese questions if we introduced the term virtuallyinto ourreason ing and defined it as follows. T wo stra ight lines ar esaid to be virtually parallel*when they meet at some realinfinite distance, H or - H . Suppose, for exam ple, we ha vethree straight lines, A, B , C, and tha t A m eets B at an in-finite distance H l f and meets C at another but still infinitedistance H j. Here we may accu rately assert tha t A, B , Carevirtuallyparallel, for, by our very definition of the wordsinfiniteand infinitesimal, it follows that the error of deviationfrom the parallel position, though theoretically real, mustfor ever remain too small for the most perfect instrument todetect, and for the most powerful notation accurately orapproximately to express in finite terms. In like ma nn er,if we suppose A and B to be two ratio s, finite, infinite orinfinitesimal, the statement that "A and B are virtually1Aequ al" means =1 + h, in which h denotes any infinites-Bimal ratio, as defined in 4. In the infinitesimal calcu lus(including the differential and integral) an equation (A = B)often asserts virtual and no t real equality ; bu t logicalaccuracy may be secured by the tacit convention that when-ever we have the statement (A B), it is to be understoodas asserting that "A is either absolutely or virtuallyequal toB ". L et A, B , C be three poin ts on th e surface of a sphereof radius R, forming the spherical triangle AB C. W he the rR be finite, infinite,-or infinitesim al, as defined in 3 , 4,AB BC CAif the ratios -=-, , -^ - be all three infinitesimal, the sum-tv K Hof the three angles A, B , C is virtuallybut not really equalto two right angles. If th e three points A, B , C be on arealplane surface,then the sum of t he three angles A, B , C isreallyequal to two right angles. Again, a finite section ABof a curve may be called virtually straight when at every

1Similarly two straigh t lines arereally parallel when their (non-exist-ent) point of intersection is at some pseudo-infinite distance, such as1 2, aoo - o r ( j o r r

For example, the statement ( 1 + o+ 2+ 2+ " " ' + 2 H = 2 ) a s 8 e r t avirtual and notabsolute equality.3 3 * M

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5 1 0 HTJGH MACCOLL :point of the curve between A and B the radioa of curvatureis infinite, and the curvature consequently infinitesimal.E ven an infinite section AB may be called virtuallystraightwhen is infinitesimal, B being any of the rad ii of cu r-vature; *for in th is case the infinity A B (infinite in rega rdto any finite unit) is infinitesimal compared with the infinityK. T he symbolic and linguistic conventions here proposedwould, I think, greatly increase the logical accuracy ofmodern geometry without in the least impairing its greatpower as a practical instrument of discovery and research.9. Metaphysicians sometimes ask whethe r spaceth eactual space of our perceptive experiencethe space filledwith the entity called matter,or with the entity calledether,or with bothis really infinite. Considering we can give nosatisfactory definition either of 'm a t t e r ' or of 'et he r, ' thequestion hardly adm its of a clear'and intelligible answer, or,a t any rate , of any answer t h a t logicians, metaphysicians, andphys icists would be all likely to accept. T he purely idealspace of the mathematician is far easier to understand, andthis space must, I think, be pronounced infinitethat is tosay, infinite in the sense of the word explained in 3forthe simple reason that the opposite supposition plunges usa t once into logical contradictions. W e can all, w ithout anyConflict of opposing concepts, imagine a sphere whose radiusis too large to be expressible (whatever be our conventionalnotation) in terms oi finite ratios alone, and that sphere is,by our very definition, infinite. And we cannot stop the re.By our definitions and conventions also, it follows thatH i + H , = H ,; that the sum of thereal infinities H, and H2make a "third real infinity H 3 greater th at either. Similarlywe get H j + Hj + H j = H 4 ,and so on for ever. B ut wecann ot, without further da ta, assert H , - H 2 = H , ; forH t - H s may = F^ or may = 0, since neither the suppositionH, - H ? + F , nor th e supposition H^ = H^, involves anyformal inconsistency.

10. It may be objected that the definitions which I herepropose of the finite, the infinite, and the infinitesimalarequite arbitrary. T o this I reply, firstly, tha t all definitionsare more or less arbitr ary; secondly, tha t, however arbitrarymy definitions may be, they are mutually consistent, and inA -p A T3 I D1 By this is meant that -ir - j - j etc., are respectively infinitesimal i i KJfor the separate points Pi, Pj, P b etc., between A and B. T he infinitiesR,, Rt, Rj, etc., are generally, though not necessarily, unequal

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SYMBOLIC SEASO NING. 5 1 1no way clash with, but, on the contrary, render more precise,the rather vague significations usually attached to the wordsin ordinary language and even in mathematics; and, lastly,th at th ey are very convenient and, if accepted, would m a-terially increase the formal accuracy of our reasoning whetherthe questions discussed be metaphysical, logical, or math-em atical. Clear working definitions of the words finite,infinite and infinitesimal are imperatively need ed; and ifthose I here propose be not found suitable, others should besub stitu ted . B y " working definitions'.' I mean definitionswhich, by their formal precision, would prevent ambiguitiesand staggering paradoxesparadoxes that may be true orfalse according to the meanings attached to the words inwhich they are expressed. In ordinary informal speech.variability of meaning is, of course, permissible, as the con-text generally prevents all am biguity. N o one, for instance,would misunderstand the m eaning of such a statem ent as" H e took infinitepains, yet his gains were infinitesimal ;bu t in logic and m athem atics th e-c ase is different. H erealso, it is true, the context as a rule prevents ambiguity, butnot always; and when it does not, the errors into whichwe fall are often serious. W e should especially remem berth at such expressions as -, - , 5 , like their reciprocals-, -, -(J 0 1 . 2 oo ,etc., are no t real ratios at all, but pure unrealities, thoughthey have their utility as symbols.11. Paradoxes also arise from the fact that our unit ofreference is not always con sta nt. A pound of tea is lighterat the equator than in London or Paris or Melbourne ; yeta pound of tea always weighs a pound, neither more norless, wherever we may weigh it, provided the scales we usebe correct. And if we took our pound and scales and weigh tsto the moon or to the planet Mars, the result, in spite of thegreatly diminished attraction , would be the same; the poundof tea would be much lighter, but so would our unit of com-parison, the metallic pound with which we weighed it. It isthe same with all spatial dimensions. We can never be surethat our units of comparison remain constant; or, rather,we may be quite sure tha t they do not. T he actual lengthof the standard yard or standardmetre varies with the tem-pera ture. T rue , so far as our experience goes, the variationowing to this cause is sli gh t; but th en our experience of th epossibilities and actualities of nature is literally infinitesimal.For aught we know to the contrary, there may be other andfar more powerful causes at work, causes which (unlike heat)act equally on all the perceptible substances ofouruniverse,



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5 1 2 HUGH MACCOLL :though not equally, perhaps, or at all, on all the substancesof other worlds beyond our ken; so that, in comparison withsome constant standard unit lying hidden in some infinitelydista nt sphere, the dimensions of everything in our visibleuniverseof the sun, of the moon, of the earth, of itsm oun tains, oceans, seas and rivers, of th e houses upon itssurface, and of the inhabitants who live in them, our ownselves included, may be rapidly diminishing,1 and in nearlythe same relative proportions, so that the actual size of eachone of us to-day may, in comparison with this constant unit,be only the hun dred th or the millionth pa rt of w hat he orshe was yesterday. Persona lly I believe the actual variationto be much less serious; but this is an opinion for which,as for other cherished convictions, I can find no logicalfoundation.12. Let us now examine the meanings of the words finite,infiniteand infinitesimal in reference to time. H ere too wehave the actual and the ideal, and, as in the case of space,the ideal is easier to deal with than th e actual. Actual tim eis measured by clocks and watches, and the correc tness ofthese is tested by observations of the m otions, or apparentmotions, of the sun, moon and stars; but the respectivemotions of these, when m utually compared, are not uniform ;so we take the apparent motion of one of them, the sun, anddividing its cycle into a certain number of equal parts, wetake one of these as our constant unit of reference. For allpractical purposes this answ ers all our n eed s; but what ofthe assumptions on which this theory of time is founded?Ho w do we know tha t these solar cycles are even approxi-mately equal? How do we know tha t, in comparison withsome other unit of time, depending, say, upon the moreuniform motion of some other heavenly body, far away inspace beyond our power ever to discover, the motions of allthe heavenly bodies, of our clocks, of our watches, of every-thing we know, including our very thoughts and sensations,may not be rapidly increasing in velocity, and in nearly thesame relative proportions, so that, when m easured by thisstandard unit, our years, days, hours, minutes and seconds,and, consequently, the duration of our lives may be infinites-imal in comparison with the years, days, hours, etc., and theduration of the lives of our fathers or grandfathers? T heparadox arising from the possible variation of our standardand unit of space has thus its counterpart in the paradox

1 Of course it is equally possible that the variation may be in the oppo-site direction.



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5 1 4 HU GH MACCOLL:

for every variable, whether it be a variable of the class 6r (astatement true now but not always) or a variable of theclass 0, (a statement false now but not always) remains stilla variable, so that (0T )', like (#,) is a formal ce rta in ty. Onthe other hand, we getfor 6\ means (0T)T, which is a formal c ertain ty, and acertainty cannot be a variable, since certainties and variablesform tw o mutually exclusive classes by definition. T hism ight also be proved in a sligh tly different way as follows.Le t A - 01= T r W e getA - &=6 ; but (A*/ = (r{f = - , .Similarly we may show th at A ', thoug h equivalent to A', isnot synonymous w ith A'. For let A - #,. W e have onthe one hand (AT = A ' - 0i - e,and on the other

- VT he statements A and A' (like ABand its denial A~B) are ofthesame degree, whereas A1(whether x stands for Tor t oreor T)or 6, or any other class of statements) is one degreehigher, and may therefore be called thereviiion ofthejudgmtntA. T wo contradictory judgm ents, A and A', are placedbefore us , and we hav e. to decide which is tru e. If wedecide in favour of the affirmative A, we say that "A istrue " and write A T ; if we decide in favour of the negativeA', we say that " A is false " and w rite A'. B ut the questionto be decided may be not merely to decide whether A is trueor false, but whether A follows necessarily from, or is incon-sistent with, our definitions or admitted and unquestioneddata. In that case we write A' when we decide th at Adoesfollow necessarily from our data; we write A11 when wedecide th at A is inconsistent w ith o ur da ta ; and we w riteA' when we decide that A neither follows from nor is in-consistent with our data. Similarly, A**,or its synonym(A*)", is a revision of the jud gm ent A* ; and so on. I t willbe noticed th at A** and AJJ, which m eans (A*)*, are quitedifferent statements, since A x and Ax are different. T hestatement A' asserts that A belongs to the class x; thestatement Axtakes thisfor granted. For example, Ai, whichmeans (A#)\ asserts that " thevariablestatement A isfalse ;whereas A*1, which m eans (A*)', asserts th at " it JBfalse thatA is variable . T he statements A and Ax are of th e same

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SYMBOLIC REASONING. 5 1 5degree ; the statem ent A* is of a degree higher. T he state -ment A1"*, since it means (A**)*, is of the first degree asregards its subject A 1";*bu t since it also m eans (A1)"7, it isof thesecond degree as regards A", and of the thirddegree asregards the root-statement A. Now , suppose A stand s torQ* . In th at case, thoug h A is the root-statement as regardsA.***, it is a statement of the second degree as regards theroot-statement Q; and A**", which is of the third degree asregards A, is of the fifth degree as regards Q ; for, expressedin terms of Q, it means Q*****.15 . L et me here 6ay a few words in reply to the logicianswho maintain that statements can only be classed as trueand false, and th at my introd uction of such classes ascertainties, imp ossibilities, and variables, and of any others thatmay concern our argument or researches, is wrong, or atany rate , outside the proper domain of logic, and especiallyof symbolic logic. T his is very much as if one argued tha tsince animals are only divisible into two classes, males andfemales, it is no business of true zoology to consider therespective charac teristics of such creatures as lions, tigers,and leopards, to say noth ing of o thers still more objection-able. All such att em pts to surroun d symbolic logic by aChinese wall of exclusion are futile.16. Another argum ent against my system is tha tvariablestatements, statements which are sometimes true and some-times false, have no real existence; that a statement if oncetru e is tru e always, and if once false is false always. B utsurely this is a mere play upon words, and it does not seemto me very accurate even as th at . A servant, in reply to aninquiry at the street door in the morning, says, and saystruly, that " M rs . Brown is not at ho m e" . T he sameservant, in reply to the same inquiry in the afternoon, saysagain, and this time, in obedience to instructions, saysfalsely, tha t " M rs. B rown is not at h o m e" . She m akesexactly the same statem ent as in the m orning, becausesheuses exactlythesame form of words; but this statement, \hisform of w ords, wh ich was t rue in th e m orning, because inthe morning itconveyed trueinformation, is false in the after-noon, because in the afternoon itconveyedfalse information.17. Let us look at the matter from another point of view.Suppose we have no data but our definitions or symbolicand linguistic conventions. L e t A, B , C respectively denotethe three statem ents " 7 is greater tha n 5 ," " 6 is greaterthan 9," a? iB greater tha n x . Is it no t clear tha t withthese meanings of the symbols we may truly and confidentlymak e the three-factor compound statem ent A'B'C*? Fo r,



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5 1 6 HUGH MACCOLL :by our very definitions of the words certain, impossible,variable, respectively represented by the symbols e, n, 6, isnot the first statement A certain, because itfollows necessarilyfrom ov.rdata,which are here limited to our definitions andlinguistic conven tions? Is not the second stateme nt Bimpossible, because it contradicts (or is inconsistent with) ourdata t And is not the third statem ent C avariable, because,though perfect ly intel l igible, it is neither certain norimpossible fT o say that C isneithertruenor falsewould be inco rrec t; forit may be either. It is true when x is greater than 1; it isfalse when x is not greater than 1.18. T o take another ca se ; suppose two men are playingdice, and th at, jus t before a throw , thre e spectators makethe three following statements, which we will denote byA, B , C: " T he num ber th at will turn up is less than 8 " (A)," T h e number that will turn up is greater than 8 " (B)," T he num ber tha t w ill turn up is 5 " (C). Since by ourdata, or tacit conventions, the only numbers possible are1, 2, 3 , 4, 5 , 6, is it not clear t h at we m ust have A'B*C#?Is not Acertain because it follows necessarily from our data ?Is not B impossible because it is inconsistent with our da ta ?And is not C avariable because it neither follows from nor isinconsistent w ith our data ? In t he language of probability,the chance ofA is 1, the chance of B is 0, and the chance ofC is neither 1 nor 0 but a proper fraction. W h at th at roper fraction is the sta tem en t C does not say ; bu t wenow it to be I. T aking the three denials A', B ', C', thechan ce of A' is 0. the chance of B ' is 1, and the chance ofC is | ; so that we have (A')'(B ')'(C)'. T his shows tha there, as always, the denial of any certainty A is an im-possibility A', the denial of any impossibility B is a certaintyB ', and the denial of any variable C is also a variable C.

19. Other paradoxes arise from the fact that each of thewords if and impliesis used in different senses. P u tt in g Aand B for two propositions, the statem ents " If A then B "and " A impb'es B , " which, in my symbolic system, I find itconvenient to treat as synonymous and as having the mean-ing which I represent symbolically by any of the threesynonymo us symbols (A : B ), (AB')"\ (A' + B )', are used bysome logicians not only in the above sense, but also in theweaker sense which I attach to the mutually synonymoussymbols (AB ')' and (A' + B ) r ; because these logicianserroneously consider my eto be equivalent to my T , and myn to my . B ut there is yet another sense in which we allsometimes use the word implies; for when we say " Aimplies B " we sometimes mean not only (A B )', that it is



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SYMBOLIC SEASONING. 5 1 7impossible for A to be true without B being also true, or theequivalent statement that the affirmation of A coupled withthe denial of B is inconsistent with our data, but also that AcontainsB , th at is to say, th at B is a particularcase of A.In this sense, I think it would be better to use the wordcontains rather than the word implies. Fo r example, we maysay that the formula (xnxn - i "+ n) contains the formula{x*x% Is) as a particular case, and similarly th at ( s V - s5)contains (6*6* = 66). Re presenting the first and most generalof these three statements by the functional symbol(x ,m,n),it follows from our definition of a logical function th a t (x , 3, 2) m ust denote the second, and t ha t (6, 3, 2) mustden ote the third . I t also follows th at {x, m, n) contains (x ,3 , 2), and tha t (x, 3, 2)contains (6, 3 , 2). Again let (A, B, C), or simply , denote the Barbara ofgeneral logic,namely, (A: B) (B: C ): (A: C),in which A, B, C may be any statements whateverstate-m ents which may or may no t have the same subject; andlet , (A, B , C), or simply den ote the B arb ara of th etraditionallogic, nam ely,( A , : B , ) ( B , : C . ) : ( A . : C . ) ,in wh ich the statem ents A, B, C are understood to have thesam e subject 8. W e may then say th at $ contains , as apa rticu lar case. Also, since and , are both certainties, wec a n a s s e r t n o t o n l y : ,, t h a t implies ,, b u t a l s o < pt: ,tha t ,implies , since any certainty ex impliesany othercertainty e9. Fo r, by definition, we have tx :e, ')() W h h t fy , yv) W e cannot however assert tha t , contains ,for it is , th at is a pa rticu lar case of , and not


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mind volume xv issue 60 1906 [doi 10.1093%2fmind%2fxv.60.504] maccoll, hugh -- iv.—symbolic...

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