1977 - A Reassessment of Boole's Theory of Logic

8/13/2019 1977 - A Reassessment of Boole's Theory of Logic

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364 JAMES W. VAN EVRA

utilise therules andoperations which hold goodin a0 1 algebra, whileonth e other hand,themathematical method renderstheoperations effectedandth eexpressions arising indealing with theproblems moreor less obscure,since they cannotbesubjected tological interpretation.5Boole's mathematicism goes so far . . . that he introduces symbols andprocedures which admit of nological interpretation,or only acomplicatedandscarcely interesting one. Thuswemeet with subtractionanddivisionandnumbers greater thanI .6Although Boole's system lends itself insomeways toeasy manipulation,wemustadmit thatitcontains. . .some defects of rigour suchas the use of theletter for the expression of existential propositions, the admission ofnumerical coefficients other than1 and 0, and the use of the operationofdivision towhich noconstant meaning hasbeen assigned in logic. In thecourseof thehalf century after thepublicationof hisLawsofThoughttheseshortcomings were allremovedby his followers.7L'alg bre de Boole souffrait d'uncertain nombrededefauts. Avant tout ellecomprenaitdes expressions enigmatiques,parexemple les coefficients 0/0ou1/0 . . . ouencore le coefficient 2/3 qui s'y recontre de temps aautre,e tc . 8Owingto his free use of algebraic operations suchas division andsubtraction,thesuccess of Boole's general methodis offset by theappearance,inhis calculations,ofnumericalandfractional coefficients and ofother expressions whose logical significance is, to say theleast, uncertain. It was thetaskof hissuccessors to get rid ofthese anomaliesand so to give thesystem /agreater appearance of being whatitclaimed,a mathematical logicand notth emere logical mathematics thatitsopponents suspected it to be.9The view of Boole's theory of logic which I present below centers on

the following points:First, specific cr it icism s such as those above, i.e., those to the effect

that Boole includes logically uninterpretable expressions in his system oflogic, are largely undeserved. They turn on a misunderstanding of whatBoole called his "general method in logic," which provides for theinclusion of purely mathematical interludes in logical transformations, as away of putting the formal power of mathem atics to work in logical contexts.The questionable expressions serve as components in such interludes only,and were not considered by Boole to be logical expressions at all.

Concentrating on the general method, there is also a way of squaringBoole's explicit rejection of division as a logical operation, with hisapparent(implicit) acceptance of the operation as a basis for interpretingcoefficients such as 0/0 and 1/0. Though the evidence is largely circumstantial, it is possible that Boole did not count such expressions as beinglogical either, but again as being parts of the general method only.Second,suggestions to the effect that Boole's successors in algebraiclogic correc ted his "mistakes" are misleading. Later algebraic logicswere indeed more system atic, complete, and generally understandable thanLaws of Thought(hereinafter 'LT'), but in an important sense, they remaininferior to it. Specifically, Boole's successors tended uncritically toassume that logic and algebra fit perfectly together in the sense that they


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share all of their operations. But that assumption creates genuine problems for logic, the main one being, ironically, the unintelligibility ofdivision used as a logical operator. While Boole's logic is far fromperfect, he displays a greater awareness of logic as an independentdiscipline, aswell as more sophistication in the use of mathematics as anaid to logic while avoiding the conflation of the two areas, than did hissuccessors in algebraic logic.

Third,much of the relatively recent criticism of Boole's logic treatshim as if he were a contemporary logician. But taking him to task for not,e.g., providing the same sophistication in proofs as one finds in contemporary logic not only overlooks the plain fact that Boole stands on the otherside of a tremendous revolution in logic, it obscures much of his trueimpor tance in the history of the subject. That importance, it seems to me,is best exemplified by his general method. Regardless of its imperfectionby today's standards, Boole's use of an independent formal device as an aidto logical processes marks him as an important figure in the transitionalepoch between the traditional, purely syllogistic logics, and the contemporary, post Fregean logics.The following discussion deals exclusively with Boole's theory as itoccurs in LT. After its publication, Boole wrote anessay (in [2], pp. 21 Iff.)containing views which radically diverge from those contained in LT.However, the status of that essay remains unclear, and there is evidencethatBoole never adopted theviewsit contains.101 Even a cursory glance at LT reveals that the formal developments itcontains occur in two separate sections of the work. On the one hand,ChapterII contains the basic logic, i.e., the fundamental logical system. Onth e other hand, Chapters V VIII deal with the process of development, andits subsidiaries, such as elimination and reduction. The remainder of thelogical portion of the work (which comprises the first fifteen chapters) isdevoted to a variety of essentially auxiliary topics, such as variant interpretations of the symbolism, and specific examples of inferences using thelogic already developed.Boole apparently considered the separation between these two parts ofthe work to have more than merely spatial significance. As will be seenbelow, he considered the later chapters to be dependent on the basis laid inChapter , but at the same time to be more than just a simple extension ofit. The two parts were, for Boole, of two fundamentally different sorts.

Chapter is entitled "Of Signs in General, and of the Signs Appropriate to the Science of Logic in Particular; also of the Laws to which thatClass of Signs are Subject." As the title suggests, the chapter consists ofa virtually self contained presentation of Boole's basic logic. Everythingneeded for a system of logic is there: a general discussion of the nature ofsigns, a specification of the part icular symbols used in logic, a collectionof examples of legitimate expressions of the system using the vocabulary,together with their interpretations,laws of the system, and a specificationof its logical coefficients.The heart of the basic logic consists of the three well known binary


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operators 'x', '+'and' ', and the two constant class symbols 0 and 1. Inmodern terms (which are not to be found in Boole's own work), the symbolsmay be understood as follows: 'x x y9 stands for the intersect ion of theclasses xandy, x + y' for a truncated version of class union in which theexpression in question is significant except in those cases in whichxandyintersect; and x y9 for what might be called logical exception (i.e., x'sexcept for the y's ), which is significant only if y is contained inx. Of theconstants , 0 stands for the null class, and inLaws of Thought,1 stands forthe class containing everything in the universe of discourse.

Equally wellknown is the relationship which Boole set up between hisbasic logic and mathematical algebra. Each of the functions serves as ananalogue of its arithmetical counterpart, and the laws of logic correspondin like fashion with expressions in mathematics. Boole circumscribes theextent of the similarity by laying particular stress on the law of idempotence, xx =x, which holds universally in the logic, but in standardalgebra, only for the values 0 and 1. This leads Boole to say that "The twosystems of elective symbols and of quantity osculate, if I may use theexpression, in the points 0 andI. 11 That is, "The laws, the axioms, andth eprocesses, of [an algebra in which the symbolsx, y, zetc., admitindifferently of the values 0 and 1 and of these values alone]willbe identical inthei r whole extent with the laws, the axioms, and the processes of analgebra of logic. Difference of interpretation alonewilldivide them."12

Toward the end of Chapter II, Boole makes it clear that while eachlogical operator has an algebraic analogue, the reverse is not the case.There the question of whether division is to be counted as a logical operation arises, and Boole meets it by saying,

theaxiom of algebraists, that both sidesof anequation may bedivided bythe same quantity, has noformal equ ivalent [inbasic logic]. I say noformalequivalent, because, in accordance with thegeneral spirit ofthese inq uiries,it is noteven sou ght todetermine whether the men tal operation which isrepresented by removing a logical symbol, z, from a combination zx>is initself ana logous with theoperation of division in arithm etic. That mentaloperationis indeed identical with what is commonly termed abstraction, andit will hereafter appear that its laws are dependent upon the laws alreadydeducedinthis chapter . 13

Inother words, division is not even considered as a possible logical operationsince abstraction can be handled with the logical apparatus already athand,which does not include division. While the question still remains asto whether Boole slips division in later on, his remarks in Chapter IIstrongly suggest that he was committed to excluding division from consideration as a logical operator.2 While Chapter II deals with Boole's basic logic, Chapters V ff. introduceafundamentally different topic. It is within these chapters that virtually allof the expressions with which the cr it ics are concerned appear, and it ishere that Boole lays out what he calls a "general method in logic."14 Themethod differs from the basic logic in the sense that Boole considers the


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earlier logic to be the basis for the generalization which the methodwillprovide:

The previous chapt er s of this work have been devoted to the investigationof the fundamental laws of the operations of the mind in reasoning; of theirdevelopment in the laws of the symbols of Logic; and of the principles ofexpression, by which that species of proposit ions called p rimar y may bere pr esent ed in the language of symbols. These inquiries have been in thestri ct est sense preliminary. They form an indispensable introduct ion to oneof the chief objects of this t reatisethe c onstruct ion of a system or methodof logic upon the basis of an exact summary of the fundamental laws ofthought.15

As the chapter unfolds, the difference between the basic logic and thegeneral method in logic becomes clear. While the basic logic wasrestricted to dealing with completely interpreted logical operations, thegeneral methodinvolves the application of more generalized formal devices(development being the primary one) in logical contexts. Boole's apparentaim was to be able to use non logical transformational devices in logicalcontexts in order to strengthen logical transformation.Indescribing the general method, Boole begins by asking ". . . whetherit is necessary to restrict the application of [symbolical laws of logicalprocesses] by the same conditions of interpretability under which knowledge of them was obtained."16 That is, (using his own example) since theexpression (x +y' is uninterpretable if x and y do not represent distinctclasses, must one only use +9when it is certain that an interpretableexpression will result? Boole goes on to say that if such restrict ion isnecessary, no general method is possible. But, "On the other hand, if suchrestriction is unnecessary, in what light are we to contemplate processeswhich appear to be uninterpretable in that sphere of thought which they aredesigned to aid?"17 Not surprisingly, Boole holds such restrict ion to beunnecessary, and then goes on to answer the questionjust posed.H ebegins by pointing out that in ordinary reasoning, there is alwaysacorrespondence between what he refers to as process and interpretation,which means that the various steps in an ordinary reasoning process areall interpretable, i.e., none are merely formal moves. Accordingly, thereare those who hold that:

as the laws or axioms which govern the use of symbols are establishedupon an investigation of those cases only in which interpret ation is possible,we have no right to extend their application to other cases in which in ter pr et a t ion is impossible or doubtful, even though (as should be admitted) suchapplication is employed in the intermediate steps of demonstration only. 18

Suchan objection, says Boole, is fallacious. "It is an unquestionable fact,"h e says, "that the validity of a conclusion arrived at by any symbolicalprocess of reasoning, does not depend on our ability to interpret the formalresults which have presented themselves in the different stages of theinvestigation."19 This statement forms the foundation of the general methodin logic. In concrete terms, Boole reduces the method to the followingrule:


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368 J A M E S W. VAN EVRA

We may infact layaside thelogical interpretation of thesym bols in [a]givenequation; convert them into quantitative symbols, susceptible only of thevalues 0 and 1; perform upon them as such all the req u isite processesofsolution;andfinally res tore tothem their logical interpretation. 20

The process Boole is here advocating is of a distinctly different sort fromthe basic logic of Chapter II. He is suggesting that any logical symbol maybe treated as its mathematical counterpart in the manner laid down inChapter II. Thenany available mathematical operation may be used on it,whether that operation is logically interpretable or not. The final (mathematical) expression in the sequence must again be one which correspondsto a logical expression. With the purely mathematical interlude lyingbetween, the sequence may then be treated as the inference of the final(logical) expression from the initial one.

Development plays a central role in this method. Boole considered itto be a process by which one can construct the bridge between logicallyinterpretable expressions in accordance with the above rule. While itperforms this logical function, Boole did not consider it to be solely alogical device at all; in fact, he initially defined it as encompassing muchmore than just logical application:

Definition Any functionf(x)inwhich x i s a logical symbol, or a symbolof quantity susceptible only of thevalues 0 and 1, is said to be developed,whenit is reduced to theform ax+b(l x)taandbbeing so de termined asto make the result equivalent toth e fun ction from which it wasderived.This definition assumes, thatit is possible tor ep resentanyfunction f(x)in the form supposed. 2

This definition covers anyfunction, logical or not, so long as the variable xmay be restrict ed to the values 1 and 0. As he puts it later on, the resultsof development are "t rue and intelligible for quantitative/ symbols (whichmeetthe restrict ions mentioned in the definition)always; for logical symbols,always wheninterpretable. 22 In other words, development need notyield logically significant expressions.

I t is at this point that some of the crit ic 's complaints can be dispatched,for some of the allegedly uninterpretable logical symbols to which theypoint can now be seen to be nothing of the sort . Thevery first example ofdevelopment which Boole provides contains several of the expressions inquestion. In the example,23 Boole sets out to develop the function ,

1 +x 2 +the result being = x+(l x). But Boole does not claim that this1 2X oexpression is logically interpretable, or is a logical expression inanysense. Rather, he makes it abundantly clear thatx is here being treated asa quantitative symbol restrict ed to the values 0 and 1, by prefacing theexample with the remark that "By the principle which has been asserted inthis chapter, it is lawful to treat AT as a quantitative symbol, susceptibleonly of the values 0 and I."2 4 The example is properly one of development,bu t definitely not of development in a logical context.

Excluding consideration of 1/0 and 0/0, to be considered separately


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below, the remaining instances of expressions in LT which the criticsclaim to be both logical and uninterpretable25 fall into one of two classes.First, they may be non-logical expressions which occur in the steps of aninference which intervene between logically interpretable expressions inaccordance with the rule of the general method, or second, like the examplejust considered, they may be constituents of expressions which serve asnon-logical examples of processes which also have logical applications. Inneither case has Boole done anything obscure, complicated, or unintelli-gible.

There remains the possibility, however, that Boole illegitimatelyintroduced division as a logical operator. For instance, in providing anexample of the development of a function of two variables, Boole uses the

1 - x 1 - x 0function , which, when developed, is = - xy + Qx(l - y) +l *- ~ y oi- (1 -x)y + (1 -x)(l -y). Neither - nor - have any standing in basic logic,hence we might expect them to be treated as purely mathematical expres-sions in accordance with the general method as outlined above. Instead, hesays that It will in the next chapter be seen that the forms 0/0 and 1/0,the former of which is known to mathematicians as the symbol of inde-terminate quantity, admit, in such expressions as the above, of a veryimportant logical interpretation.''26

But, in a manner entirely typical of the period, Boole's use of logical'is incurably vague and ambiguous. He may have meant that these symbolsare , in effect, to be added to basic logic. If this is the proper interpreta-tion, criticisms to the effect that Boole introduced division into logic arejustified. But there is an equal likelihood that these expressions wereintended to be restricted to the general method, i.e., were meant to apply to,but not be a part of, the expressions of the basic logic. In my view, thelatter is the correct interpretation.

The latter interpretation can best be understood by considering anotherof his examples. Beginning with the statement Men not mortals do notexist, symbolized y(l - x) = 0, Boole wishes to provide a reversedefinition of mortals in terms of men. The initial equation yields, mathe-matically, y -yx =0, oryx =y. Of this result he says,

Were thisanordina ry algeb raic equation,weshould, in thenext p lace, divideboth sidesof it byy. But it has been remarked in ChapterIIthattheopera-tionof division cannotbeperformed withthesymbols with whichwe are nowengaged. Our recourse, then is to express theoperation, and develop theresultby the [general methodinlogic].27Although this passage is somewhat unclear, owing to the fact that Booledoes not say in what the difference between performance and expressionconsists, presumably he is reaffirming the principle of the general method,whereby division may be used as a step in the transformation process, aslong as the process ends with a logically significant expression. Thus expressing division yields x = y/y which, again, is not interpretable inbasic logic. Boole then develops this equation, the result being

x =y +0/0 (1 - y).


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This expression, too, seems to be uninterpretable in basic logic, for itcontains 0/0, which is not a part of basic logic. But Boolegives0/0 thefollowing "logical interpretation":

Let[a] coefficient be in the form 0/0. Now, as in arithmetic, the symbol0/0 represents anindefinite number, except when otherwise determined bysome special circumstance, analogy would suggest that in the system of thiswork the same symbol should represent anindefinite class.28

Inthe particular example under consideration, this interpretation results inth e following meaning being assigned to x =y +0/0 (1 3;)': Mortalsconsist of all men together with an indefinite remainder (some, none, or all)of mortals who are not men.

Before proceeding further, note should be taken of Boole's use of'some'. Symbolized V, he first introduces its use in Chapter IV. Thereherefers to it as an elective of a peculiar sort : ". . . it is indefinite in allrespects but this, that it contains some individuals of the class to whoseexpression it is prefixed . . ."2 9 Thus 'vx' means "somex or, x is notempty." While Boole's intent is clear enough, i.e., to provide a device forth e expression of existential propositions, the device itself suffers from afatal flaw, namely, the impossibility of a class whose only defining characteristic is that it have members. Theflaw is due largely to the restrictiven a t u r eof algebraic logics generally.

With 'some' thus understood, the question now arises, does Boolesurreptitiously slip division into his logic via the expression 0/0? I aminclined to take his disclaimer concerning division seriously, andsuggestthathe does not. There are twoways in which the "indefinite rem ainder"represented by / 0' might be understood. '0/ 0' might be conceived of asan elective standing for a truly indefinite class, i.e., a class with a literallyindefinitely large membership. Were this the intended interpretation, / 0'would indeed have to be counted as another (and quite peculiar) variety ofelective. Also, on such an interpretation, division would become (implicitly)apart of basic logic.

But it is not at all clear that Boole intended 0/0 to function as a distinctcoefficient in basic logic. On the contrary, his use of the symbol inChapterVIsuggests that he thought of 0/0 as being, in effect, a higher levelvariable ranging over the coefficients already available in basic logic.Specifically, Boole seems to use / 0' not to stand directly for an indefiniteclass, but as a way of indicating the irrelevance of class size, in certaincases, for the validity of inferences in which they occur. For instance,concerningthe above example, he asks, ". . . what remainder of not men isimplied by the premise ["men not mortal do not exist "]?" His answer isthat ". . . whether those beings which are not men areall, some, or none,of them mortal, the t ru th of the premise that all men are mortal willbeequally unaffected, and therefore the expression here indicates thatsome,all, ornone of the class to whose expression it is affixed must be taken."30Although this explanation of the use of / 0' is less than totally clear, I takehim to be saying, in effect, that the class is indefinite only at a higher


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level. That is, he is sayingthat the class is composed of some, or none, orall of the mortals, and that there are no other alternatives. Hence at thebasic level,the proposition in question is perfectly determinate: it simplyfalls under one of the three coefficients which the system already provides.T he indefiniteness occurs at a higher level, i.e., at the level at which it isdetermined which of the three possible propositions the class actuallycontains. In other words, '0/ 0' is not simply an addition to V, V, and ' ,but is a way of indicating that class size, though definite, need not bespecified.On this interpretation of '0/ 0', it need not be considered to be a part ofbasic logic at all. Rather, it remains a part of the general method only.However, unlike the other constituents of the general method, it is neithersimply a purely mathematical part of the method, nor does it have a uniquelogical counterpart. Instead, it operates at the levelof the general methodas a variable which ranges over the three logical coefficients.

In summary, in this section I have shown that although there are roughspots in it, and although it defies systematically clear understanding,Boole's logic does not merit much of the criticism aimed ri it. Chargesthat Boole includes logically uninterpretable expressions in his logic havebeenfor the most part cleared, in the sense that most of these expressionswere understood by Boole to be parts of the general method only, and notlogical expressions. While the situation is less clear concerning division,t h e r e is at least a way of understanding Boole's use of expressions such as / 0, which is consistent with his claim that he does not introduce divisionas an operation in (basic) logic.3 Did Boole's successors in algebraic logic really remove the "shortcomings" and "anomalies" from his logic? In the sense that the logics of,e.g., Schroder, Peirce and Whitehead were more fully and systematicallydeveloped than Boole's, the answer is "ye s. " Also, in the later algebraiclogics, one finds fewer of the relatively minor technical gaffes of the sortwhich appear in Boole's work. Thus Peirce and Jevons showed how therestrictionwhich Boole placed on V could be ignored, and the equivocationon V which occurs in LT is not found in later algebraic logics.

Ina different sense, however, the matter is not as clear. The algebraiclogicians in Boole's posterity tended to overlook Boole's device thatpermits the use of mathematical operations in logical contexts, whileavoiding the general conflation of mathematics and logic. Instead, they tendto treat logic and mathematics as being so related that every operation inthe one has its unique counterpart in the other, despite the fact that thisposes serious problems for logic. They seem to be more interested, thatis, in forcing the "fi t" between logic and mathematics than in the problemsthis creates for logic.Nowhere is this tendency clearer than in the treatment divisionreceived at the hands of Boole's logical descendants. While Boole, asindicated earlier, explicitly rejected division as a logical operation in LT,an don the present interpretation avoided even the implicit introduction ofth eoperation into basic logic through such coefficients as 0/0 and 1/0, his


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successors show no ret icence at introducing the operation direct ly intologic. The remainder of this section consists of a few brief examples ofhow division was handled by some of these successo rs. In each case, theresult of admitting logic as a basic operation is the same, namely, thegeneration of fundamentally indeterminateclasses into (basic) logic.

In a paper entitled "On an Improvement in Boole's Calculus of Logic,"C.S. P eir ce sets out to ". . . exhibit Boole's system in a modified form ."After introducing some minor notational changes, he tu rns his attention tothetopic of logical division:

Leta bbereadalogicallydividedbyb,and bedefinedby thecondition thatIfb,x=a x =a b

xis not fully determined by this condition. It will vary fromatoa+bandwillbeuninterpretable ifais notwhollycontained under b.The inclusion of division in logic has the unfortunate result that if

treated as logical abstraction, it yields logically indeterm inate classeswithin what amounts to basic logic itself, which is the result Boole seemingly avoids. That is, x a b is problematic because there is no determinationof bwith respect tox. F rom the equation, we know thatx containsa,but how many members of bare members ofx, if any at all, is left completely undetermined by the operation. Since the analogous indeterminacyin Boole's system was seemingly restricted to a higher level, the sameproblem did not arise there.

Schroder'sDer Operationskreis des Logikkalkuls consists in large partin an attempt to clean up Boole's logic (". . . Nicht deutungsfahigenSymbolen wie 2, 1, 1/3, 1/0 gerechnet werden mu ss."32) In developing hisown calculus of logic, he recognizes four basic operations:

Auchin demkalkulder Logikgibt es, wie in derArithmetik,4SpeciesoderGrundrechnungsarten,welchejedoch,wiesich zeigenwird,endgultigaufdreiverschiedenartige Elementaroperationen reducirt werden ko nnen. Nichtshindert,jene 4 Grundoperationen mit denseIbenNamen zu benennen undmittelst derselben Rechenzeichenauszudr cken,wie sie in der Arithmetikgebrauchlichsind.33

The four operations which Schr der recognizes are (as he sets them out):1) DieMultiplikation, genannt 1') DieAddition oder Collektive

Determination. Zusammenfassung (Collektion).2) D ieDivisionoderAbstraction. 2') DieSubtaktionoderException

(Ausschliessung).34

Schr 'der's approach is somewhat different from P eir ce's in that heattempts to avoid actively using division in his logic (in [12] he speaks ofthe "inverse" operations (i.e., division and subtraction) as being primarilyof "theoretical interest"3 5). Nevertheless, he still considers them to befull fledged operations of the calculus, even if they are "keine unbedingtausfuhrbaren operationen, " due to the fact that unless the usual rest r ic tions are followed, meaningless expressions occur.


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By incorporating division into logic, Schroder meets the same fate asP e i r c e . Beginning with c = a i , Schroder determines the value of c interms ofaandbin the following way:

Als Werth voncfindet sich nun in der genanntenWeise:a rb =abi + ub

a (& +u) b + uabfur ein unbestimmtes u. . , 36

Immediately thereafter, he refers to the class u as ". . . eine volligbeliebige, arbitrare. Here again, the result of using division as a logicalopera toris the generation of fundamentally indeterminate classes.

Of Boole's successors, John Venn was particularly familiar with hiswork. Venn's Symbolic Logic contains perhaps the most complete discussionof Boole's work to be found in the works of any of his successors. OfVenn's work, for instance, Whitehead remarked, "The task of givingthorough consistency to Boole's ideas and notation, with the slightestpossible change, was performed by Venn in his 'Symbolic Logic' "3 7 Earlyin the book Venn comments that Boole ". . . attempts no explanation whatsoever of his use of thesign of division in logic,"38 and then goes on to askwhether there can be a logical interpretation for the operation which is theinverse of multiplication.

is view, like that of Peirce and Schroder, is that division has a placein logic proper. Venn defines x/y' in terms of what he calls "rest ric tio n"in the sense that x/y*is regarded as identifying a class which, when theyrestrictionis placed on it, yields x. The class is hence composed of allx,and,of all things which are " . . . neitherxnor y>it may take as much oras little as it pleases, that is, a perfectly indefinite propor tion."39 Shortlythereafter hesays that it is essential thatx/ y be regarded as identifying aclass, even though it may be ". . . indefinite in regard to its limits . . ., butit is never anythingelse than a true logical class or conceivably assignablegroup of individuals."40

While Boole never gives reasons for his exclusion of division as alogical operation, it is tempting to think that it was occasioned by a desiretoavoid such completely indeterminateclasses in logic. But regardless ofBoole's grounds for rejecting division, suggestions to the effect that Boole'ssuccessors did a better job of developing algebraic logic are not entirelyconvincing. Later logics may have been more tidy, but they lack an import an t variety of sophistication found in LT. That is, as the precedingdiscussion indicates, while later logicians were intent on mathematicizinglogic, Boole's approach is much more sensitive to the requirements oflogic as an independent discipline.4 There remains the question of Boole's importance within the history oflogic. The task of fixing his proper historical place has been largelyobscured by cr itics who measure his achievements, and shortcomings, bycontemporary standards. Instances of this sort of criticism can be foundamong the examples at the beginning of the paper; another is Michael


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Dum mett's review of [2]. In his review, he ch ara cte rize s Boole's achieve-ments by saying,As had Leibniz two centuries earlier, he devised a general theory of classesunder Boolean operations, a theory which of course contained the traditionaltheory of the syllogism . This move gained its imp ortance for logic rat he rfrom the novelty ofany extension of logical theory than from the magnitude ofthe extension itself; and anyone unacquainted with Boole's works will receivean unpleasant sur pris e when he discove rs how ill-constructed the theoryactually was and how confused his explanations of it. 41

Dummett then proceeds to recite the standard criticisms, i.e., those con-cerning the difficulty of symbolizing statements in algebraic logic, theproblem s surrounding V , '+', and '- ', etc. He also critic ize s Boole for notproving that, in effect, the general method is always truth-preserving, andthat the process of development really does, as Boole claims it does, followfrom Taylor's theorem.Such cr itic ism is at be st uninformative. Boole is simply not a contem -po rary logician, and m easuring him by our c urr en t s tandard s ser ve s onlyto undersc ore that fact. Since Boole stands on the other side of a majorrevolution in the science of logic, holding him up to standards used on ourside of that revolution is singula rly unproductive. That Bo ole's logic is notas rigo rou sly developed a s o urs is undeniably tr ue , but also not verysignificant. Conversely, what is significant are attempts to fix, e.g., therelationsh ip between B oole 's work and the logical environment during theperiod which just preceded the full impact of Frege-Principia logic.Instead of following the m ore rec en t v arie ty of c rit ici sm , I have con-cen trated on a feature of Boole's logic which makes him, I think, an im po r-tant con tributor to the transition al period in logic which occupied much ofthe nineteenth century. His development of the general method, rega rd le ssof its impe rfections by our stan da rds, goes well beyond the simple forma l -ization of ordinary language, which had chara cterized logic from A risto tle'sday. In the method, we see one of the first imp ortant a ttem pts to bringmathem atical methods to bear on logic while retaining the ba sic indepen-dence of logic from m athem atics. In this s ens e, the method con stitutes afurther step away from the dependence on ordin ary language which hadcharacterized earlier logics.

Boole thus stand s with a foot in each of two logical epochs; the generalmethod links him with contemporary logic, and the still-present pull of thesyllogism with the older epoch. In the transition, m oreov er, Boole's r oleis pivo tal. After, him , algeb raic logic, by overly str es sin g the connectionbetween logic and mathematics, headed toward extinction by losing sight oflogic as an independent discipline altogether.

NOTES1. By R. Feys in Boole as a logic ian, Proceedings of the Royal Irish Academy,vol . 57, Sec. A, No. 6, pp. 97-106.


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2. E.g . , Boole cannot corr ect ly be cal led the ' fa ther of modern l o g ic ' / ' MichaelDummett in [11], p. 205.3 . A par t icu lar ly engaging ass ess m en t of Boole ' s genera l contr ibution to logic wasgiven by Hugh Mac Coll:

[The Mathematical Analysis of Logic and Laws of Thought] excited muchadmiration in the mathematical world, and, i t may be added, caused nosm all t re pidation among logician s, who saw the ir hi therto inviolatete rr i t or y now for the f irs t t im e invated by a foreign powe r, and withweapons which they had too much rea so n to dre ad. ( Sym bolical re as on -i n g , Mind, vol. V (1880), p. 46.)4. [5], p . 113. As early as 1864, Jevon s displayed a cle ar understa nding of Boo le'sge ne ral method in logi c, explained below . Thus in [6], (176, pp. 67-68), heaccurate ly descr ibes B oole ' s pro ce ss , bu t re je cts i t as no t correspond ing tothe laws and condit ions of thought in reali ty.5. [7], p. 115.6. [3], p. 298.7. [8], p. 422.8. [9], p. 171 .9. [4], p. 543.

10. Boole never published the essay in question and Rhees ([2], p. 12) quotes DeMo rgan as saying that Boole would not have wished su ch an es sa y publishedposthumou sly, since i t might give the im pre ssio n that LT had been reje cted .Also , in the ess ay , Boole s ee m s to acc ept d ivision as a logical operation; how-eve r, in a lat er published work Differential Equations, Suppleme ntary volum e,Mac Millan & Co., C am bridg e (1865)) he again de nies (p. 377) that in ve rs esym bols stand for o pe rat ion s. In addition, Venn was of the opinion that Boolenever included inverse operations in his logic. Cf. [15], p. 77n.)

11. [2], p. 119.12. [1], pp. 37-3 8.13. [1], pp. 36-3 7.14. Boole re fe rs to the pr oc es s in a num ber of ways, and never gives i t a definitenam e. I have chosen ge ne ra l method in lo gi c as a m att er of convenience.Boole uses the term in [1], p . 70, when he says that if logical symbols wereres t r ic ted to log ically in terpretab le contexts only , . . . the inquiry after ageneral method in Logic would be a hopeless quest .15. [1], p. 66.16. [1], p. 67.17. Ibid. Jevo ns held suc h a view, though i t is unlikely that Boole wa s re fer rin g toh im.18. [1], p. 67.19 . [1], pp. 67- 68.20. [1], p . 70. Boole i tal iciz ed the ent ire pas sag e.


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21. [1], p. 72.22. [1], p. 77.23. [1], p. 73.24. [1], p. 72.25. Afairly complete list of these as they oc cu r in Chs. V XV ofLaws of T hought is

as follows: (in each pai r the first num ber is the page, and the expre ssionfollowing is the supposedly unint erp ret able symbol oc cu rring ther e) 94, 1; 95,\\97, i; 115, 2; 123, 2; 125, 2; 131, 2.

26. [1], p. 74.27. [1], p. 89.28. Ibid29. [1], p. 63.30. [1], pp. 89 90.31. [12], 3.6 (p. 6). H ere 'a, b' stands for the logical multiplication of a and b.

'a =b' stands for the 'numerical ident ity" of theclasses aand b.32. [13], p. iii.33. [13], p. 2.34. Ibid35. Cf. [14], Vol. I, pp . 478 ff.36. [13], p. 30.37. [16], p. 115.38. [15], p. viii.39. [15], pp . 77, 78.40. [15], p . 79.41. [11], p. 205.

R F R N S

[1] Boole, G eorge, An I nvestigation of the Laws of Thought,D over Publications,Inc., NewYork (1951). A rep rin t of the London edition of 1854.

[2] Boole, G eorge, Studies inLogic andProbability edited by R. Rhees, Watts&Co., London (1952).

[3] Boche ski, I. M.,A History ofFormal Logic, University of No tre D ame P re ss,Notre D ame, Indiana (1961). A revised translation (by I. Thomas) of FormateLogik, erlag Karl Alber, F rie bu rg, Munich (1956).

[4] H eath, P . L., "Bo o le , " in TheEncyclopedia ofPhilosophy The MacMillan Co.andThe F ree P re ss, NewYork; Collier M acM illan, London (1967).


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[5] Jevons, W. Stanley, The Principles of Science, a Treatise onLogic and Scientific Method, Dover Publicatio ns, Inc ., New York (1958). A re pr int of theLondon edition of 1874.

[6] Jevons, W. Stanley, "P u r e Lo gic " (1864) in Pure Logic and Other MinorWorks, Burt F ranklin, New York (1971). A reprin t of the London edition of1890.

[7] Jorgensen, Jorgen, A Treatise of Formal Logic, Russell and Ru ssell, NewYork (1962). A reprin t of the Copenhagen edition of 1931.

[8] Kneale, William, and Martha Kneale, The Development of Logic, Oxford Universit y P ress, Oxford, England (1962).

[9] Kotarbi ski, Tadeusz, Le ons sur Vhistoire de la logique, Presses universitair es de France (1964). A tr anslation of Wykiady z dziejbw logiki, PWN,Warsaw (1964).

[10] Lewis, C. I., A Survey of Symbolic Logic, Dover P ublications, Inc., New York(1960). A revision of the University of California P ress edition of 1918.

[11] D ummet t, Michael, Review of [2], The Journal of Symbolic Logic, vol. 24(1959), pp. 203 209.

[12] Pei rc e, Charles S., Collected Papers of Charles Sanders Peirce, edited byCharles H artshorne and Paul Weiss, H arvard University P res s, Cambridge(1965).

[13] Schroder, E., Der operationskreis des logikkalkuls, Teubner, Leipzig (1877).[14] Schroder , E., Vorlesungen uber die algebra der logik, 3 vols., Chelsea Pub

lishing Co., Bronx, NewYork. A re pri nt of the Leipzig edition of 1890 1905.[15] Venn, John, Symbolic Logic, Chelsea Publishing Co., Bronx, NewYork (1971).

A reprin t of the London edition (2nd) of 1894.[16] Whitehead, A. N., A Treatise on Universal Algebra, Hafner Publishing Co.,

NewYork (1960). A reprin t of the Cambridge edition of 1897.

Universityof WaterlooWaterloo,Ontario,Canada

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1977 - A Reassessment of Boole's Theory of Logic