abeles - toward a visual proof system - lewis carroll method of trees

Log. Univers.c© 2012 Springer Basel AGDOI 10.1007/s11787-012-0049-6 Logica Universalis

Toward A Visual Proof System:Lewis Carroll’s Method of Trees

Francine F. Abeles

Abstract. In the period 1893–1897 Charles Dodgson, writing as LewisCarroll, published two books and two articles on logic topics. Manuscriptmaterial first published in 1977 together with letters and diary entriesprovide evidence that he was working toward a visual proof system forcomplex syllogistic propositional logic based on a mechanical tree methodthat he devised.

Mathematics Subject Classification. Primary 03F03; Secondary 01A55.

Keywords. Tree proofs, Dodgson.

1. Introduction

In his first book, The Game of Logic (1887) [17], Charles L. Dodgson, writingas Lewis Carroll (1832–1898), created a diagrammatic system to solve syllo-gisms. This visual logic method which employs triliteral and biliteral diagramsis a proof system for categorical syllogisms that we now know is sound andcomplete. The soundness of a proof system ensures that only true conclusionscan be deduced. Conversely, its completeness guarantees that all true conclu-sions can be deduced. But he did not use the method as a proof system beyondsyllogisms. For more complex arguments, he settled on another visual method,the Method of Trees. Dodgson died before he could publish this work in partII of his book on symbolic logic. W. W. Bartley, III (1934–1990) discoveredgalley proofs and reconstructed the material, publishing it in 1977. Dodgson’sMethod of Trees is a sound, complete, and decidable proof system for soriteseswhich are complex syllogistic arguments. The decidability of a proof systemrefers to the existence of an effective method to determine whether a givenproposition is logically valid. The concepts of soundness, completeness, anddecidability of proof systems were first applied in the twentieth century. (Fora more complete development of these ideas, see [2,4].)

His second book, one that went into four editions in the year of its pub-lication, Symbolic Logic, Part I, provided a bridge between what he created in

F. F. Abeles Log. Univers.

The Game of Logic and what he accomplished but never published in the sec-ond part of his symbolic logic book [14]. In Symbolic Logic, Part I (1896), in thesection titled, Appendix, Addressed to Teachers, Carroll indicated some of thetopics he planned for part II of his book. These include “[T]he very puzzlingsubjects of Hypotheticals, Dilemmas, and Paradoxes.” [10, p. 229] Dodgsonconsidered hypotheticals and paradoxes—he often used the term dilemmas forthese, as intriguing subjects to work on. And earlier, he had displayed an inter-est in mathematical fallacies which he defined as an “argument which deceivesus, by seeming to prove what it does not really prove. . . ” [10, p. 129] Dodgsonwas generally interested in the quality of arguments, particularly those thatcould confuse. Paradoxes fall in this category because they appear to provewhat is known to be false. And paradoxes certainly challenged Dodgson tocreate ingenious methods to solve them, particularly his tree method.

For example, working on the Liar Paradox in 1894, Carroll included twoversions of it in part II of symbolic logic. The first is, “The Problem of the FiveLiars,” and the second is, “The Salt and Mustard Problem.” [10, pp. 352–354]

In a diary entry on 24 September 1895 Dodgson wrote,

I got, from Adamson, [Charles Stewart Adamson (1867–1897), fellowof St. John’s] a “Liar” Problem in which one says “Three of theothers lie, three speak truly.” I have made some of this new kind,viz. “Liar Problems without personalities.” They seem to need apeculiar kind of Tree. [41, p. 216]

The Liar Paradox, invented by the Greek philosopher Eubulides in thefourth century B.C.E., is important in logic because it deals with the conceptsof truth and falseness on which the fundamental idea of the validity of a logicalargument is based.

In subsequent sections of this paper, I will discuss Carroll’s two articlesin the journal, Mind, “A Logical Paradox,” published in 1894, and “Whatthe Tortoise said to Achilles,” published the following year. The first of these,the “Barbershop Paradox,” is really a fallacy. And “What the Tortoise Saidto Achilles,” Dodgson called a puzzle. [10, p. 470] He used the term puzzle todescribe cunning logical arguments as well as paradoxes. I will argue that thesetwo articles deal with important issues that Dodgson wanted to resolve as heworked toward a visual proof system for soriteses in the period, 1893–1897.From letters and diary entries, we know that he began working with his treemethod in October of 1894, and was working with it in February of 1897. Hebegan his work with hypothetical propositions, i.e., conditionals, in Februaryof 1893, and was engaged with it in December of 1896.

2. The Method of Trees

In July 1894, Dodgson devised his Method of Trees to test the validity of highlycomplicated multiliteral statements that he called soriteses. A form of syllogis-tic reasoning, the argument of a sorites is arranged so that the predicate of anyone of the propositions becomes the subject of the next one, and the conclusion

Toward A Visual Proof System

brings the subject of the first proposition together with the predicate of thelast proposition. In Dodgson’s time, this type of reasoning still was consideredto be the archetype of correct reasoning. The object is to obtain the strongest(complete) possible conclusion from the premises. For Dodgson’s contempo-raries, the central problem of the logic of classes, known as the eliminationproblem, was to determine the maximum amount of information obtainablefrom a given set of propositions. In his 1854 book, An Investigation of theLaws of Thought [13], George Boole (1815–1864) made the solution to thisproblem considerably more complex when he provided the mechanism of apurely symbolic treatment which allowed propositions to have any numberof terms, thereby introducing the possibility of an overwhelming number ofcomputations. Dodgson’s tree method is a way of reasoning efficiently fromsoriteses, and it precedes the work of twentieth century logicians who devel-oped it further. His tree method is a mechanical test of validity through areductio ad absurdum argument for a large part of propositional logic. (Thistopic is developed in greater detail in [5]; also see [8] and [9].) As I. H. Anellisnoted, “[T]he tree method has a history . . . there are syllogistic versions ofthe trees of Beth tableaux to be found in the work from the last century ofCharles Dodgson.” [8, p. 62].

To use the tree method to test the validity of arguments, or equivalentlythe consistency of a set of suitable sentences, we list the basic inference rulesfor arbitrary propositions (sentences) S, T .

In rules 2, 4, 8, the symbols below the line are connected by “or;” in rules 5and 10 there are two sets of these. In rules 3, 7, 9 the two symbols below theline are connected by “and.” Note that Dodgson considered the three state-ments: no x are y; some x exist and none of them are y; all x are not y, to beequivalent

The tree method is a direct extension of truth tables, and migrating totrees from the tables is easy to do. (For a complete discussion of this topic,see [6].) Using truth tables to verify inconsistency is straight forward, butvery inefficient, as anyone who has worked with truth tables involving eight


or more cases knows. Instead, the truth tree method examines sets of casessimultaneously, thereby making it efficient to test the validity of argumentsinvolving a very large number of sentences by hand or with a computer. Totest the validity of an argument consisting of two premises and a conclusion,equivalently determining whether the set of the two premise sentences andthe denial of the conclusion sentence is inconsistent, by the method of truthtables involving say, three terms, requires calculating the truth values in eightcases to determine whether or not there is any case where the values of allthree terms are true. But a finished closed tree establishes that validity ofthe argument by showing there are no cases in which the three sentences aretrue. However, if any path in a finished tree cannot be closed, the argument isinvalid because an open path represents a set of counterexamples.

3. The Froggy Problem

The Appendix, Addressed to Teachers in the fourth edition of his book on sym-bolic logic also contains eight problems that Carroll posed to hint at what hisreaders could expect to see in the second part of the book. Though unnamed,“The Froggy Problem” is the third of these eight. [15, pp. 188–189] No solutionto it was ever recorded until 2010 when Graham Hawker published his solu-tion. An overview of it follows. The reader is encouraged to read the completesolution [27]

Hawker employed Dodgson’s tree method to determine the complete con-clusion to this problem, i.e. determining all the relations among the four ret-inends, the word Dodgson used for all the terms appearing in the conclusionthat are deducible from its premises which contain eliminands, the word heused for the terms that are eliminated in arriving at the conclusion.

Premises. [16, pp. 188–189]

1. When the day is fine, I tell Froggy “You’re quite the dandy, old chap!”;2. Whenever I let Froggy forget that 10 pounds he owes me, and he begins

to strut about like a peacock, his mother declares “He shall not go outa–wooing!”;

3. Now that Froggy’s hair is out of curl, he has put away his gorgeous waist-coat;

4. Whenever I go out on the roof to enjoy a quiet cigar, I’m sure to discoverthat my purse is empty;

5. When my tailor calls with his little bill, and I remind Froggy of that 10pounds he owes me, he does not grin like a hyena;

6. When it is very hot, the thermometer is high;7. When the day is fine, and I’m not in the humor for a cigar, and Froggy is

grinning like a hyena, I never venture to hint that he’s quite the dandy;8. When my tailor calls with his little bill and finds me with an empty purse,

I remind Froggy of that 10 pounds he owes me;9. My railway shares are going up like anything!


10. When my purse is empty, and when, noticing that Froggy has got hisgorgeous waistcoat on, I venture to remind him of that 10 pounds heowes me, things are apt to get rather warm;

11. Now that it looks like rain, and Froggy is grinning like a hyena, I can dowithout my cigar;

12. When the thermometer is high, you need not trouble yourself to take anumbrella;

13. When Froggy has his gorgeous waistcoat on, but is not strutting aboutlike a peacock, I betake myself to a quiet cigar;

14. When I tell Froggy that he’s quite a dandy, he grins like a hyena;15. When my purse is tolerably full, and Froggy’s hair is one mass of curls,

and when he is not strutting about like a peacock, I go out on the roof;16. When my railways shares are going up, and when it’s chilly and looks

like rain, I have a quiet cigar;17. When Froggy’s mother lets him go a–wooing, he seems nearly mad with

joy, and puts on a waistcoat that is gorgeous beyond words;18. When it is going to rain, and I am having a quiet cigar, and Froggy is

not intending to go a–wooing, you had better take an umbrella;19. When my railway shares are going up, and Froggy seems nearly mad with

joy, that is the time my tailor always chooses for calling with his littlebill;

20. When the day is cool and the thermometer low, and I say nothing toFroggy about his being quite the dandy, and there’s not the ghost of agrin on his face, I haven’t the heart for my cigar!

Dictionary [10, 340]

1. Universe: “Cosmophases”;2. E = this;3. a = Froggy’s hair is out of curl;4. b = Froggy intends to go a–wooing;5. c = Froggy is grinning like a hyena;6. d = Froggy’s mother permits him to go a–wooing;7. e = Froggy seems nearly mad with joy;8. h = Froggy is strutting about like a peacock;9. k = Froggy is wearing a waistcoat that is gorgeous beyond words;

10. l = I go out on my roof;11. m = I remind Froggy of the 10 pounds he owes me;12. n = I take a quiet cigar;13. r = I tell Froggy that he’s quite the dandy;14. s = It is going to rain;15. t = It is very hot;16. v = My purse is empty;17. w = My railway shares are going up;18. z = My tailor calls with his little bill;19. A = The thermometer is high;20. B = You had better take an umbrella;


To begin he assumes the negative of the conclusion, i.e. Ea′b′d, and goes on toshow this assumption leads to a contradiction. The process requires 54 stepsand the height of the tree, beginning at the root and ending at the leaves, is 6.The symbolic statement of the conclusion, no Ea′d are b′ translates to: ifFroggy’s hair was a mass of curls and he had his mother’s permission he wouldgo a-wooing today.

Dodgson defined the term, Cosmophase, as “the state of the Universe atsome particular moment: and I regard any Proposition, which is true at thatmoment, as an Attribute of that Cosmophase.” [10, p. 481] Dodgson was notthe first to employ the concept of a universe, but it was a feature his con-temporary, John Venn (1834–1923) did not think important enough to botherwith. (See [38–40]) Yet it is an idea that is essential in depicting the universeof discourse, a key concept in modern logic that was introduced in 1847 byAugustus De Morgan (1806–1871) in his book, Formal Logic [23], also dis-cussed by Boole [12, pp. 15–16] and developed further by him in 1854. [13, pp.42– 43].

In computer science, a database has a state which is a value for each ofits elements. A trigger can test a condition that can be specified by a whenclause, i.e. a certain action will be executed only if the rule is triggered and thecondition holds when the triggering event occurs. Curiously, Dodgson’s defi-nition of a Cosmophase, although it is not recognized as a word in theOxfordEnglish Dictionary, fits nicely into this modern framework. In the database for“The Froggy Problem,” the elements are the propositions in the Dictionary,and the when conditionals are the 17 of the 20 premises that begin with theword, when.

Dodgson first began to handle soriteses involving hypotheticals in the sec-ond edition of his symbolic logic book where he included as the first problemin the section, Appendix, Addressed to Teachers, one whose 12 propositionsbegin with, whenever. And in another problem, the third, where six proposi-tions begin with if. (The third edition is rare and I have been unable to consultit.)

4. Superfluous Premises

Problem 2 in the Appendix, Addressed to Teachers, though unnamed, is knownas “The Pork Chop Problem”. [10, pp. 331–337] In an exchange of letters inOctober and November of 1896 to John Cook Wilson (1849–1915), WykehamProfessor of Logic at Oxford, Dodgson modified the original eighteen pre-mise version containing superfluous premises to one that appears with fifteenpremises. Bartley includes both versions as well as their solutions by the treemethod.

In a letter dated November 13, 1896, to his mathematically talented sis-ter, Louisa, Dodgson wrote, “Mr. Cook Wilson says he has found a Rule [fordetecting superfluous premises]. . . I have had very little success in devisingmodes of finding out whether there are superfluous Premisses.” [10, 345n].

In another letter to Louisa dated November 18, 1896, Dodgson wrotein reference to the verification of a tree that she was working on, “It often


happens, in a Tree, that we tack on Attributes that are of no use furtherdown, so that the quoted Premiss is, in that place, superfluous. You mayalways know such a Premiss, while verifying, by finding that it wo’n’t [sic]eliminate anything. When this happens, the simple rule is to ignore that Pre-miss, even if it’s a Branch one. . . . My concluding remark will, I expect, be asurprise to you.You haven’t got the Complete Conclusion: and you’ve used foursuperfluous Premisses! [10, pp. 348–350].

5. Hypotheticals and Sequences

In August 1894, Dodgson submitted his paper, “What the Tortoise said toAchilles,” to G. F. Stout (1860–1944), the editor of Mind. In a letter datedAugust 24 Stout asked Dodgson, “Is there any difference between affirmingA and affirming the truth of A?” The very next day Carroll responded, “Ofcourse affirming a Proposition, and affirming that the Proposition is true arethe same thing: but if ‘A’ represents a long enunciation, it is very convenient tobe allowed to say ‘A is true’, instead of quoting it at full length . . . in a Hypo-thetical, the truth of the Protasis, the truth of the Apodosis, & the validity ofthe sequence, are 3 distinct Propositions.” He gave this example,

If I grant

(1) All men are mortal, and Socrates is a man. but not(2) The sequence “If all men are mortal, and if Socrates is a man, then Soc-

rates is mortal” is valid. Then I do not grant(3) Socrates is mortal. [10, pp. 471–472]

Three weeks later in a letter dated September 17, 1894 and addressed,“Dear Sir,” most likely John Alexander Stewart (1846–1933), White’s Profes-sor of Moral Philosophy at Oxford, Dodgson explicitly distinguished the termshypothetical and sequence. “I thought Hypotheticals never asserted facts, butmerely sequences.” [Unpublished ms., Berol Collection [11]].

In two diary entries, the first on 11 December 1894; the second 10 dayslater he recorded,

I am giving all my time to Logic, and have at last got a workabletheory of Hypotheticals—to represent “a & b” by “ab′

0 † a1 † b′1”

meaning by “0”,“cannot exist”, and by “1”, “can exist.” [41, p. 184]My night’s thinking over the very puzzling subject of “Hypoth-

eticals” seems to have evolved a new idea—that there are two kinds,(1) where the Protasis [premise] is independent of the Hypothetical,(2) where it is dependent on it. [41, pp. 185–186]

It seems clear that between August and December of 1894, Dodgson may havebeen considering a direction that was more formally developed later by HughMacColl (1837–1909), as early as 1896–1897, and expanded in his 1906 book,Symbolic Logic and Its Applications [31], where he defined strict implication,in which the content of the antecedent and consequent have a bearing on thevalidity of the conditional, 20 years before modal logic began to be placed on


a modern footing beginning with the work of the American philosopher andlogician, Clarence Irving Lewis (1883–1964).

But in Bartley’s edition of Carroll’s Symbolic Logic, we can see that upuntil August 1894, Dodgson developed a formal logic, i.e. a logic in which thevalidity of an argument is a matter of form only, not of interpretation. Dodgsonbelieved that deductions are independent of the meanings of the terms beingreasoned about. For him, a correct mathematical argument is a certainty, thatis, an argument is true once a proof of it has been given, without appealing tothe meanings of the terms in its premises and conclusion. (For a more completedevelopment, see [3].)

Dodgson published a number of articles dealing with the topic of hy-potheticals. Three papers appeared in 1892, two of which are the Eighth paperon Logic, in two versions, and the third is the Eighth and Ninth Papers onLogic. Notes. An example of a sequence problem solved by the tree methodthat Dodgson gave in the first version of the Eighth paper on Logic is,(1) If all a are b, no c are d;(2) If no a are b, and if some c are not d, some e are not f ;(3) If some a are b and some not, and if some g are h, no e are f .

Prove that, if some c are d and some not, and if all e are f , no g are h.[10, p. 323]

Dodgson also created an elaborate problem in sequences with the name,“The Problem of the School-Boys” that is related to a privately printed piecefrom 1892 called, “A Challenge to Logicians.” Bartley discusses the connec-tions between them extensively. [10, pp. 326–331]

What The Tortoise Said To Achilles. Dodgson’s concern with superfluouspremises may have been the underlying motivation for his article, “What theTortoise Said to Achilles,” which appeared in Mind in 1895 [20]. This humorouspiece deals with an important problem about inference in logic that Dodgsonwas the first to recognize, i.e. the rule permitting a conclusion to be drawnfrom premises cannot be treated as a further premise without generating aninfinite regress.

In “What the Tortoise Said to Achilles,” the tortoise dictates these threestatements to Achilles who writes them down. [1, pp. 188–191]

A. Things that are equal to the same thing are equal to each other.B. The two sides of this triangle are two things that are equal to the same.Z. The two sides of this triangle are equal to each other.

{There is an error on p. 106 in the statement of Z in [1]}The tortoise points out to Achilles that someone could refuse to accept

Z for one of two reasons. The first because she might deny the truth of thepremises A and B; second because she might deny the validity of the inferencefrom the true premises to the conclusion Z.

The tortoise now asks Achilles to imagine that he, the tortoise, believesin the second reason for refusal and challenges Achilles to convince him tologically accept Z as true. In other words, the Tortoise accepts the truth of Aand of B, but denies the inference: C. If A and B are true, Z must be true.


And even if the tortoise can be convinced Cis true and to use it as a premisein a new argument: D. If A and B and C are true, Z must be true, the tortoisecould still refuse to accept Z because he does not accept the further inferenceD, etc., ad infinitum.

Recall that Zeno’s paradox involves a race between Achilles, who couldrun ten times as fast as the tortoise who is given a 100 yard start, where therace continues until Achilles either overtakes the tortoise or resigns. It’s clearthat when Achilles has run the first 100 yards, the tortoise has advanced 10yards. When Achilles has run those 10 yards, the tortoise has gone one moreyard, etc. To overtake the tortoise, Achilles must advance over an infinite num-ber of successive distances. So Achilles can never overtake the tortoise. Theparadox, more appropriately the fallacy, occurs because of the false assump-tion that a set of distances infinite in number is also infinite in total length. Inother words, the two contestants are not assumed to be running with uniformspeeds where half the distance run would occur in half the time, a quarterof the distance in a quarter of the time, etc. If this assumption is invoked,Achilles could and would overtake the tortoise.

Zeno’s Paradox appears as the fifth of six classical puzzles in Book XXIof part II of Dodgson’s symbolic logic where he remarked that “Achilles andthe Tortoise” is not a paradox, but a “mathematical Fallacy, and involves thefalse assumption that a series of distances, infinite as to number, is also infiniteas to total length.” [10, p. 438].

In a letter dated 14 December 1896 answering an earlier one from CookWilson bearing on the principles of logical inference involved in “What theTortoise Said to Achilles,” Dodgson disagreed with Wilson’s interpretation ofa further conclusion of a given [different] syllogism which concerns the use ofa rule of inference, as a premise. [10, p. 475].

Ivor Grattan-Guinness observes that the tortoise’s argument in “Whatthe Tortoise Said to Achilles” is the reverse of Zeno’s. In Zeno’s, there is aninfinite number of steps that (it is alleged) can be completed in a finite time.Whereas in “What the Tortoise Said to Achilles” there are three steps that(seem to) require an infinite amount of time. Grattan-Guinness [26, pp. 167–168] adds that the infinite intermediate sequence of steps C, D,. . . implyingthat Z can never be deduced from A and B contradicts the fact that deduc-tions like this one are always being made. [26, pp. 167–168] (For different views,see ([25] and [36]).

Barbershop Paradox. In “What the Tortoise said to Achilles,” Dodgson’s focusis on the nature of the sequences. In the “Barbershop Paradox,” his focus ison the process of reasoning from the premises to the conclusion. The unsettlednature of the topic of hypotheticals during Dodgson’s lifetime is apparent atthe beginning of the Note that Carroll appended to his article dealing with theBarbershop Paradox, titled, “A Logical Paradox,” published in Mind in 1894.

This paradox. . . is, I have reason to believe, a very real difficultyin the Theory of Hypotheticals. The disputed point has been forsome time under discussion by several practised logicians, to whom


I have submitted it; and the various and conflicting opinions, whichmy correspondence with them has elicited, convince me that thesubject needs further consideration, in order that logical teachersand writers may come to some agreement as to what Hypotheticalsare, and how they ought to be treated. [10, p. 438]

Dodgson struggled with a number of issues surrounding hypotheticals in thisNote. He asks several questions, the first being whether a hypothetical can belegitimate when its premise is false; the second being whether two hypothe-ticals whose forms are if A then B; if A then not-B can be compatible.

In 1892 William E. Johnson (1858–1931) had published the first of threepapers in Mind titled “The Logical Calculus” where he distinguished the term,conditional from the term, hypothetical. Dodgson, like most logicians of his timedid not make this distinction, and used the term hypothetical for both situa-tions. Johnson’s view was that a conditional expresses a relation between twophenomena, while a hypothetical expresses a relation between two propositionsof independent import. So a conditional connects two terms, while a hypothet-ical connects two propositions. [28, p. 17] John Neville Keynes (1852–1949),with whose work Dodgson was quite familiar, agreed with Johnson’s view.Venn, however, although he, too, knew Johnson’s work held a very differentview of hypotheticals, contending that because they are of a non-formal nature,they really should not be considered part of symbolic logic. [38, p. 243]

Earlier versions of the “Barbershop Paradox,” show the change in theway Dodgson represented conditionals. In the earlier versions, he expressed ahypothetical proposition in terms of classes, i.e. if A is B, then C is D. Onlylater did he designate A, B, C, and D as propositions.

Also we know that Dodgson had written to several logicians abouthypotheticals, including John Venn. A letter dated 11 August, 1894 fromDodgson to Venn resulted in Venn including a version of the Barbershop Para-dox in the second edition (1884) of his book, Symbolic Logic on p. 442. Keynes[30, pp. 273–274] included a version of the Barbershop Paradox in editions ofhis book; and the philosopher of logic, Francis H. Bradley (1846–1924) dis-cussed it in correspondence. [29, pp. 92–95].

The Barbershop Paradox can be stated this way. There are three men,Allen, Brown, Carr, in a house who may go in and out, providing all never goout at the same time. Also, Allen never goes out without Brown. Under theseconditions, can Carr ever go out? Symbolically,

1. If A is true, B is true.2. C is true, then if A is true B is not true.

Bertrand Russell (1872–1970) gave what is now the generally accepted con-clusion to this problem in his 1903 book, The Principles of Mathematics. If prepresents: Carr is out; q represents: Allen is out; r represents: Brown is out,then the Barbershop Paradox can be written as: (1) q implies r; (2) p impliesthat q implies not-r. Russell asserts that the only correct inference from (1)and (2) is: if p is true, q is false, i.e. if Carr is out, Allen is in. [37, p. 18].


When working with conditionals, Dodgson employs material implicationin which the connection between the antecedent and the consequent of the con-ditional [if (antecedent), then (consequent)] is formal, i.e. it does not dependon their truth values, a result of Boole’s logic, to argue for Carr leaving theshop but that he was uncomfortable with it.

A version of the Barbershop Paradox that was not recognized as such byBartley, Question 14122, was published in February 1899 in The EducationalTimes after Dodgson’s death, and was reprinted in Mathematical Questionsand Solutions the next year [18,19]. Two different solutions appeared thatsame year, one by Harold Worthington Curjel (1868–1945), a member of theLondon Mathematical Society, the other by Hugh MacColl. [22,32] (For a moredetailed discussion of the Barbershop Paradox, see [35] and [36].)

6. Conclusion

In the history of mathematics Dodgson stands out as the only mathematicianwho has achieved world-wide fame as a literary figure. However, his mathe-matical writing often reflected his literary bent, resulting in a prose style thatseparated him from other mathematicians of his time. And his almost obsessiveconcern with exactness introduced a certain stiffness into many of his seriousmathematical works. But the humor he used is infectious and infuses many ofhis mathematical works, particularly those on logic. That this was his reputa-tion is apparent in reviews of Symbolic Logic, Part I appearing during his lifetime. (See, for example, MacColl’s review in The Athenaeum in 1896 [33], andan anonymous review in The Educational Times the same year [9].) And thequotations that continue to be taken by modern authors, reinforces this view.For example, in chapter III of Book XII in part II of Symbolic Logic, instead ofjust exhibiting the tree piecemeal for a particular problem he gives a “solilo-quy” as he works it through, accompanied by “stage directions” showing whathe is doing [10, pp. 289–292]. This approach enables the reader to constructthe tree for himself and be entertained in the process.

Perhaps the most serious criticism of Carroll’s logic writings is that theydeal only with syllogistic logic, the basis of the system of logical reasoningthat prevailed in England up to the first quarter of the 20th century. AlthoughBoole and his followers understood that they were just algebraicizing logic, i.e.rewriting syllogisms in a new notational system rather than inventing a newlogical calculus, nevertheless, they correctly claimed that all valid argumentscannot be reduced to these traditional forms. Dodgson, however, was intenton demonstrating that syllogistic logic permitted much more general reasoningthan what was commonly believed.

Dodgson’s place in the development of symbolic logic in England wasunknown until Bartley published his edition of Carroll’s symbolic logic book,including his reconstruction from newly found manuscript material of Carroll’sunpublished second part of that book. Dodgson had indicated he planned athird part, but it has never been found, and probably was never written.Bartley’s publication changed Carroll’s reputation as a logician. In the past


35 years other logicians have added to what Bartley began, writing extensivelyabout Dodgson’s place in the development of symbolic logic in England. (See,for example, [34], and [24].) In this paper I have taken a further step by dem-onstrating that the direction of Dodgson’s work in symbolic logic in the period1893 to 1897 was toward the development of a visual proof system for complexsyllogistic propositional logic based on the mechanical tree method that hedevised.

Acknowledgment

The author wishes to thank Professor M. Wagner for creating the LaTexversion of this paper.

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Francine F. AbelesDepartment of Mathematics and Department of Computer ScienceKean UniversityUnion, NJ 07083USAe-mail: [email protected]

Received: January 27, 2012.

Accepted: May 2, 2012.

abeles - toward a visual proof system - lewis carroll method of trees

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