Lewis Carroll's Formal Logic

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Lewis Carroll’s Formal Logic

FRANCINE F. ABELES

Department of Mathematics & Computer Science, Kean University, Union, NJ 07083, USA

Received 10 November 2003 Revised 2 September 2004

Charles L. Dodgson’s reputation as a significant figure in nineteenth-century logic was firmly establishedwhen the philosopher and historian of philosophy William Warren Bartley, III published Dodgson’s ‘lost’book of logic, Part II of Symbolic Logic, in 1977. Bartley’s commentary and annotations confirm that Dodg-son was a superb technical innovator. In this paper, I closely examine Dodgson’s methods and their evolutionin the two parts of Symbolic Logic to clarify and justify Bartley’s claims. Then, using more recent publica-tions and unpublished letters, I argue that Dodgson approached the elimination problem in class logic dif-ferently than his contemporaries, and in doing so, anticipated several important concepts and techniques inautomated deductive reasoning. These materials also provide additional insight into his reasons for writingthis book.

1. Introduction

Charles L. Dodgson began writing his magnum opus, Symbolic Logic in the 1870sunder his pseudonym, Lewis Carroll. The first part was published in 1896, two yearsbefore his death. Two more parts were projected but never appeared. WilliamWarrenBartley, III discovered the galley proofs of Part II, with the assistance of Morton N.Cohen, and the second part appeared for the first time in 1977 in his edition ofCarroll’s Symbolic Logic. In these two parts Dodgson developed a formal logic inwhich he set down intuitively valid formal rules for making inferences. Since incarrying out an inference a variety of possible steps are available, Dodgson used hisrules to determine which steps were the most promising. The set of techniques hecreated to mechanize logical reasoning is the subject of this paper.

A comparison of the two parts reveals the progress Dodgson made toward anautomated approach to the solution of multiple connected syllogistic problems(soriteses), and puzzle problems bearing intriguing names like ‘The Problem ofGrocers on Bicycles’, and ‘The Pigs and Balloons Problem’. His approach led him toinvent the Method of Trees in 1894, discussed in part 3 of this paper. AlthoughDodgson worked with a restricted form of the logic of classes and used ratherawkward notation and odd names, he introduced important methods thatforeshadowed modern concepts and techniques in automated reasoning. Some ofthese are: truth trees, binary resolution, unit preference and set of support strategies,and refutation completeness.

The central problem of the logic of classes is known as the elimination problem: todetermine the maximum amount of information obtainable from a given set ofpropositions. In his 1854 publication of An Investigation of the Laws of Thought,George Boole made the solution to this problem considerably more complex when heprovided the mechanism of a purely symbolic treatment which allowed propositionsto have any number of terms, thereby introducing the possibility of an overwhelmingnumber of computations. Several British logicians who offered improvements wereprimarily: William Stanley Jevons who introduced a logical alphabet for class logic in1869, and John Venn who in 1880 constructed a graphical method capable of

HISTORY AND PHILOSOPHY OF LOGIC, 26 (FEBRUARY 2005), 33–46

History and Philosophy of LogicISSN 0144-5340 print/ISSN 1464-5149 online # 2005 Taylor & Francis Ltd

http://www.tandf.co.uk/journals DOI: 10.1080/01445340412331311947

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satisfactorily representing at most four simply connected regions. Dodgson improvedon Venn’s method by inventing a novel procedure whereby ten sets can be easilyrepresented. More importantly, Dodgson chose a different route when he developedhis main approaches to deal with the elimination problem in the second part ofSymbolic Logic.

Bartley made several claims about the importance of Dodgson’s work: ‘Part IIof Symbolic Logic reveals Carroll as a more interesting technical innovator than hadhitherto been supposed. . . .’ (1977, p. 28) ‘Even on the technical level, one finds inPart II what one might expect of a first-class logician working just seven yearsbefore Russell published The Principles of Mathematics.’ (1977, p. 30) In this paperwe will examine Dodgson’s methods in Symbolic Logic closely, tracing theirevolution from Part I to Part II, with the aim of clarifying and justifying Bartley’sclaims.

2. The elimination problem in Part I of Symbolic LogicEssential to Dodgson’s approach is his idiosyncratic algebraic notation which he

called the Method of Subscripts. A short glossary will give the reader the essentials(Bartley 1977, pp. 119–120). Letters are used for terms which can represent classes orattributes.

In Part II of Symbolic Logic letters are used to represent statements too. Thesubscript 0 on a letter denotes the negation of the existence of the object; the subscript1 denotes the object’s existence. When there are two letters in an expression, it doesnot matter which of them is first or which of them is subscripted because eachsubscript takes effect from the beginning of the expression. For example, xy0 can beread as ‘no x are y’, or as ‘no y are x’, while x1 y0 means all x are not y. Clearly, thesetwo expressions are equivalent (see Figure 1).

Dodgson’s first formal technique, the Method of Underscoring, permits droppingtwo terms of opposite sign (one is the negative of the other) in propositions that areconnected by ‘and.’ For example, in xm0 { ym’0, m and m’ can be eliminated.(Dodgson used a double underscore on the second term of the pair.) When he actuallyused this method to obtain a complete conclusion to a sorites involving two terms, heomitted the subscripts. The terms that are eliminated in the premises he calledEliminands; the terms that are retained and appear in the conclusion, he calledRetinends. He called an intermediate conclusion a partial conclusion. The nextexample exhibits the formal working out of a sorites using this method (Bartley 1977,pp. 138–139).

Premises 1 2 3 4 5 6 7k1l’0 { dh’0 { a1c0 { b1e’0 { k’h0 { b’l0 { d’1c’0

Each premise is connected by ‘and’. To solve the sorites, first order the premises sothat their terms can be eliminated suitably. One such order is: 1,5,2,6,4,7,3. Workingwith premises 1 and 5, k and k’ are eliminated, leaving l’h. ‘Anding’ this partialconclusion (PC) together with premise 2 produces l’d (by eliminating h and h’).‘Anding’ this PC with premise 6 produces db’ (by eliminating the l, l’ pair). ‘Anding’this PC with premise 4 gives de’ (by eliminating the b, b’ pair). ‘Anding’ this PC withpremise 7 gives e’c’ (by eliminating the d, d’ pair). Finally, ‘anding’ this PC withpremise 3 produces the conclusion e’a (by eliminating the c, c’ pair). Since a is an entity(in premise 3), Dodgson ‘ands’ a1 to the conclusion to give the complete conclusiona1e’0.

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Dodgson’s Method of Diagrams is the second and last formal technique hepresented in Part I. Essentially, it’s an improvement on Venn’s diagrammatic method.As Trenchard More, Jr 1959 showed, Venn diagrams can be constructed for anynumber (greater than three) of simply connected regions bounded by Jordan curves,but not easily. Anthony J. Macula 1995 proved that Dodgson’s method works easilyfor ten sets or more. However, Dodgson did not provide any examples beyond threesets in Part I or Part II. Here is one of his examples, a triliteral diagram for thecomplete conclusion of the syllogism:

No philosophers are conceited. No x are mSome conceited persons are not gamblers. Some m are y’Some persons, who are not gamblers, are not philosophers. Some y’ are x’

Figure 1. Glossary

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Premise 1 states no xm exist, so we indicate the xm compartment is empty byplacing a ‘0’ in each of its cells. Premise 2 states some my’ exist, so we indicate the my’compartment is occupied by placing a ‘1’ in its only available cell. The informationthis gives about x and y is that the x’y’ compartment is occupied, i.e. some x’y’ exist.Hence the complete conclusion is: some persons who are not philosophers are notgamblers. This diagram (without the 0s and 1) is Figure 2 (Bartley 1977, p. 93)

In Part II of Symbolic Logic, Dodgson generalized his Method of Diagrams. Afour set diagram involves sixteen cells (boxes): a square divided into four equalsquares and two equal rectangles, one placed vertically and the other horizontally inthe large square. This diagram represents the sets S1, S2, S3, S4 and their complements.To construct an eight set diagram involving 256 boxes, place in each of the sixteenboxes of the four set diagram a smaller version of itself. S5 will be everything on theleft side of any one of these small four set diagrams, and its complement will beeverything on the right side of any one of these small four set diagrams. S6 will be thetop half of all the small four set diagrams, and its complement will be the bottom halfof all the small four set diagrams. Everything inside the vertical rectangle of any one ofthe small four set diagrams is S7, and its complement is everything outside any one ofthese vertical rectangles. Everything inside the horizontal rectangle of any one of thesesmall four set diagrams is S8, and its complement is everything outside any one of thehorizontal rectangles (Macula 1995, pp. 270–271; Bartley 1977, pp. 244–246.).

3. The elimination problem in Part II of Symbolic LogicDodgson introduced two additional methods of formal logic in Part II of Symbolic

Logic. The first, a direct approach to the solution of multiliteral soriteses, he calledbarred premises which is an extension of his underscoring method. Briefly, a barredpremise is one in which a term t occurs in one premise and its negative t’ occurs in twoor more premises, and conversely. For example, if a premise contains the term a and

Figure 2. A Triliteral Diagram

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the two eliminand terms bc, then abc is a nullity implying that a has the pair ofattributes: bc’ or b’c or b’c’, i.e. a is barred by the nullity from having attributes bc.Dodgson extended this idea to what he called a barred group: when a term t occurs intwo or more premises and t’ also occurs in two or more premises. His rule for workingwith barred premises requires that all the premises barring a given premise be usedfirst. We will call these definitions and the rule for working with them his Method ofBarred Premises. It is an early formal technique to guide the order of use of thepremises of a sorites to arrive at the conclusion.

Here is an example, ‘The Pigs and Balloons Problem’ (Bartley, 1977, pp. 378–380).Since this problem is in the first figure (a formula for a syllogism having two premisesthat are nullities with unlike eliminands, the same terms having opposite signs,yielding a nullity where both retinends keep their signs), Dodgson is able to create aRegister of Attributes showing the eliminands (a term that appears in both rows of theRegister, i.e. in positive and in negative form in two premises). When a term appearsin both rows and in one row in more than two premises, we have the case of barredpremises. All other terms are retinends (see Figure 3).

There are eight eliminands: b, c, d, e, f, g, h, l; four retinends: a, j, k, m; and threebarred premises: premise 5 which is barred by premises 2 and 7 (5 contains f’, 2 and 7contain f); 7 which is barred by 1,6, and 9 (7 contains c while 1, 6, and 9 each contains

Figure 3. The Pigs and Balloon Problem

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c’); and 8 (containing g’) is barred by 4 and 6 (containing g). Dodgson’s solution to thisundated problem is: No wise young pigs go up in balloons (kmaj’0), and it is producedby working with the premises in the order 1,4,6,8,9,7,3,2,5. We see that premises 2 and7 are used before premise 5; 1 and 6 are used before 7; and 4 and 6 are used before 8,according to his rule of operating with barred premises that all the premises barring agiven premise must be used first.

When did Dodgson first use this method? Certainly earlier than 16 July 1894 whenhe wrote in his diary that he had worked a problem of 40 premises with many bars‘like a genealogy, each term proving all its descendants’ (Bartley 1977, p. 279) This isthe date when he constructed his fourth formal method which he called the Method ofTrees. It is the earliest modern use of what is essentially a truth tree to reasonefficiently in multiliteral soriteses in the logic of classes. In effect, it is a mechanical testof validity using a formal reductio ad absurdum argument.

Below is a simple example of a sorites solved by the tree method (Bartley 1977, pp.292–296; discussed in Abeles 1990) The object is to prove bd’l is a nullity by assumingit is an entity and reasoning to a contradiction (see Figure 4).

From the Register we see there are six eliminands, and three retinends: bd’l whichwe select as the root of the tree and assume it is an entity (some existing thing has theattributes b and l but not d). Using premise 4, bd’l must also have attribute h (becauseb and l do not have h’). Since any premise now will force the tree to divide, we choosethe first one and use it to try to establish contradictions that will deny that bd’l is anentity. By premise 1, bd’l must have either attributes k’ or attributes m’km. (These arethe only two contradictories of mk, since m’k’ is subsumed by them.) We can eliminatem in mk’ because it appears only in premise 1. If it is k’, then by premises 3 and 7, bd’lmust also have the attributes aa’, a contradiction. We add this fact to the tree as7,3.aa’.

But if bd’l has m’k, we note that m’ occurs only in premise 5; c is there and wealready have k. So for m’ to be incompatible with ck implies that of the possible pairsck’, c’k, we must have the one with c’. We add this fact to the tree as 5.c’. By premises 2and 6, bd’l must also have attributes ee’, another contradiction, so we add 6,2.ee’ to thetree. The tree is now complete, so we can conclude that bd’l cannot be an entity, itmust therefore be a nullity. The tree is illustrated in Figure 5 (Bartley 1977, p. 295).

The numbers in the tree to the left of the terms are those of the premises used, theorder being downward and right to left. Premise 1 causes a branch (disjunction)signified by the 1 placed under the middle of it. Ao’ indicates a contradiction, whichDodgson called an absurdity. Inside the square brackets is a reference number

Figure 4. A Sorites Problem

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signifying the completion of a separate sorites when the tree is verified, a process I willnow explain.

The verification of a tree, which can be done for a sorites in the first figure only,begins top down with branches headed by single letters before using branches headedby two letters, etc. Each branch is expressed as a simple sorites with its (partial)conclusion which is given a reference number. These lists are then combined andshould produce the complete conclusion, the root of the tree. For the tree in figure 5,the first sorites is: 3, 7 } hk’b0. We number its partial conclusion 8 (in brackets). Thesecond sorites is: 2, 6, 5, 1, 8, 4 } d’bl0. The complete list is: 3, 7, [8]; 2, 6, 5, 1, 8, 4 }bd0’l. This bottom up trace of the tree obeys the rule for using barred premises: 2 and 6are used before 5; 1, 3, and 6 are used before 4; 3 and 7 are used before 1 and 5.Dodgson considered the tree method to be superior to the barred premises method.He wrote:

We shall find that the Method of Trees saves us a great deal of the trouble entailedby the earlier process. In that earlier process we were obliged to keep a carefulwatch on all the Barred Premisses so as to be sure not to use any such premiss untilall its ‘Bars’ had appeared in that Sorites. In this newMethod, the Barred Premisesall take care of themselves. (Bartley 1977, p. 287)

Dodgson used just three figures to designate all the classical syllogisms. Thesecond figure consists of a nullity and an entity with like eliminands (both positive orboth negative) as premises, whose conclusion is an entity in which the nullity-retinendchanges sign. The third figure is composed of two nullities having like eliminands,yielding an entity in which both retinends change their signs. But only for first figureproblems can the Method of Barred Premises be used to verify the tree, i.e. establishthe order of the premises directly. The reason is that first figure problems use onlypropositions called Universal Negatives. Such problems are equivalent to proposi-tions called Universal Affirmatives. For example, the statement no x are y isequivalent to the statement all x are y’.

Figure 5. A Tree Solution to the Sorites Problem

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Dodgson created puzzles involving entity premises (a second figure problem) asearly as October 1892. One of these, known as the Problem of the School Boys,requires the completion of the puzzle’s twelfth and last sentence (Bartley 1977, pp.326–331). Using the conditions in this incomplete sentence as the root of a tree, all itsbranches end in the same terms which are the completion of the twelfth sentence.

Another puzzle involving disjunctive statements of the type: A is B or C or D; ABis C or H can be cast in the form of a first figure problem. Applying the tree method,one branch remains open, i.e. does not end in a contradiction. Hence the aggregate ofthe retinends cannot be proved to be a nullity (Bartley 1977, pp. 374–376).

On 16 July 1894 Dodgson connected his tree method with his earlier work, theMethod of Diagrams. He wrote, ‘It occurred to me to try a complex Sorites by themethod I have been using for ascertaining what cells, if any, survive for possibleoccupation when certain nullities are given’ (Bartley 1977, p. 279) In an unpublishedletter to his mathematically inclined sister, Louisa, answering questions she had raisedabout one of his problems that she was attempting to solve, he wrote:

As to your 4 questions, . . . The best way to look at the thing is to suppose the Re-tinends to be Attributes of the Univ. Then imagine a Diagram, assigned to thatUniv., and divided, by repeated Dichotomy, for all the attributes, so as to have2n Cells, for n Attributes. (A cheerful Diagram to draw, with, say, 50 Attributes!There would be about 1000,000,000,000,000 Cells.) If the Tree vanishes, it showsthat every Cell is empty,. . . . (Dodgson, Weaver Collection, 13 November 1896)

And on 4 August 1894 he connected his tree method with his Method ofUnderscoring, writing in his diary, ‘I have just discovered how to turn a genealogyinto a scored Sorites’ (Abeles 1990, p. 30) It appears he planned to do further workwith this method and its natural extensions, barred premises, and barred groups. In anunpublished letter whose first page is missing, probably from late in 1896 or early in1897, he wrote, most probably to Louisa:

I have been thinking of that matter of ‘Barred Groups’ . . .. It belongs to a mostfascinating branch of the Subject, which I mean to call ‘The Theory of Inference’. . .. Here is one theorem. I believe that, if you construct a Sorites, which will elim-inate all along, and will give the aggregate of the Retinends as a Nullity, and if youintroduce in it the same letter, 2 or 3 times, as an Eliminand, and its Contradictorythe same number of times, and eliminate it each time it occurs, you will find, if yousolve it as a Tree, that you don’t use all the Premisses! (Dodgson, Weaver Collec-tion, undated)

The difficulty of establishing a theorem to determine superfluous premisestroubled him. It was a problem he was unable to solve. In an unpublished letter dated25 September 1896 to the Oxford logician, John Cook Wilson, in connection with asorites problem Dodgson wrote:

What you say about ‘superfluous Premisses’ interests me very much. It is a matterthat beatsme, at present . . . &, if you can formulate any proof enabling you to say‘all the Premises are certainly needed to prove the Conclusion,’ I shall be very gladto see it. (Dodgson, Sparrow Collection, 25 September 1896. Courtesy Morton N.Cohen)

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4. Automated deduction

The beginning of the automation of deduction goes back to the 1920s with thework of Thoralf Skolem who studied the problem of the existence of a modelsatisfying a given formula, and introduced functions to handle universal andexistential quantifiers. Other logicians such as David Hilbert, Wilhelm Ackermann,Leopold Lowenheim, Jacques Herbrand, Emil Post, and a little later, AlonzoChurch, Kurt Godel, and Alan Turing introduced additional important ideas. One ofthe most important ideas, a consequence of Hilbert’s metamathematical framework,was the notion that formalized logic systems can be the subject of mathematicalinvestigation. But it was not until the 1950s that computer programs, using a tree asthe essential data structure, were used to prove mathematical theorems. The focus ofthese early programs was on proofs of theorems of propositional and predicate logic.Describing the 1957 ‘logic machine’ of Newell, Shaw, and Simon, Martin Davis 1983noted that a directed path in a tree gave the proof of a valid argument where itspremises and conclusion were represented as nodes, and an edge joining two premisenodes represented a valid derivation according to a given set of rules for deriving theproofs.

The modern tree method, as a decision procedure for classical propositional logicand for first order logic, originated in Gerhard Gentzen’s work on natural deduction,particularly his formulation of the sequent calculus known as LK. But the route wasnot a direct one, the main contributors being Evert Beth, Richard Jeffrey, and thecontemporary logicians, Jaako Hintikka and Raymond Smullyan. In 1955, Bethpresented a tableau method he had devised consisting of two trees that would enable asystematic search for a refutation of a given (true) sequent. A tree is a left-sided Bethtableau in which all the formulae are true. The rules for decomposing the tree, i.e. theinference rules, are equivalent to Gentzen’s rules in his sequent calculus.

Many modern automated reasoning programs employ a reductio ad absurdumargument, while other reasoning programs that are used to find additionalinformation do not seek to establish a contradiction. In 1985, one of Dodgson’spuzzle problems, the Problem of the School Boys, was modified by Ewing Lusk andRoss Overbeek 1985 to be compatible with the direct generation of statements (inclausal form) by an automated reasoning program. Their program first produced aweaker conclusion before generating the same stronger conclusion Dodgson producedusing his tree method.

Several of the methods Dodgson used in his Symbolic Logic contain kernels ofconcepts and techniques that later were employed in automatic theorem proving. Hisonly inference rule, underscoring, which takes two propositions, selects a term in eachof the same subject or predicate having opposite signs, and yields another proposition,is an example of binary resolution, the most important of the early proof methods inautomated deduction.

Bartley had this to say about Dodgson’s tree method for reaching validconclusions from sorites and puzzle problems:

Carroll’s procedure bears a striking resemblance to the trees employed . . .accord-ing to a method of ‘Semantic Tableaux’ published in 1955 by the Dutch logician,E. W. Beth. The basic ideas are identical. (Bartley 1977, p. 32)

Dodgson was the first person in modern times to apply a mechanical procedure,his tree method, to demonstrate the validity of the conclusion of certain complex

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problems. Although he did not take the next step, attaching the idea of inconsistencyto the set of premises and conclusion, nevertheless, his method for multiliteralsyllogisms in the first figure is a formal test for inconsistency that qualifies as a finiterefutation of the set of eliminands and retinends. His construction of a tree uses oneinference rule (algorithm), binary resolution, and he guides the tree’s development witha restriction strategy, now known as a set of support, that applies binary resolution ateach subsequent step of the deduction only if the preceding step has been deducedfrom a subset of the premises and denial of the conclusion, i.e. from the set ofretinends. This strategy improves the efficiency of reasoning by preventing theestablishment of fruitless paths. And this tree test is both sound and complete, i.e. if theinitial set of the premises and conclusion is consistent, there will be an open paththrough the tree; if there is an open path in the finished tree, the initial set of thepremises and conclusion is consistent, respectively.

Before creating his tree method, Dodson used his Method of Barred Premises toguide the generation of the most promising (ordered) lists of the premises and partialconclusions to produce the complete conclusion of a sorites. He realized that toomany of these lists would not lead to a proper conclusion, so he abandoned thisapproach in favor of his tree method. But modern automated reasoning programs canuse a direct approach, suitably guided to prevent the proving of spurious partialresults that are irrelevant to obtaining the complete result. The solution by EwingLusk and Ross Overbeek 1985 to Dodgson’s ‘Salt and Mustard Problem’ and by A G.Cohn 1989 to the same problem five years later using a many sorted logic illustrate thepower of two of these programs.

When Dodgson used the Method of Barred Premises to verify a tree, he guided thegeneration of the ordered lists by employing an ordering strategy known now as unitpreference which selects first the propositions with the fewest number of terms. In hisown words:

[W]hen there are two Branches, of which one is headed by a single Letter, and theother by a Pair, to take the single Letter first, turn it into a Sorites, and record itsPartial Conclusion: then take the double-Letter Branch: turn it also into a Sorites.(Bartley 1977, p. 295)

He also employed a rule to eliminate superfluous premises (those premises that donot eliminate anything) when verifying a tree. His rule was to ignore such a premise,even if it caused a branching of the tree. But in the absence of more powerful inferencerules and additional strategies he had no way to approach the solution of thesemultiliteral problems more efficiently.

5. The setting for Symbolic LogicWhy did Dodgson write his symbolic logic books under his pseudonym? Bartley

suggests a combination of motives: he wanted the material to appeal to a widegeneral audience, particularly to young people, a task made easier using the wideacclaim accorded him as the writer, Lewis Carroll. Then too, there was thefinancial motive of achieving the greater revenue that books authored by LewisCarroll would generate than books by the mathematician Charles Dodgson. By1896 Dodgson was very much concerned about his mortality and the responsibilityhe bore for the future care of his family, especially his unmarried sisters. But there

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were more reasons why he wanted the exposure his pseudonym would offer.Dodgson was a deeply religious man and considered his mathematical abilities as agift to be used in the service of G-d. In a letter to Louisa dated 28 September 1896he wrote:

[W]hereas there is no living man who could (or at any rate would take the troubleto) & finish, & publish, the 2nd Part of the Logic. Also I have the Logic book in myhead. . ..So I have decided to get Part II finished first . . .. The book will be a greatnovelty, & will help, I fully believe, to make the study of Logic far easier than itnow is: & it will, I also believe, be a help to religious thoughts, by giving clearnessof conception & of expression, which may enable many people to face, & conquer,many religious difficulties for themselves. So I do really regard it as work for God.(Bartley 1977, pp. 370–371)

Dodgson was both a popularizer and educator of mathematics. Shortly before thepublication of The Game of Logic in February 1887, where he first introduced histriliteral diagram he began teaching logic classes at Oxford high schools, continuingthis activity off and on through July 1896. Dodgson hoped the book would appeal toyoung people as an amusing mental recreation. The objective of the ‘game’, playedwith a board and counters, is to solve syllogisms. He found this book, and SymbolicLogic, Part I even more so, useful in teaching students between the ages of 12 and 20.He believed his own book on symbolic logic was far superior to those in current use. In1894, answering a letter from a former child friend, Mary Brown, now aged thirty-two, he wrote:

You ask what books I have done . . .. At present I’m hard at work (and have beenfor months) on my Logic-book. (It really has been on hand for a dozen years: the‘months’ refer to preparing for the Press.) It is Symbolic Logic, in 3 Parts—andPart I is to be easy enough for boys and girls of (say) 12 or 14. I greatly hope itwill get into High Schools, etc. I’ve been teaching it at Oxford to a class of girlsat the High School, another class of the mistresses(!), and another class of girlsat one of the Ladies’ Colleges. (Cohen 1979, p. 1031)

In a letter dated 25 November 1894 to his sister, Elizabeth, he wrote:

One great use of the study of Logic (which I am doing my best to popularise)would be to help people who have religious difficulties to deal with, by makingthem see the absolute necessity of having clear definitions, so that, before enteringon the discussion of any of these puzzling matters, they may have a clear idea whatit is they are talking about. (Cohen 1979, p. 1041)

Dodgson was adamant about having clear definitions for teaching and discussingtopics to support distinctions among the different meanings concepts may have. Tothis end, in June 1895, he sent to several logicians, including Cook Wilson, Venn andJohn Alexander Stewart, a noted editor of Aristotle, a sheet titled, ‘LogicalNomenclature’, asking the recipient to choose from a list of terms (or to supplyanother term) the one he considered best for that set of words (Dodgson, BerolCollection, 18 June 1895). Of the eight sets, the last three deal with propositions. Forexample, number six asks the recipient to select the most appropriate term to denote a

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proposition that is the conclusion of an invalid argument from the set: inconsequent,incorrect, unlawful, illegal, illegitimate, illogical, wrong, invalid (Venn chose‘invalid’.)

Arguably, Dodgson’s formulation of formal logic came late in his life as theculmination of his publications on Euclid’s geometry in the 1860s and 1870s.Logical arguments using rules of inference are a major component of bothgeometry and formal logic. In mathematics generally, and in geometry particularly,one begins with a set of axioms and certain inference rules to infer that if oneproposition is true, then so is another proposition. To Dodgson, geometry andlogic shared the characteristic of certainty, a quality that always interested him.Exactly one month before he died, in an unpublished letter Dodgson wrote toStewart criticizing a manuscript Stewart had given to him for his opinion, hecommented:

Logic, under that view, would become to me, a science of such uncertainty that Ishd [should] take no further interest in it. It is its absolute certainty which at pre-sent fascinates me. (Dodgson, Berol Collection, 14 December 1897)

But by the early 1890s he had shifted his focus away from the truth given bygeometrical theorems to the validity of logical arguments. In the Preface (p. xi) to thethird edition (1890) of Curiosa Mathematica Part I, A New Theory of Parallels, hepointed out that the validity of a syllogism is independent of the truth of its premises,giving this example:

I have sent for you, my dear Ducks, said the worthy Mrs. Bond, ‘to enquire withwhat sauce you would like to be eaten?’ But we don’t want to be killed!’ cried theDucks. ‘You are wandering from the point’ was Mrs. Bond’s perfectly logical reply.

He believed that mental activities and mental recreations, like games andparticularly puzzles, are enjoyable and confer a sense of power to those who make theeffort to solve them. In an Appendix to the fourth edition of Symbolic Logic, Part Iaddressed to teachers, he wrote:

I claim, for Symbolic Logic, a very high place among recreations that have the nat-ure of games or puzzles . . .. Symbolic Logic has one unique feature, as comparedwith games and puzzles, which entitles it, I hold, to rank above them all . . .. Hemay apply his skill to any and every subject of human thought; in every one ofthem it will help him to get clear ideas, to make orderly arrangement of his knowl-edge, and more important than all, to detect and unravel the fallacies he will meetwith in every subject he may interest himself in. (Bartley 1977, pp. 45–46)

The statements of almost all the problems in both parts of Symbolic Logic areamusing to read. This attribute derives from the announced purpose of the books, topopularize the subject. But Dodgson naturally incorporated humor into much of hisserious mathematical writing, infusing this work with the mark of his literary genius.In Book (chapter) XII of part two of Symbolic Logic, instead of just exhibiting the treepiecemeal for a particular problem he gives a ‘soliloquy’ as he works it through,accompanied by ‘stage directions’ showing what he is doing to enable the reader toconstruct the tree for himself, amusingly.

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6. Conclusion

Although Dodgson’s concepts and methods of formal logic are an anticipation ofseveral ideas and techniques that developed in the second half of the twentiethcentury, some of his ideas, such as his use of existential import, were backwardlooking. However, Dodgson may not have held this idea as a philosophical belief.Bartley observed that existential import is implicit in Dodgson’s Method ofSubscripts. Using this notation, Dodgson had no other way to separate subject frompredicate, e.g. xy1z’0 which expresses all xy are z, implies that there are some xy(because of the subscript 1). But we can interpret this as either no xy are not z, or all xyare z which are equivalent in modern logic usage.

However, by 1897 Dodgson may have been rethinking his use of existentialimport. Bartley cites a diary entry from 1896, and an undated letter to CookWilson as evidence (Bartley 1977, pp. 34–35.) However, there is even moreevidence, incomplete in Bartley’s book, to support this break with the idea ofexistential import. Book (chapter) XXII of Part II contains Dodgson’s solutions toproblems posed by other logicians. One of these solutions, to a problem posed byAugustus DeMorgan that concerns the existence of their subjects, appears in anunaddressed letter dated 15 March 1897 (Bartley 1977, pp. 480–481) FromDodgson’s response to this letter six days later, we now know it was sent to hissister, Louisa, responding to her solution of the problem. In this unpublished letter,Dodgson suggested:

[I]f you take into account the question of existence, and assume that each Proposi-tion implies the existence of its Subject, & therefore of its Predicate, then you cer-tainly do get differences between them: each implies certain existences not impliedby the others. But this complicates the matter: & I think it makes a neater problemto agree (as I shall propose to do in my solution of it) that the Propositions shallnot be understood to imply existence of these relationships, but shall only be un-derstood to assert that, if such & such relationships did exist, then certain resultswould follow. (Dodgson, Berol Collection, 21 March 1897)

Bartley’s claims about the importance of Dodgson’s work in logic are certainlyjustified. Symbolic Logic includes the earliest modern mechanical deduction systemknown, a significant work from someone with a lifelong interest in mechanicalmethods and equipment. Over a twenty year period, between 1867 and 1888, Dodgsonhad acquired a substantial calculating machine, a newly invented electric pen, and anearly model of the typewriter. It is ironic that the celebration, which continues to beaccorded to Lewis Carroll, the author of Alice in Wonderland and Through theLooking Glass, generated the expectation that all his subsequent writings essentiallywould be humorous pieces. Arguably, this expectation influenced the reception of thesecond part of Symbolic Logic. Although the ingenuity of the puzzles and examplesDodgson created were applauded, Bartley’s claims about the significance ofDodgson’s work were questioned so that its value in the development of logic wasnot fully appreciated when the book was first published more than twenty-five yearsago.

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