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Why Make Things Simple When You Can Make Them Complicated? An Appreciation of Lewis Carroll’s Symbolic Logic

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Abstract

Lewis Carroll (1832–1898) published a system of logic in the symbolic tradition that developed in his time. Carroll’s readers may be puzzled by his system. On the one hand, it introduced innovations, such as his logic notation, his diagrams and his method of trees, that secure Carroll’s place on the path that shaped modern logic. On the other hand, Carroll maintained the existential import of universal affirmative Propositions, a feature that is rather characteristic of traditional logic. The object of this paper is to untangle this dilemma by exploring Carroll’s guidelines in the design of his logic, and in particular his theory of existential import. It will be argued that Carroll’s view reflected his belief in the social utility of symbolic logic.

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Notes

  1. Carroll’s real name was ’Charles Lutwidge Dodgson’. Carroll signed with his real name most of his mathematical works but signed his logical writings with his literary pseudonym (see [50]). Consequently, we will refer to him as ‘Lewis Carroll’ in this paper.

  2. This function of formal languages relates to logical systems which belong to the tradition of Calculus. Some logicians such as Charles S. Peirce, Gottlob Frege, Giuseppe Peano, Bertrand Russell, and others, designed languages rather to perform an analysis of logical inferences [11].

  3. It must be said here that some symbolic logicians, such as Venn, shared to some extent Cook Wilson’s view on the artificiality of the problems they were addressing. This led some of them to dismiss the utility of the first logic machines which were designed around 1880 [36].

  4. This kind of Propositions was implicit in many earlier logical notations. It was also explicitly developed by Franz C. Brentano whose work was known to British logicians in Carroll’s time [41]. The resemblance between Carroll and Brentano’s forms is noted in [25, p. 264; 75, p. 63]

  5. Note that symbol ‘\(\dag \)’ stands for ‘and’ and is placed between the Premises, while stands, roughly, for ‘imply’ and is placed between the Premises and the Conclusion. Also, an accent stands for negation; for instance, \(x'\) represents not-x.

  6. An elegant online demonstrator designed by Mark R. Richards shows how to solve syllogistic problems with Carroll’s diagrams. It is found at: https://lewiscarrollresources.net/gameoflogic/.

  7. Carroll apparently invented this method on 16 July 1894. That day, he noted in his journal developing and using successfully what he called the “Genealogical method” [81, p. 155].

  8. Interestingly, when Carroll communicated his method of trees in 1896 to his colleague Cook Wilson, the latter revealed using “for years” a similar procedure. Cook Wilson named his technique the method of “hanging plants”, which roughly are inverted trees [27].

  9. Carroll’s first paradox, commonly known as the Barbershop paradox, appeared in 1894 in the journal Mind [20]. It led to a large debate for over a decade among British logicians regarding the nature of hypotheticals [46]. For a discussion of this paradox, see [59]. Carroll’s second paradox, commonly known as the paradox of inference, appeared in 1895 in the same journal [21]. It did not get any responses in Carroll’s lifetime, but was widely commented by twentieth century logicians and philosophers. For a discussion, see [12, 53].

  10. This view is already found in Carroll’s journal on 15 December 1884, when he reported being “Again in logical difficulties: it now seems inevitable to make “all x is y” assert existence of x.” [80, p. 156].

  11. Carroll apparently tested this view in 1894, as shown by his journal entry of 19 June that year: “I have been testing the theory that the Copula in Logic does not assert the existence of the subject, but that “all x are y” only means “if any x exist, all of them are y.” It works out right with “Barbara” etc., but fails in “Darapti,” which, with this interpretation, gives no conclusion” [81, p. 151].

  12. Some authors suggested that Carroll may have changed his theory of existential import in subsequent parts of Symbolic Logic, which he never managed to complete [2, 9]. It is true that Carroll added to the section on existential import in the fourth edition of his treatise an opening note stating: “Note that the rules, here laid down, are arbitrary, and only apply to Part I of my “Symbolic Logic.”’ [22, p. 19]. However, it is unknown what change he may have made in subsequent parts.

  13. It must be explained that this abridged notation rather expresses the conjunction of “Some x exist” and “No x are not-y”. Carroll simply noted that it was not necessary to explicitly mark the superfluous information contained in the Proposition “Some x are y”. The fact that “Some x exist” suffices for the purpose. Indeed, given that “Some x exist” and knowing that “No x are not-y”, it necessarily follows that “Some x are y” [22, p. 72] (see [32, pp. 40–41]).

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Acknowledgements

Earlier versions of this paper were partly presented at several events over the years, notably at the ‘History of Logic Meeting’ organized by the British Society for the History of Mathematics in Honour of Ivor Grattan-Guinness (London, 27 May 2017). The preparation of this paper benefited from the support of the TalTech internal grant SSGF21021.

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  1. Ragnar Nurkse Department of Innovation and Governance, Tallinn University of Technology, Tallinn, Estonia

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Moktefi, A. Why Make Things Simple When You Can Make Them Complicated? An Appreciation of Lewis Carroll’s Symbolic Logic. Log. Univers. 15, 359–379 (2021). https://doi.org/10.1007/s11787-021-00286-1

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