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# MODERN LOGIC: THE BOOLEAN PERIOD: CARROLL

The contributions of Lewis Carroll (Charles L. Dodgson, 18321898) to logic consist of several pieces published between 1887 and 1899. The Game of Logic (London, 1887) is a book written for young people to teach them to reason logically by solving syllogisms using diagrams and colored counters. His diagrammatic method is a visual logic system that we know now to be sound and complete.

In Symbolic Logic, Part I (London, 1896) Carroll developed two formal methods to solve syllogisms and

sorites. The first is the Method of Underscoring that is dependent on his idiosyncratic algebraic notation that he called the Method of Subscripts. The second is his Method of Diagrams, which he extended to handle more than three terms (classes), but without providing examples. However, his diagrammatic system is an improvement over that of his contemporary, John Venn, because first, unlike Venn's system, Carroll's can handle existential statements. Second, as A. Macula showed in 1995, diagrams for ten terms (sets) or more can be drawn more easily than Venn diagrams for a large number of sets. Finally, the diagrams are self-similar and can be generated by a linear iterative process. Carroll used his method to reduce the nineteen or more valid forms of inference codified by medieval Aristotelian logicians first to fifteen forms and then to just three formulas.

Carroll published two pieces in the journal Mind. The first, "A Logical Paradox" (N. S. vol. 3, 1894, 436438) is an example of hypothetical propositions. W. W. Bartley III remarks in the second edition of his book, Lewis Carroll's Symbolic Logic (1986, p. 505) that for about eighty years eminent logicians and philosophers failed to see this problem as little more than a routine exercise in Boolean algebra. Of the eleven questions Dodgson sent to The Educational Times (ten on mathematical topics) the substance of one, Question 14122, (February 1, 1899, vol. lii, p. 93) on his logical paradox, had appeared as a "Note" to his 1894 Mind article. H. MacColl and H. W. Curjel provided (different) solutions. The second piece in Mind, "What the Tortoise Said to Achilles" (N. S. vol. 4, 1895, 278280) is a humorous example of an important problem about logical inference that Carroll was perhaps the first to recognize: the rule allowing a conclusion to be drawn from a set of premises cannot itself be treated as an additional premise without generating an infinite regress.

We see in Bartley's 1986 publication of Carroll's lost book, Symbolic Logic, Part II, that Carroll introduced two additional methods of formal logic. The first, the method of barred premises, a direct approach to the solution of problems involving multiliteral statements is an extension of his Method of Underscoring. The second and most important, the Method of Trees, a mechanical test of validity using a reductio ad absurdum argument, is the earliest modern use of a truth tree to reason in the logic of classes. It uses one inference rule (binary resolution) and a restriction strategy (set of support) to improve the efficiency of the construction. His tree method is a sound and complete formal logic system for sorites.

## Bibliography

Abeles, Francine F. "Lewis Carroll's Formal Logic." History and Philosophy of Logic 26 (2005): 3346.

Bartley, William W., III, ed. Lewis Carroll's Symbolic Logic. 2nd ed. New York: Clarkson N. Potter, 1986.

Beth, Evert W. The Foundations of Mathematics. Amsterdam: North Holland, 1965.

Macula, Anthony J. "Lewis Carroll and the Enumeration of Minimal Covers." Mathematics Magazine 69 (1995): 269274.

Wos, Larry, et al. Automated Reasoning. Englewood Cliffs, NJ: Prentice-Hall, 1984.

Francine F. Abeles (2005)