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Peirce’s Role in the History of Logic: Lingua Universalis and Calculus Ratiocinator

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The Road to Universal Logic

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Abstract

This chapter discusses central issues of Peirce’s conception of logic, comparing his work with the works of other logicians of the time, in particular Boole, Venn, Schröder, and Frege. It presents a detailed analysis of Peirce’s approach to notation, including pasigraphy and Peirce’s claw.

For a detailed account of the full panoply of characteristics of modern logic claimed to have been due to Frege, but which occur in the classical Boole–Schröder calculus, many of them due to Peirce, see [4].

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Notes

  1. 1.

    We know, for example, that there are inferences that fail in traditional logic in the presence of the empty set that are valid in the class calculus and modern propositional logic. For a discussion of the issue with respect to the traditional versus Boolean squares of opposition, see, e.g., Wu [144, 145].

  2. 2.

    Esquisable [20] understands Leibniz’s characteristica, however, to be a formal combinatorial system, in which symbols (such as numbers as the quintessential symbol) represent symbolic knowledge (cognitio symbolica) that is subject to manipulation in accordance with combinatorial rules. Though symbolic knowledge is entirely blind (cognitio caeca; cognitio suppositiva), it differs from intuitive knowledge (cognitio intuitiva) by its complexity and symbolic mediation and formal representation. Leibniz’s earliest treatment of symbolic knowledge is found in his “Meditationes de cogitationes, veritate et ideis” of November 1684; see Leibniz [52, pp. 585–592]. Legris [50] examines Frege’s use of this Leibnizian concept to evaluate Frege’s Begriffsschrift notation. Legris [51] also uses the analysis of the cognitio symbolica of Esquisable [20] to examine the extent to which Boole, Schröder, and Frege developed mathematical logic as the attainment of Leibniz’s program, and concludes that both Schröder’s algebra of relatives and Frege’s Begriffsschrift share the goal of simultaneously constructing both a language and a calculus, and that, as a consequence, van Heijenoort [129] and Hintikka [40] made too sharp a distinction between logic as calculus and logic as language. Legris [50] examines Frege’s use of this Leibnizian concept to evaluate Frege’s Begriffsschrift notation. See also Patzig [60] for an account of Frege’s and Leibniz’s respective conceptions of the lingua characteristica (or lingua charactera) and their relationship. Patzig [60, p. 103] notes that Frege wrote of the idea of a lingua characteristica along with calculus ratiocinator, using the term “lingua characteristica” for the first time only in 1882 in “über den Zweck der Begriffsschrift” (in print in [24]) and then again in “über die Begriffsschrift des Herrn Peano und meine einige” [27], in the Begiffsschrift [21] terming it a “formal language” – “Formelsprache.” In the foreword to the Begriffsschrift, Frege wrote only of a lingua characteristica or “allgemeine Charakteristik” and a calculus ratiocinator, but not of a charcteristica universalis. Peckhaus [75] likewise argues that both logic as calculus and logic as language are to be found in the work of Schröder. Other aspects of demonstrating the extent to which Schröder’s work satisfied the criteria defined for a modern mathematical logic include Peckhaus [71, 72, 73, 74, 75] and Thiel [124, 125], and Peckhaus [76] is an extensive sustained survey of Schröder’s work. Frege evidently borrowed the term “Begriffsschrift” from the characterization [127, 4] by Friedrich Adolf Trendelenburg (1802–1872) of Leibniz’s general characteristics. The term had already been used by Wilhelm von Humboldt (1767–1835) in the treatise “Ueber die Buchstabenschrift und ihren Zusammenhang mit dem Sprachbau” of 1824 (see [43]) on the letter script and its influence on the construction of language, and by Franz B. (also Frantisek Bolemír) Kvet (1825–1864) in Leibnitz’ens Logik. Trendelenburg’s work was known to, and cited, by both Frege in the Begriffsschrift [21, V] and by Schröder in both Der Operationskreis des Logikkalkuls [114, VI] and in the Vorlesungen über die Algebra der Logik [116, I, 38].

  3. 3.

    Korte [46, 183], however, rejects the nearly universal claims, not only by Frege, but by van Heijenoort and Sluga, among others, that the Begriffsschrift is, indeed, a language, but not a calculus. Korte’s claim is argued upon the basis of Frege’s logicism.

  4. 4.

    Peirce also divides the branches of semiotics into “pure grammar,” “logic proper,” and “pure rhetoric,” the latter his alternative name for methodeutic (q.v. [97, 99]).

  5. 5.

    The standard mode of reference in Peirce scholarship for citing material from Peirce’s Collected Papers is by volume number and paragraph; thus, e.g., 4.239 cites volume IV, paragraph 239 of the Collected Papers.

  6. 6.

    My emphasis.

  7. 7.

    See, e.g., Thiel [126] for a discussion of the relation of syntax and semantics according to Frege; see Anellis [4] for an analysis of the relation between these dichotomies and where Peirce’s work in particular stood. Anellis [4, 261–262] argues that, even if Peirce had no explicit and formal definition of formal system, it is present and at least implicit in his work, for example in his [82] “On the Logic of Number” providing an informal axiomatization of number theory.

  8. 8.

    For details of Russell’s and Peirce’s accounts and criticisms of one another’s contributions to logic, see Anellis [1995]; for discussion of the views of Peirce and his adherents towards Russell’s work in logic, see [3].

  9. 9.

    In a review published in Mind in January 1892 [48] of the first volume of Schröder’s Vorlesungen über die Algebra der Logik, Christine Ladd-Franklin remarked on Peirce’s view that “for the purposes of Logic, there is no difference between the transitive relation for terms and the transitive relation for propositions” [48, 128] and discussed some respects in which Schröder’s views differed from Peirce’s.

  10. 10.

    In 1891, in his “Principii di logica matematica,” [62, n. 5] remarked on the connection of his “sign of deduction” and notations used by others, including that of Peirce.

  11. 11.

    See also [31] for a survey and analysis of Wiener’s thesis.

  12. 12.

    Schröder announced the shift in his attitude towards formal logic as a result of having read Peirce’s [83] “The Logic of Relatives.” Schröder’s also explained his shift in letters to Felix Klein (1849–1925) (see Peckhaus [71, 198–202] for transcriptions of the letters).

  13. 13.

    Van Heijenoort [129, 325, n. 3] understood Frege [27, 371] to have asserted that “Boole’s logic is a calculus ratiocinator, but no lingua charaterica; Peano’s mathematical logic is in the main a lingua characterica and subsidiarily, also a calculus ratiocinator, while my Begriffsschrift intends to be both with equal stress.”

  14. 14.

    In a much broader sense, Peirce would also include induction and deduction as means of inference along with deduction.

  15. 15.

    Similar, because it would be incorrect to understood existential graphs are basically just the entitative graphs with quantification.

  16. 16.

    For details on computational aspects Peirce’s X-frame notation and applications, see Clark [15] and Zellweger [146].

  17. 17.

    In its original conception, as explicated by the medieval philosophers, the logica utens was a practical logic for reasoning in specific cases, and the logica docens a teaching logic, or theory of logic, concerning the general principles of reasoning. These characterizations have been traced back at least to the Logica Albertici Perutilis Logica (ca. 1360); see [1] of Albertus de Saxonia (1316–1390) and then his school in the 15th century, although the actual distinction can be traced back to the Summulae de dialectica of Johannes Buridanus (ca. 1295 or 1300–1358 or 1360) (see Buridanus [13]). See, e.g., Bíard [7] for Buridan’s distinction, and Ebbesen on Albert.

  18. 18.

    Peano’s assessment of Schröder’s system is found in Peano [63, 68].

  19. 19.

    Peirce tended to conflate Russell and Whitehead even with respect to Russell’s Principles of Mathematics, even prior to the appearance of the co-authored Principia Mathematica [139], presumably because of their earlier joint work “On Cardinal Numbers” [139] in the American Journal of Mathematics, to which Russell contributed the section on, a work with which Peirce was already familiar.

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Acknowledgement

The author is grateful to Nathan Houser for suggestions that considerably improved the text and helped clarify a number of points.

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  1. Peirce Edition Project, Indiana University, Indianapolis, USA

    Irving H. Anellis

  2. Purdue University at Indianapolis, Indianapolis, USA

    Irving H. Anellis

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  1. City University of New York, New York, New York, USA

    Arnold Koslow

  2. Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil

    Arthur Buchsbaum

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Anellis, I.H. (2015). Peirce’s Role in the History of Logic: Lingua Universalis and Calculus Ratiocinator . In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15368-1_5

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