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LINGUA CHARACTERICA AND CALCULUS RATIOCINATOR: THE LEIBNIZIAN BACKGROUND OF THE FREGE-SCHRÖDER POLEMIC

Part of: History of mathematics and mathematicians Mathematical Logic and Foundations Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  29 June 2020

JOAN BERTRAN-SAN MILLÁN
JOAN BERTRAN-SAN MILLÁN*
Affiliation:
INSTITUTE OF PHILOSOPHY CZECH ACADEMY OF SCIENCES JILSKÁ 1 110 00PRAGUECZECH REPUBLICE-mail: [email protected]
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Abstract

After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal.

In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme.

Keywords

FregeSchröderLeibnizlingua charactericacalculus ratiocinatorconcept-scriptalgebra of logic

MSC classification

Primary: 01A55: 19th century
Secondary: 03-03: Historical (must also be assigned at least one classification number from Section 01) 03A05: Philosophical and critical
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 2 , June 2021 , pp. 411 - 446
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© Association for Symbolic Logic, 2020

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References

BIBLIOGRAPHY

Badesa, C. (2004). The Birth of Model Theory. Princeton: Princeton University Press.10.1515/9781400826186CrossRefGoogle Scholar
Badesa, C. & Bertran-San Millán, J. (2017). Function and argument in Begriffsschrift . History and Philosophy of Logic, 38, 316–341.10.1080/01445340.2017.1354173CrossRefGoogle Scholar
Bertran-San Millán, J. (2018). Frege’s Begriffsschrift and Logicism. In Arazim, P., & Lávička, T., editors, The Logica Yearbook 2017. London: College Publications, pp. 239–253.Google Scholar
Blanchette, P. (2012). Frege’s Conception of Logic. Oxford: Oxford University Press.10.1093/acprof:oso/9780199891610.001.0001CrossRefGoogle Scholar
Boole, G. (1854). An Investigation of the Laws of Thought. Cambridge: Macmillan. Reedition in (New York: Dover, 1858).Google Scholar
Brady, G. (2000). From Peirce to Skolem. A Neglected Chapter in the History of Logic, Volume 4 of Studies in the History and Philosophy of Mathematics. Amsterdam: Elsevier.Google Scholar
Brown, F. M. (2009). George Boole’s deductive system. Notre Dame Journal of Formal Logic, 50, 303–330.10.1215/00294527-2009-013CrossRefGoogle Scholar
Cajori, F. (1929). A History of Mathematical Notations, Volume II: Notations mainly in Higher Mathematics. Chicago: Open Court.Google Scholar
Corcoran, J. (2003). Aristotle’s prior analytics and Boole’s laws of thought . History and Philosophy of Logic, 24, 261–288.10.1080/01445340310001604707CrossRefGoogle Scholar
Dascal, M. (1987). Leibniz. Language, Signs and Thought: A collection of essays. Amsterdam: John Benjamins.10.1075/fos.10CrossRefGoogle Scholar
de Maimieux, J. (1797). Pasigraphie, premiers élémens du nouvel art-science d’écrire et d’imprimer en une langue de manière à être lu et entendu dans toute autre langue sans traduction. Bureau de la Pasigraphie: Paris.Google Scholar
Demopoulos, W. (1994). Frege, Hilbert, and the Conceptual structure of model theory. History and Philosophy of Logic, 15(2), 211–225.10.1080/01445349408837233CrossRefGoogle Scholar
Dummett, M. (1973). Frege. Philosophy of Language. London: Duckworth.Google Scholar
Euler, L. (1755). Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. Saint Petersburg: Academia Imperialis Scientiarum Petropolitanae.Google Scholar
Frege, G. (1879a). Anwendungen der Begriffsschrift. Lecture at the January 24, 1879 meeting of Jenaischen Gesellschaft für Medizin und Naturwissenschaft . Jenaische Zeitschrift für Naturwissenschaft, 13, 29–33. Reedition in (Frege, 1964, pp. 89–93). English translation by T. W. Bynum in (Frege, 1972, pp. 204–208).Google Scholar
Frege, G. (1879b). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Louis Nebert. Reedition in (Frege, 1964, pp. 1–88). English translation by T. W. Bynum in (Frege, 1972, pp. 101–203).Google Scholar
Frege, G. (1880–1881). Booles rechnende Logik und die Begriffsschrift. Originally unpublished. Edition in (Frege, 1969, pp. 9–52). English translation by P. Long and R. White in (Frege, 1979, pp. 9–46).Google Scholar
Frege, G. (1882a). Booles logische Formelsprache und meine Begriffsschrift. Originally unpublished. Edition in (Frege, 1969, pp. 53–59). English translation by P. Long and R. White in (Frege, 1979, pp. 47–52).Google Scholar
Frege, G. (1882b). Über den Zweck der Begriffsschrift. Lecture at the January 27, 1882 meeting of Jenaischen Gesellschaft für Medizin und Naturwissenschaft. Published in 1882 in Jenaische Zeitschrift für Naturwissenschaft, 16, 1–10. Reedition in (Frege, 1964, pp. 97–106). English translation by T. W. Bynum in (Frege, 1972, pp. 90–100).Google Scholar
Frege, G. (1882c). Über die wissenschaftliche Berechtigung einer Begriffsschrift. Zeitschrift für Philosophie und philosophische Kritik, 81, 48–56. Reedition in (Frege, 1964, pp. 106–114). English translation by T. W. Bynum in (Frege, 1972, pp. 83–89).Google Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik: eine logisch-matematische Untersuchung über den Begriff der Zahl. Breslau: Wilhelm Koebner.Google Scholar
Frege, G. (1897). Über die Begriffsschrift des Herrn Peano und meine eigene. Lecture at the Königlich Sächsische Gesellschaft der Wissenschaften zu Leipzig of July 6, 1896. Published in 1897 in Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig: Mathematisch-physische Klasse, 48, 361–378. Reedition in (Frege, 1967, pp. 220–233). English translation by V. H. Dudman in (Frege, 1984, pp. 234–248).Google Scholar
Frege, G. (1964). Begriffsschrift und andere Aufsätze. Hildesheim: Georg Olms.Google Scholar
Frege, G. (1966). Logische Untersuchungen. Göttingen: Vandenhoeck und Rupert.Google Scholar
Frege, G. (1967). Kleine Schriften. Hildesheim: Georg Olms.Google Scholar
Frege, G. (1969). Nachgelassene Schriften. Hamburg: Felix Meiner.Google Scholar
Frege, G. (1972). Conceptual Notation and Related Articles. Oxford: Clarendon Press.Google Scholar
Frege, G. (1979). Posthumous Writings. Chicago: University of Chicago Press.Google Scholar
Frege, G. (1984). Collected Papers on Mathematics, Logic, and Philosophy. Oxford: Blackwell.Google Scholar
Gabbay, D. M., & Woods, J. (2004). Handbook of the History of Logic, Volume 3: The Rise of Modern Logic: From Leibniz to Frege. Amsterdam: Elsevier North Holland.Google Scholar
Gauss, C. T. (1812). Disquisitiones generales circa seriem infinitam $1+\frac{\unicode{x3b1} \unicode{x3b2}}{1\cdot \gamma }x+\frac{\alpha \left(\alpha +1\right)\beta \left(\beta +1\right)}{1\cdot 2\cdot \gamma \left(\gamma +1\right)}\mathrm{xx}+$ etc. Pars 1. Göttingen: Commentationes Societatis Regiae Scientiarum recentiores 2.Google Scholar
Goldfarb, W. (1979). Logic in the twenties: The nature of the quantifier. The Journal of Symbolic Logic, 44, 351–368.CrossRefGoogle Scholar
Haaparanta, L. (2006). Compositionality, contextuality and the philosophical method. Acta Philosophica Fennica, 78, 289–301.Google Scholar
Haaparanta, L. (2009a). The Development of Modern Logic. Oxford: Oxford University Press.CrossRefGoogle Scholar
Haaparanta, L. (2009b). The Relations between Logic and Philosophy, 1874–1931. See Haaparanta (2009a), pp. 222–262.10.1093/acprof:oso/9780195137316.003.0025CrossRefGoogle Scholar
Hailperin, T. (1981). Boole’s Algebra isn’t Boolean Algebra. Mathematics Magazine, 54, 172–184.CrossRefGoogle Scholar
Hailperin, T. (1986). Boole’s Logic and Probability. Amsterdam: North Holland.Google Scholar
Hailperin, T. (2004). Algebraic Logic 1685–1900. See Gabbay & Woods (2004), pp. 323–388.Google Scholar
Heis, J. (2012). Attempts to Rethink Logic. In Wood, A. W. & Hahn, S. S., editors, The Cambridge History of Philosophy of the Nineteenth Century (1790–1870). Cambridge: Cambridge University Press, pp. 95–132.10.1017/CHO9780511975257.008CrossRefGoogle Scholar
Heis, J. (2013). Frege, Lotze, and Boole. In Reck, E. H., editor, The Historical Turn in Analytic Philosophy. New York: Palgrave Macmillan, pp. 113–138.Google Scholar
Heis, J. (2014). The Priority Principle from Kant to Frege. Noûs, 48, 268–297.10.1111/nous.12053CrossRefGoogle Scholar
Hintikka, J. (1988). On the development of the model-theoretic viewpoint in logical theory. Synthese, 77, 1–36.CrossRefGoogle Scholar
Hintikka, J. (1997a). Introduction. See Hintikka (1997b), pp. ix–xxii.Google Scholar
Hintikka, J. (1997b). Lingua Universalis vs Calculus Ratiocinator. Dordrecht: Kluwer.CrossRefGoogle Scholar
Huntington, E. V. (1904). Sets of independent postulates for the algebra of logic. Transactions of the American Mathematical Society, 5, 288–309.10.1090/S0002-9947-1904-1500675-4CrossRefGoogle Scholar
Jané, I. (2005). Review of C. Badesa, The Birth of Model Theory: Löwenheim Theorem in the Frame of the Theory of Relatives. Philosophia Mathematica, 13, 91–106.Google Scholar
Jourdain, P. E. B. (1914). Preface. In L’Algèbre de la Logique. Paris: Gauthier-Villars, pp. iii–x. English translation by L. G. Robinson in [Chicago: Open Court, 1914].Google Scholar
Kluge, E.-H. W. (1977). Frege, Leibniz et alii . Studia Leibnitiana, 9, 266–274.Google Scholar
Kluge, E.-H. W. (1980). Frege, Leibniz and the Notion of an Ideal Language. Studia Leibnitiana, 12, 140–154.Google Scholar
Korte, T. (2010). Frege’s Begriffsschrift as lingua characterica . Synthese, 174, 283–294.CrossRefGoogle Scholar
Kremer, M. (2006). Logicist responses to Kant: (Early) Frege and (Early) Russell. Philosophical Topics, 34, 163–188.CrossRefGoogle Scholar
Leibniz, G. W. (1679–1686). Ad Specimen Calculi Universalis. Originally unpublished. Edition in (Leibniz, 1999, n. 69, pp. 280–288). English translation by G.H.R. Parkinson in (Leibniz, 1966, pp. 33–39).Google Scholar
Leibniz, G. W. (1679–1686). Ad Specimen Calculi Universalis addenda. Originally unpublished. Edition in (Leibniz, 1999, n. 70, pp. 289–296). English translation by G.H.R. Parkinson in (Leibniz, 1966, pp. 40–46).CrossRefGoogle Scholar
Leibniz, G. W. (1686c). Generales Inquisitiones de Analysi Notionum et Veritatum. Originally unpublished. Edition in (Leibniz, 1999, n. 165, pp. 739–788). English translation by G.H.R. Parkinson in (Leibniz, 1966, pp. 47–87).Google Scholar
Leibniz, G. W. (1688–1689). Fundamenta calculi ratiocinatoris. Originally unpublished. Edition in (Leibniz, 1999, n. 192, pp. 917–922). English translation by M. Dascal in (Dascal, 1987, pp. 181–188).Google Scholar
Leibniz, G. W. (1765). Oeuvres philosophiques latines & françoises de feu Mr. De Leibnitz. Amsterdam, Leipzig: Jean Schreuder.Google Scholar
Leibniz, G. W. (1840). Opera philosophica quae exstant latina, gallica, germanica omnia. Berlin: G. Eichleri.Google Scholar
Leibniz, G. W. (1966). Logical Papers. Oxford: Clarendon Press.Google Scholar
Leibniz, G. W. (1999). Sämtliche Schriften und Briefe, Volume 4 of VI. Berlin: Deutschen Akademie der Wissenschaften.Google Scholar
Lenzen, W. (2004a). Calculus Universalis – Studien zur Logik von G. W. Leibniz. Paderborn: Mentis.CrossRefGoogle Scholar
Lenzen, W. (2004b). Leibniz’s Logic. See Gabbay & Woods (2004), pp. 1–83.Google Scholar
Lewis, C. I. (1918). A Survey of Symbolic Logic. Berkeley: University of California Press.Google Scholar
Löwenheim, L. (1911). Rezension von Ernst Schröder, Abriss der Algebra der Logik, Zweiter Teil. Archiv der Mathematik und Physik III. Reihe, 17, 71–73.Google Scholar
Löwenheim, L. (1915). Über Mögichkeiten im Relativkalkül. Mathematische Annalen, 76, 447–470. English translation by S. Bauer-Mengelberg in (van Heijenoort, 1967a, pp. 228–251).CrossRefGoogle Scholar
Malink, M., & Vasudevan, A. (2016). The Logic of Leibniz’s Generales Inquisitiones de Analysi Notionum et Veritatum . The Review of Symbolic Logic, 4, 686–751.CrossRefGoogle Scholar
Mancosu, P., Zach, R., & Badesa, C. (2009). The development of mathematical logic from Russell to Tarski, 1900–1935. See Haaparanta (2009a), pp. 318–470.CrossRefGoogle Scholar
Mitchell, O. H. (1883). On a new algebra of logic. See Peirce (1883b), pp. 72–106.CrossRefGoogle Scholar
Patzig, G. (1969). Leibniz, Frege und die sogenannte ‘lingua characteristica universalis’. Studia Leibnitiana. Supplementa, 3, 103–112.Google Scholar
Peckhaus, V. (1991). Ernst Schröder und die, pasigraphischen systeme’ von Peano und Peirce. Modern Logic, 1, 174–205.Google Scholar
Peckhaus, V. (1994). Wozu Algebra der Logik? Ernst Scrhöders nach einer universalen Theorie der Verknüpfungen. Modern Logic, 4, 357–381.Google Scholar
Peckhaus, V. (1996). The axiomatic method and Ernst Schröder’s algebraic approach to logic. Philosophia Scientiae, 1, 1–15.Google Scholar
Peckhaus, V. (2004a). Calculus ratiocinator versus characteristica universalis? The two traditions in logic, revisited. History and Philosophy of Logic, 25, 3–14.CrossRefGoogle Scholar
Peckhaus, V. (2004b). Schröder’s logic. See Gabbay & Woods (2004), pp. 557–610.Google Scholar
Peckhaus, V. (2014a). Ernst Schröder on pasigraphy. Revue d’Histoire des Sciences, 67, 207–230.CrossRefGoogle Scholar
Peckhaus, V. (2014b). Leibniz’s Influence on 19th Century Logic. In Zalta, E., editor, The Stanford Encyclopedia of Philosophy (Spring 2014 Edition). Available at: https://plato.stanford.edu/archives/spr2014/entries/leibniz-logic-influence/.Google Scholar
Peirce, C. S. (1870). Description of a notation for the logic of relatives, resulting from an amplification of the conceptions of Boole’s calculus on logic. Memoirs of the American Academy of Arts and Sciences, 9, 317–378.CrossRefGoogle Scholar
Peirce, C. S. (1880). On the algebra of logic. American Journal of Mathematics, 3, 15–57. Reedition in (Peirce, 1989, pp. 163–209).CrossRefGoogle Scholar
Peirce, C. S. (1883a). The logic of relatives. See Peirce (1883b), pp. 187–203. Reedition in (Peirce, 1989, pp. 453–466).Google Scholar
Peirce, C. S. (1883b). Studies in logic by members of the John Hopkins University. Philadelphia: John Benjamins.Google Scholar
Peirce, C. S. (1885). On the algebra of logic: A contribution to the philosophy of notation. American Journal of Mathematics, 7, 180–202. Reedition in (Peirce, 1993, pp. 162–190).CrossRefGoogle Scholar
Peirce, C. S. (1989). Writings of Charles S. Peirce, Vol. 4. Bloomington: Indiana University Press, pp. 1879–1884.Google Scholar
Peirce, C. S. (1993). Writings of Charles S. Peirce, Vol. 5. Bloomington: Indiana University Press, pp. 1884–1886.Google Scholar
Rescher, N. (1954). Leibniz’s Interpretation of his Logical Calculi. The Journal of Symbolic Logic, 19, 1–13.CrossRefGoogle Scholar
Schirn, M. (1984). Sluga über Freges These der Priorität von Urteilen gegenüber Begriffen. Archiv für Geschichte der Philosophie, 66, 194–215.CrossRefGoogle Scholar
Schröder, E. (1873). Lehrbuch der Arithmetik und Algebra für Lehrer und Studirende, Volume 1: Die sieben algebraischen Operationen. Leipzig: G. Teubner.Google Scholar
Schröder, E. (1877). Der Operationskreis des Logikkalkuls. Leipzig: G. Teubner.Google Scholar
Schröder, E. (1880). Rezension: G. Frege, Begriffsschrift . Zeitschrift für Mathematik und Physik, 25, 81–94. English translation by T. W. Bynum in (Frege, 1972, pp. 218–231).Google Scholar
Schröder, E. (1890). Vorlesungen über die Algebra der Logik (exakte Logic), Vol. 1. Leipzig: G. Teubner.Google Scholar
Schröder, E. (1891). Vorlesungen über die Algebra der Logik (exakte Logic), Vol. 2 (1. Abteilung). Leipzig: G. Teubner.Google Scholar
Schröder, E. (1895). Vorlesungen über die Algebra der Logik (exakte Logic) , Vol. 3. Leipzig: G. Teubner. Partial English translation by G. Brady in (Brady, 2000, pp. 207–427).Google Scholar
Schröder, E. (1898a). Die selbständige Definition der Mächtigkeiten $0,1,2,3$ und die explizite Gleichzahligkeitsbedingung. Nova Acta Leopoldina. Abhandlungen der Kaiserlich Leop.-Carol. Deutschen Akademie der Naturforscher, 71, 364–376.Google Scholar
Schröder, E. (1898b). Ueber zwei Definitionen der Endlichkeit und G. Cantor’sche Sätze. Nova Acta Leopoldina. Abhandlungen der Kaiserlich Leop.-Carol. Deutschen Akademie der Naturforscher, 71, 301–362.Google Scholar
Schröder, E. (1899). On Pasigraphy. Its present State and the Pasigraphic Movement in Italy. The Monist, 9, 246–262.Google Scholar
Schröder, E. (1905). Vorlesungen über die Algebra der Logik (exakte Logic), Vol. 2 (2. Abteilung). Leipzig: G. Teubner.Google Scholar
Sluga, H. (1980). Gottlob Frege. The Arguments of the Philosophers. London: Routledge.Google Scholar
Sluga, H. (1987). Frege Against the Booleans. Notre Dame Journal of Formal Logic, 28, 80–98.CrossRefGoogle Scholar
Tappenden, J. (1995). Extending knowledge and ‘Fruitful Concepts’: Fregean themes in the foundations of mathematics. Noûs, 29, 427–467.CrossRefGoogle Scholar
Tarski, A. (1941). On the Calculus of Relations. The Journal of Symbolic Logic, 6, 73–89.CrossRefGoogle Scholar
Thiel, C. (1977). Leopold Löwenheim: life, work, and early influence. In Gandy, R. & Hyland, M., editors, Logic Colloquium 76: Proceedings of a conference held in Oxford in July 1976. Oxford: North Holland, pp. 235–252.Google Scholar
Thiel, C. (1994). Schröders zweiter Beweis für die Unabhängigkeit der zweiten Subsumtion des Distributivgesetzes im logischen Kalkül. Modern Logic, 4, 382–391.Google Scholar
Trendelenburg, F. A. (1856). Über Leibnizens Entwurf einer allgemeinen Charakteristik . Berlin: F. Dümmler. Reedition in (Trendelenburg, 1867, pp. 1–47).Google Scholar
Trendelenburg, F. A. (1867). Historische Beiträge zur Philosophie, Volume 3: Vermischte Abhandlungen. Berlin: G. Bethge.Google Scholar
van Heijenoort, J. (1967a). From Frege to Gödel, a Source Book in Mathematical Thought. Cambridge: Harvard University Press.Google Scholar
van Heijenoort, J. (1967b). Logic as Calculus and Logic as Language. Synthese, 17, 324–330.CrossRefGoogle Scholar
Wiener, N. (1913). A Comparison Between the Treatment of the Algebra of Relatives by Schröder and that by Whitehead and Russell. Ph.D. thesis, Harvard University.Google Scholar
Wilson, M. (2010). Frege’s mathematical setting. In Potter, M., & Ricketts, T., editors, The Cambridge Companion to Frege. Cambridge: Cambridge University Press, pp. 379–412.CrossRefGoogle Scholar
Zermelo, E. (1908). Mathematische Logik. Vorlesung gehalten von Prof. Dr. E. Zermelo zu Göttingen im S. S. 1908. Lecture notes by K. Grelling. Nachlaß Zermelo, C 129/224, Universitätsbibliothek Freiburg im Breisgau.Google Scholar
Schröder, E. (1909). Abriss der Algebra der Logik. Edited by Müller, K. E.. Leipzig: Teubner.Google Scholar