Abstract
The “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with systems of relevance logic and paraconsistent logic. In this paper it will be argued that McCall’s historical analysis is fundamentally mistaken since it doesn’t take into account two versions of connexivism. While “humble” connexivism maintains that connexive properties (like the condition that no proposition implies its own negation) only apply to “normal” (e.g., self-consistent) antecedents, “hardcore” connexivism insists that they also hold for “abnormal” propositions. It is shown that the overwhelming majority of the forerunners of connexive logic were only “humble” connexivists. Their ideas concerning (“humbly”) connexive implication don’t give rise, however, to anything like a non-classical logic.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Angell, R. B. (1962). A propositional logic with subjunctive conditionals. The Journal of Symbolic Logic, 27(3), 327–343.
Annas, J., & Barnes, J. (2000). Sextus Empiricus. Cambridge University Press.
Bocheński, J. M. (1961). A history of formal logic. Chelsea Publ.
Burks, A. W., & Copi, I. M. (1950). Lewis Carroll’s barber shop paradox. Mind, 59, 219–222.
Burleigh, W. (1955). De Puritate Artis Logicae Tractatus Longior, ed. by P. Boehner. E. Nauwelaerts/F. Schöningh.
Caramuel y Lobkowitz, J. (1681). Leptotatos Latine Subtilissimus. Conrad.
Couturat, L. (1901). La logique de Leibniz. F. Alcan; reprint Olms, 1985.
Crupi, V., & Iacona, A. (2020). The evidential conditional. Erkenntnis, publ. online, (December 31)
De Rijk, L. M. (1967). Logica Modernorum - a contribution to the history of early Terminist logic. Van Gorcum.
De Rijk, L. M. (Ed.). (1970). Petrus Abaelardus Dialectica. Van Gorcum.
Estrada-González, L., & Ramírez-Cámara, E. (2020). A Nelsonian response to ‘the most embarrassing of all twelfth-century arguments’. History and Philosophy of Logic, 41, 101–113.
Goodman, N. (1947). The problem of counterfactual conditionals. The Journal of Philosophy, 94, 113–128.
Iacona, A. (2019). Strictness and Connexivity. Inquiry 63, publ. online Oct. 20.
Iwakuma, Y. (1993). Parvipontani’s thesis ex Impossibili Quidlibet sequitur: Comments on the sources of the thesis from the twelfth century. In K. Jacobi (Ed.), Argumentationstheorie – Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns (pp. 123–151). Brill.
Johnston, S. (2019). Per se modality and natural implication – An account of Connexive logic in Robert Kilwardby. Logic and Logical Philosophy, 28, 449–479.
Kapsner, A. (2019). Humble Connexivity. Logic and Logical Philosophy, 28, 513–536.
Kneale, W., & Kneale, M. (1962). The development of logic. Clarendon Press.
Leibniz, G. W. (1686). In F. Schupp (Ed.), Generales Inquisitiones de Analysi Notionum et Veritatum. Meiner 1982.
Lenzen, W. (1984). Leibniz und die Boolesche Algebra. Studia Leibnitiana, 16, 87–103.
Lenzen, W. (1986). ‘Non est’ non est ‘Est non’ – Zu Leibnizens Theorie der Negation. Studia Leibnitiana, 18, 1–37.
Lenzen, W. (1987). Leibniz’s Calculus of strict implication. In J. Srzednicki (Ed.), Initiatives in logic (pp. 1–35). Martinus Nijhoff.
Lenzen, W. (2004). Calculus Universalis – Studien zur Logik von G. W. Leibniz. Mentis.
Lenzen, W. (1990). Das System der Leibnizschen Logik. De Gruyter.
Lenzen, W. (2014). Leibniz: Logic. In J. Fieser & B. Dowden (Eds.), Internet encyclopedia of philosophy. http://www.iep.utm.edu/leib-log/
Lenzen, W. (2019). Leibniz’s Laws of consistency and the philosophical foundations of Connexive logic. Logic and Logical Philosophy, 28, 537–551.
Lenzen, W. (2020). A critical examination of the historical origins of Connexive logic. History and Philosophy of Logic, 41, 16–35.
Lenzen, W. (2020). Kilwardby’s 55th lesson. Logic and Logical Philosophy, 29, 485–504.
Lenzen, W. (2021). The third and fourth stoic account of conditionals. In M. Blicha & I. Sedlár (Ed.), The Logica yearbook 2020 (pp. 127–146). College Publications.
Lenzen, W. (2021). Abaelards Logik. Brill/mentis.
Lenzen, W. (2021). What follows from the impossible: Everything or nothing? Forthcoming in History and Philosophy of Logic.
Lewis, D. (1973). Counterfactuals. Blackwell.
MacColl, H. (1878). The Calculus of equivalent statements (second paper). In Proceedings of the London Mathematical Society (Vol. S1-9, pp. 177–186).
Mares, E. (2020). Relevance Logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (winter 2020 edition) https://plato.stanford.edu/archives/win2020/entries/logic-relevance/
Martin, C. (1968). Williams’ machine. The Journal of Philosophy, 83, 564–572.
Martin, C. (1987). Embarrassing arguments and surprising conclusions in the development of theories of the conditional in the twelfth century. In J. Jolivet & A. de Libera (Eds.), Gilbert de Poitiers et ses Contemporains (pp. 377–400). Bibliopolis.
Martin, C. (1991). The logic of negation in Boethius. Phronesis, 36, 277–304.
Martin, C. (2004). Logic. In J. E. Brower & K. Guilfoy (Eds.), The Cambridge companion to Abelard (pp. 158–199). Cambridge University Press.
McCall, S. (1966). Connexive implication. The Journal of Symbolic Logic, 31(3), 415–433.
McCall, S. (2012). A history of Connexivity. In D. M. Gabbay, F. J. Pelletier, & J. Woods (Eds.), Handbook of the history of logic, Vol. 11, logic: A history of its central concepts (pp. 415–449). Elsevier
Migne, J. P. (Ed.). (1860). Manlii Severini Boetii Opera omnia. Garnier.
Nasti De Vincentis, M. (2006). Conflict and connectedness: Between modern logic and history of ancient logic. In E. Ballo & M. Franchella (Eds.), Logic and philosophy in Italy (pp. 229–251). Polimetrica International Scientific Publisher.
Nelson, E. (1930). Intensional relations. Mind, 39, 440–453.
O’Toole, R., & Jennings, R. (2004). The Megarians and the stoics. In D. Gabbay & J. Woods (Eds.), Handbook of the history of logic, Vol. 1, Greek, Indian and Arabic logic (pp. 397–522). Elsevier.
Parkinson, G. H. R. (1966). Leibniz – Logical Papers. Clarendon Press.
Patzig, G. (1959). Aristotle and syllogisms from false Premisses. Mind, 68, 186–192.
Pizzi, C. (1977). Boethius’ thesis and conditional logic. Journal of Philosophical Logic, 6, 283–302.
Pizzi, C., & Williamson, T. (1997). Strong Boethius’ thesis and consequential implication. Journal of Philosophical Logic, 26, 569–588.
Priest, G. (1999). Negation as cancellation, and Connexive logic. Topoi, 18(1), 141–148.
Priest, G., Tanaka, K., & Weber, Z. (2018). Paraconsistent Logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (summer 2018 edition). https://plato.stanford.edu/archives/sum2018/entries/logic-paraconsistent/
Ramsey, F. P. (1931). General propositions and causality. In D. H. Mellor (Ed.), F. P. Ramsey: Philosophical papers (pp. 145–163). Cambridge University Press 1990.
Routley, R., & Montgomery, H. (1968). On systems containing Aristotle’s thesis. Journal of Symbolic Logic, 33, 82–96.
Sanford, D. (1989). If P, then Q – Conditionals and the foundations of reasoning. Routledge.
Sigwart, C. (1871). Beiträge zur Lehre vom hypothetischen Urtheile. H. Laupp.
Stalnaker, R. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies in logical theory (pp. 98–112). Blackwell.
Strobach, N., & Malink, M. (Eds.). (2005). Aristoteles Analytica Priora Buch II. De Gruyter.
Thom, P., & Scott, J. (Eds.). (2015). Robert Kilwardgy Notule libri Priorum. Oxford University Press.
Wansing, H. (2020). Connexive Logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2020 ed.) https://plato.stanford.edu/archives/spr2020/entries/logic-connexive/
Weidemann, H. (1997). Aristoteles über Schlüsse aus falschen Prämissen. Archiv für Geschichte der Philosophie, 79, 202–211.
Whity, A. (2017) A principled account of Boethian hypothetical syllogisms. https://www.academia.edu/4273408/A_Principled_Account_of_Boethian_Hypothetical_Syllogisms
Wilks, I. (2007). Peter Abelard and his contemporaries. In D. M. Gabbay & J. Woods (Eds.), Handbook of the history of logic, Vol. 2 mediaeval and renaissance logic (pp. 83–155). Elsevier.
Wright, T. (Ed.). (1863). Alexandri Neckham – De Naturis Rerum Libri Duo. Longman, Roberts, and Green.
Acknowledgements
My thanks are to two anonymous referees whose comments and criticisms greatly helped to improve this paper. In particular, they warned me to avoid the impression that this paper was aiming at criticizing the whole enterprise of connexive logic or, even worse, the whole enterprise of non-classical logic including relevance logic, paraconsistent logic, etc. The aim of the paper is only to give a thorough analysis of the historical texts in order to determine which classical author defended which type of (“humble” or “hardcore”) connexivity.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Financial and non-financial interests
The author has no relevant financial or non-financial interests to disclose.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lenzen, W. Rewriting the History of Connexive Logic. J Philos Logic 51, 525–553 (2022). https://doi.org/10.1007/s10992-021-09640-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-021-09640-6