Abstract
In this paper, we zoom in on some important features of Chrysippus’ logic by framing them in a natural deduction system for relevant logic. Our view is alternative to other recent reconstructions of Chrysippus’ view based on the sequent calculus. We show how, adding suitable elimination rules (akin to the five indemonstrables) that are missing in the Chrysippean system, one automatically obtains classical logic as a kind of deductive upper bound.
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Notes
- 1.
By “proof rule” in the context of this paper we mean a rule that takes proofs as premisses and conclusion, as opposed to an “inference rule” that takes propositions as premisses and conclusion.
- 2.
On this point see Bobzien (2019).
- 3.
What is essentially the same argument in a slightly different setting presented in Hitchcock (2006).
- 4.
According to most scholars, the Stoics did no allow inferences with less than two premisses; see Bobzien (2019).
References
Acerbi, F. (2008). Conjunction and disjunction in Euclid’s elements. Histoire, Épistémologie, Langage, 30, 21–47.
Anderson, A.R., & Belnap, N. D. (1975). Entailment: The logic of relevance and necessity. Princeton University Press.
Becker, O. (1957). Zwei Untersuchungen zur antiken Logik. Harrassowitz.
Bobzien, S. (1996). Stoic syllogistic. Oxford Studies in Ancient Philosophy, 40, 133–192.
Bobzien, S. (1997). The Stoics on hypotheses and hypothetical arguments. Phronesis, 42(3), 299–312.
Bobzien, S. (2003). Stoic logic. In B. Inwood (Ed.), The Cambridge companion to Stoic philosophy. Cambridge University Press.
Bobzien, S. (2019). Stoic sequent logic and proof theory. History and Philosophy of Logic, 40(3), 234–265.
Bobzien, S., & Dyckhoff, R. (2019). Analyticity, balance and non-admissibility of cut in Stoic logic. Studia Logica, 107(2), 375–397.
Bocheński, J. M. (1970). A history of formal logic. Chelsea Publishing Company.
Casari, E. (2017). La logica stoica. A cura di Enrico Moriconi. ETS.
D’Agostino, M., & Mondadori, M. (1994). The taming of the cut. Classical refutations with analytic cut. Journal of Logic and Computation, 4, 285–319.
Diogenes Laertius. (2013). Lives of eminent philosophers (T. Dorandi, Ed.). Cambridge University Press.
Frede, M. (1974). Stoic vs. Aristotelian syllogistic. Archiv für Geschichte der Philosophie, 56(1), 1–32.
Hitchcock, D. (2006). The peculiarities of Stoic logic. In A. D. Irvine & K. A. Peacock (Eds.), Mistakes of reason: Essays in honour of John Woods (pp. 224–242). University of Toronto Press.
Hülser, K. (1987–1988). Die Fragmente zur Dialektik der Stoiker: Neue Sammlung der Texte mit deutscher Übersetzung und Kommentaren (4 vols.). Frommann-Holzboog.
Jennings, R. E. (1994). The genealogy of disjunction. Oxford University Press.
Kneale, W. C., & Kneale, M. (1962). The development of logic. Clarendon Press.
Lewis, C. I., & Langford, C. H. (1932). Symbolic logic. Dover.
Mates, B. (1953). Stoic logic. University of California Press.
Mignucci, M. (1993). The Stoic themata. In K. Döring & T. Ebert (Eds.), Dialektiker und Stoiker: Zur Logik der Stoa und ihrer Vorläufer (pp. 217–238). Steiner.
Milne, P. (1995). On the completeness of non-philonian Stoic logic. History and Philosophy of Logic, 16(1), 39–64.
Mueller, I. (1978). An introduction to Stoic logic. In J. M. Rist (Ed.), The Stoics. University of California Press.
O’Toole, R. R., & Jennings R. E. (2004). The Megarians and the Stoics. In M. D. Gabbay & J. Woods (Eds.), Handbook of the history of logic, Volume I. Greek, Indian and Arabic logic (pp. 397–522). Elsevier.
Read, S. (1989). Relevant logic: A philosophical examination of inference. Blackwell.
Sextus Empiricus. (1996). The skeptic way: Sextus Empiricus’ outlines of Pyrrhonism (Translated, with Introduction and Commentary, by B. Mates). Oxford University Press.
Sextus Empiricus (2005). Against the logicians (R. Bett, Ed.). Cambridge University Press.
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D’Agostino, M., Piazza, M. (2022). Chrysippus’ Logic in a Natural Deduction Setting. In: Ademollo, F., Amerini, F., De Risi, V. (eds) Thinking and Calculating. Logic, Epistemology, and the Unity of Science, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-030-97303-2_4
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