Abstract
The paper presents a contra-classical dialectic logic, inspired and motivated by Hegel s dialectics. Its axiom schemes are
Thus, in a sense, this dialectic logic is a kind of “mirror image“ of connexive logic. The informal interpretation of ‘\(\rightarrow \)’ emerging from the above four axiom schemes is not of a conditional (or implication); rather, it is the relation of determination in the presence of truth-value gaps: \(\varphi \rightarrow \psi \) is read as \(\varphi \) determines \(\psi \), namely, necessarily, if \(\varphi \) is true, then \(\psi \) is either true or false, not gappy. As far as I know, such a connective has not been considered before in the literature.
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Notes
A more general definition appeals to consequences (arguments) instead of merely theorems. Also, Humberstone [5] excludes logics the connectives of which can be translated to classical logic. For the current needs, the simpler definition suffices.
At this stage, only a implication-negation fragment is considerd, without conjunction and/or disjunction.
Or negations thereof, assuming double-negation equivalence.
There is also a sketchy presentation of logics related to Hegel in a manuscript by Ricardo Arturo Nicolás-Francisco [6], not related to the axiom H.
Note the possibility of vacuous determination in case \(\varphi \) is not true.
This is the same as a C-frame for Wansing’s connexive logic C [7].
It can be shown that persistent via support of truth/falsity extends to all formulas.
References
Da Costa, N. C., & Wolf, R. G.: Studies in paraconsistent logic I: The dialectical principle of the unity of opposites. Philosophia 9, 189–217 (1980)
Estrada-González, L.: The Bochum plan and the foundations of contra-classical logics. CLE e-Prints, 19(1), (2020)
Ficara, E.: Hegel and the consequentia mirabilis. Hist. Philos. Logic 39(4), 357–364 (2018). https://doi.org/10.1080/01445340.2018.1441679
Francez, N.: A View of Connexive Logics. College Publications, London (2021)
Humberstone, L.: Contra-classical logics. Australas. J. Philos. 78(4), 438–474 (2000)
Nicolás-Francisco, R.A.: A study of two dialectical logics. In: Romero, J.R.M., Galindo, M.O., Cortés, C.Z., Parra, P.P. (eds.) LANMR 2020–Thirteenth Latin American Workshop on Logic/Languages, Algorithms and New Methods of Reasoning, pp. 26–35. CEUR workshop proceedings, Aachen University (2020)
Wansing, H.: Connexive modal logic. In Schmidt, R., Pratt-Hartmann, I., Reynolds, M., Wansing, H. (eds.) Advances in Modal Logic, Vol. 5, pp.367–383. College Publications, King’s College, London (2005)
Wansing, H.: Connexive logic. In Zalta, E.N. (ed.), The Stanford Encyclopedia of Philosophy. Summer 2022 edn (2022). https://plato.stanford.edu/archives/spr2020/entries/logic-connexive/
Acknowledgements
I thank Elena Ficara for some discussions of Hegel’s logic, and Luis Estrada-González for some comments of a preliminary version of the paper.
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Francez, N. A Dialectic Contra-Classical Logic. Log. Univers. 17, 221–229 (2023). https://doi.org/10.1007/s11787-023-00324-0
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DOI: https://doi.org/10.1007/s11787-023-00324-0