Abstract
A very common twofold view in contemporary philosophy is that classical logic is the correct view of logical consequence and that possibility conforms to classical logic in the sense that ‘possible worlds’ — whatever else they may be — are closed under classical logic. These two views are assumed in this paper. My aim in this paper is to show that a very natural ‘paraconsistent’ (and also ‘paracomplete’) consequence relation is involved in the given view of possible worlds and logical consequence.
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Notes
The extension to first-order languages (and first-order logic) is straightforward but present purposes demand only the simpler propositional level. Exactly what makes vocabulary ‘logical’ is a hard question, and one on which I herein remain silent, except to assume the standard batch of vocabulary as logical. Gil Sagi’s work highlights some of the background difficulties Sagi (2017, 2018).
An alternative, briefly discussed in Section 5, is ‘egalitarian’ with respect to the roles of duo-point worlds.
Call a connective extensional iff its evaluation at a point never demands going beyond that point. (Example: typical modal connectives are not extensional.) The logical connectives canvassed here are extensional in just that sense, even if one needs to look at different components of a point to evaluate the connectives.
Those familiar with the Valerie Routley & Richard Routley ‘Star’ account of logical negation Routley and Routley (1972), some history and background to which is given by Dunn (2019), should note that the duo-points account has the feel of the Star account but is not the Star account. The Star account is not extensional (as understood above); it involves a shift of points whereas the duo-points account involves only an ‘internal shift’ (of sorts).
For a useful history of the given consequence relation see Mike Dunn’s account Dunn (2019), and for further history and discussion of FDE see the Omori–Wangsing collection Omori and Wansing (2019).
Concerning the metaphysical questions flagged in Section 2.3 above, note that the construction provides a natural way of understanding what Berto & Jago call ‘FDE worlds’ Berto and Jago (2013).
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Acknowledgements
Thanks to Franz Berto, A. J. Cotnoir, Franca D’Agostini, Michael DeVito, Elena Ficara, Daniel Nolan, Luis Estrada González, Graham Priest, Elisángela Ramírez, Greg Restall, Gill Russell, Kevin Scharp, Zach Weber and participants in a super-special seminar at St Andrews. Thanks especially to Nikolaj Pedersen for helpful feedback.
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Beall, J. From possible worlds to paraconsistency: on the inevitability of paraconsistent entailment. AJPH 1, 26 (2022). https://doi.org/10.1007/s44204-022-00028-0
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DOI: https://doi.org/10.1007/s44204-022-00028-0