Completeness for counter-doxa conditionals – using ranking semantics

Review of Symbolic Logic 12 (4):861-891 (2019)
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Abstract

Standard conditionals $\varphi > \psi$, by which I roughly mean variably strict conditionals à la Stalnaker and Lewis, are trivially true for impossible antecedents. This article investigates three modifications in a doxastic setting. For the neutral conditional, all impossible-antecedent conditionals are false, for the doxastic conditional they are only true if the consequent is absolutely necessary, and for the metaphysical conditional only if the consequent is ‘model-implied’ by the antecedent. I motivate these conditionals logically, and also doxastically by properties of conditional belief and belief revision. For this I show that the Lewisian hierarchy of conditional logics can be reproduced within ranking semantics, provided we slightly stretch the notion of a ranking function. Given this, acceptance of a conditional can be interpreted as a conditional belief. The epistemic and the neutral conditional deviate from Lewis’ weakest system $V$, in that ID or even CN are dropped, and new axioms appear. The logic of the metaphysical conditional is completely axiomatised by $V$ to which we add the known Kripke axioms T5 for the outer modality. Related completeness results for variations of the ranking semantics are obtained as corollaries.

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Raidl Eric
University Tübingen

Citations of this work

Definable Conditionals.Eric Raidl - 2020 - Topoi 40 (1):87-105.
The Implicative Conditional.Eric Raidl & Gilberto Gomes - 2023 - Journal of Philosophical Logic 53 (1):1-47.
Full & Partial Belief.Konstantin Genin - 2019 - In Richard Pettigrew & Jonathan Weisberg (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 437-498.

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References found in this work

Williamson on Counterpossibles.Berto Francesco, David Ripley, Graham Priest & Rohan French - 2018 - Journal of Philosophical Logic 47 (4):693-713.
Impossible Worlds: A Modest Approach.Daniel Nolan - 1997 - Notre Dame Journal of Formal Logic 38 (4):535-572.
Remarks on counterpossibles.Berit Brogaard & Joe Salerno - 2013 - Synthese 190 (4):639-660.
Counterpossibles.Timothy Williamson - 2018 - Topoi 37 (3):357-368.

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