Abstract
Standard modal logic for alethic modalities analyses modalities as ranging over all possible worlds (the Leibnizian universe). This leaves very little room in the space of worlds to entertain impossible things. My proposal is to liberate the Leibnizian universe and reinforce the relative aspect of possibility; worlds are possible with respect to some worlds, and impossible for others. The central idea is to isolate relative possibility (Kripke) from conditionality (Lewis/Stalnaker). To accommodate counterpossibles, I provide a dialetheic conditional modal logic, a theory that is dialetheic at every level, in the logic as well as in the set theory behind it.
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Notes
- 1.
If you’re not familiar with bisimulation, think of the picture as showing that any formula that can be invalidated at \(\texttt {w}\) can also be invalidated at \(\texttt {w}'\) (see Priest 2008, Sect. 3.7.5); anything impossible at \(\texttt {w}\) is also impossible at \(\texttt {w}'\), because \(\texttt {w}\) doesn’t see any more worlds than \(\texttt {w}'\).
- 2.
- 3.
Repetition intended, to stress that dialetheic logic is used in the object and meta levels.
- 4.
I continue to represent models with equivalence classes for \(\texttt {R}\) and Lewis’ systems of spheres \(\texttt {S}\), as I find them more convivial, with apologies if you don’t agree.
- 5.
The proof of soundness and completeness is entirely standard, and I won’t bother you with the details.
- 6.
Williamson formulates pos as the logically equivalent
, but this formulation will be better for us. - 7.
For a thorough criticism of Williamson on counterpossibles, see Berto et al. (2017), who also propose a conditional modal logic similar to mine, but using a distinction between possible and impossible worlds instead of my more general notion of relative possibility. I don’t like the twentieth-century distinction between possible and impossible worlds, or between good and bad worlds in general, as discussed in Girard and Weber (2015).
- 8.
For a counterexample to nec, take a model with two worlds \(\texttt {w}\) and \(\mathtt {v}\) such that \(\mathtt {v}\not \in \texttt {R}(w)\), \(\mathtt {V}(p) =\{\mathtt {v}\}\), \(\mathtt {v}\in \texttt {S}(\texttt {w}, \llbracket p \rrbracket )\) but \(\mathtt {v}\not \in \mathtt {V}(q)\). Then \(\Box (p \rightarrow q)\) is vacuously true at \(\texttt {w}\) whereas
is false. For a counterexample to pos, take a model with three worlds \(\texttt {w}, \mathtt {u}\) and \(\mathtt {v}\) such that \(\mathtt {V}(p) = \{\mathtt {u}, \mathtt {v}\}\), \(\mathtt {V}(q) = \{\mathtt {v}\}\), \(\mathtt {u}\in \texttt {R}(\texttt {w})\), and \(\mathtt {v}\in \texttt {S}(w, \llbracket p \rrbracket )\). - 9.
From Blackburn et al. (2001, Definition 3.2) A formula \(\varphi \) defines a class of frames K if for all frames \(\mathtt {F}\), \(\mathtt {F}\in \texttt {K}\) iff \(\varphi \) is valid in \(\mathtt {F}\).
- 10.
The proofs of definability are straightforward and left to the reader.
- 11.
A condition that implies but is not equivalent to POS is the converse of NEC: \((\texttt {R}(\texttt {w})\cap \texttt {X}) \subseteq \texttt {S}(\texttt {w}, \texttt {X})\). If we assumed this stronger condition in combination with NEC, we would reduce conditionals to strict implication, which would be a bad idea (cf., David 1973, Sect. 1.2).
- 12.
‘So that is our basic procedure for handling the antecedent of a counterfactual implicated in a case of extra-mathematical explanation: hold as much fixed as you can within mathematics compatible with the twiddle, without inducing a contradiction’, Baron et al. (2017), p.11.
- 13.
VC is Lewis’s favourite counterfactual logic. I choose it here without a strong philosophical commitment. Assume the limit assumption for simplicity. The choice is entirely modular and you can pick your favourite conditional logic if you don’t like VC; no problem.
- 14.
The axiom pos is canonical for POS in a standard Henkin completeness proof adapted from Chellas (1975). Check or proceed without concern.
- 15.
\(\texttt {w}\in \llbracket \Box \lnot \varphi \rrbracket ^{+} \Rightarrow \texttt {R}(\texttt {w}) \subseteq \llbracket \lnot \varphi \rrbracket ^{+} \Rightarrow \texttt {R}(\texttt {w}) \subseteq \llbracket \varphi \rrbracket ^{-}\Rightarrow \texttt {w}\in \llbracket \Diamond \varphi \rrbracket ^{-} \Rightarrow \texttt {w}\in \llbracket \lnot \Diamond \varphi \rrbracket ^{+}\).
- 16.
Since the problem is about the duality of modalities, it arises just the same in a formulation with three truth values.
- 17.
As I want to keep the focus on modalities and conditionals, I won’t go into motivating the propositional logic here, and refer you to (Weber et al. 2016) instead.
- 18.
I add the constant \(\bot \) to be able to identify triviality. Unlike in twentieth-century logic, it cannot be defined by a contradictory formula.
- 19.
With apologies to Kabay (2008).
- 20.
Assume that \(\texttt {w}\in \llbracket \Diamond \bot \rrbracket \), then \(\texttt {R}(\texttt {w}) \cap \llbracket \bot \rrbracket \ne \emptyset \), so there exists a world \(\mathtt {x}\) such that \(\mathtt {x}\in \texttt {R}(\texttt {w}) \wedge \mathtt {x}\in \llbracket \bot \rrbracket \). But \(\mathtt {x}\in \llbracket \bot \rrbracket \) implies that \(\mathtt {x}\in \Box \varphi \) for every \(\varphi \), so \(\texttt {R}(\mathtt {x}) \subseteq \llbracket \varphi \rrbracket \) for every \(\varphi \). Furthermore, \(\mathtt {x}\in \texttt {R}(\texttt {w})\) implies by 5 that \(\texttt {w}\in \texttt {R}(\mathtt {x})\). So there is a world \(\mathtt {x}\) such that \(\texttt {w}\in \texttt {R}(\mathtt {x}) \wedge \texttt {R}(\mathtt {x})\subseteq \llbracket \varphi \rrbracket \) for every \(\varphi \), so \(\texttt {w}\in \llbracket \varphi \rrbracket \) for every \(\varphi \). Therefore \(\texttt {w}\in \llbracket \bot \rrbracket \).
, but this formulation will be better for us.
is false. For a counterexample to References
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Acknowledgements
Graham, you’ve been a source of inspiration for several scholars that have known or read you, and you are a model of perseverance against adversity. Above all, you are a good friend, and I’m very pleased to be able to contribute this paper in your honour. I would like to thank Zach Weber, Fred Kroon, Joy Britten and Michael Hillas for valuable comments on a previous draft of this paper.
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Girard, P. (2019). Dialetheic Conditional Modal Logic. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_14
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