Abstract
Since Montague’s work it is well known that treating a single modality as a predicate may lead to paradox. In their paper “No Future”, Horsten and Leitgeb (2001) show that if the two temporal modalities are treated as predicates paradox might arise as well. In our paper we investigate whether paradoxes of multiple modalities, such as the No Future paradox, are genuinely new paradoxes or whether they “reduce” to the paradoxes of single modalities. In order to address this question we develop a notion of reducibility based on a version of Smoryński Diagonalized Operator Logic. We show that there are reducible multimodal paradoxes as well as irreducible paradoxes of interaction. In particular, we show the No Future paradox to be an irreducible paradox according to our notion of reducibility.
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Notes
“Paradoxes” like Fitch’s paradox form a notable exception to this claim. For an overview of the unimodal paradoxes obtained by appeal to diagonalization we refer the reader to Egré [4].
The modal operator logic G L (“Gödel-Löb”) forms a notable exception to this rule. Sambin and De Jongh showed independently that G L has the fixed-point property, i.e. that we can find fixed-points for all modal formulas.
Egré [4] already uses modal operator logics with fixed-points to provide a systematization and an overview of the (uni)modal paradoxes.
Without the restriction to boxed propositional variables the approach would be trivialized. Diagonalizing the formula ¬p would lead directly into contradiction. Smoryński [13] (pp. 72/73) points out that the restriction to boxed formulas is also well motivated.
This suggests that an operator formulation of DML along the lines of DOL or Alberucci and Facchini’s [1] modal μ ~-calculus may be preferable. Still, in many respects the present formulation in terms of fixed-point constants proves sufficient and is somewhat easier to handle. It may also be preferable not to call DML a logic but rather a theory, however with the mentioned caveat we stick to the original terminology.
For ease of presentation we simply write δ instead of the correct δ ¬◊■p throughout the proof.
(■Nec) is just the rule of necessitation for the ■-operator.
Again we omit the subscript to the fixed point constant δ ¬□■p throughout the proof.
A referee pointed to the fact that the reducibility of a paradox could also be spelled out in terms of conservativity: a paradox is reducible, if and only if, \(\mathcal {S}^{F}\) in \(\mathcal {L}_{\Box \blacksquare }^{F}\) is \(\mathcal {L}_{\Box }^{F}\cup \mathcal {L}_{\blacksquare }^{F}\)-conservative over \(\mathcal {S}^{F}\) in \(\mathcal {L}_{\Box }^{F}\cup \mathcal {L}_{\blacksquare }^{F}\). While this is true, spelling out the criterion in this way does not provide further insight since we always start with \(\mathcal {S}^{F}\) inconsistent in \(\mathcal {L}_{\Box \blacksquare }^{F}\). As a consequence whenever \(\mathcal {S}^{F}\) is inconsistent in \(\mathcal {L}_{\Box }^{F}\cup \mathcal {L}_{\blacksquare }^{F}\), \(\mathcal {S}^{F}\) in \(\mathcal {L}_{\Box \blacksquare }^{F}\) will be a \(\mathcal {L}_{\Box }^{F}\cup \mathcal {L}_{\blacksquare }^{F}\)-conservative extension. Whenever \(\mathcal {S}^{F}\) in \(\mathcal {L}_{\Box }^{F}\cup \mathcal {L}_{\blacksquare }^{F}\) is consistent, then \(\mathcal {S}^{F}\) in \(\mathcal {L}_{\Box \blacksquare }^{F}\) will not be a conservative extension. For a discussion of the notion of conservative extension in the setting of multimodal logics see, for example, Williamson [15].
The reader acquainted with provability logic and the fixed-point theorem for G L by De Jongh and Sambin might note that the semantic result is actually stronger as G L is not valid on all converse well-founded frames.
For the notion of ‘rank’ cf. Boolos [3], pp. 94-95.
For assume to the contrary that there is a world w from which an infinite R-path starts. Define a subset X of W as the set of those worlds that can be reached in finitely many steps from w and which lie on the infinite R-path. By converse well-foundedness there needs to be an R-maximal element in X which contradicts the initial assumption. This also implies that a dead end is always reached in finitely many steps.
See Blackburn et al. [2] for more on Sahlquist formulas and how to compute the characteristic first-order property defined by a Sahlquist formula.
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Acknowledgments
We thank Catrin Campbell-Moore and Jan Heylen for comments on drafts of this paper and an anonymous referee for a number of helpful suggestions. We also thank audiences in Munich, Paris, Leuven, Amsterdam, Buenos Aires and Irvine for helpful suggestions. Our work has been carried out within the DFG funded research project “Syntactical Treatments of Interacting Modalities”. The project is hosted by the Munich Center for Mathematical Philosophy which in turn enjoys support by the Alexander von Humboldt Foundation.
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Stern, J., Fischer, M. Paradoxes of Interaction?. J Philos Logic 44, 287–308 (2015). https://doi.org/10.1007/s10992-014-9319-5
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DOI: https://doi.org/10.1007/s10992-014-9319-5