Journal of Philosophical Logic 44 (3):287-308 (2015)

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 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
Keywords Modal paradox  Multimodal paradoxes  Paradoxes of interaction  Syntactical treatments of modalities  Modal logic  Liar paradox  Theories of truth  Fixed-points  Fixed-point logic
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DOI 10.1007/s10992-014-9319-5
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References found in this work BETA

Modal Logic.Patrick Blackburn, Maarten de Rijke & Yde Venema - 2002 - Cambridge University Press.
The Logic of Provability.George S. Boolos - 1993 - Cambridge University Press.
A Paradox Regained.D. Kaplan & R. Montague - 1960 - Notre Dame Journal of Formal Logic 1 (3):79-90.
Self-Reference and Modal Logic.George Boolos & C. Smorynski - 1988 - Journal of Symbolic Logic 53 (1):306.

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