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Syntactical Treatments of Interacting Modalities

Subject Area Theoretical Philosophy
Term from 2011 to 2018
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 196767730
 
Final Report Year 2019

Final Report Abstract

In standard modal logic, modalities such as ‘necessary’ or ‘possible’ are expressed as sentential operators. This project investigated the logic and semantics of modalities when they are expressed as predicates of sentences or propositions, and in particular the logic and semantics of languages with multiple modalities, such that different modal predicates can interact with each other. Within the project, several salient formal systems for the treatment of modal predicates have been developed. On the one hand, the semantic fixed-point constructions of Kripke for type-free truth predicates, generalized by Halbach and Welch for modal predicates, has been refined. On the other hand, several axiomatic systems as well as sequent systems have been developed, both in classical versions as exemplified by Stern’s systems MKF and MFS, but also in the form of partial systems, such as MPKF and the MG3-systems in Fischer’s habilitation thesis. The development of the systems followed a careful preparation in the first phase of the project in which criteria of adequacy have been outlined. Especially the role of the predicates within the semantic ascent has been investigated. One of the findings is that the expressive function of those predicates can be measured by means of so-called speed-up results. For a better understanding of the relationship between the axiomatic and semantic approaches, a systematic investigation of the criteria of adequacy mentioned before has been carried out. Amongst others, the notion of N-categoricity has been introduced: a fruitful and theoretically successful notion that has already had some impact in the ‘formal theories of truth’ community. Additionally, semi-formal proof systems have been developed, which are useful tools in bridging the gap between the semantic approaches and the formal sequent systems. These semi-formal systems have also been developed as frameworks for possible further proof-theoretic analyses for several sequent systems for modal predicates. One of the main obstacles in the development of those systems are wellknown paradoxes, such as the knower paradox. A reasonable step towards a better understanding of the paradoxes has been provided through a systematic investigation of the relation of multimodal paradoxes versus paradoxes of single modalities. Stern and Fischer (2015) introduce a fruitful notion of reducibility in order to distinguish genuine paradoxes of interaction from reducible ones. The project’s results have already been applied in various logico-philosophical studies. For example, Stern (2018) uses the findings obtained during the project to shed light on Gödel’s famous Disjunction concerning absolutely undecidable mathematical problems, while Nicolai (2018) has extended Stern’s result on modal theories (Stern, 2014, 2016) to the supervaluational setting. Campbell-Moore (2015) uses Stern’s strategy to obtain plausible axiomatizations for truth and self-referential probability, building on previous work by Leitgeb (2012).

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