Classical and Fuzzy Two-Layered Modal Logics for Uncertainty: Translations and Proof-Theory

Authors

Paolo Baldi^{1, *,} , Petr Cintula^2, , Carles Noguera^3,

¹Department of Philosophy, University of Milan, Milan, Italy

²Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic

³Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Prague, Czech Republic

^*Corresponding author. Email: paolo.baldi@unimi.it

DOI

10.2991/ijcis.d.200703.001How to use a DOI?

Keywords

Mathematical fuzzy logic; Logics of uncertainty; Łukasiewicz logic; Probability logics; Two-layered modal logics; Hypersequent calculi

Abstract

This paper is a contribution to the study of two distinct kinds of logics for modelling uncertainty. Both approaches use logics with a two-layered modal syntax, but while one employs classical logic on both levels and infinitely-many multimodal operators, the other involves a suitable system of fuzzy logic in the upper layer and only one monadic modality. We take two prominent examples of the former approach, the probability logics Prlin and Prpol (whose modal operators correspond to all possible linear/polynomial inequalities with integer coefficients), and three logics of the latter approach: PrŁ, PrŁ△ and PrPŁ△ (given by the Łukasiewicz logic and its expansions by the Baaz–Monteiro projection connective △ and also by the product conjunction). We describe the relation between the two approaches by giving faithful translations of Prlin and Prpol into, respectively, PrŁ△ and PrPŁ△, and vice versa. We also contribute to the proof theory of two-layered modal logics of uncertainty by introducing a hypersequent calculus HPrŁ for the logic PrŁ. Using this formalism, we obtain a translation of Prlin into the logic PrŁ, seen as a logic on hypersequents of relations, and give an alternative proof of the axiomatization of Prlin.

Copyright

© 2020 The Authors. Published by Atlantis Press B.V.

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Paolo Baldi
AU  - Petr Cintula
AU  - Carles Noguera
PY  - 2020
DA  - 2020/07/17
TI  - Classical and Fuzzy Two-Layered Modal Logics for Uncertainty: Translations and Proof-Theory
JO  - International Journal of Computational Intelligence Systems
SP  - 988
EP  - 1001
VL  - 13
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200703.001
DO  - 10.2991/ijcis.d.200703.001
ID  - Baldi2020
ER  -