Modal Logics of Uncertainty with Two-Layer Syntax: A General Completeness Theorem

0431413 - ÚI 2015 RIV DE eng C - Conference Paper (international conference)
Cintula, Petr - Noguera, Carles
Modal Logics of Uncertainty with Two-Layer Syntax: A General Completeness Theorem.
Logic, Language, Information, and Computation. Heidelberg: Springer, 2014 - (Kohlenbach, U.; Barceló, P.; de Queiroz, R.), s. 124-136. Lecture Notes in Computer Science, 8652. ISBN 978-3-662-44144-2. ISSN 0302-9743.
[WoLLIC 2014. International Conference /21./. Valparaíso (CL), 01.09.2014-04.09.2014]
R&D Projects: GA ČR GAP202/10/1826
EU Projects: European Commission(XE) 247584 - MATOMUVI
Institutional support: RVO:67985807 ; RVO:67985556
Keywords : two-level modal logic * logics of uncertainty * theory of probability * weakly implicative logics * Kripke frames
Subject RIV: BA - General Mathematics; BB - Applied Statistics, Operational Research (UTIA-B)

Modal logics with two syntactical layers (both governed by classical logic) have been proposed as logics of uncertainty following Hamblin's seminal idea of reading the modal operator P(A) as 'probably A', meaning that the probability of a formula A is bigger than a given threshold. An interesting departure from that (classical) paradigm has been introduced by Hajek with his fuzzy probability logic when, while still keeping classical logic as interpretation of the lower syntactical layer, he proposed to use Lukasiewicz logic in the upper one, so that the truth degree of P(A) could be directly identified with the probability of A. Later, other authors have used the same formalism with different kinds of uncertainty measures and other pairs of logics, allowing for a treatment of uncertainty of vague events (i.e. also changing the logic in the lower layer). The aim of this paper is to provide a general framework for two-layer modal logics that encompasses all the previously studied two-layer modal fuzzy logics, provides a general axiomatization and a semantics of measured Kripke frames, and prove a general completeness theorem.
Permanent Link: http://hdl.handle.net/11104/0235975