In recent years, Pawlak’s classical framework of rough set theory has been generalized in various ways. Particularly relevant to the present volume is the fact that some of these generalisations have been building on algebraic and logical insights and techniques, and involve modal algebras based on general (i.e. not necessarily distributive) lattices on the algebraic side, and the following two types of relational structures on the side corresponding to approximation spaces:

- structures based on formal contexts from Rudolf Wille’s Formal Concept Analysis;

- structures based on reflexive (directed) graphs.

It is precisely this second type of structures that have been recently proposed as a basic framework for modelling relative entropy in information theory. This proposal pivots on the following core ideas:

- in many concrete situations, the indiscernibility relations of approximation spaces can be assumed to be reflexive, but neither symmetric nor transitive;

- in Pawlak’s original framework, the upper and lower approximations arising from any given indiscernibility relation reflect an external perspective on (in)discernibility, which is captured algebraically by the notion of Boolean algebra with modal operators and its canonically associated classical modal logic S5.

Specifically, formulas in this logic denote arbitrary subsets of a given approximation spaces, and modal operators serve to identify those special formulas that denote definable sets (and hence, definable approximations of arbitrary sets).

Contrasting the well-known, external perspective just outlined, recently, an internal perspective on indiscernibility has also been provided, which is particularly useful in all situations in which the indiscernibility in question is induced by relative entropy, i.e. an inherent boundary to knowability (due e.g. to perceptual, theoretical, evidential or linguistic limits).

In these cases, rather than generating modal operators, the indiscernibility relation can also generate a complete general (i.e. not necessarily distributive) lattice, the associated logic of which is much weaker than classical propositional (modal) logic. Formulas in this logic denote the definable sets by default, and hence they represent ‘all there is to know’, i.e. the theoretical horizon to knowability, given the information entropy encoded into the indiscernibility relation.

Weaker logics than classical propositional logic have higher standards of truth, and the logic mentioned above is no exception: this weaker propositional logic behaves as an evidential logic in which formulas can be true, or false, or neither, depending on whether there is evidence in support, or against, or neither. Hence, as the principle of excluded middle fails at a meta-theoretic level, we will refer to this (evidence-based) notion of truth as hyper-constructive, rather than classical. The seminal papers in which these ideas are discussed are [1,2]. Some applications of these same ideas in a many-valued context are discussed in the papers [3,4].

Much work needs to be done to fully develop this research program, even if we only consider the logic side of it.

Thematically appropriate contributions to this research program include (among many others):

- Gödel translation on graph-based frames with a Block-Esakia-style theorem;

- Parametric correspondence involving graph-based frames;

- Dempster Shafer theory on graph-based frames;

- A Goldblatt Thomason characterization theorem for graph-based frames;

- Algebraic proof theory on graphs-based frames;

- Lawvere/allegory-based semantics of non-distributive first order logic.

However, other facets of this program go beyond logic and include a broad range of disciplines spanning from formal modeling and probabilistic and statistical reasoning to formal epistemology.

The ambitious aim of this volume is to collect cutting-edge results and insights in the formal modelling of entropy in rough set theory, so that these results and insights can be systematically connected to each other.

Reference

1. Conradie, M.; Craig, A.; Palmigiano, A.; Wijnberg, N.M. Modelling informational entropy. arXiv 2019, arXiv:1903.12518.
2. Conradie, W.; Palmigiano, A.; Robinson, C.; Wijnberg, N. Non-distributive logics: from semantics to meaning. arXiv 2020, arXiv:2002.04257.
3. Conradie, W.; Craig, A.; Palmigiano, A.; Wijnberg, N.M. Modelling competing theories. arXiv 2019, arXiv:1905.11748.
4. Conradie, W.; Palmigiano, A.; Robinson, C.; Tzimoulis, A.; Wijnberg, N.M. Modelling socio-political competition. arXiv 2019, arXiv:1908.04817. 

Prof. Dr. Alessandra Palmigiano
Dr. Willem Conradie
Dr. Yiyu Yao
Guest Editors

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• entropy
• rough set theory
• formal concept analysis
• evidential logics
• non-distributive logics
• formalizations of statistical reasoning
• knowability and uncertainty