Rough concepts

Abstract

The present paper proposes a novel way to unify Rough Set Theory and Formal Concept Analysis. Our method stems from results and insights developed in the algebraic theory of modal logic, and is based on the idea that Pawlak's original approximation spaces can be seen as special instances of enriched formal contexts, i.e. relational structures based on formal contexts from Formal Concept Analysis.

Non-distributive logics: from semantics to meaning

“…Moreover, this is a Sahlqvist formula and corresponds on Kripke frames (W, R) to reflexivity or, equivalently, ∆ ⊆ R. Since L is not sentential, the closest approximation to the classical formula ϕ → ϕ is the L -sequent ϕ ⊢ ϕ, which turns out (cf. [15,Proposition 4.3]) to correspond on polarity-based structures F as above to the first-order condition R ⊆ I. That is, for every object a and feature x, if aR x (i.e.…”

“…Again, the L -sequent ϕ ⊢ ϕ turns out (cf. [15,Proposition 4.3]) to correspond on polarity-based structures F to the first-order condition 6 that reads: for every object a and feature x, if agent i thinks that a has feature x, then (agent i must recognize a as an example of what i understands as an x-object, i.e. as a member of i's understanding of the formal context generated by feature x, and hence) agent i must attribute to a also all the features that, according to i, are shared by all x-objects.…”

“…On polarity-based structures F, the L -sequent ϕ ⊢ ϕ (cf. [15,Proposition 4.3]) corresponds to the first-order condition I ⊆ R , indicating that, whenever an object has a feature, the agent knows this.…”

Slanted canonicity of analytic inductive inequalities

“…The procedure described below serves both to compute the first order correspondent of the given inequality in various semantic settings, as discussed e.g. in [13,11,8,14], and to compute the shape of the analytic structural rules corresponding to the given inequality, as discussed in [26,6].…”

Non Normal Logics: Semantic Analysis and Proof Theory

“…Propositions 2,10,14,17); (c) based on these semantic relationships, introduce multi-type normal logics into which the original non-normal logics can be embedded via suitable translations (cf. Section 4) following a methodology which was successful in several other cases [18,19,20,21,29,9,28,30,33,48]; (d) retrieve wellknown dual characterization results for axiomatic extensions of monotone modal logic and conditional logics as instances of general algorithmic correspondence theory for normal (multi-type) LE-logics applied to the translated axioms (cf. Section B); (e) extract analytic structural rules from the computations of the first-order correspondents of the translated axioms, so that, again by general results on proper display calculi [31] (which, as discussed in [2], can be applied also to multi-type logical frameworks) the resulting calculi are sound, complete, conservative and enjoy cut elimination and subformula property.…”