Logics for Rough Concept Analysis

Greco, Giuseppe; Jipsen, Peter; Manoorkar, Krishna; Palmigiano, Alessandra; Tzimoulis, Apostolos

doi:10.1007/978-3-662-58771-3_14

Cited by 20 publications

(28 citation statements)

References 18 publications

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“…Of course, the interpretation and use of graph-based structures proposed in the present paper does not exclude the possibility of other interpretations and uses, as is suggested by the fact that the 'companion' polarity-based semantics for lattice-based modal logic has been used to provide different interpretations of the lattice-based modal logic, including one in which lattice-based modal logic is viewed as an epistemic logic of categories [6,7] and one [5,19] in which the same logic is viewed as the logic of rough concepts, where polarity-based semantics is used as an encompassing framework for the integration of rough set theory [22] and formal concept analysis [17], and as a basis for further developments such as a Dempster-Shafer theory of concepts [16].…”

Section: Introductionmentioning

confidence: 99%

Modelling Informational Entropy

Conradie

Craig

Palmigiano

et al. 2019

Logic, Language, Information, and Computation

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By 'informational entropy', we understand an inherent boundary to knowability, due e.g. to perceptual, theoretical, evidential or linguistic limits. In this paper, we discuss a logical framework in which this boundary is incorporated into the semantic and deductive machinery, and outline how this framework can be used to model various situations in which informational entropy arises.Keywords: Lattice-based modal logic · Epistemic logic · Concept lattice · Graphbased semantics · Polarity-based semantics ⋆ The research of the third author is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054, and a Delft Technology Fellowship awarded in 2013. PreliminariesNotation. We let ∆ U denote the identity relation on a set U, and we will drop the subscript when it causes no ambiguity. The superscript (·) c denotes the relative complement of the subset of a given set. Hence, for any binary relation R ⊆ S × T , we let R c ⊆ S × T be defined by (s, t) ∈ R c iff (s, t) R. For any such R and any S ′ ⊆ S and T ′ ⊆ T , we let R[S ′ ] := {t ∈ T | (s, t) ∈ R for some s ∈ S ′ } and R −1 [T ′ ] := {s ∈ S | (s, t) ∈ R for some t ∈ T ′ }, and write R[s] and R −1 [t] for R[{s}] and R −1 [{t}], respectively. Any such R gives rise to the semantic modal operators R , [R] : P(T ) → P(S ) s.t. R W := R −1 [W] and [R]W := (R −1 [W c ]) c for any W ⊆ T . For any T ⊆ U × V, and any U ′ ⊆ U and V ′ ⊆ V, letKnown properties of this construction (cf. [14,) are collected below.. 4. T (1) [U ′ ] = T (1) [T (0) [T (1) [U ′ ]]] and T (0) [V ′ ] = T (0) [T (1) [T (0) [V ′ ]]]. 5. T (0) [ V] = V ′ ∈V T (0) [V ′ ] and T (1) [ U] = U ′ ∈U T (1) [U ′ ].

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Section: Introductionmentioning

confidence: 99%

Modelling Informational Entropy

Conradie

Craig

Palmigiano

et al. 2019

Logic, Language, Information, and Computation

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“…Consequently, their corresponding modal axiomatic principles p ⊢ p and p ⊢ p do not commonly occur in the literature either. However, as stated in Proposition 5.8 and .9, in the present setting these principles characterize a natural and meaningful class of relations (which we refer to as 'sub-delta', by extension), which, as noticed in [38], has already cropped up in the literature on rough formal concept analysis [46], and is potentially useful for applications (see Subsections 5.4 and 5.5 below). Definition 9.…”

Section: By Definitionmentioning

confidence: 87%

Rough concepts

Conradie,

Frittella,

Manoorkar

et al. 2019

Preprint

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“…However, all the results mentioned above straightforwardly apply also to properly displayable multi-type calculi, which have been recently introduced to extend the scope and benefits of proper display calculi also to a wide range of logics that for various reasons do not fall into the scope of the analytic inductive definition. These logics crop up in various areas of the literature and include well known logics such as linear logic [30], dynamic epistemic logic [20], semi De Morgan logic [26,27], bilattice logic [28], inquisitive logic [21], non normal modal logics [3], the logics of classes of rough algebras [25,24]. Interestingly, the multi-type framework can be…”

Section: Discussionmentioning

confidence: 99%

Syntactic completeness of proper display calculi

Chen¹,

Greco²,

Palmigiano³

et al. 2021

Preprint

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A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e. those calculi that support general and modular proof-strategies for cut elimination), and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analiticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e. those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (aka unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so that the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination and subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e. by showing that a (cut free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far this proof strategy for syntactic completeness has been implemented on a case-by-case base, and not in general. In this paper, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut free derivation can be effectively generated which has a specific shape, referred to as pre-normal form.

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Logics for Rough Concept Analysis

Cited by 20 publications

References 18 publications

Modelling Informational Entropy

Modelling Informational Entropy

Rough concepts

Syntactic completeness of proper display calculi

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