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Labelled calculi for the logics of rough concepts

Ineke van der Berg, Andrea De Domenico, Giuseppe Greco, Krishna Manoorkar, Alessandra Palmigiano, Mattia Panettiere

Abstract

We introduce sound and complete labelled sequent calculi for the basic normal non-distributive modal logic L and some of its axiomatic extensions, where the labels are atomic formulas of the first order language of enriched formal contexts, i.e., relational structures based on formal contexts which provide complete semantics for these logics. We also extend these calculi to provide a proof system for the logic of rough formal contexts.

Labelled calculi for the logics of rough concepts

Abstract

We introduce sound and complete labelled sequent calculi for the basic normal non-distributive modal logic L and some of its axiomatic extensions, where the labels are atomic formulas of the first order language of enriched formal contexts, i.e., relational structures based on formal contexts which provide complete semantics for these logics. We also extend these calculi to provide a proof system for the logic of rough formal contexts.
Paper Structure (16 sections, 7 theorems, 8 equations, 1 table)

This paper contains 16 sections, 7 theorems, 8 equations, 1 table.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Preliminaries
  4. Basic normal non-distributive modal logic and some of its axiomatic extensions
  5. Relational semantics of $\mathcal{L}$-logics
  6. The logic of rough formal contexts
  7. Relational labelled calculi for $\mathcal{L}$-logics
  8. Labelled calculus $\mathbf{R.L}$ for the basic $\mathcal{L}$-logic
  9. Relational calculi for the axiomatic extensions of the basic $\mathcal{L}$-logic
  10. The relational calculus $\mathbf{R.L}\rho$ for the $\mathcal{L}$-logic of rough formal contexts
  11. Properties of $\mathbf{R.L}\rho$ and $\mathbf{R.L}\Sigma$
  12. Soundness
  13. Syntactic completeness of the basic calculus and its axiomatic extensions
  14. Conclusions
  15. Soundness of the basic calculus
  16. ...and 1 more sections

Key Result

lemma thmcounterlemma

For any sets $U, V$, $U'$ and $V'$, and for any families of sets $\mathcal{V}$ and $\mathcal{U}$,

Theorems & Definitions (13)

  • lemma thmcounterlemma
  • proposition thmcounterproposition
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • ...and 3 more