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Qualitative reasoning in a two-layered framework

Marta Bilkova, Sabine Frittella, Daniil Kozhemiachenko, Ondrej Majer

TL;DR

This paper provides two-layered logics built over the classical propositional logic using a unary belief modality $\Be$ that connects the inner layer to the outer one where the reasoning is formalised by means of G\"{o}del logic.

Abstract

The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events. The paper is divided into two parts. In the first part, we formalise reasoning with the qualitative counterparts of capacities, belief functions, and probabilities, within the framework of two-layered logics. Namely, we provide two-layered logics built over the classical propositional logic using a unary belief modality $\Be$ that connects the inner layer to the outer one where the reasoning is formalised by means of Gödel logic. We design their Hilbert-style axiomatisations and prove their completeness. In the second part, we discuss the paraconsistent generalisations of the logics for qualitative uncertainty that take into account the case of the available information being contradictory or inconclusive.

Qualitative reasoning in a two-layered framework

TL;DR

This paper provides two-layered logics built over the classical propositional logic using a unary belief modality that connects the inner layer to the outer one where the reasoning is formalised by means of G\"{o}del logic.

Abstract

The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events. The paper is divided into two parts. In the first part, we formalise reasoning with the qualitative counterparts of capacities, belief functions, and probabilities, within the framework of two-layered logics. Namely, we provide two-layered logics built over the classical propositional logic using a unary belief modality that connects the inner layer to the outer one where the reasoning is formalised by means of Gödel logic. We design their Hilbert-style axiomatisations and prove their completeness. In the second part, we discuss the paraconsistent generalisations of the logics for qualitative uncertainty that take into account the case of the available information being contradictory or inconclusive.
Paper Structure (18 sections, 20 theorems, 70 equations, 2 figures)

This paper contains 18 sections, 20 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weaker uncertainty measures
  3. Two-layered logics for uncertainty
  4. Two-dimensional treatment of inconsistent information
  5. Plan of the paper
  6. Preliminaries: bi-Gödel logic
  7. Qualitative reasoning about uncertainty
  8. Qualitative uncertainty in a two-layered framework
  9. Correspondence theory for weak uncertainty measures
  10. Logics of qualitative probabilities
  11. Paraconsistent logics of monotone comparative belief
  12. Belnap--Dunn logic
  13. Paraconsistent expansions of $\mathsf{biG}$
  14. Language and semantics of $\mathsf{MCB}$ and $\mathsf{NMCB}$
  15. Axiomatisation
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $W\neq\varnothing$ and let further $\preccurlyeq$ be a linear preorder on $2^W$. Consider the following conditions on $\preccurlyeq$ for all $X,Y,Z\subseteq W$. Then, it holds that

Figures (2)

  • Figure 1: De Morgan lattice $\mathbf{4}$
  • Figure 2: $[0,1]^{\Join}$ --- continuous extension of $\mathbf{4}$. $(x,y)\!\leq_{[0,1]^{\Join}}\!(x',y')$ iff $x\!\leq\!x'$ and $y\!\geq\!y'$

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2: Language and semantics of $\mathsf{biG}$
  • Remark 2.1
  • Definition 2.3: $\mathcal{H}\mathsf{biG}$
  • Remark 2.2
  • Theorem 3.1: Qualitative characterisations of uncertainty measures
  • Definition 3.2: Language and semantics of $\mathsf{QG}$
  • Remark 3.1
  • Example 3.1: Comparing certainty in $\mathsf{QG}$
  • Example 3.2
  • ...and 61 more