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The logic behind desirable sets of things, and its filter representation

Gert de Cooman, Arthur Van Camp, Jasper De Bock

TL;DR

This paper develops a unifying filter-theoretic view of coherent sets of desirable sets of things (SDS), connecting imprecise-probability inference to propositional logic over a lattice of events. It establishes order-isomorphisms between families of (finitely) coherent SDSes (and SDFSes) and (principal or proper) filters on bounded distributive lattices of events, enabling conjunctive representations as limits inferior or intersections of simpler conjunctive models ${K_{D}}$ derived from coherent SDTs $D$. By working through two key examples—propositional logic (Lindenbaum algebra) and coherent choice with gambles (Levi’s E-admissibility)—the paper shows how the abstract framework subsumes standard logics and decision rules, linking to Campell–Moore-type credal representations when Archimedean/mixing conditions hold. The results provide powerful, interpretable representations that reduce complex conservative inferences to tractable combinations of basic, conjunctive components, while also clarifying the essential role of finitary closures and the potential pitfalls in infinitary settings.

Abstract

We identify the (filter representation of the) logic behind the recent theory of coherent sets of desirable (sets of) things, which generalise coherent sets of desirable (sets of) gambles as well as coherent choice functions, and show that this identification allows us to establish various representation results for such coherent models in terms of simpler ones.

The logic behind desirable sets of things, and its filter representation

TL;DR

This paper develops a unifying filter-theoretic view of coherent sets of desirable sets of things (SDS), connecting imprecise-probability inference to propositional logic over a lattice of events. It establishes order-isomorphisms between families of (finitely) coherent SDSes (and SDFSes) and (principal or proper) filters on bounded distributive lattices of events, enabling conjunctive representations as limits inferior or intersections of simpler conjunctive models derived from coherent SDTs . By working through two key examples—propositional logic (Lindenbaum algebra) and coherent choice with gambles (Levi’s E-admissibility)—the paper shows how the abstract framework subsumes standard logics and decision rules, linking to Campell–Moore-type credal representations when Archimedean/mixing conditions hold. The results provide powerful, interpretable representations that reduce complex conservative inferences to tractable combinations of basic, conjunctive components, while also clarifying the essential role of finitary closures and the potential pitfalls in infinitary settings.

Abstract

We identify the (filter representation of the) logic behind the recent theory of coherent sets of desirable (sets of) things, which generalise coherent sets of desirable (sets of) gambles as well as coherent choice functions, and show that this identification allows us to establish various representation results for such coherent models in terms of simpler ones.
Paper Structure (28 sections, 45 theorems, 142 equations, 2 tables)

This paper contains 28 sections, 45 theorems, 142 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Sets of desirable (sets of) things: an overview
  3. Desirable things
  4. Desirable sets of things
  5. Desirable sets of things: the finitary case
  6. Conjunctive models
  7. Sets of desirable finite sets of things
  8. Towards a representation with filters
  9. The basic representation lattices
  10. A brief primer on (inference with) filters
  11. Filter representation for finitely coherent SDSes
  12. Filter representation for coherent SDSes
  13. Finitary SDSes
  14. Filter representation for finitely coherent SDFSes
  15. Example: propositional logic
  16. ...and 13 more sections

Key Result

Proposition 1

The partially ordered set $\langle\mathbf{D},\subseteq\rangle$ is a complete lattice. For any non-empty family $S_{i}$, $i\in I$ of elements of $\mathbf{D}$, we have for its infimum and its supremum that, respectively, $\inf_{i\in I}S_{i}=\bigcap_{i\in I}S_{i}$ and $\sup_{i\in I}S_{i}={\mathop{\math

Theorems & Definitions (89)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Definition 1: Conjunctivity
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7: Conjunctivity
  • Theorem 8
  • Proposition 9
  • ...and 79 more