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Knowability as continuity: a topological account of informational dependence

Alexandru Baltag, Johan van Benthem

TL;DR

This work studies knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting and investigates epistemic independence in topological terms and shows that it is compatible with functional (but non-continuous) dependence.

Abstract

We study knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting. We also investigate epistemic independence in topological terms and show that it is compatible with functional (but non-continuous) dependence. We then proceed to study a stronger notion of knowability based on uniformly continuous dependence. On the technical logical side, we determine the complete logics of languages that combine general functional dependence, continuous dependence, and uniformly continuous dependence.

Knowability as continuity: a topological account of informational dependence

TL;DR

This work studies knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting and investigates epistemic independence in topological terms and shows that it is compatible with functional (but non-continuous) dependence.

Abstract

We study knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting. We also investigate epistemic independence in topological terms and show that it is compatible with functional (but non-continuous) dependence. We then proceed to study a stronger notion of knowability based on uniformly continuous dependence. On the technical logical side, we determine the complete logics of languages that combine general functional dependence, continuous dependence, and uniformly continuous dependence.
Paper Structure (53 sections, 32 theorems, 174 equations, 3 tables)

This paper contains 53 sections, 32 theorems, 174 equations, 3 tables.

Table of Contents

  1. Introduction and preview
  2. Dependence in the world
  3. Dependence, knowledge and information flow
  4. Variables as wh-questions
  5. Functional dependence and propositional determination
  6. From idealized sharp values to empirical measurement
  7. Topological models for empirical inquiry and knowledge
  8. Empirical variables as topological maps
  9. Propositionalization: empirical questions as topologies
  10. Continuity as information-carrying dependence
  11. Special case: conditional knowability of a proposition
  12. Further notions: full independence versus topological independence
  13. Special topologies
  14. 'Knowing how': from plain continuity to uniform continuity
  15. Plan of this paper
  16. ...and 38 more sections

Key Result

Proposition 2.1

(Heine-Cantor Theorem) If $F:(\mathbb{D}, d_\mathbb{D})\to (\mathbb{E}, d_\mathbb{E})$ is a map from a compact metric space to another metric space, then $F$ is (globally) continuous iff it is (globally) uniformly continuous. If $F:(\mathbb{D}, d_\mathbb{D})\to (\mathbb{E}, d_\mathbb{E})$ is a map

Theorems & Definitions (73)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 63 more