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Filling in the semantics for intuitionistic conditional logic

Brendan Dufty, Jim de Groot

Abstract

We prove completeness results for a wide variety of intuitionistic conditional logics. We do so by first using a canonical model construction obtain completeness with respect to descriptive conditional frames, and then introducing the fill-in method to transfer this to classes of conditional frames without extra structure. The fill-in method closes the gap between descriptive conditional frames, which do not have a canonical underlying frame, and conditional frames.

Filling in the semantics for intuitionistic conditional logic

Abstract

We prove completeness results for a wide variety of intuitionistic conditional logics. We do so by first using a canonical model construction obtain completeness with respect to descriptive conditional frames, and then introducing the fill-in method to transfer this to classes of conditional frames without extra structure. The fill-in method closes the gap between descriptive conditional frames, which do not have a canonical underlying frame, and conditional frames.

Paper Structure

This paper contains 17 sections, 24 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Conditionals and conditional logic
  3. Conditionals in an intuitionistic setting
  4. Our contributions
  5. Intuitionistic normal modal logic
  6. Intuitionistic logic
  7. Adding a normal modality
  8. Intuitionistic conditional logic
  9. The axioms and rules of intuitionistic conditional logic
  10. Frames
  11. Canonical model construction
  12. The fill-in method
  13. The empty fill-in
  14. The cautious fill-in
  15. Outlook
  16. ...and 2 more sections

Key Result

Theorem 2.8

Suppose $\varphi \in \mathcal{L}_{\Box}$ is a formula whose Gödel translation $t(\varphi)$ is Sahlqvist. Then $\varphi$ is d-persistent.

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • proof : Proof sketch
  • Definition 3.1
  • ...and 61 more