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Relational semantics for flat Heyting-Lewis Logic

Jim de Groot, Tadeusz Litak

Abstract

We introduce relational semantics for "flat Heyting-Lewis logic" $\mathsf{HLC}^{\flat}$. This logic arises as the extension of intuitionistic logic with a Lewis-style strict implication modality that, contrary to its "sharp" counterpart $\mathsf{HLC}^{\sharp}$, does not turn meets into joins in its first argument. We prove completeness and the finite model property for $\mathsf{HLC}^{\flat}$ and for several extensions with additional axioms.

Relational semantics for flat Heyting-Lewis Logic

Abstract

We introduce relational semantics for "flat Heyting-Lewis logic" . This logic arises as the extension of intuitionistic logic with a Lewis-style strict implication modality that, contrary to its "sharp" counterpart , does not turn meets into joins in its first argument. We prove completeness and the finite model property for and for several extensions with additional axioms.

Paper Structure

This paper contains 17 sections, 24 theorems, 16 equations, 1 table.

Table of Contents

  1. Introduction
  2. Intuitionistic strict implication, sharply and flatly
  3. Syntax and axioms of the base flat system
  4. The sharpening
  5. Algebraic semantics
  6. Relational semantics for flat Heyting-Lewis logic
  7. Flat frames
  8. Upward-flat frames
  9. Relation to sharp semantics
  10. Canonical models and completeness
  11. Completeness and the fmp for axiomatic extensions
  12. Reusing the full canonical model
  13. Modifying the full canonical model
  14. Open subframes and extension stability
  15. Conclusions and future work
  16. ...and 2 more sections

Key Result

Proposition 2.2

For $\mathrm{Ax} \subseteq \mathcal{L}_{\sto}$ and uniform substition $\sigma$, the following rules are admissible in $\mathsf{HLC^{\flat}} \oplus \mathrm{Ax}$:

Theorems & Definitions (63)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • ...and 53 more