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Sequent Calculi for some subintuitionistic Logics

Fatemeh Shirmohammadzadeh Maleki

Abstract

This Paper investigate sequent calculi for certain weak subintuitionistic logics. We establish that weakening and contraction are height-preserving admissible for each of these calculi, and we provide a syntactic proof for the admissibility of the cut rule for most of these sequent calculi. We also demonstrate the equivalence of these calculi to their corresponding axiomatic systems, thereby confirming their soundness and completeness with respect to neighbourhood semantics.

Sequent Calculi for some subintuitionistic Logics

Abstract

This Paper investigate sequent calculi for certain weak subintuitionistic logics. We establish that weakening and contraction are height-preserving admissible for each of these calculi, and we provide a syntactic proof for the admissibility of the cut rule for most of these sequent calculi. We also demonstrate the equivalence of these calculi to their corresponding axiomatic systems, thereby confirming their soundness and completeness with respect to neighbourhood semantics.

Paper Structure

This paper contains 20 sections, 56 theorems, 128 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Subintuitionistic logics
  3. Syntax
  4. Semantics
  5. The system GWF
  6. Admissibility of structural rules
  7. Equivalence with the axiomatic system and syntactic completeness
  8. Sequent calculi for some subintuitionistic logics between WF and F
  9. The system ${\sf GWF_{N } }$
  10. The system ${\sf GWF_{N_{2} } }$
  11. The system ${\sf GWF \widehat{\sf D}}$
  12. The system ${\sf GWF \widehat{\sf C}}$
  13. Sequent calculus for some other subintuitionistic logics
  14. The system ${\sf GWFI }$
  15. The system ${\sf GWFC }$
  16. ...and 5 more sections

Key Result

Theorem 2.3

(Weak Deduction theorem, FD Theorem 2.19)

Figures (4)

  • Figure 1: The Hilbert-style system for ${\sf WF}\xspace$
  • Figure 2: Lattice of some subintuitionistic logics
  • Figure 3: Lattice of some subintuitionistic logics
  • Figure 4: Lattice of some subintuitionistic logics

Theorems & Definitions (107)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • Definition 3.1
  • ...and 97 more