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Axiomatization of Boolean Connexive Logics with syncategorematic negation and modalities

Tomasz Jarmuzek, Jacek Malinowski, Aleksander Parol, Nicolo Zamperlin

TL;DR

This work provides a formal presentation of mentioned properties and axiom schemata that allow to incorporate them into Hilbert-style calculi and presented axiomatic systems are provided with proofs of soundness, completeness and decidability.

Abstract

In the article we investigate three classes of extended Boolean Connexive Logics. Two of them are extensions of Modal and non-Modal Boolean Connexive Logics with a property of closure under an arbitrary number of negations. The remaining one is an extension of Modal Boolean Connexive Logic with a property of closure under the function of demodalization. In our work we provide a formal presentation of mentioned properties and axiom schemata that allow us to incorporate them into Hilbert-style calculi. The presented axiomatic systems are provided with proofs of soundness, completeness, and decidability. The properties of closure under negation and demodalization are motivated by the syncategorematic view on the connectives of negation and modalities, which is discussed in the paper.

Axiomatization of Boolean Connexive Logics with syncategorematic negation and modalities

TL;DR

This work provides a formal presentation of mentioned properties and axiom schemata that allow to incorporate them into Hilbert-style calculi and presented axiomatic systems are provided with proofs of soundness, completeness and decidability.

Abstract

In the article we investigate three classes of extended Boolean Connexive Logics. Two of them are extensions of Modal and non-Modal Boolean Connexive Logics with a property of closure under an arbitrary number of negations. The remaining one is an extension of Modal Boolean Connexive Logic with a property of closure under the function of demodalization. In our work we provide a formal presentation of mentioned properties and axiom schemata that allow us to incorporate them into Hilbert-style calculi. The presented axiomatic systems are provided with proofs of soundness, completeness, and decidability. The properties of closure under negation and demodalization are motivated by the syncategorematic view on the connectives of negation and modalities, which is discussed in the paper.
Paper Structure (6 sections, 27 theorems, 9 equations)

This paper contains 6 sections, 27 theorems, 9 equations.

Table of Contents

  1. An introduction to Boolean Connexive Logics
  2. Motivation: syncategorematic negation and modalities
  3. Closure under multiple negations
  4. Closure under demodalization
  5. Further work: a syncategorematic approch to negation and modalities
  6. Acknowledgements.

Key Result

Theorem 1.4

Let $\text{R} \in \mathbf{R}$, then:

Theorems & Definitions (62)

  • Definition 1.1: Model for $\mathcal{L}_{\textsf{BCL}}$
  • Definition 1.2: Truth-conditions in model for $\mathcal{L}_{\textsf{BCL}}$
  • Definition 1.3: Validity for $\mathsf{For}_{\textsf{BCL}}$
  • Theorem 1.4: Theorem 5.1 from JarmuzekMalinowski2019
  • Theorem 1.6: Theorem 6.1 from JarmuzekMalinowski2019
  • Definition 1.8: Model for $\mathcal{L}_{\textsf{MBCL}}$
  • Definition 1.9: Truth-conditions in model for $\mathcal{L}_{\textsf{MBCL}}$
  • Definition 1.10: Validity for $\mathsf{For}_{\textsf{MBCL}}$
  • Theorem 1.11: Theorem 3.1 from JarmuzekMalinowski2019a
  • Theorem 1.13: Theorem 4.1 from JarmuzekMalinowski2019a
  • ...and 52 more