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Regular non-normal modal classicalities

Alfredo Roque Freire, Manuel António Martins

TL;DR

A novel investigation into the consistency operator ($\circ$), traditionally associated with paraconsistent logics, is presented as a means of capturing non-normal modal classicalities within the Kripke framework and its applicability beyond paraconsistency into classical and modal logics is extended.

Abstract

We present a novel investigation into the consistency operator ($\circ$), traditionally associated with paraconsistent logics, as a means of capturing non-normal modal classicalities within the Kripke framework. By semantically reinterpreting $\circ$ as an operator that distinguishes top and bottom values from other values in the algebra, we extend its applicability beyond paraconsistency into classical and modal logics. We introduce the logic $\mathcal{B}_4^\circ$, a four-valued Boolean logic augmented with the consistency operator, and provide a sound and complete axiomatization. Building on this foundation, we extend the semantics to the modal domain using the many-logics modal logic (MLML) framework. Specifically, we construct Kripke frames based on an eight-valued Boolean algebra that contains three distinct four-valued subalgebras, each representing a different world type. Our analysis reveals that the resulting modal logic exhibits a normal local consequence relation alongside a non-normal global consequence relation. Consequently, the characterization of frame properties --such as transitivity, reflexivity, and Euclideanness --deviates from modal logic $K$, requiring novel semantic tools. We further identify new modal formulas in an extended language that capture previously unavailable kinds of accessibility, leading to frame characterizations unattainable in traditional modal frameworks.

Regular non-normal modal classicalities

TL;DR

A novel investigation into the consistency operator (), traditionally associated with paraconsistent logics, is presented as a means of capturing non-normal modal classicalities within the Kripke framework and its applicability beyond paraconsistency into classical and modal logics is extended.

Abstract

We present a novel investigation into the consistency operator (), traditionally associated with paraconsistent logics, as a means of capturing non-normal modal classicalities within the Kripke framework. By semantically reinterpreting as an operator that distinguishes top and bottom values from other values in the algebra, we extend its applicability beyond paraconsistency into classical and modal logics. We introduce the logic , a four-valued Boolean logic augmented with the consistency operator, and provide a sound and complete axiomatization. Building on this foundation, we extend the semantics to the modal domain using the many-logics modal logic (MLML) framework. Specifically, we construct Kripke frames based on an eight-valued Boolean algebra that contains three distinct four-valued subalgebras, each representing a different world type. Our analysis reveals that the resulting modal logic exhibits a normal local consequence relation alongside a non-normal global consequence relation. Consequently, the characterization of frame properties --such as transitivity, reflexivity, and Euclideanness --deviates from modal logic , requiring novel semantic tools. We further identify new modal formulas in an extended language that capture previously unavailable kinds of accessibility, leading to frame characterizations unattainable in traditional modal frameworks.
Paper Structure (11 sections, 22 theorems, 17 equations, 2 figures)

This paper contains 11 sections, 22 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A semantical abstraction of the consistency operator
  3. Extending the use of consistency operator to classical logic
  4. Non normal global modal consequence relation
  5. Outline of the paper
  6. The 4-ball-valued Boolean logic
  7. The logic of 4-ball-Boolean Algebra
  8. Many-logics modal logic
  9. four-eight logics
  10. Characterization of frames in the (8-4) modal logic
  11. Conclusions and further work

Key Result

Theorem 2.1

For every set of formulas $\Gamma \cup \{\varphi\}\subseteq \mathrm{Form}(\mathcal{L}^\circ)$,

Figures (2)

  • Figure 1: The Boolean algebra of 8 values and its sub algebras with 4 values.
  • Figure 2: Example of out of bubble frame

Theorems & Definitions (62)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1: Sound and complete
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.1
  • proof
  • Definition 2.5
  • Lemma 2.2
  • proof
  • ...and 52 more