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A basic system of paraconsistent Nelsonian logic of conditionals.

Grigory K Olkhovikov1

  • 1Department of Philosophy I, Ruhr University Bochum, Universitätsstraße 150, Bochum, 44780 Germany.

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|December 10, 2024
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Summary

This study introduces a new conditional logic, denoted as PCL, based on paraconsistent logic. PCL exhibits significant connections with intuitionistic and modal logic systems, expanding the landscape of non-classical logics.

Keywords:
Conditional logicConstructive logicModal logicParaconsistent logicStrong completenessStrong negation

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Area of Science:

  • Logic
  • Formal Semantics
  • Mathematical Foundations

Background:

  • Conditional logics are crucial for reasoning under uncertainty and counterfactuals.
  • Paraconsistent logics handle contradictions without leading to triviality, offering alternatives to classical logic.
  • Existing intuitionistic and modal logics provide frameworks for non-classical reasoning.

Purpose of the Study:

  • To define a Kripke semantics for a conditional logic derived from Nelson's logic of strong negation.
  • To axiomatize the minimal system generated by this Kripke semantics, named PCL.
  • To explore and map the relationships between PCL, intuitionistic conditional logic (IC), and a specific modal logic (M).

Main Methods:

  • Development of a Kripke semantics tailored for a conditional logic based on a paraconsistent propositional logic.
  • Axiomatization of the minimal conditional logic system (PCL) induced by the defined semantics.
  • Comparative analysis of PCL by examining embeddings into related intuitionistic and modal logic systems.

Main Results:

  • A novel conditional logic, PCL, is formally defined and axiomatized.
  • PCL demonstrates strong connections with the intuitionistic conditional logic (IC) and the modal logic (M).
  • The study elucidates these connections through the analysis of logical embeddings between the systems.

Conclusions:

  • The defined Kripke semantics provides a sound and complete foundation for the paraconsistent conditional logic PCL.
  • PCL serves as a bridge, linking paraconsistent, intuitionistic, and modal logical frameworks.
  • This research expands the understanding of non-classical logics and their interrelations.