A basic system of paraconsistent Nelsonian logic of conditionals.

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.