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Expanding Logics of Perfect Paradefinite Algebras with an Implication Connective


Core Concepts
The authors investigate how to conservatively expand the logics of perfect paradefinite algebras with an implication connective, while ensuring the resulting logic is self-extensional.
Abstract
The paper builds on previous work on perfect paradefinite algebras and their associated logics of formal inconsistency and undeterminedness. The authors explore different approaches to adding an implication connective to these logics: They first consider logics with a classical-like implication, but these fail to be self-extensional. They then focus on expanding the perfect paradefinite algebra with a Heyting-style implication, based on the relative pseudo-complement. This leads to self-extensional SET-SET and SET-FMLA logics, which are shown to be closely related to Moisil's symmetric modal logic. The authors provide detailed axiomatizations for these new implicative logics, study their algebraic semantics, and investigate properties like interpolation and amalgamation. The main contribution is the systematic investigation of implicative expansions of the logics of perfect paradefinite algebras, while ensuring the resulting logics are self-extensional.
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Deeper Inquiries

What other approaches could be explored to add an implication connective to the logics of perfect paradefinite algebras while preserving self-extensionality

One approach to adding an implication connective to the logics of perfect paradefinite algebras while preserving self-extensionality could involve exploring different types of implication operators. For example, instead of using a Heyting implication as studied in the paper, one could investigate the possibility of incorporating a relevance implication or a fuzzy implication. These alternative implication connectives have different properties and may interact with the underlying algebra in unique ways, potentially leading to interesting results in terms of self-extensionality and logical behavior. Another approach could be to consider hybrid logics that combine different types of implication connectives within the same logic. By introducing multiple implication operators with distinct semantics, it may be possible to create a more nuanced and expressive logical system that captures a wider range of reasoning patterns and relationships between propositions. This approach could provide a richer framework for reasoning in the context of perfect paradefinite algebras.

How do the implicative expansions studied in this paper relate to other paraconsistent and paracomplete logics with implications, such as Nelson's logic or the logics of formal inconsistency

The implicative expansions studied in this paper, particularly in the context of perfect paradefinite algebras, can be related to other paraconsistent and paracomplete logics with implications, such as Nelson's logic or the logics of formal inconsistency. These logics share the common goal of dealing with inconsistencies and undeterminedness in a formal and systematic way. By adding an implication connective, the logics of perfect paradefinite algebras extend their expressive power and provide a framework for reasoning about implication in the context of formal inconsistency and undeterminedness. Nelson's logic, known for its paraconsistent nature, deals with contradictions in a way that allows for reasoning in the presence of inconsistent information. The implicative expansions in the logics of perfect paradefinite algebras can offer insights into how implication interacts with inconsistency and how logical systems can handle contradictory information while maintaining coherence. The logics of formal inconsistency, on the other hand, focus on reasoning in the presence of formal contradictions. By adding an implication connective to these logics, one can explore how implications are interpreted and utilized in contexts where the truth values of propositions may be uncertain or contradictory. This extension can provide a deeper understanding of the interplay between implication, inconsistency, and undeterminedness in logical systems.

The authors mention connections to Moisil's symmetric modal logic. Are there further connections to modal logics that could be explored in this context

The connections to Moisil's symmetric modal logic mentioned in the paper suggest a link between the logics of perfect paradefinite algebras and modal logics that exhibit symmetry properties. Further exploration could involve investigating how modal operators, such as necessity and possibility, can be integrated into the logics of perfect paradefinite algebras to create modal extensions of these logics. By introducing modalities, one can capture notions of necessity, possibility, and temporal logic within the framework of perfect paradefinite algebras, leading to a more comprehensive and versatile logical system. Additionally, exploring connections to other modal logics, such as S5 modal logic or intuitionistic modal logics, could provide insights into the modal properties of the logics of perfect paradefinite algebras. By studying the correspondence between modal operators and the algebraic structures of perfect paradefinite algebras, one can uncover new relationships between modal logic and the logics of formal inconsistency and undeterminedness. This exploration may lead to the development of novel modal logics tailored to the specific characteristics of perfect paradefinite algebras.
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