Reconfiguring Logical Space: A Semantics for Possibility, Proof, and Paradox

This paper introduces a novel semantic framework for "reconfiguring logical space," aiming to provide a unified and robust treatment of possibility, proof, and paradox. Traditional logical space, often conceptualized as a monolithic structure of possible worlds, has proven highly effective for modal logic but encounters significant challenges when accommodating the intricate dynamics of formal proof and the emergence of logical and semantic paradoxes. We propose a stratified or multi-dimensional logical space, wherein different layers or perspectives allow for a more nuanced interpretation of accessibility relations, consequence operations, and truth conditions. This reconfigured space enables the containment and analysis of paradoxes without resorting to overly restrictive classical logical principles, while simultaneously providing a rich environment for modeling the constructive aspects of proof theory. The methodology involves formally defining such a stratified structure and demonstrating how it can resolve classic paradoxes like the Liar, and offer a coherent account of modal operators and provability within a single overarching semantic system. The results reveal that this re-conceptualization offers a more flexible and expressive foundation for logic, paving the way for a deeper understanding of the interplay between these fundamental logical concepts.