Two-dimensional Kripke Semantics II: Stability and Completeness
<|reference_start|>Two-dimensional Kripke Semantics II: Stability and Completeness: We revisit the duality between Kripke and algebraic semantics of intuitionistic and intuitionistic modal logic. We find that there is a certain mismatch between the two semantics, which means that not all algebraic models can be embedded into a Kripke model. This leads to an alternative proposal for a relational semantics, the stable semantics. Instead of an arbitrary partial order, the stable semantics requires a distributive lattice of worlds. We constructively show that the stable semantics is exactly as complete as the algebraic semantics. Categorifying these results leads to a 2-duality between two-dimensional stable semantics and categories of product-preserving presheaves, i.e. models of algebraic theories in the style of Lawvere.<|reference_end|>
@article{kavvos2024two-dimensional,
    title={Two-dimensional Kripke Semantics II: Stability and Completeness},
    author={G. A. Kavvos},
    journal={arXiv preprint arXiv:2406.03578},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.03578},
    primaryClass={cs.LO math.CT math.LO}
}