Four negations and the spectral presheaf

TL;DR

This paper introduces biquasiintuitionistic logic with four negations, extends spectral presheaves to orthocomplemented lattices, and explores their algebraic and topos-theoretic properties in quantum mechanics.

Contribution

It develops a new logical framework with four negations and generalizes spectral presheaves to orthocomplemented lattices, linking logic, algebra, and quantum structures.

Findings

Biquasiintuitionistic logic with four negations is formulated.

Spectral presheaves are extended to arbitrary orthocomplemented lattices.

A no-go theorem shows spectral presheaves cannot model relevance logic.

Abstract

Using Vakarelov's theory of lattice logics with negation, we introduce the (co)quasiintuitionistic logic, and prove its soundness and completeness with respect to the class of (co)quasiintuitionistic algebras. Combining these algebras together, we obtain biquasiintuitionistic algebras and the biquasiintuitionistic logic. Their further extension with the Skolem algebra structure defines Akchurin algebras and the respective logic, which is a product of biquasiintuitionistic and biintuitionistic logics, featuring four distinct negations. Next we generalise the framework of spectral presheaves (which is a main object in the Butterfield--Isham--D\"{o}ring topos theoretic approach to quantum mechanics) to arbitrary complete orthocomplemented lattices, and show that the orthocomplementation determines two negation operators on the spectral presheaf (one paraconsistent, another paracomplete), equipping the set of all closed-and-open subpresheaves of a spectral presheaf with the structure of a biquasiintuitionistic algebra. Combined with the generic Skolem (i.e. Heyting and Brouwer) algebra structure of this set, this gives a particular instance of an Akchurin algebra. We also show that the underlying orthocomplemented lattice can be reconstructed as an internal object of the spectral presheaf, resulting as the image of a double coquasiintuitionistic (resp., quasiintuitionistic) negation monad (resp., comonad). Finally, we prove a no-go theorem for the claim that the spectral presheaf is a model of a dialectical (or any other) relevance logic.