Four negations and the spectral presheaf

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 corresponding logic. Their further extension with the Skolem algebra structure defines Akchurin algebras and the respective logic. Next we generalise the framework of spectral presheaves (which is a main object in the Butterfield--Isham--Döring 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, which is a sound model of a product of biquasiintuitionistic and biintuitionistic logics, featuring four distinct negations. 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.

• The paper shows that clopen subpresheaves of the spectral presheaf over any complete orthocomplemented lattice form an Akchurin algebra with four distinct negations and a complete algebraic model of biQInt ⊗ biInt.
• The paper uses outer and inner daseinisation, Stone spectra, adjunctions, and Lindenbaum constructions to establish soundness and completeness while extending key spectral-presheaf results beyond orthomodular lattices.
• The paper reconstructs the original lattice from fixed points of the presheaf’s negation-induced monad and comonad, and proves that its paraconsistent structure cannot model standard relevance logics without collapsing to Boolean logic.

The paper develops a lattice-logical framework in which a single algebraic structure carries four distinct negation operators, and shows that the spectral presheaf — central to the Butterfield–Isham–Döring topos-theoretic formulation of quantum mechanics over an arbitrary complete orthocomplemented lattice $L$ — is naturally such a structure. The work also proves a no-go theorem against interpreting the spectral presheaf as a model of relevance logic.

Background: negations, Vakarelov algebras, and the spectral presheaf

The paper adopts Vakarelov's algebraic approach to logics with negation, cataloguing eighteen equational conditions on a map $\lnot:K\to K$ over a bounded lattice. Combinations of these yield subminimal (Vakarelov), quasiintuitionistic, coquasiintuitionistic, intuitionistic, cointuitionistic, paraconsistent, and paracomplete negations, with involutions and orthocomplementations as special cases. A quasiintuitionistic algebra is a bounded distributive Vakarelov algebra whose negation satisfies antitone contraposition, double negation introduction ($x \le \lnot\lnot x$), and paraconsistency-type condition $x\land\lnot x \le y$; its $(\wedge,\vee)$-dual is a coquasiintuitionistic algebra. Combining both negations gives biquasiintuitionistic algebras; further requiring the underlying lattice to be a Skolem (Heyting–Brouwer) algebra yields the Akchurin algebra, named after I.A. Akchurin's 1974 proposal to analyse quantum state spaces via Grothendieck topoi. The paper notes that Heyting/Brouwer/Skolem algebras force distributivity, and that adding either of two De Morgan-type axioms to a De Morgan algebra collapses it to a boolean lattice.

On the geometry side, the construction replaces the category $\mathcal{V}(\mathcal{N})$ of commutative W*-subalgebras of a von Neumann algebra by the poset $\mathtt{C}$ of complete boolean sublattices of $L$, defining $\Sigma^L$ via Stone spectra rather than Gel'fand spectra. This generalises Cannon–Döring from orthomodular to arbitrary complete orthocomplemented lattices; notably, several key results of Cannon–Döring are shown not to depend on orthomodularity. The set $\mathcal{O}(\Sigma^L)$ of clopen subpresheaves forms a complete distributive lattice, with meets/joins computed via interior/closure in Stone topology. Outer daseinisation $\lnot:K\to K$0 admits a right adjoint, and inner daseinisation $\lnot:K\to K$1 (new at this generality) admits a left adjoint.

Four negations and soundness

Two negations arise on $\lnot:K\to K$2 by transporting orthocomplementation through each adjunction: $\lnot:K\to K$3, which is paraconsistent (and fully so for the original Hilbert-space case per Döring–Eva–Özawa), and the dual $\lnot:K\to K$4, which is paracomplete and previously unstudied. Combined with the Heyting and Brouwer negations inherent in the Skolem structure, this yields four distinct negations. Main structural results:

Structure	Logic modelled	Status
$\lnot:K\to K$5	coQInt	complete coquasiintuitionistic algebra
$\lnot:K\to K$6	QInt	complete quasiintuitionistic algebra
both negations	biQInt	complete biquasiintuitionistic algebra
full Skolem + both	biQInt ⊗ biInt	complete Akchurin algebra

Soundness/completeness is proved via Lindenbaum-algebra constructions for sequent calculi extending Goldblatt/Vakarelov-style lattice logics. Thus $\lnot:K\to K$7 models the product logic $\lnot:K\to K$8 — a strong claim, since the four negations coexist without collapsing into boolean structure precisely because neither paraconsistent nor paracomplete negation is an involution satisfying all De Morgan laws.

Internal lattices and reconstruction of $\lnot:K\to K$9

A quasiintuitionistic negation acts as a closure operator (monad); a coquasiintuitionistic one as an interior operator (comonad). Their fixed-point subsets carry induced orthocomplemented lattice structures, generalising McKinsey–Tarski's theorem that the closed elements of a closure algebra form a boolean lattice. This provides a generalised Kolmogorov–Glivenko theorem relating QInt/coQInt to internal orthologic, at the cost of restricting to formulas lacking meets (resp. joins).

For the spectral presheaf, the two internal orthocomplemented lattices are mutually isomorphic and each isomorphic to the underlying lattice $x \le \lnot\lnot x$0. Consequently, $x \le \lnot\lnot x$1 is uniquely reconstructible from the internal logic of its own spectral presheaf, as the image of a double-negation comonad/monad — a categorical analogue of Stone duality at the level of the internal logic.

No-go result for relevance logics

The paper refutes Eva's unproven claim that $x \le \lnot\lnot x$2 with the paraconsistent negation models the dialectical relevance logic DKQ. The argument: any relevance logic in the standard hierarchy (B through R, plus E, FDE, quantified extensions) has as its extensional reduct a De Morgan algebra; but the paraconsistent (or paracomplete) negation on $x \le \lnot\lnot x$3 is an involution iff it makes the whole structure a boolean lattice, which contradicts paraconsistency. This decisively separates the paraconsistent behaviour here (driven by non-involutive orthocomplementation transported through daseinisation) from the De Morgan-style paraconsistency of relevance logics.

Limitations and open questions

The completeness results concern algebraic semantics only; relational/frame semantics are asserted but deferred to companion papers still in preparation. The generality beyond orthomodular lattices relies on dropping orthomodularity being harmless for the specific propositions cited, which the authors verify case-by-case rather than systematically. Whether the Akchurin structure supports a proof-theoretic or operational reading in physics — e.g., whether the four negations correspond to distinct physical operations — remains open, as does the extension to more general base categories announced for subsequent work.

Conclusion

The paper establishes that the clopen subobjects of a spectral presheaf form a canonical instance of a new algebraic structure carrying four logically independent negations, proves the corresponding sequent calculi sound and complete, reconstructs the base lattice from internal logic, and closes off the relevance-logic interpretation. It thereby sharpens the logical identity of the topos-theoretic quantum framework while leaving open the physical interpretation of its multi-negational structure (2603.08699).