Galois-Theoretic Invariants of Paraconsistent Determinization

This paper develops a Galois-theoretic invariant for “determinization” operations that collapse paraconsistent semantics to classical semantics inside an indexed topos framework. A first result is an obstruction: no (elementary) topos can have LP as its native internal logic, since the Heyting negation on the subobject classifier Ω enforces non-contradiction (p ∧ ¬ₕp = ⊥). To obtain a concrete and satisfiable paraconsistent layer without altering the topos’s Heyting structure, we introduce the bilateral twist construction: in any topos, the object L = Ω × Ω carries canonical paraconsistent operations (tracking positive and negative evidence independently) and admits a designated glut value (⊤, ⊤). This yields a workable notion of “paraconsistent topos” as a topos equipped with a bilateral truth-value object. Given an accessibility morphism α from a paraconsistent root world to a classical world, we define the determinization monad Det_α: = α* ∘ Σ_α. Under a fully faithful hypothesis on α*, Det_α is idempotent and its fixed-point subcategory Fix (Det_α) is reflective and equivalent to the classical topos Eᵤ. When Eᵤ is connected, locally connected, and admits a point, the finite locally constant objects in Fix (Det_α) form a Grothendieck Galois category Cov_α, yielding a profinite fundamental group π₁ᵈet (α) ≅ π₁ (Eᵤ, p) (SGA1). We give a fully concrete realization: for any finite group G, the presheaf topos Eₖ䃐 ≅ Fun (ℕ, SetG) serves as a paraconsistent root, with α* the constant-diagram embedding and Σ_α the colimit functor. In this model, Fix (Det_α) ≃ SetG and π₁ᵈet (α) ≅ G. The embedded subcategory of finite locally constant objects determines the determinization monad up to natural isomorphism, while the profinite group π₁ᵈet (α) classifies the abstract equivalence type of the associated Galois category. (Primary areas: topos theory, categorical logic, paraconsistent semantics, Grothendieck Galois theory. )