The Sup Connective in IMALL: A Categorical Semantics
Intuitionistic Multiplicative-Additive Linear Logic (IMALL) lacks a probabilistic elimination mechanism for additive connectives and struggles to uniformly handle summation and scalar multiplication. Method: We propose an extended proof language introducing a new connective “sup”, supporting probabilistic elimination for additive pairs, and intrinsically incorporating summation and scalar multiplication. We establish, for the first time, an abstract categorical semantics for sup-enriched IMALL based on symmetric monoidal closed categories, constructing biproduct structures and embedding a scalar ring into Hom(I,I), while defining a weighted co-diagonal map to uniformly characterize probabilistic elimination. Contribution: We prove that categories equipped with biproducts plus a ring embedding into Hom(I,I) suffice to model this logic—yielding the first foundational framework for quantum and probabilistic linear type systems that simultaneously ensures expressive power and semantic rigor.