The Modal–Dependence Calculus (MDC): A Minimal Modal Framework for Invariance and Dependence

This paper develops the Modal–Dependence Calculus (MDC), a minimal extension of quantified S5 that formalizes ontological dependence using only modal resources and a world-relative existence predicate. Dependence is defined as asymmetric modal entailment, allowing MDC to capture directionality, priority, and explanatory structure without primitive grounding operators. Nine axioms (A0–A8) impose structural constraints on dependence—well-foundedness, invariance-directedness, and modal asymmetry—from which three central results follow: (i) every contingent entity stands in a dependence chain terminating in something invariant, (ii) such chains terminate in a necessary element, and (iii) any invariant necessary terminus is unique within the adopted framework. MDC thereby derives a form of structural monism as a theorem rather than an assumption. The framework is explicitly conditional and diagnostic: it does not assert that its axioms must hold, but delineates the structural conditions any adequate account of ontological dependence must either accept or reject.