The Fluid Axiom

Modal logic reasons about necessity and possibility through Kripke structures: discrete graphs where nodes are possible worlds and edges are accessibility relations. World A can access world B; from A, the proposition holding at B is possible. If it holds at all accessible worlds, it is necessary. The framework is powerful but combinatorial — the number of worlds is finite and fixed, and each accessibility relation is a binary edge.

Sulc replaces the graph with a flow. In Fluid Logic, each modal operator is backed by a neural stochastic differential equation. Applying a possibility operator to a proposition means integrating the SDE forward: starting from the current state, the stochastic trajectory explores a continuous manifold of accessible states, and the proposition is evaluated along the path. Necessity requires the proposition to hold across the distribution of trajectories. Nested modal formulas — “it is necessarily possible that X” — compose SDEs into a single differentiable graph, each layer of modality adding another stochastic integration.

The key structural move is replacing accessibility relations with transition densities. In Kripke semantics, world A either accesses world B or it does not — binary. In Fluid Logic, state A reaches state B with a probability determined by the SDE’s drift and diffusion coefficients. The accessibility relation becomes graded, continuous, and learnable. The neural network parameterizing the SDE learns which states are reachable from which, and how easily, from data.

This is not merely a computational trick. The SDE formulation gives modal operators a new interpretation: they are entropic risk measures. Necessity corresponds to worst-case evaluation over the reachable distribution. Possibility corresponds to best-case. The distinction between the two operators is the distinction between pessimistic and optimistic evaluation of stochastic outcomes — a connection that is invisible in discrete Kripke structures but falls out naturally from the continuous formulation.

The deeper point: logic and dynamics are the same structure viewed at different resolutions. Kripke semantics discretizes dynamics into a graph. Fluid Logic recovers the dynamics. The axioms of modal logic — reflexivity, transitivity, symmetry of the accessibility relation — become properties of the SDE: whether the drift has fixed points, whether trajectories are reversible, whether the stationary distribution is symmetric. The axioms were always dynamical constraints. The graph was always an approximation.