What is Stochastic Supervenience?

What is Stochastic Supervenience? Youheng Zhang
https://orcid.org/0000-0002-2363-487X School of Marxism, University of Electronic Science and Technology of China, Chengdu, Sichuan, China zhangyouheng@whut.edu.cn Abstract: Standard formulations of supervenience typically construe the determinandum as a point- value strictly fixed by base-level states. Yet, in scientific domains ranging from statistical mechanics to deep learning, basal structures frequently determine law-governed families of probability measures rather than single outcomes. This paper develops a general framework for stochastic supervenience, representing this dependence via Markov kernels that map base states to higher-level distributions. I formulate axioms securing law-like fixation, non-degeneracy, and directional asymmetry, demonstrating that classical deterministic supervenience is recovered as the Dirac boundary case of this broader topological space. To render these metaphysical commitments empirically tractable, the paper integrates information-theoretic diagnostics—such as normalized mutual information, divergence spectra, and tail sensitivity. These indices serve to distinguish genuine structural stochasticity from epistemic noise, stratify degrees of distributional multiple realization, and diagnose intervention-salient macro- organization. The resulting framework offers a conservative extension of physicalist dependence, reconciling the priority of the base level with the structured uncertainty ubiquitous in the special sciences. Keywords: stochastic supervenience; Markov kernels; multiple realizability; probabilistic dependence 1 Introduction Classical discussions of supervenience standardly proceed within a familiar tripartite schema. Following Kim (1990), supervenience involves covariance, dependence, and non-reducibility. Weak supervenience states that, within a single possible world, identity of the base properties guarantees identity of the higher-level properties. Strong and global supervenience reinforce modal stability and determinacy, respectively, by securing cross-world property matching and by requiring an isomorphism in the overall distribution of properties across worlds (Kim 1984; 1987). Mainstream discussions remain within this schematic framework, concentrating on its interpretation and application (e.g., Chen 2011; Hoffmann and Newen 2007; Kovacs 2019; Leuenberger 2009; Moyer 2008; Napoletano 2015; Shagrir 2002; 2013). Classical theories further share a background presupposition: what is determined—the determinandum— 2

is a value (or set of values) of higher-level properties. When we confront cases in which outputs are systematically indeterministic yet plainly tightly constrained by underlying structure and law, this framework proves rigid. Faced with phenomena that are probabilistic in a structured way but not mere noise, one is pushed either (i) to retain determinism (and thereby lose a stratified layer of uncertainty) or (ii) to embrace strong forms of irreducibility or emergence. Yet in many central scientific domains the underlying structures and laws do not appear to fix a single point value outcome, but rather to fix a stable family of probability distributions. Examples include:  the Born probabilities of quantum measurement outcomes (see Born 1955; Wigner 1963; Brukner 2017);  power laws and species abundance distributions in ecological and social complex systems (Hubbell 2001, 32-45; Newman 2005; Bettencourt et al. 2007);  the direct modeling of conditional and generative distributions in statistical learning and deep generative architectures (Vapnik 1999; Gal and Ghahramani 2016; Kingma et al. 2014). These cases suggest that if we continue to insist on pointwise functional determination, we cannot distinguish between (1) an ontic, law-governed probabilistic structure and (2) residual epistemic ignorance. Work in the philosophy of physics has developed sophisticated accounts of how such probability distributions arise from underlying dynamics. Myrvold (2021) offers a hybrid theory of probabilities, treating Gibbsian and Boltzmannian probability measures as central theoretical posits that mediate between micro-dynamics and macroscopic regularities. Wallace and Frigg (2021), as well as Wallace (2019), analyze Gibbsian statistical mechanics in detail, showing how ensemble measures and invariant distributions can underwrite experimental practice and connect microscopic and thermodynamic descriptions. These approaches elucidate the origin and empirical role of statistical-mechanical probabilities, but they do not explicitly formulate the dependence between base-level states and higher-level probabilistic structures in supervenience-style terms. The framework of stochastic supervenience proposed here is meant to complement such accounts by articulating that dependence as a law-like mapping from base states to families of higher-level pro