Determinacy Is Not Emergence: Clarifying Dynamical Indeterminism, Causal Emergence, and Probabilistic Futures

Abstract

Three ideas are often discussed in neighboring vocabularies but answer different questions: whether a physical theory is dynamically deterministic, whether a system exhibits causal emergence across scales, and whether an observer should represent the future probabilistically. This note separates these issues and records the consequences of doing so. Dynamical determinism is defined relative to a specified theory and a complete state: the successor law is deterministic when the conditional successor measure is a Dirac measure. Causal emergence, by contrast, concerns whether a macro-scale description can possess stronger or more informative causal relations than a chosen micro-scale description under a specified causal-emergence formalism. Probabilistic future representation concerns an observer's or model's distribution over possible trajectories and can arise even when the underlying dynamics are deterministic, through incomplete information, coarse-graining, chaos, measurement limits, or computational limits. Minimal finite-state examples illustrate two points already compatible with the existing literature: causal emergence can occur in a deterministic system, and stochastic successor laws need not generate higher-scale causal organization. These examples are used as clarifications rather than as novel theorems. The main conclusion is therefore negative but useful: dynamical indeterminism is not an explanation of emergence, and a probabilistic model of the future is not by itself evidence that physical reality has an ontically open future. The quantum-mechanical implications remain interpretation-dependent and are treated only as an application of the distinction.