Probabilistic Modal Logic for Quantum Dynamics

Traditional quantum mechanics provides predictive accuracy but lacks a clear framework for articulating the epistemic status of quantum systems, particularly during measurement. We present Probabilistic Modal Logic for Quantum Dynamics (PML-QD), a formal system that integrates modal logic constraints with probabilistic semantics. Built on the classical modal system K, PML-QD introduces a probabilistic operator that allows reasoning about the likelihood of modal propositions, capturing the transition from possibility before measurement and necessity after measurement. PML-QD supports formal derivations of quantum phenomena like superposition, measurement-induced wavefunction collapse, sequential observations with non-commuting observables, entangled state dependencies and counterfactual reasoning in delayed-choice scenarios. Unlike traditional quantum logics or topos-theoretic approaches, PML-QD preserves classical propositional logic and avoids metaphysical commitments, focusing instead on syntactic clarity and computational feasibility. Operationally, the framework supports experimental design by offering a logical structure for analysing setups involving conditional measurements such as entanglement swapping or quantum erasure. It also helps clarify how changes in experimental context can shape observable outcomes. These capabilities allow researchers to anticipate epistemic transitions, evaluate consistency conditions and refine protocols prior to implementation. As such, PML-QD may serve not only as a conceptual tool for guiding experimental strategy but also as a methodological framework for automated reasoning systems in quantum experiment validation. Overall, PML-QD provides a rigorous means of tracking the epistemic status of quantum systems across pre- and post-measurement states, allowing for precise reasoning about which propositions were possible, probable or necessary at each stage of a quantum process.