QBISM LOGIC

This paper aims to provide a rigorous logical foundation for QBism, an interpretation of quantum theory that treats quantum states as personal probability assignments. The author seeks to formalize the "QBist picture"—where an agent updates their expectations based on prospective actions and experienced outcomes—into a mathematically tractable logical system. The approach involves constructing a "well-formed core datum" consisting of an admissible prior space, a family of measurements, and specific kernels for Born rule probabilities and state updates. By introducing a guarded dynamic language, the author models how agents revise their commitments after experiencing measurement outcomes.

The paper establishes a global reduction theorem, proving that any formula containing dynamic operators (representing updates) can be reduced to a non-dynamic equivalent. For systems that are effectively semialgebraic, the author demonstrates that this fragment can be translated into first-order formulas over real closed fields. This translation effectively reduces the problem of logical validity to first-order reasoning, yielding a sound, complete, and decidable recursive calculus. Furthermore, the paper shows that standard finite-dimensional quantum theory realizes this framework when a symmetric informationally complete (SIC) reference measurement is present. In this realization, the SIC image is shown to satisfy the consistency bounds and lower polar conditions characteristic of qplex geometry.

By formalizing QBism as a logical system, this work bridges the gap between the philosophical interpretation of quantum theory and formal methods in computer science and logic. It provides a precise, automated way to reason about the evolution of an agent's beliefs in a quantum context. The ability to decide the validity of statements about histories and posterior states offers a new tool for analyzing the consistency of QBist models and provides a clear mathematical bridge between abstract probabilistic frameworks and the standard Hilbert space formalism of quantum mechanics.