Theory of Semantics - Formal Axiomatic Framework

Classical information theory and contemporary predictive processing models provide rigorous accounts of signal transmission and statistical optimization, yet they fundamentally fail to formalize semantic meaning. We propose a comprehensive mathematical framework in which meaning is not a property of symbols, nor is it reducible to the minimization of prediction error. Instead, meaning emerges as the vanishing of residual interpretive tension during the topological closure between a quaternion-valued signal and an observer’s configured state. We model the observer as a coarse-grained topological invariant embedded in the division ring of quaternions H, allowing us to capture the non-commutative geometry of interpretation. Crucially, we extend this formalism beyond passive reception. We introduce formal mechanisms for endogenous symbol generation and source attribution, providing rigorous mathematical definitions for thinking, dreaming, hallucinating, and expressing. The dynamics are governed by a strictly event-driven, sequential composition of three nested causal operators: intrinsic biological constraint, volitional agency (modeled as a non-commutative torque), and signal-dependent physical equilibrium. We prove a Gödelian incompleteness theorem using a universal probe, demonstrating that no finite observer can achieve topological completeness without volitional reconfiguration.