From Mathematical Principles of Logic to a Calculus of Uncertainty

Classical formal logic provides a rule-governed foundation for truth, derivability, admissibility, andsymbolic transformation. However, binary truth values are often insufficient for computationalsystems that must represent partial support, unresolved state, decay, convergence, and changingevidence. This paper develops a five-state calculus of uncertainty grounded in the formal lineageof Hilbert and Ackermann, Church, and Turing. The proposed calculus represents logical states astagged potential values over a normalized implementation space p ∈ [0, 1] and an equivalent signedtheoretical space s(t) ∈ [−1, 1]. The five named regions are false, almost false, undetermined, almosttrue, and true.The central claim of the calculus is that uncertainty need not be treated as failure, noise, or vagueincompleteness. Instead, uncertainty can be represented as computable state. The undeterminedstate is not interpreted as half true, but as the center of a signed state space. Certainty is defined asdistance from this center, while uncertainty is maximal at the center and minimal at the poles. Logicaloperators are defined over potential values and lifted back into named states through threshold-basedclassification.Beyond static classification, the calculus treats propositions as state trajectories. A proposition maymove toward truth, collapse toward falsity, decay toward uncertainty, oscillate under competingevidence, or converge toward a settled pole. The framework therefore introduces dynamic quantitiessuch as velocity, acceleration, distance traveled, directional accumulation, certainty, uncertainty,convergence, and admissibility. These quantities allow two propositions with the same instantaneouslabel to be distinguished by their histories and motion.A minimal LambdaScript reference implementation is provided to demonstrate that the calculus canbe expressed as executable symbolic definitions rather than as informal terminology. The result is acompact computational substrate for uncertainty-aware state validation, intended as a stepping stonetoward Beyond Binary systems and neurocontextual lateral propagation-diffusion architectures.