EPiC: A Four-Valued Evidential Constraint Calculus for First-Order Reasoning

This article introduces the Evidence Propagation Calculus (EPiC), an operational framework for first-order reasoning built on a simple but productive observation: familiar inference patterns such as Modus Ponens and Modus Tollens behave like the movement of evidential markers across a structured graph. Positive evidence at an antecedent propagates forward to the consequent; negative evidence at a consequent propagates backward. When both markers coexist at a node, the system is locally inconsistent but not operationally broken. To make this observation precise, EPiC grounds reasoning in a four-valued evidential domain V={N,T,F,B}, where N denotes absence of evidence, T positive evidence, F negative evidence, and B their coexistence. Each logical connective is assigned a local evidential table, and inference is treated uniformly as the progressive restriction of admissible configurations under an evidential order: inadmissible values are eliminated, minimal surviving values are selected as the next effective evidential states, and the resulting restrictions propagate across shared variables. Compound formulas are decomposed into families of local unary and binary constraints through auxiliary variables, making the propagation process explicit and structurally uniform. Within this setting, Modus Ponens, Modus Tollens, and polarity-switching negation are not postulated as primitive rules. They emerge as derived consequences of the same local table calculus. The framework distinguishes different operational routes of justification. In some cases, positive support reaches the target formula directly through successive local restrictions. In others, propagation first stabilizes the relevant components and the target occurrence is then fixed by the corresponding connective table. Consistency is not a second basic notion of justification but a distinguished property of certain justified outcomes. The article establishes local and global soundness, conservativity over the classical fragment, and a conditional adequacy result. It further develops a translation between decomposed formulas and informational graphs, with a reverse reconstruction theorem for well-formed graphs. The result is a unified operational account of first-order reasoning situated between model-theoretic and proof-theoretic approaches, in which semantics, propagation, and graphical structure are mutually supporting rather than independently layered.