Constraint Logic: A Conservative Non-Classical Logic System Based on the Uniqueness of Constraints

This paper proposes Constraint Logic (CL), a conservative non-classical logic system with “constraint” as its sole primitive concept. Grounded in a bounded distributive lattice (constraint lattice), CL a priori delineates the boundary of cycle-free formulas via a purely syntactic, recursively defined constraint dependency set, and characterizes the semantic legitimacy of a proposition by its unique minimal constraint set. Only valid propositions with a determinate constraint set are assigned classical binary truth values; cyclic formulas without a determinate constraint set are excluded from reasoning entirely. This framework resolves all classical self-referential paradoxes at their root, without modifying classical connectives, introducing truth gaps, or accepting true contradictions. We fully present the formal syntax, lattice-valued semantics, axiomatic system, and inference rules of CL, rigorously prove the existence and uniqueness of the unique minimal constraint set for cycle-free formulas, the soundness and weak completeness of the system, and a general resolution theorem for self-referential sentences. CL faithfully and conservatively embeds classical propositional logic as its empty-constraint fragment, fully preserving all core laws and reasoning capabilities of classical logic. This manuscript is a preprint version and has been submitted for peer review.