Abstract

Universal statements in formal logic often contain unstated assumptions that lead to logical inconsistencies when subjected to transformation, particularly contraposition. This paper introduces a formal requirement for Domain Restriction {D_r}, ensuring logical validity in universal statements by explicitly declaring all necessary category constraints. Four major insights structure this analysis: 1. Incomplete Statements and Hidden Assumptions – Universal statements implicitly assume category constraints that must be explicitly stated to prevent logical collapse. 2. Subjects and Properties – A subject can have a property, but a property without an explicitly defined subject leads to logical ambiguity. 3. Invalid Relationships – Many-to-Many relationships are inherently invalid in universal logic, requiring restructuring into Many-to-One or One-to-Many relationships. 4. The Rigor of Relevant Domain {D_r} – A necessary filtering mechanism to restrict logical transformations to valid categories and prevent logical vacuums. We formalize the Relevant Domain {D_r} as a necessary precondition for universal statements, resolving paradoxes like Hempel’s Raven Paradox while ensuring that logical transformation preserves category integrity and meaningful comparison.