Throughout the twentieth century, the classical logic of Frege and Russell dominated the field of formal logic. But, as Ross Brady argues here, persistent logical paradoxes may be better solved by a new type of weak relevant logic. Brady's universal logic, intended to analyze naive set/class theories, is modeled on the properties of set-theoretic containment and driven by an inference connective understood as "meaning containment".

Universal Logic begins with an overview of classical and strong relevant logic and discusses the limitations of both in analyzing many historically significant paradoxes, including Russell's Paradox and the Heterologicality Paradox. By formulating a weak quantified relevant logic, this is the first text to demonstrate how such set-theoretic and semantic paradoxes can be solved in a systematic way that is conceptualized independently of the paradoxes themselves. An important part of this system is the two-sorted treatment of classicality, where the Law of Excluded Middle and Disjunctive Syllogism apply to classical sentences.