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LogAB: A first-order, non-paradoxical, algebraic logic of belief

LogAB is a family of logics of belief. It holds a middle ground between the expressive, but prone to paradox, syntactical first-order theories and the often inconvenient, but safe, modal approaches. In this report, the syntax and semantics of LogAB are presented. LogAB is algebraic in the sense that it is a language of only terms; there is no notion of a formula, only proposition-denoting terms. The domain of propositions is taken to be a Boolean algebra, which renders classical truth conditions and definitions of consequence and validity theorems about LogAB structures. LogAB is shown to be sufficiently expressive to accommodate complex patterns of reasoning about belief while remaining paradox-free. A number of results are proved regarding paradoxical self-reference. They are shown to strengthen previous results, and to point to possible new approaches to circumventing paradoxes in syntactical theories of belief.