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

摘要

Log_AB 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 Log_AB are presented. Log_AB 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 Log_AB structures. Log_AB 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.