Hyperboolean Algebras and Hyperboolean Modal Logic
Valentin Goranko, Dimiter Vakarelov · Journal of Applied Non-Classical Logics · 1999
Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a hyperboolean modal logic) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show that it lacks the finite model property. The method of axiomatization hinges upon the fact that a “difference” operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research.