Is there an intrinsic, hidden modal logic within ordinary (abstract) Boolean algebras?
I do not question whether interior or closure algebras are Boolean algebras. Nor do I seek for an explanation of what modal logic is or does. My question is rather motivated by the observation that probability (clearly a modal concept) is defined on a non-modal algebra (an ordinary Boolean algebra which forms the algebraic semantics of the classical, two-valued, propositional calculus).
Perhaps it is simply because interior or closure algebras are not required. After all, mathematicians need not worry about the philosophical nature of probability. Or, perhaps, it is because using (e.g.) a closure algebra would bring more inconveniences than advantages when it comes to the development of the mathematical theory of probability? Or, perhaps, it is because there is already a (hidden) modality embedded into ordinary Boolean algebras (maybe yet to be discovered)?