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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)?

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I don't think there is an hidden modal logic in ordinary logic. Modal logic requires the introduction of a new operator (and so it is based on, but distinct from ordinary logic), just as probabilities require the introduction of a measure associated with propositions through new axioms. At most modal logic can be trivialized into ordinary logic by assuming $ \Box p \eq p $. Similarly, probabilities can be trivialized into ordinary logic by assuming that they can take only two values (0 or 1).

Probability is a modal concept but I don't know if it can be represented with modal logic, because you'd need a measure on possible worlds, whereas modal logic only has an accessibility relation (something is possible or not, not "more or less possible").

Finally the modal aspect of probabilities depends on the interpretation. For example frequentist interpretations don't need possible worlds and modal concepts: probabilities are just a ratio between a number of events of a type and a class of reference.

You can think of probability calculus and interior algebra as a mathematical machinery. The modal interpretation comes afterwards.

Note that modal logic can be used in proof theory (which is related to logic), but I don't think that's what you were after.

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