S4's accessibility relation encodes iterated epistemic or metaphysical possibility, a hyperintensional structure that extensional sort distinctions cannot fully capture without circularity.

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Key Terms

Epistemic: "Epistemic" relates to knowledge—how we know things, what counts as knowledge, and whether we can trust what we believe to be true. It comes from the Greek word for knowledge and is used to describe questions about the reliability and validity of our beliefs and understanding. For example, "epistemic humility" means acknowledging the limits of what you can actually know for certain.
Modal logic(logic): A system of logic that deals with concepts like possibility, necessity, and what could or must be true.
S4(as used in the statement): A specific system of modal logic with particular rules about how possibility and necessity work; it's named S4 just like a product model number.
accessibility relation(Modal logic; used to relativize the truth of necessity operators to contextually relevant worlds): A binary relation R on the set of possible worlds W such that R holds between world w and world w' iff w' is possible given the facts of w

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circularity(as used in logic and epistemology)

A logical problem where an explanation depends on the very thing it's trying to explain, going in a circle rather than providing genuine support.

extensional(logic and philosophy of probability)

A property of relations (like material implication) that Nagel claimed Reichenbach's probability implication lacks

hyperintensional(Applied to explanation and the connective 'because')

A context is hyperintensional when expressions within it cannot be substituted with necessary equivalents salva veritate — i.e., substituting one necessarily equivalent expression for another can change the truth value of the containing sentence.

metaphysical possibility(Distinguished from mathematical possibility to argue that some mathematically consistent results are ruled out by the nature of concrete reality)

What is possible in the concrete world, which is a more restrictive domain than mathematical possibility

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S4's accessibility relation encodes iterated epistemic or metaphysical possibility, a hyperintensional structure that extensional sort distinctions cannot fully capture without circularity.

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