The many-sorted translation preserves compactness only when modal operators are interpreted over first-order definable accessibility relations, but S4 permits second-order frame conditions.

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Accessibility relations(in modal logic and epistemology): A formal way of mapping out which situations or possibilities an agent (like a person) can distinguish or imagine from their current position—basically, what they can 'see' or consider as real options.
Many-sorted translation(in formal logic and model theory): A method of converting or translating logical statements into a different form that distinguishes between different types or categories of objects.
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.
Second-order frame conditions(in formal logic): Rules that describe the structure of logical systems in a more complex way, talking about properties and relationships between properties themselves, not just individual objects.
compactness(Explained here as a consequence of derivations using only finitely many premises): The model-theoretic property whereby a set of sentences is satisfiable if and only if every finite subset of it is satisfiable
first-order definable(Used to explain why vague quantifiers resist complete axiomatization in FOL.): A property of a quantifier or predicate such that its semantics can be fully expressed within the syntax and inference rules of first-order logic.
modal operators: unary connectives

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