David Lewis and Timothy Williamson argue that necessitism—the thesis that necessarily everything necessarily exists—provides independent metaphysical grounds for BF's truth across all modal systems.

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BF(Quantified modal logic): The Barcan Formula, a controversial principle provable in SQML but invalid in KQML
David Lewis(the philosopher who created this theory): An influential American philosopher (1941-2001) who developed Counterpart Theory as a way to understand how we talk about objects in different possible worlds.
Metaphysical grounds(The deeper philosophical foundation that would justify an idea): The fundamental reasons or explanations rooted in the basic nature of reality itself (rather than just in how things appear or work practically).
Modal systems(the systems across which the thesis applies): Different frameworks in logic for reasoning about what is possible, impossible, necessary, and contingent (things that could be different).
Necessarily (in logic/philosophy)(describing what must be the case logically): Something that must be true in all possible situations or circumstances, rather than just happening to be true in our world.
Timothy Williamson(The other philosopher co-authoring the argument in this statement): A prominent British philosopher known for work on knowledge, logic, and philosophy of language.
necessitism(Philosophy of modality; a logical truth of SQML): The view that everything that exists exists necessarily — both possibilia and actually existing things alike are necessary beings, such that there are no worlds from which they are altogether absent.

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