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    Carmelics

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    Inverse View

    It is not the case that The inference from 'a polynomial algorithm exists for one NP-complete problem' to 'polynomial algorithms exist for all NP problems' treats existence as a transferable property, but Kripkean possible-worlds semantics demands we specify in which worlds and under which interpretations such algorithms exist.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Mathematical truths are necessary, not contingent. P=NP has a determinate truth-value independent of possible worlds; Kripkean semantics doesn't apply.
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    • 2.The NP-completeness reduction IS the inference justification: it logically entails that solving one implies solving all. No modal ambiguity needed.
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    • 3.Invoking possible-worlds semantics for computational complexity conflates metaphysical modality with epistemic uncertainty. They are distinct problems.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.P=NP is contingent, not necessary: in some possible worlds it's true, others false. Claiming the inference holds universally ignores modal distinctions.
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    • 2.Kripkean semantics requires rigid designation of mathematical objects across worlds. 'Polynomial algorithm' lacks fixed reference without specifying interpretive context.
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    • 3.The inference commits a modal fallacy: from 'possibly (∃P: solves NP-complete in poly-time)' to 'necessarily (∀Q∈NP: solvable in poly-time)' without justification.
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