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    The inference from 'a polynomial algorithm exists for one... — Carmelics
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    Challenges→The existence of a polynomial time algorithm for any single NP-complete problem would entail the existence of polynomial time algorithms for all problems in NP.

    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.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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    Reasons Against

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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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    Key Terms

    Kripkean possible-worlds semantics(in philosophy of language and logic)
    A framework developed by philosopher Saul Kripke for understanding meaning and truth by imagining different possible scenarios or 'worlds' where things could be different, and checking what's true in each one.
    NP problems(in computer science)
    A broad class of difficult math and logic problems where checking a proposed solution is quick, even if finding solutions is hard.
    NP-complete problem(Computational complexity theory)
    A problem that is among the most difficult in NP, to which all other problems in NP are polynomial-time reducible, such that a polynomial time solution for any one would yield polynomial time solutions for all problems in NP.
    Polynomial algorithm(in computer science)
    A step-by-step procedure for solving a problem that doesn't take an absurdly long time—roughly, the time it takes grows at a reasonable rate as the problem gets bigger.
    Saul Kripke(in 20th-century philosophy)
    An influential modern philosopher who developed new ways of thinking about meaning, necessity, and possibility by using the idea of 'possible worlds.'
    Transferable property(in logic and reasoning)
    A characteristic that, if it's true for one thing, automatically must be true for something related to it.
    existence(Kant's analysis in the Critique of Pure Reason as applied to the ontological argument)
    Not a real predicate or positive determination; it does not add to or enlarge the concept of a subject.
    inference(Nyāya epistemology)
    A component of epistemology in Nyāya philosophy; a veritable inference yields knowledge about the world and must have premises that are themselves known
    possible worlds(Leibniz's modal semantics, anticipating contemporary possible-worlds semantics)
    Worlds that have existence in a tenuous sense; fictional worlds used to characterize the nature of possibles that are never actualized

    Connections

    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    Invoking possible-worlds semantics for computational complexity conflates metaph...Kripkean semantics requires rigid designation of mathematical objects across wor...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Mathematical truths are necessary, not contingent. P=NP has a determinate truth-...
    P=NP is contingent, not necessary: in some possible worlds it's true, others fal...
    +3 moreShow less
    The NP-completeness reduction IS the inference justification: it logically entai...The existence of a polynomial time algorithm for any single NP-complete problem ...The inference commits a modal fallacy: from 'possibly (∃P: solves NP-complete in...