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    Putnam's model-theoretic argument establishes that formal... — Carmelics
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    Challenges→There exists a nonstandard model M in which all polynomial-time computable functions are total but the exponential function is not total.

    Putnam's model-theoretic argument establishes that formal consistency alone cannot fix intended reference, so 'polynomial-time computable' in M may not refer to the same class as in the standard model.

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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
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    • 1.Putnam showed that any formal system admits multiple models satisfying identical axioms, so syntax alone underdetermines which model is 'intended'.
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    • 2.Computational predicates like 'polynomial-time computable' depend on interpretation of primitive terms; formal consistency permits non-standard domains.
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    • 3.Without extra-formal constraints (use, causation, intentionality), we lack principled grounds to exclude non-standard interpretations of 'P vs NP' claims.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Putnam's argument targets reference-fixing for general terms; 'polynomial-time computable' has clear extensional definition in standard models of arithmetic.
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    • 2.Model-theoretic indeterminacy affects existence claims but doesn't show meaning itself is indeterminate—we grasp 'polynomial-time' via algorithm structure.
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    • 3.Mathematical practice demonstrates robust inter-model agreement on computational classes; this convergence suggests determinate reference despite formal underdetermination.
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    Key Terms

    Formal consistency(what the argument contrasts with explanatory success)
    Whether a system of logic or mathematics follows its own rules without contradiction—if the rules don't contradict each other, the system is consistent.
    Model (in logic)(in formal logic and semantics)
    An imaginary scenario or description of a possible world where certain statements are true or false—used to test whether logical arguments work.
    Putnam, Hilary(as a philosopher cited for natural kind essentialism)
    A 20th-century philosopher who developed the idea that the meaning of words like 'water' depends on what water actually is in nature, not just on how it appears to us.
    intended reference(What the argument says consistency alone cannot determine)
    What a word is actually supposed to point to or mean in the real world—the thing you have in mind when you use a word.
    model-theoretic argument(Philosophy of language; challenge to theories that determine reference via truth-maximization)
    An argument, advanced by Hilary Putnam, showing that there exist many different assignments of reference to subsentential expressions of a language that make all utterances of that language true, thereby underdetermining reference.
    polynomial-time computable(Used as an example of a phrase that might refer to different things in different models)
    A mathematical term describing problems that a computer can solve relatively quickly (within a reasonable amount of time that increases predictably as the problem gets bigger).
    standard model(Contrasted with non-standard models that also satisfy the theory's axioms)
    The intended interpretation of an arithmetical theory, namely the structure of the natural numbers.

    Connections

    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    Computational predicates like 'polynomial-time computable' depend on interpretat...Mathematical practice demonstrates robust inter-model agreement on computational...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Model-theoretic indeterminacy affects existence claims but doesn't show meaning ...
    Putnam showed that any formal system admits multiple models satisfying identical...
    +3 moreShow less
    Putnam's argument targets reference-fixing for general terms; 'polynomial-time c...There exists a nonstandard model M in which all polynomial-time computable funct...Without extra-formal constraints (use, causation, intentionality), we lack princ...