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    Bolzano anticipated Gentzen's Hauptsatz in a certain sense — Carmelics
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    Bolzano anticipated Gentzen's Hauptsatz in a certain sense

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Bolzano's semantic notion of entailment requires that from the smallest number of premises the largest number of conclusions follow, none of which has simple components not also contained in the premises
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    • 2.Gentzen's characterization of normal proofs holds that they do not make a detour and do not introduce concepts not contained in the final result
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    • 3.Bolzano's and Gentzen's criteria are structurally parallel: both require that inferential steps not introduce extraneous conceptual content
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Bolzano's criterion operates at the semantic level of conceptual containment between propositions-in-themselves, not at the level of proof syntax.
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    • 2.Gentzen's Hauptsatz (Cut-Elimination) is a syntactic, proof-theoretic result about the eliminability of the cut rule from formal derivations.
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    • 3.Structural parallelism between a semantic constraint and a syntactic theorem does not constitute anticipation without shared proof-theoretic mechanism or intent.
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    Reason against 2 of 2
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    • 1.Bolzano's notion of Abfolge concerns the objective grounding relations among truths, not the normalization of formal inference procedures.
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    • 2.Gentzen's motivation was to establish consistency and the subformula property within a formal calculus, a project unintelligible in Bolzano's pre-formalist framework.
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    • 3.Attributing anticipation conflates Bolzano's metaphysics of grounding with Gentzen's proof-theoretic program, obscuring genuinely distinct intellectual projects.
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    Related

    Attributing anticipation conflates Bolzano's metaphysics of grounding with Gentz...Bolzano's and Gentzen's criteria are structurally parallel: both require that in...Bolzano's criterion operates at the semantic level of conceptual containment bet...Bolzano's notion of Abfolge concerns the objective grounding relations among tru...
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    Bolzano's semantic notion of entailment requires that from the smallest number o...Gentzen's Hauptsatz (Cut-Elimination) is a syntactic, proof-theoretic result abo...Gentzen's characterization of normal proofs holds that they do not make a detour...Gentzen's motivation was to establish consistency and the subformula property wi...Structural parallelism between a semantic constraint and a syntactic theorem doe...

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    Source

    AI-extracted1/3 agreementValid
    SEP: bolzano
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    The relation of entailment between (sets of) conceptual truths has the characteristic feature that, according to it, from the smallest number of premisses the largest number of conclusions will follow none of which has simple components which are not also contained in its premisses (WL II, 386). Ideas such as these sound akin to Gerhard Gentzen’s characterization of normal proofs: “They do not make a detour. They do not introduce concepts which are not contained in its final result and must ther
    Extraction notes

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    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit