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    Many-sorted logic cannot be considered a proper (strict) ... — Carmelics
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    Home/Philosophy of Language
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    Many-sorted logic cannot be considered a proper (strict) extension of first-order logic

    Philosophy of Language
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    1 reason for
    2 reasons against

    Reasons For

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    Reason for
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    • 1.Lindström (1969) proves that first-order logic is the strongest logic possessing simultaneously Compactness and Löwenheim-Skolem properties
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    • 2.First-order logic is the strongest logic where Löwenheim-Skolem holds and its set of valid sentences is recursively enumerable
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    • 3.Many-sorted logic also possesses these properties (Compactness and Löwenheim-Skolem) as corollaries of its completeness theorem
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Many-sorted logic has strictly greater expressive power than single-sorted FOL when sort predicates cannot be first-order definable in the intended structure.
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    • 2.Wang (1952) and Oberschelp (1962) demonstrate that many-sorted logic captures distinctions between sort domains that require additional axioms or expanded signatures in single-sorted FOL to replicate.
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    • 3.A logic that requires non-trivial translation overhead with signature expansion to simulate another logic's native expressiveness constitutes a genuine, not merely apparent, extension of that logic.
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    Reason against 2 of 2
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    • 1.Lindström's theorem characterizes logics up to expressive equivalence over unrestricted signatures, but many-sorted logic operates over typed signatures that are not freely interchangeable with single-sorted ones.
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    • 2.The Lindström-style collapse of many-sorted logic into FOL depends on encoding sorts as unary predicates, which presupposes a metatheoretic translation that alters the intended semantics of sort-partitioned domains.
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    Philosophy of LanguageProof of definition segments

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    Related

    A logic that requires non-trivial translation overhead with signature expansion ...First-order logic is the strongest logic where Löwenheim-Skolem holds and its se...Lindström (1969) proves that first-order logic is the strongest logic possessing...Lindström's theorem characterizes logics up to expressive equivalence over unres...
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    Many-sorted logic also possesses these properties (Compactness and Löwenheim-Sko...Many-sorted logic has strictly greater expressive power than single-sorted FOL w...The Lindström-style collapse of many-sorted logic into FOL depends on encoding s...Wang (1952) and Oberschelp (1962) demonstrate that many-sorted logic captures di...

    Similar

    Many-sorted logic cannot be considered a proper extension of first-ord...98%Many-sorted logic is in the same situation as one-sorted first-order l...88%Many-sorted logic reduces to classical one-sorted first-order logic.87%Many-sorted logic, like first-order logic, is undecidable87%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-many-sorted
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    , determining validity, or equivalently, testing for satisfiability of given formulas) for many-sorted logic is undecidable. So, we are in the same situation encountered in one-sorted first-order logic. Of course, if a calculus is to be helpful it would never allow erroneous reasonings: it is not going to drive us from true hypotheses to false conclusions. It must be a sound calculus. Further, it is highly desirable that all the consequences of a set \(\Gamma\) of hypotheses could be derived fr
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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