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    The meaning of a formula (the proposition expressed) does... — Carmelics
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    Home/Philosophy of Language
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    The meaning of a formula (the proposition expressed) does not represent a reality distinct from the linguistic system in which the formula occurs

    Philosophy of LanguageTruth & Knowledge
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    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.The Curry-Howard correspondence allows rephrasing the intuitionist position as: the proposition expressed by a formula of Heyting Arithmetic is the type of its proofs
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    • 2.The notion of 'type' here derives from lambda-calculus, not a straightforward synonym for 'set' or 'species'
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    • 3.Lambda-calculus is a formal system with rich interconnections with programming and computer science, with readings in which instances of types are purely syntactic, proof-theoretic entities
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Frege's context principle distinguishes sense (Sinn) from reference (Bedeutung), where sense is objective but not reducible to any formal system.
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    • 2.The sense of a mathematical formula can remain constant across mutually untranslatable formal systems, indicating sense transcends any particular linguistic encoding.
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    • 3.If meaning were exhausted by position within a formal system, co-referential terms in different systems could never be recognized as such, yet mathematicians routinely identify them.
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    Reason against 2 of 2
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    • 1.Gödel's incompleteness theorems demonstrate that arithmetical truth outruns provability in any consistent formal system, as shown by the truth of the Gödel sentence.
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    • 2.If the proposition expressed by a formula were constituted entirely by its role within a formal system, no proposition could be true yet unprovable within that system—contradicting Gödel's result.
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    • 3.The Curry-Howard correspondence maps proofs to types within a system, but cannot account for truths that resist formalization within that system, leaving extra-systemic mathematical reality intact.
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    Related

    Frege's context principle distinguishes sense (Sinn) from reference (Bedeutung),...Gödel's incompleteness theorems demonstrate that arithmetical truth outruns prov...If meaning were exhausted by position within a formal system, co-referential ter...If the proposition expressed by a formula were constituted entirely by its role ...
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    Lambda-calculus is a formal system with rich interconnections with programming a...The Curry-Howard correspondence allows rephrasing the intuitionist position as: ...The Curry-Howard correspondence maps proofs to types within a system, but cannot...The notion of 'type' here derives from lambda-calculus, not a straightforward sy...The sense of a mathematical formula can remain constant across mutually untransl...

    Similar

    Speakers typically mean to convey propositions other than the one lite...81%The linguistic meaning of a formula is often ambiguous, and identifyin...81%Sense cannot be identified with the proposition that expresses it.78%Sentence meaning does not simply combine with facts to determine truth...78%

    Source

    AI-extracted1/3 agreementValid
    SEP: formalism-mathematics
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    It seems to be Kreisel who introduced the slogan ‘formulae as types’, with Martin-Löf responsible for the more widespread ‘propositions as types’ slogan (See again, Wadler, 2015). In the philosophical context, ‘proposition’ is often used to mean something like the meaning of a sentence, i.e. of a formula of a certain sort. Using this terminology, a widespread intuitionist position is that that the proposition expressed by a formula is the set (or species, for the intuitionist) of all proofs of t
    Extraction notes

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

    Details

    Type
    claim
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    1 edit