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    Home/Original/inverse
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    Inverse View

    It is not the case that Dependent type theories (e.g., Martin-Löf type theory) are not merely extensions of Howard's original correspondence but constitute distinct foundational frameworks.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Dependent type theories preserve the core Curry-Howard mapping: types remain propositions and terms remain proofs, just with richer structure.
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    • 2.Extensions maintaining underlying principles (propositions-as-types, proofs-as-programs) are naturally called extensions, not distinct frameworks.
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    • 3.Calling dependent types a 'distinct framework' rather than extension obscures their historical and conceptual continuity with Howard's original insight.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Dependent types introduce type families indexed by terms, fundamentally changing proof structure in ways Howard's correspondence doesn't address.
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    • 2.Martin-Löf type theory includes identity types and inductive families, enabling entirely new classes of proofs impossible in simple type theory.
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    • 3.The computational behavior of dependent types creates novel proof-relevant distinctions that constitute a genuinely different foundational framework.
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