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The Completeness of Heyting First-order Logic
"... Restricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃x: A.F (x) is understood as disjoint union, are the project ..."
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Restricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃x: A.F (x) is understood as disjoint union, are the projections, and these do not preserve firstorderedness. This note shows, however, that the Curry-Howard theory is conservative over Heyting’s system. The meaning of the title becomes clear if we take the intuitionistic meaning of the logical constants to be given in Curry-Howard type theory, CH. This is the basic theory of types built up by means of ∀ and ∃, as outlined in [Howard, 1980] and presented in some detail by Martin-Löf, for example in [Martin-Löf, 1998], as part of his intuitionistic theory of types. The quantifier ∀ denotes in this system the operation of taking cartesian products ∀x:A.F (x) = Πx:AF (x)
