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The Completeness of Heyting Firstorder Logic
"... Restricted to firstorder formulas, the rules of inference in the CurryHoward type theory are equivalent to those of firstorder predicate logic as formalized by Heyting, with one exception: ∃elimination in the CurryHoward theory, where ∃x: A.F (x) is understood as disjoint union, are the project ..."
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Restricted to firstorder formulas, the rules of inference in the CurryHoward type theory are equivalent to those of firstorder predicate logic as formalized by Heyting, with one exception: ∃elimination in the CurryHoward 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 CurryHoward 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 CurryHoward 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 MartinLöf, for example in [MartinLöf, 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)