On Natural Deduction in Classical First-Order Logic: Curry-Howard Correspondence, Strong Normalization and Herbrand’s Theorem (original) (raw)
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Intuitionistic first-order logic extended with a restricted form of Markov's principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possible witnesses. As corollary, we derive that the logic is Herbrand constructive: whenever it proves any existential formula, it proves also an Herbrand disjunction for the formula. Finally, using the techniques just introduced, we also provide a new computational interpretation of Arithmetic with Markov's principle. 1998 ACM Subject Classification F.4.1 Proof Theory Markov's Principle was introduced by Markov in the context of his theory of Constructive Recursive Mathematics (see [15]). Its original formulation is tied to Arithmetic: it states that given a recursive function f : N → N, if it is impossible that for every natural number n, f (n) = 0, then there exists a n such that f (n) = 0. Markov's original argument for justifying it was simply the following: if it is not possible that for all n, f (n) = 0, then by computing in sequence f (0), f (1), f (2),. . ., one will eventually hit a number n such that f (n) = 0 and will effectively recognize it as a witness. Markov's principle is readily formalized in Heyting Arithmetic as the axiom scheme ¬¬∃α N P → ∃α N P where P is a primitive recursive predicate [14]. When added to Heyting Arithmetic, Markov's principle gives rise to a constructive system, that is, one enjoying the disjunction and the existential witness property [14] (if a disjunction is derivable, one of the disjoints is derivable * Funded by the Austrian Science Fund FWF Lise Meitner grant M 1930–N35 † Funded by the Vienna Science Fund WWTF project VRG12-004
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