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On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard Correspondence for Markov's Principle in First-Order Logic and Arithmetic

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Federico Aschieri, Matteo Manighetti

Logic in Computer Science · Submitted 16 Dec 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1612.05457 (version 2)
Pith IDTDA6Z3QT, Pith's permanent record of this paper and its review
CitationFederico Aschieri, Matteo Manighetti. "On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard Correspondence for Markov's Principle in First-Order Logic and Arithmetic." arXiv:1612.05457 (2016). https://pith.science/paper/TDA6Z3QT
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,935 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.85 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; .

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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. background 1.00 0.04 0.13 asserted 1.00
s1 We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle. contribution 1.00 0.94 0.09 asserted 0.94
s2 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. method 0.94 0.92 0.13 asserted 1.00
s3 We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possible witnesses. result 0.94 0.92 0.16 asserted 1.00
s4 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. result 0.92 0.94 0.38 asserted 1.00
s5 Finally, using the techniques just introduced, we also provide a new computational interpretation of Arithmetic with Markov's principle. contribution 0.88 0.95 0.17 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

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Original abstract

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.

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