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Glivenko sequent classes and constructive cut elimination in geometric logics

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Abstract

A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation based on a notion of embeddability of derivations. Additionally, conservativity for classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes.

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Notes

  1. Barr’s theorem is often alleged to achieve more in that it also allows to eliminate uses of the axiom of choice. That such formulations of Barr’s theorem should be taken with caution is demonstrated in [25] where internal versus external addition of the the axiom of choice is considered and it is shown that the latter preserves conservativity whereas the former does not.

  2. Barr’s Theorem corresponds to Orevkov’s first class.

  3. By “constructive” here we mean not relying on classical logical principles such as excluded middle or linearity of ordinals but we do not mean acceptable in all schools of constructive mathematics.

  4. See [25, §7] for a different proof, based on constructive ordinals, of cut elimination in infinitary logic. The proof in [2] does not use ordinals, but it is inherently classical in that it uses a one-sided calculus based on De Morgan’s dualities.

  5. Even if all proofs in [7] make no use of non-constructive assumptions about ordinals, we prefer to avoid completely the assumption of total ordering.

  6. Derivations can thus be represented as (infinite) trees, where the nodes are the sequents in the derivation, and a nodes that corresponds to a premiss of a rule is an immediate successor of the node that corresponds to the conclusion of such rule. Therefore, a node that corresponds to the conclusion of a rule with \(\beta \) premisses has \(\beta \) immediate successors.

  7. This example is due to Parlamento and Previale [24].

  8. One may be mislead here by assuming that the correspondence between branches implies that the two derivations have the same structure. However, this is not the case as the correspondence is not required to be injective nor surjective.

  9. Since the number of nodes of the tree is at most countable, one may also define an encoding such that the correspondence is unique. This however would require more effort and we would lose the property that every infinite sequence has an initial segment that indexes a branch of the tree.

  10. Orevkov [23] proved optimality for the finitary case by classifying the other possible classes of sequents and exhibiting for each of them a sequent that is classically but not intuitionistically derivable.

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Acknowledgements

Special thanks are due to Peter Schuster for precious comments and helpful discussions on various points. The authors are grateful to Mario Piazza and Tarmo Uustalu for their interest and valuable suggestions. We are also grateful to an anonymous referee for detailed and insightful comments, and to the audience of the conferences LAP 2020, LAP 2021, and TYPES 2021, where the paper was presented. The first two authors are members of the “Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni” (GNSAGA of the Istituto Nazionale di Alta Matematica (IndAM).

Funding

Partial financial support was received from the Academy of Finland, research project no. 1308664.

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Authors and Affiliations

  1. Department of Computer Science, University of Verona, Strada le Grazie 15, 37134, Verona, Italy

    Giulio Fellin

  2. Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146, Genoa, Italy

    Sara Negri

  3. Department of Philosophy and Communication Studies, University of Bologna, Via Zamboni 38, 40126, Bologna, Italy

    Eugenio Orlandelli

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  1. Giulio Fellin
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  2. Sara Negri
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  3. Eugenio Orlandelli
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Correspondence to Eugenio Orlandelli.

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This paper is a revised and extended version of the conference paper [7]. The latter presented only the constructive cut elimination for classical and intuitionistic geometric logics based on Brouwer’s Bar Induction. The main novelties of this paper are (i) that also minimal geometric logic is considered, (ii) that the notion of “proof-embeddability” is here introduced and transfinite inductions on ordinals are replaced by Noetherian induction with proof-embeddability, and (iii) that proofs of conservativity for the infinitary Glivenko classes are given.

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Fellin, G., Negri, S. & Orlandelli, E. Glivenko sequent classes and constructive cut elimination in geometric logics. Arch. Math. Logic 62, 657–688 (2023). https://doi.org/10.1007/s00153-022-00857-z

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