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Geometric Rules in Infinitary Logic

In Ofer Arieli & Anna Zamansky (eds.), Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Springer Verlag. pp. 265-293 (2021)

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  1. Infinitary logic.John L. Bell - 2008 - Stanford Encyclopedia of Philosophy.
    Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...)
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  • The Gödel-McKinsey-Tarski embedding for infinitary intuitionistic logic and its extensions.Matteo Tesi & Sara Negri - 2023 - Annals of Pure and Applied Logic 174 (8):103285.
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  • Glivenko sequent classes and constructive cut elimination in geometric logics.Giulio Fellin, Sara Negri & Eugenio Orlandelli - 2023 - Archive for Mathematical Logic 62 (5):657-688.
    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 (...)
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  • On the Proof Theory of Infinitary Modal Logic.Matteo Tesi - 2022 - Studia Logica 110 (6):1349-1380.
    The article deals with infinitary modal logic. We first discuss the difficulties related to the development of a satisfactory proof theory and then we show how to overcome these problems by introducing a labelled sequent calculus which is sound and complete with respect to Kripke semantics. We establish the structural properties of the system, namely admissibility of the structural rules and of the cut rule. Finally, we show how to embed common knowledge in the infinitary calculus and we discuss first-order (...)
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