Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules

Kürbis, Nils

doi:10.1007/s11229-021-03418-8

Cited by 3 publications

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“…A constructive proof of this result as a consequences of a normalisation theorem for Milne's system is in[17]. Similar results for Negri and von Plato's intuitionistic system with general introduction and elimination rules are proved in[18].…”

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Bilateral Inversion Principles

Kürbis

2022

Electron. Proc. Theor. Comput. Sci.

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This paper formulates a bilateral account of harmony that is an alternative to one proposed by Francez. It builds on an account of harmony for unilateral logic proposed by Kürbis and the observation that reading the rules for the connectives of bilateral logic bottom up gives the grounds and consequences of formulas with the opposite speech act. I formulate a process I call 'inversion' which allows the determination of assertive elimination rules from assertive introduction rules, and rejective elimination rules from rejective introduction rules, and conversely. It corresponds to Francez's notion of vertical harmony. I also formulate a process I call 'conversion', which allows the determination of rejective introduction rules from assertive elimination rules and conversely, and the determination of assertive introduction rules from rejective elimination rules and conversely. It corresponds to Francez's notion of horizontal harmony. The account has a number of features that distinguishes it from Francez's.

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Bilateral Inversion Principles

Kürbis

2022

Electron. Proc. Theor. Comput. Sci.

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Two Kinds of Difficulties

Tranchini

2024

Trends in Logic

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Two distinct kinds of cases, going back to Crabbé and Ekman, show that the Tennant-Prawitz criterion for paradoxicality overgenerates, that is, there are derivations which are intuitively non-paradoxical but which fail to normalize. We argue that a solution to “Ekman’s paradox” consists in restricting the set of admissible reduction procedures to those that do not yield a trivial notion of identity of proofs. We then discuss a different kind of solution, due to von Plato, and recently advocated by Tennant, consisting in reformulating natural deduction elimination rules in general (or parallelized) form. Developing intuitions of Ekman we show that the adoption of general rules has the consequence of hiding redundancies within derivations. Once reductions to get rid of the hidden redundancies are devised, it is clear that the adoption of general elimination rules offers no remedy to the overgeneration of the Prawitz-Tennant analysis. In this way, we indirectly provide further support for our own solution to Ekman’s paradox.

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A classical first-order normalization procedure with $$\forall $$ and $$\exists $$ based on the Milne–Kürbis approach

Shangin

2023

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Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules

Cited by 3 publications

References 21 publications

Bilateral Inversion Principles

Bilateral Inversion Principles

Two Kinds of Difficulties

A classical first-order normalization procedure with $$\forall $$ and $$\exists $$ based on the Milne–Kürbis approach

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