Admissible and derivable rules in intuitionistic logic

Rozière, Paul

doi:10.1017/s0960129500000165

Cited by 15 publications

(10 citation statements)

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“…This conjecture was later proved by Iemhoff in the fundamental [6]. Rozière in his Ph.D. thesis [9] reached the same conclusion with a substantially different technique, independently of Visser and Iemhoff. These works elegantly settled the problem of identifying and building admissible rules.…”

Section: Introductionmentioning

confidence: 63%

Admissible Tools in the Kitchen of Intuitionistic Logic

Condoluci¹,

Manighetti²

2018

Electron. Proc. Theor. Comput. Sci.

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The usual reading of logical implication A → B as " if A then B " fails in intuitionistic logic: there are formulas A and B such that A → B is not provable, even though B is provable whenever A is provable. Intuitionistic rules apparently don't capture interesting meta-properties of the logic and, from a computational perspective, the programs corresponding to intuitionistic proofs are not powerful enough. Such non-provable implications are nevertheless admissible, and we study their behaviour by means of a proof term assignment and related rules of reduction. We introduce V, a calculus that is able to represent admissible inferences, while remaining in the intuitionistic world by having normal forms that are just intuitionistic terms. We then extend intuitionistic logic with principles corresponding to admissible rules. As an example, we consider the Kreisel-Putnam logic KP, for which we prove the strong normalization and the disjunction property through our term assignment. This is our first step in understanding the essence of admissible rules for intuitionistic logic.

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Section: Introductionmentioning

confidence: 63%

Admissible Tools in the Kitchen of Intuitionistic Logic

Condoluci¹,

Manighetti²

2018

Electron. Proc. Theor. Comput. Sci.

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“…The positive answer to Friedman's problem was given by V. V. Rybakov [44] (see also [43]). The fact that the rules corresponding to formulas ( * ) constitute the basis of admissible in IPC rules was proved in [18], where these rules are called Visser's rules (since they were reintroduced by A. Visser [47]).…”

Section: The Following Axiomsmentioning

confidence: 98%

A mind of a non-countable set of ideas

Citkin

2008

LLP

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“…In this section we briefly explain these proofs in general, the next three sections provide the technical details. The proofs make use of a connection between valuations and substitutions that goes back to Prucnal [29,31] and are crucial in the work of Ghilardi [10] and Rozière [32,33].…”

Section: Related Workmentioning

confidence: 99%

Unification in Intermediate Logics

Iemhoff¹,

Rozière²

2015

J. symb. log.

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This paper contains a proof theoretic treatment of some aspects of unification in intermediate logics. It is shown that many existing results can be extended to fragments that at least contain implication and conjunction. For such fragments the connection between valuations and most general unifiers is clarified, and it is shown how from the closure of a formula under the Visser rules a proof of the formula under a projective unifier can be obtained. This implies that in the logics considered, for the n-unification type to be finitary it suffices that the m-th Visser rule is admissible for a sufficiently large m. At the end of the paper it is shown how these results imply several well-known results from the literature.

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Admissible and derivable rules in intuitionistic logic

Cited by 15 publications

References 5 publications

Admissible Tools in the Kitchen of Intuitionistic Logic

Admissible Tools in the Kitchen of Intuitionistic Logic

A mind of a non-countable set of ideas

Unification in Intermediate Logics

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