Derivability of admissible rules

Mint︠s︡, Grigori

doi:10.1007/bf01084082

Cited by 50 publications

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“…Mints [16] introduced a particular rule that is admissible in IPC, but the implication from its antecedent to its conclusion is most certainly not a theorem. This rule was shortly thereafter generalised by Citkin [4] into something similar to the rule below, when one instantiates n as 2.…”

Section: Semantics For Admissible Rulesmentioning

confidence: 99%

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Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility

Goudsmit

2016

Stud Logica

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Abstract.Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.

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Section: Semantics For Admissible Rulesmentioning

confidence: 99%

“…He showed that any finite model of all admissible rules necessarily is of an extremely restricted shape, making use of a particular generalisation of a rule introduced by Mints [16].…”

Section: Introductionmentioning

confidence: 99%

Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility

Goudsmit

2016

Stud Logica

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“…Kuznetsov observed that the rule (¬¬p → p) → (p ∨ ¬p)/((¬¬p → p) → ¬p) ∨ ((¬¬p → p) → ¬¬p) is also admissible for IPC, but not derivable. Another example of an admissible for IPC not derivable rule was found in 1971 by G. Mints (see [26]). …”

Section: Introductionmentioning

confidence: 99%

Multiple Conclusion Rules in Logics with the Disjunction Property

Citkin

2015

Logical Foundations of Computer Science

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Abstract. We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics.

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“…A nice source of inspiration can be found in the rule below, which has been studied in several incarnations before. Its admissibility for singleton covers with n = 2 in IPC was discussed by Mints (1976). Skura (1989) considered this rule, also with only singleton covers but for arbitrary n, and proved that IPC is the sole intermediate logic which admits them all.…”

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confidence: 99%

A Note on Extensions: Admissible Rules via Semantics

Goudsmit

2013

Logical Foundations of Computer Science

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Any intermediate logic with the disjunction property admits the Visser rules if and only if it has the extension property. This equivalence restricts nicely to the extension property up to n.In this paper we demonstrate that the same goes even when omitting the rule ex falso quod libet, that is, working over minimal rather than intuitionistic logic. We lay the groundwork for providing a basis of admissibility for minimal logic, and tie the admissibility of the Mints-Skura rule to the extension property in a stratified manner.Keywords: admissible rules, minimal logic, disjunction property, extensions of Kripke modelsThe admissible rules of a theory are those rules under which the theory is closed. Derivable rules are admissible. For classical propositional logic, this is the whole story. For intuitionistic propositional logic (IPC) -and minimal logic -it is not.Friedman (1975, Problem 40) conjectured admissibility for IPC to be decidable, as has been confirmed by Rybakov (1984). De Jongh and Visser conjectured that the Visser rules form a basis of admissibility for IPC, that is to say, all admissible rules of IPC become derivable after adjoining the Visser rules. Rozière (1992) and Iemhoff (2001b) independently confirmed this. Again independently, Skura (1989) demonstrated that IPC is the sole intermediate logic that admits a restricted form of the Visser rules.At the Pisa Proof Theory workshop of 2012 George Metcalfe gave a tutorial on admissible rules. As has become standard practice, Metcalfe mentioned Lorenzen (1955) as the first place where admissible rules where studied an sich. Jan von Plato objected that Johansson (1937) already discussed them. Odintsov and Rybakov (2012) proved admissibility for minimal logic to be decidable. In this paper we lay the groundwork for studying all admissible rules of Johansson's minimal logic, with the eventual goal of providing an explicit basis of admissibility. This paper aims to provide uniformity to some of the literature regarding admissible rules for logics above minimal logic. We make several observations, many of which not elsewhere available in the generality stated here. Although this paper contains novel results, most notably the semantic characterization of admissibility for an adaptation of the rules studied by Skura, its main purpose is to provide a unified approach to the study of admissible rules over minimal logic.

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Derivability of admissible rules

Cited by 50 publications

References 3 publications

Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility

Finite Frames Fail: How Infinity Works Its Way into the Semantics of Admissibility

Multiple Conclusion Rules in Logics with the Disjunction Property

A Note on Extensions: Admissible Rules via Semantics

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