On unification and admissible rules in Gabbay–de Jongh logics

Goudsmit, Jeroen P.; Iemhoff, Rosalie

doi:10.1016/j.apal.2013.09.003

Cited by 13 publications

(28 citation statements)

References 25 publications

(24 reference statements)

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“…The characterization that is to be given at the end of this section suggests a particularly nice schema of admissible rules, namely the de Jongh rules. With some additional bookkeeping one can use this characterization to prove, for instance, that the de Jongh rules form a basis of the Gabbay-de Jongh logics of Gabbay and de Jongh (1974), as has been done in Goudsmit and Iemhoff (2012). Here we show how these results are actually more general, in that they help towards providing a basis of admissibility of minimal logic.…”

Section: Extensions Of Modelsmentioning

confidence: 75%

“…Let us mention three small results. The first is an immediate corollary of Lemma 3 and 4 of Goudsmit and Iemhoff (2012), the other two follow from the first. The final corollary comes in handy in proving that our canonical model works well, and it is also crucial in Lemma 4.…”

Section: Definition (Saturated Set)mentioning

confidence: 89%

“…This is one of the many places where one could introduce further bookkeeping by restricting the formulae considered to some set, for instance the set of atomic formulae. For details regarding this one can consult Goudsmit and Iemhoff (2012), this is the last we speak of it.…”

Section: Lemma (Forcing Of Extensions)mentioning

confidence: 99%

“…Below we cite one of the main results of Goudsmit and Iemhoff, 2012, Theorem 1. The main point is that this perfectly characterizes when a model has an extension satisfying a given theory, and that this characterization works for logics extending minimal logic.…”

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confidence: 99%

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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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Section: Extensions Of Modelsmentioning

confidence: 75%

Section: Definition (Saturated Set)mentioning

confidence: 89%

Section: Lemma (Forcing Of Extensions)mentioning

confidence: 99%

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confidence: 99%

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A Note on Extensions: Admissible Rules via Semantics

Goudsmit

2013

Logical Foundations of Computer Science

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“…Corollary 5 [14] Any fragment of T n that contains conjunction and implication has finitary n + 1/9-unification. If it does not contain disjunction it has unitary n + 1/9-unification under the same condition.…”

Section: Implications Only)mentioning

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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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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On unification and admissible rules in Gabbay–de Jongh logics

Cited by 13 publications

References 25 publications

A Note on Extensions: Admissible Rules via Semantics

A Note on Extensions: Admissible Rules via Semantics

Unification in Intermediate Logics

Multiple Conclusion Rules in Logics with the Disjunction Property

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