On Rules

Iemhoff, Rosalie

doi:10.1007/s10992-015-9354-x

Cited by 15 publications

(13 citation statements)

References 44 publications

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“…Let us observe that a rule r ∶= Γ A is admissible for logic L (in symbols, Γ ∼ L A) if any substitution that refutes A, refutes at least one member of Γ. If Γ = ∅, then rule Γ A is admissible for L if and only if A ∈ Th(L) [8]. For m-rules, an m-rule Γ ∆ is admissible for − L (in symbols Γ ∼ L ∆) if every substitution σ that refutes all formulas from ∆, refutes at least one formula from Γ [8].…”

Section: Preliminariesmentioning

confidence: 99%

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Deductive Systems with Multiple-Conclusion Rules and the Disjunction Property

Citkin¹

2019

Axioms

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Section: Preliminariesmentioning

confidence: 99%

“…If Γ = ∅, then rule Γ A is admissible for L if and only if A ∈ Th(L) [8]. For m-rules, an m-rule Γ ∆ is admissible for − L (in symbols Γ ∼ L ∆) if every substitution σ that refutes all formulas from ∆, refutes at least one formula from Γ [8]. Thus, the rule ▾ ▴ is not admissible in any logic.…”

Section: Preliminariesmentioning

confidence: 99%

Deductive Systems with Multiple-Conclusion Rules and the Disjunction Property

Citkin¹

2019

Axioms

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“…For an overview of the area of admissible rules, the reader is referred to the literature, in particular to Rybakov's monograph [12]. For a brief overview of the main results in this area on intermediate and modal logics, see [6].…”

Section: Consequence Relationsmentioning

confidence: 99%

“…Many intermediate and modal logics and fragments thereof are known to have nonderivable admissible rules, and for several an explicit basis for the admissible rules is known. We refer the reader to (the references in) [6,12]. Example 5 Interestingly, in modal and intermediate logics certain rule schemes seem generic in that they cannot be admissible without being a basis.…”

Section: Basesmentioning

confidence: 99%

Consequence Relations and Admissible Rules

Iemhoff

2015

J Philos Logic

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This paper contains a detailed account of the notion of admissibility in the setting of consequence relations. It is proved that the two notions of admissibility used in the literature coincide, and it provides an extension to multi-conclusion consequence relations that is more general than the one usually encountered in the literature on admissibility. The notion of a rule scheme is introduced to capture rules with side conditions, and it is shown that what is generally understood under the extension of a consequence relation by a rule can be extended naturally to rule schemes, and that such extensions capture the intuitive idea of extending a logic by a rule.

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“…Roughly speaking, an inference rule is admissible for a deductive system (or a logic) S if it does not produce new theorems when added to S [43,45,46,48,49,66,78]. Clearly, every derivable rule is admissible but the converse does not need to hold.…”

Section: Introductionmentioning

confidence: 99%

Almost structural completeness; an algebraic approach

Dzik

Stronkowski

2016

Annals of Pure and Applied Logic

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A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included.Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.

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On Rules

Abstract: This paper contains a brief overview of the area of admissible rules with an emphasis on results about intermediate and modal propositional logics. No proofs are given but many references to the literature are provided.

Cited by 15 publications

References 44 publications

Deductive Systems with Multiple-Conclusion Rules and the Disjunction Property

Deductive Systems with Multiple-Conclusion Rules and the Disjunction Property

Consequence Relations and Admissible Rules

Almost structural completeness; an algebraic approach

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