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On the Admissible Rules of Intuitionistic Propositional Logic
 Journal of Symbolic Logic
, 2001
"... We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give semantic criteria for admissibility. 1 Introduction The admissible rules of a theory are th ..."
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Cited by 23 (4 self)
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We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give semantic criteria for admissibility. 1 Introduction The admissible rules of a theory are the rules under which the theory is closed. It is wellknown that, in contrast to classical propositional logic, intuitionistic propositional logic IPC, has admissible rules which are not derivable. Probably the first nonderivable admissible rule known for this logic is the rule :A ! (B C)=(:A ! B) (:A ! C) stated by Harrop (1960). Extensions of this rule which are as well admissible but not derivable followed [Mints 76] [Citkin 77] but the question whether there were other admissible rules for IPC than the ones known remained open. In 1975 Friedman posed the problem whether it is decidable if a rule is an admissible rule for IPC or not. In 1984 this question was answered by Rybakov in the ...
Tenacious Tortoises: A formalism for argument over rules of inference
 Computational Dialectics (ECAI 2000 Workshop
, 2000
"... As multiagent systems proliferate and employ different and more sophisticated formal logics, it is increasingly likely that agents will be reasoning with different rules of inference. Hence, an agent seeking to convince another of some proposition may first have to convince the latter to use a rule ..."
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Cited by 10 (7 self)
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As multiagent systems proliferate and employ different and more sophisticated formal logics, it is increasingly likely that agents will be reasoning with different rules of inference. Hence, an agent seeking to convince another of some proposition may first have to convince the latter to use a rule of inference which it has not thus far adopted. We define a formalism to represent degrees of acceptability or validity of rules of inference, to enable autonomous agents to undertake dialogue concerning inference rules. Even when they disagree over the acceptability of a rule, two agents may still use the proposed formalism to reason collaboratively. 1
Proof theory for admissible rules
, 2006
"... The admissible rules of a logic are the rules under which the set of theorems of the logic is closed. In this paper a Gentzenstyle framework is introduced for defining analytic proof systems that derive the admissible rules of various nonclassical logics. Just as Gentzen systems for derivability t ..."
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Cited by 6 (1 self)
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The admissible rules of a logic are the rules under which the set of theorems of the logic is closed. In this paper a Gentzenstyle framework is introduced for defining analytic proof systems that derive the admissible rules of various nonclassical logics. Just as Gentzen systems for derivability treat sequents as basic objects, for admissibility, sequent rules are basic. Proof systems are defined here for the admissible rules of classes of both modal logics, including K4, S4, and GL, and intermediate logics, including Intuitionistic logic IPC, De Morgan (or Jankov) logic KC, and logics BCn (n = 1, 2,...) with bounded cardinality Kripke models. With minor restrictions, proof search in these systems terminates, giving decision procedures for admissibility in the corresponding logics.
A Tableau Method for Checking Rule Admissibility in S4
, 2009
"... Rules that are admissible can be used in any derivations in any axiomatic system of a logic. In this paper we introduce a method for checking the admissibility of rules in the modal logic S4. Our method is based on a standard semantic ground tableau approach. In particular, we reduce rule admissibil ..."
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Cited by 3 (3 self)
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Rules that are admissible can be used in any derivations in any axiomatic system of a logic. In this paper we introduce a method for checking the admissibility of rules in the modal logic S4. Our method is based on a standard semantic ground tableau approach. In particular, we reduce rule admissibility in S4 to satisfiability of a formula in a logic that extends S4. The extended logic is characterised by a class of models that satisfy a variant of the cocover property. The class of models can be formalised by a welldefined firstorder specification. Using a recently introduced framework for synthesising tableau decision procedures this can be turned into a sound, complete and terminating tableau calculus for the extended logic, and gives a tableaubased method for determining the admissibility of rules.
Admissible rules of modal logics
"... We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL.3), by extending the methods of S. Ghilardi and R. Iemhoff. We also investigate the notion of admissible multiple conclusion rules. ..."
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Cited by 2 (0 self)
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We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL.3), by extending the methods of S. Ghilardi and R. Iemhoff. We also investigate the notion of admissible multiple conclusion rules.
A Modal Analysis of some Principles of the Provability Logic of Heyting Arithmetic
 In Proceedings of AiML’98
, 1998
"... In this paper four principles and one scheme of the Provability Logic of (intuitionistic) Heyting Arithmetic HA are studied from a modal logical point of view. These are, besides the wellknown principles K;K4 and L of the Provability Logic of Peano Arihtmetic, the principle 2(A B) ! 2(A 2B) and ..."
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In this paper four principles and one scheme of the Provability Logic of (intuitionistic) Heyting Arithmetic HA are studied from a modal logical point of view. These are, besides the wellknown principles K;K4 and L of the Provability Logic of Peano Arihtmetic, the principle 2(A B) ! 2(A 2B) and the scheme 2::(2A ! W 2B i ) ! 2(2A ! W 2B i ). Intuitionistic modal logic is introduced. And for all principles and for the scheme considered (seperately and together) frame completeness is proved, and the finite model property is established. 1 Introduction In contrast to the situation in classical Peano Arithmetic PA, the provability logic, to be defined in the next section, of intuitionistic Heyting Arithmetic HA appears to be a rich collection of principles. The full provability logic of HA is not known (yet?), but some principles have already been found [Leivant 75, Visser 94], among which are the following. K 2(A ! B) ! (2A ! 2B) K4 2A ! 22A L 2(2A ! A) ! 2A (Lob's princip...
Canonical rules
, 2009
"... We develop canonical rules capable of axiomatizing all systems of multipleconclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, ..."
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Cited by 1 (0 self)
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We develop canonical rules capable of axiomatizing all systems of multipleconclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumptionfree rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multipleconclusion rules or (finitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.
Hypersequent Systems for the Admissible Rules of Modal and Intermediate Logics
"... Abstract. The admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In a previous paper by the authors, formal systems for deriving the admissible rules of Intuitionistic Logic and a class of modal logics were defined in a prooftheoretic framework where ..."
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Cited by 1 (1 self)
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Abstract. The admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In a previous paper by the authors, formal systems for deriving the admissible rules of Intuitionistic Logic and a class of modal logics were defined in a prooftheoretic framework where the basic objects of the systems are sequent rules. Here, the framework is extended to cover derivability of the admissible rules of intermediate logics and a wider class of modal logics, in this case, by taking hypersequent rules as the basic objects. 1
A(nother) characterization of Intuitionistic Propositional Logic
, 2000
"... In [Iemho] we gave a countable basis V for the admissible rules of IPC. Here we show that there is no proper superintuitionistic logic with the disjunction property for which all rules in V are admissible. This shows that, relative to the disjunction property, IPC is maximal with respect to its ..."
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In [Iemho] we gave a countable basis V for the admissible rules of IPC. Here we show that there is no proper superintuitionistic logic with the disjunction property for which all rules in V are admissible. This shows that, relative to the disjunction property, IPC is maximal with respect to its set of admissible rules. This characterization of IPC is optimal in the sense that no nite subset of V suces. In fact, it is shown that for any nite subset X of V , for one of the proper superintuitionistic logics Dn constructed by De Jongh and Gabbay (1974) all the rules in X are admissible. Moreover, the logic Dn in question is even characterized by X : it is the maximal superintuitionistic logic containing Dn with the disjunction property for which all rules in X are admissible. Finally, the characterization of IPC is proved to be eective by showing that it is eectively reducible to an eective characterization of IPC in terms of the Kleene slash by De Jongh (1970). 1 Intro...