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A New Logical Characterisation of Stable Models and Answer Sets
 In Proc. of NMELP 96, LNCS 1216
, 1997
"... This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of "hereandthere". We show that on logic programs equilib ..."
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Cited by 49 (13 self)
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This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of "hereandthere". We show that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz. We thereby obtain a very simple characterisation of answer set semantics as a form of minimal model reasoning in N2, while equilibrium logic itself provides a natural generalisation of this semantics to arbitrary theories. We discuss briefly some consequences and applications of this result. 1 Introduction By contrast with the minimal model style of reasoning characteristic of several approaches to the semantics of logic programs, the stable model semantics of Gelfond and Lifschitz [8] was, from the outset, much closer in spirit to the styles of reasoning found in othe...
Splittings and the finite model property
 Journal of Symbolic Logic
, 1993
"... An old and conjecture of modal logics states that every splitting of the major systems K4, S4 and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof tec ..."
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Cited by 13 (6 self)
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An old and conjecture of modal logics states that every splitting of the major systems K4, S4 and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely that they preserve the finite model property in the following sense. Whenever an extension Λ has fmp so does the splitting Λ / f of Λ by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive. 1 Splittings and the finite model property 2
Normal Monomodal Logics Can Simulate All Others
 Journal of Symbolic Logic
, 1999
"... This paper shows that nonnormal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new ..."
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This paper shows that nonnormal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic. Normal monomodal logics can simulate all others 1 This paper is dedicated to our teacher, Wolfgang Rautenberg x1. Introduction. A simulation of a logic by a logic \Theta is a translation of the expressions of the language for into the language of \Theta such that the consequence relation defined by is reflected under the translation by the consequence relation of \Theta. A wellknown case is provided by the Godel translation, which simulates intuitionistic logic by Grzegorczyk's logic (cf. [11] and [5]). Such simulations not only yield technical results but may also ...
An almost general splitting theorem for modal logic
 Studia Logica
, 1990
"... Given a normal (multi)modal logic Θ, a characterization is given of the finitely presentable algebras A whose logics LA split the lattice of normal extensions of Θ. This is a substantial generalization of [Rautenberg, 1980; Rautenberg, 1977] in which Θ is assumed to be weakly transitive and A to ..."
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Cited by 10 (4 self)
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Given a normal (multi)modal logic Θ, a characterization is given of the finitely presentable algebras A whose logics LA split the lattice of normal extensions of Θ. This is a substantial generalization of [Rautenberg, 1980; Rautenberg, 1977] in which Θ is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by [Blok, 1978] that for all cyclefree and finite A LA splits the lattice of normal extensions of K. Although we firmly believe it to be true, we have not been able to prove that if a logic Λ splits the lattice of extensions of Θ then Λ is the logic of an algebra finitely presentable over Θ; in this respect our result remains partial. A
Relevant Deduction. From Solving Paradoxes Towards a General Theory
 Erkenntnis
, 1991
"... Abstract: This paper presents an outline of a new theory of relevant deduction which arose from the purpose of solving paradoxes in various fields of analytic philosophy. In distinction to relevance logics, this approach does not replace classical logic by a new one, but distinguishes between releva ..."
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Abstract: This paper presents an outline of a new theory of relevant deduction which arose from the purpose of solving paradoxes in various fields of analytic philosophy. In distinction to relevance logics, this approach does not replace classical logic by a new one, but distinguishes between relevance and validity. It is argued that irrelevant arguments are, although formally valid, nonsensical and even harmful in practical applications. The basic idea is this: a valid deduction is relevant iff no subformula of the conclusion is replaceable on some of its occurrences by any other formula salva validitate of the deduction. The paper first motivates the approach by showing that four paradoxes seemingly very distant from each other have a common source. Then the exact definition of relevant deduction is given and its logical properties are investigated. An extension to relevance of premises is discussed. Finally the paper presents an overview of its applications in philosophy of science, ethics, cognitive psychology and artificial intelligence.
On logics with coimplication
 Journal of Philosophical Logic
, 1998
"... This paper investigates (modal) extensions of HeytingBrouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the God ..."
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Cited by 7 (1 self)
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This paper investigates (modal) extensions of HeytingBrouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the Godelembedding of intuitionistic logic into S4, itisshown that all (modal) extensions of HeytingBrouwer logic can be embedded into tense logics (with additional modal operators). An extension of the BlokEsakiaTheorem is proved for this embedding. 1
Diamonds are a Philosopher's Best Friends. The Knowability Paradox and Modal Epistemic Relevance Logic (Extended Abstract)
 Journal of Philosophical Logic
, 2002
"... Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the ..."
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Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the antirealist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.
Logic and Control: How They Determine the Behaviour of Presuppositions
 Logic and Information Flow
, 1994
"... Presupposition is one of the most important phenomena of nonclassical logic as concerns the applications in philosophy, linguistics and computer science. The literature on presuppositions in linguistics and analytic philosophy is rather rich (see [8] and the references therein), and there have been ..."
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Cited by 7 (3 self)
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Presupposition is one of the most important phenomena of nonclassical logic as concerns the applications in philosophy, linguistics and computer science. The literature on presuppositions in linguistics and analytic philosophy is rather rich (see [8] and the references therein), and there have been numerous attempts in philosophical logic to solve problems arising in in connection with presuppositions such as the projection problem. In this essay I will introduce a system of logics with control structure and elucidate the relation between contextchange potential, presupposition projection and threevalued logic. For a definition of what presuppositions are consider these three sentences. (1) Hilary is not a bachelor. (2) The present king of France is not bald. (3) limn→ ∞ an � 4 Each of these sentences is negative and yet there is something that we can infer from them as well as from their positive counterparts; namely the following. (1 † ) Hilary is male.
Subdirectly irreducible modal algebras and initial frames
 Studia Logica
, 1999
"... initial frames ..."
Characterising Combinational Timing Analyses in Intuitionistic Modal Logic
, 2000
"... The paper presents a new logical specification language, called Propositional Stabilisation Theory (PST), to capture the stabilisation behaviour of combinational inputoutput systems. PST is an intuitionistic propositional modal logic interpreted over sets of waveforms. The language is more economic ..."
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Cited by 5 (3 self)
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The paper presents a new logical specification language, called Propositional Stabilisation Theory (PST), to capture the stabilisation behaviour of combinational inputoutput systems. PST is an intuitionistic propositional modal logic interpreted over sets of waveforms. The language is more economic than conventional specification formalisms such as timed Boolean functions, temporal logic, or predicate logic in that it separates function from time and only introduces as much syntax as is necessary to deal with stabilisation behaviour. It is a purely propositional system but has secondorder expressiveness. One and the same Boolean function can be represented in various ways as a PST formula, giving rise to different timing models which associate different stabilisation delays with different parts of the functionality and adjust the granularity of the datadependency of delays within wide margins. We show how several standard timing analyses can be characterised as algorithms computing c...