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 "here-and-there". 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...

This paper shows that non--normal 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 well--known 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 ...

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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 anti-realist 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.

This paper investigates (modal) extensions of Heyting-Brouwer 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 Godel-embedding of intuitionistic logic into S4, itisshown that all (modal) extensions of Heyting-Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok-Esakia-Theorem is proved for this embedding. 1

Presupposition is one of the most important phenomena of non-classical 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 context-change potential, presupposition projection and three-valued 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.

The paper presents a new logical specification language, called Propositional Stabilisation Theory (PST), to capture the stabilisation behaviour of combinational input-output 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 data-dependency of delays within wide margins. We show how several standard timing analyses can be characterised as algorithms computing c...

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This chapter considers theorem proving for discrete temporal logics. We are interested in deciding or at least enumerating the formulas of the logic which are valid, that is, are true in all circumstances. Most of the techniques for temporal theorem-proving have been extensions for methods developed for classical logics but completely novel techniques have also been developed. Initially we concentrate on discrete linear-time temporal logics, describing axiomatic, tableau, automata and resolution based approaches. The application of these approaches to other temporal logics is discussed. 1 Introduction Readers of this handbook will be aware of the wide variety of useful tasks which require reasoning about time. There are many applications of temporal reasoning tasks to problems of knowledge changing, to planning, to processing natural language, to managing the interchange of information, and to developing complex systems. There are a wide variety of temporal logics available in which s...

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