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This report continues the study of quantified equilibrium logic, QEL, introduced in [25, 26], and its monotonic base logic, here-and-there. We present a slightly modified version of QEL where the so-called unique name assumption or UNA is... more
This report continues the study of quantified equilibrium logic, QEL, introduced in [25, 26], and its monotonic base logic, here-and-there. We present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternative axiom set for first-order here-and-there. The new system appears to be simpler as well as making it easier to derive some simple semantic ...
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The study of abstract properties of nonmonotonic inference has thrown up a number of general conditions on inference relations that are often thought to be desirable, and sometimes even essential, for an adequate system of nonmonotonic... more
The study of abstract properties of nonmonotonic inference has thrown up a number of general conditions on inference relations that are often thought to be desirable, and sometimes even essential, for an adequate system of nonmonotonic reasoning. However, several of the key conditions on inference that have been proposed in the literature make explicit reference to the classical concept of logical consequence, and there is a general tendency to focus attention on inference operations that are supraclassical in the sense of ...
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Logic programming research has been based largely on Horn-clause logic, because definite programs can be interpreted efficiently using SLD-resolution. However, there are several reasons to extend the ideas and concepts of Hornclause logic... more
Logic programming research has been based largely on Horn-clause logic, because definite programs can be interpreted efficiently using SLD-resolution. However, there are several reasons to extend the ideas and concepts of Hornclause logic programming to more general formulas. The paper offers a framework for discussing questions of constructivity and completeness that arise in the field of clause logic programming. Constructive properties of different calculi are investigated and their relation to a certain family of constructive logics ...
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As a doxastic counterpart to epistemic logic based on S5 we study the modal logic KSD that can be viewed as an approach to modelling a kind of objective and fair belief. We apply KSD to the problem of minimal belief and develop an... more
As a doxastic counterpart to epistemic logic based on S5 we study the modal logic KSD that can be viewed as an approach to modelling a kind of objective and fair belief. We apply KSD to the problem of minimal belief and develop an alternative approach to nonmonotonic modal logic using a weaker concept of expansion. This corresponds to a certain minimal kind of KSD model and yields a new type of nonmonotonic doxastic reasoning.
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This paper is devoted to logical aspects of two closely related semantics for logic programs: the partial stable model semantics of Przymusinski [20] and the well-founded semantics of Van Gelder, Ross and Schlipf [24]. For many years the... more
This paper is devoted to logical aspects of two closely related semantics for logic programs: the partial stable model semantics of Przymusinski [20] and the well-founded semantics of Van Gelder, Ross and Schlipf [24]. For many years the following problem remained open: Which (non-modal) logic can be regarded as yielding an adequate foundation for these semantics in the sense that its minimal models (appropriately defined) coincide with the partial stable models of a logic program? Initial work on this problem was undertaken by ...
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Equilibrium logic is an approach to nonmonotonic reasoning that generalises the stable model and answer set semantics for logic programs. We present a method to implement equilibrium logic and, as a special case, stable models for logic... more
Equilibrium logic is an approach to nonmonotonic reasoning that generalises the stable model and answer set semantics for logic programs. We present a method to implement equilibrium logic and, as a special case, stable models for logic programs with nested expressions, based on polynomial reductions to quantified Boolean formulas (QBFs). Since there now exist efficient QBF-solvers, this reduction technique yields a practically relevant approach to rapid prototyping. The reductions for logic programs with nested expressions ...
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We introduce and study a modal logic wK4f that is related to the idea of minimal belief in much the same way as its strengthening, S4F, has been shown to be related to the idea of minimal knowledge. wK4f can be obtained by adding a... more
We introduce and study a modal logic wK4f that is related to the idea of minimal belief in much the same way as its strengthening, S4F, has been shown to be related to the idea of minimal knowledge. wK4f can be obtained by adding a weakened version of axiom F to the modal logic wK4. We show that, like S4F, wK4f is sound and complete with respect to the class of all minimal topological spaces ie topological spaces with only three open sets. We describe the rooted frames of wK4f by quadruples of natural numbers. Finally we ...
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The objective of the present paper is to introduce a new approach to the study of the logical structure of scientific theories. We shall consider, in broad outline, a basic framework for handling theories and for analyzing a number of key... more
The objective of the present paper is to introduce a new approach to the study of the logical structure of scientific theories. We shall consider, in broad outline, a basic framework for handling theories and for analyzing a number of key metatheoretical concepts. The framework itself may be formally captured within a suitable system of set theory. However, its major innovative and heuristic thrust derives from some notions employed in modern abstract logic; in particular, it encourages the application of model-theoretic concepts and ...
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QHT is a first-order super-intuitionistic logic that provides a foundation for answer set programming (ASP) and a useful tool for analysing and transforming non-ground programs. We recall some properties of QHT and its nonmonotonic... more
QHT is a first-order super-intuitionistic logic that provides a foundation for answer set programming (ASP) and a useful tool for analysing and transforming non-ground programs. We recall some properties of QHT and its nonmonotonic extension, quantified equilibrium logic (QEL). We show how the proof theory of QHT can be used to extend to non-ground programs previous results on the completeness of θ-subsumption. We also establish a reduction of QHT to classical logic and show how this can be used to obtain and extend ...
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Equilibrium logic, introduced in [20], is a conservative extension of answer set semantics for logic programs to the full language of propositional logic. In this paper we initiate the study of first-order variants of equilibrium logic.... more
Equilibrium logic, introduced in [20], is a conservative extension of answer set semantics for logic programs to the full language of propositional logic. In this paper we initiate the study of first-order variants of equilibrium logic. In particular, we focus on a quantified version QN 5 of the propositional many-valued logic N 5 of here-and-there with strong negation, and define the condition of equilibrium via a minimal model construction. We verify Skolem forms and Herbrand theorems for QN 5 and show that, like its propositional counterpart, the ...
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We apply tableau methods to the problem of computing entailment in the nonmonotonic system of equilibrium logic, a generalisation of the inference relation associated with the stable model and answer set semantics for logic programs. We... more
We apply tableau methods to the problem of computing entailment in the nonmonotonic system of equilibrium logic, a generalisation of the inference relation associated with the stable model and answer set semantics for logic programs. We describe tableau calculi for the nonclassical logics underlying equilibrium entailment, namely here-and-there with strong negation and its strengthening classical logic with strong negation. A further tableau calculus is then presented for computing equilibrium entailment. This makes use of a new ...
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We discuss equilibrium logic, first presented in Pearce (1997), as a system of nonmonotonic reasoning based on the nonclassical logic N 5 of here-and-there with strong negation. Equilibrium logic is a conservative extension of answer set... more
We discuss equilibrium logic, first presented in Pearce (1997), as a system of nonmonotonic reasoning based on the nonclassical logic N 5 of here-and-there with strong negation. Equilibrium logic is a conservative extension of answer set inference, not only for extended, disjunctive logic programs, but also for significant extensions such as the programs with nested expressions described by Lifschitz, Tang and Turner (forthcoming). It provides a theoretical basis for extending the paradigm of answer set programming beyond current ...
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Equilibrium logic is an approach to non-monotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case ofnested logic programs, where arbitrary Boolean... more
Equilibrium logic is an approach to non-monotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case ofnested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs intoquantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of second-order logic, and its formulas are usually referred to asquantified Boolean formulas(QBFs). We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some QBF such that the latter is valid preci...
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In this note I wish to consider an interesting problem arising in the theory of truthlikeness or verisimilitude, and to comment on a curious argument of Oddie [I981] that relates to the problem. Briefly put, the problem is this: Should... more
In this note I wish to consider an interesting problem arising in the theory of truthlikeness or verisimilitude, and to comment on a curious argument of Oddie [I981] that relates to the problem. Briefly put, the problem is this: Should measures of truthlikeness be invariant under translation? That is, should the degree of truthlikeness of statement 4 or a theory T agree with the degree of truthlikeness of the translation of, or T in another (equivalent) language or conceptual framework? Oddie's argument is designed to justify a negative reply to this ...
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The doctrines of scientific realism have enjoyed a close and enduring, if not always harmonious, association with Tarski's semantic conception of truth and theories of formal semantics generally. From its inception... more
The doctrines of scientific realism have enjoyed a close and enduring, if not always harmonious, association with Tarski's semantic conception of truth and theories of formal semantics generally. From its inception Tarski's theory received unqualified support from some realists, like Karl Popper, who saw it as legitimizing the use of semantic notions in epistemology and the philosophy of science. Realist theses like'theoretical as well as observational terms in science refer to real entities'; and'scientific laws consist of ...
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78 DAVID PEARCE arbitrary function f of T such as appears in the structure x listed above, and try to make precise the notion of a method of determination for f in T or, equivalently, to say what is meant by an f-determining model of T. A... more
78 DAVID PEARCE arbitrary function f of T such as appears in the structure x listed above, and try to make precise the notion of a method of determination for f in T or, equivalently, to say what is meant by an f-determining model of T. A first formulation of C is then provided, followed by a series of possible objections to C accompanied in each case by suggestions for adjusting C so as to meet the objection. In conclusion, the authors make it clear they feel that an informally presented revised criterion, incorporating all the required modifications, ...
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There are three fundamental components in the methodology of theory reconstruction associated with what has become known as the structuralist account of empirical theories. The first of these is an insistence upon axiomatization by means... more
There are three fundamental components in the methodology of theory reconstruction associated with what has become known as the structuralist account of empirical theories. The first of these is an insistence upon axiomatization by means of a set-theoretic predicate; the second may be identified with the formal apparatus for describing the logical structure and empirical content of theories, as developed by JD Sneed and his collaborators. The third position, forcefully argued by Wolfgang Stegmiiller is roughly the view that any ...
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Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti. cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set... more
Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti. cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's. rst-order logic as a basis for de. ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove ...
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Schroeder-Heister and Schaefer (1989) provide a valuable analysis of certain logical relations that may hold between scientific languages and theories. In particular, they define two purely syntactic relations, representation and... more
Schroeder-Heister and Schaefer (1989) provide a valuable analysis of certain logical relations that may hold between scientific languages and theories. In particular, they define two purely syntactic relations, representation and commensurability, which they compare and contrast with a model-theoretic relation of reduction, as explicated within the so-called structuralist view of theories developed by Sneed, Stegmiiller and others. The main methodological consequence they draw from this analysis is a defense of Stegmiiller's ...
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We revive the idea that a deductive-nomological explanation of a scientific theory by its successor may be defensible, even in those common and troublesome cases where the theories concerned are mutually incompatible; and limiting,... more
We revive the idea that a deductive-nomological explanation of a scientific theory by its successor may be defensible, even in those common and troublesome cases where the theories concerned are mutually incompatible; and limiting, approximating and counterfactual assumptions may be required in order to define a logical relation between them. Our solution is based on a general characterization of limiting relations between physical theories using the method of nonstandard analysis.
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It is well known that every normal modal logic L has an algebraic counterpart based on boolean algebras with operators. The axioms of the logic are translated into the equations of the corresponding algebraic theory. All in all we get a... more
It is well known that every normal modal logic L has an algebraic counterpart based on boolean algebras with operators. The axioms of the logic are translated into the equations of the corresponding algebraic theory. All in all we get a variety VL of Boolean algebras with operators determined by these equations. In the literature this variety is often referred to as the variety of L-algebras. The variety of L-algebras contains a special algebra, the free algebra on countable generators, also called the Lindenbaum- Tarski algebra of the logic L. The Lindenbaum- ...
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Summary In hisProgress and its Problems, Laudan dismisses the problem of incommensurability in science by endorsing two general assertions. The first claims there are actually no incommensurable pairs of theories or research traditions;... more
Summary In hisProgress and its Problems, Laudan dismisses the problem of incommensurability in science by endorsing two general assertions. The first claims there are actually no incommensurable pairs of theories or research traditions; the second maintains that his problem-solving model of scientific progress would be able rationally to appraise even incommensurable pairs of theories or traditions (are compare them for their progressiveness). I argue here that Laudan fails to provide a plausible defence of either ...
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Research Interests: Philosophy and Erkenntnis
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Summary This project deals with the study of mathematical foundations, as well as deduction and programming, for several types of non-classical logics, especially multiple-valued logics and intermediate logics, with the aim of providing... more
Summary This project deals with the study of mathematical foundations, as well as deduction and programming, for several types of non-classical logics, especially multiple-valued logics and intermediate logics, with the aim of providing means for intelligent management of information. The classical problem of deduction is, given explicit knowledge expressed in a given formalism together with a set of inference rules, to deduce the implicit knowledge which could be relevant for applications. Specifically, we are interested in ...
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Madrid, November 2006 11 soundness for K so to prove soundness for the system K we must show t The K axiom is true in all models. Given a model〈 W, R, V〉, it suffices to show that if (a) 2 (p→ q) and 2p are true in a world w, then also... more
Madrid, November 2006 11 soundness for K so to prove soundness for the system K we must show t The K axiom is true in all models. Given a model〈 W, R, V〉, it suffices to show that if (a) 2 (p→ q) and 2p are true in a world w, then also (b) 2q is true in w. Suppose (a) holds. Then by (V 2), p→ q and p are true in all w′ such that R (w, w′), so by (V→) so is q. Therefore by (V 2), 2q is true in w. t The transformation rules US, MP and N are validity preserving, ie when applied to formulas true in all models, they lead to formulas true in all ...
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