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Modulated Fibring and the Collapsing Problem
, 2001
"... Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse ..."
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Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem. 1
Fibring Logics with Topos Semantics
, 2002
"... The concept of fibring is extended to higherorder logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the metatheorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich ..."
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Cited by 11 (6 self)
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The concept of fibring is extended to higherorder logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the metatheorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich logics. This result is extended to weaker logics in the cases where fibring preserves conservativeness of HOLenrichments. Soundness is shown to be preserved by fibring without any further assumptions.
NonTruthFunctional Fibred Semantics
, 2001
"... wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are n ..."
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Cited by 7 (4 self)
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wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are non{truth{functional (ntf) rooms . For simplicity, we shall only work at this level of abstraction. As shown in [3], everything can be smoothly lifted to the fully edged indexed case. In the sequel, AlgSig' denotes the category of algebraic many sorted signatures with a distinguished sort ' (for formulae) and morphisms preserving it. Given one such signature , we denote by Alg() the category of {algebras and {algebra homomorphisms, and by cAlg() the class of all pairs hA; i with A a<
Completeness Results for Fibred Parchments Beyond the Propositional Base
 Recent Trends in Algebraic Development Techniques  Selected Papers, volume 2755 of Lecture Notes in Computer Science
, 2003
"... In [6] it was shown that fibring could be used to combine institutions presented as cparchments, and several completeness preservation results were established. However, their scope of applicability was limited to propositionalbased logics. Herein, we extend these results to a broader class of ..."
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Cited by 4 (3 self)
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In [6] it was shown that fibring could be used to combine institutions presented as cparchments, and several completeness preservation results were established. However, their scope of applicability was limited to propositionalbased logics. Herein, we extend these results to a broader class of logics, possibly including variables, terms and quantifiers.
Heterogeneous fibring of deductive systems via abstract proof systems
, 2005
"... Fibring is a metalogical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have ..."
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Fibring is a metalogical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have different presentations (e.g. one presented by a Hilbert calculus and the other by a sequent calculus), has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semidecidability is investigated.
On constructing fibred tableau calculus for BDI logics
 In 9th Pacific Rim International Conference on Artificial Intelligence (PRICAI06
, 2006
"... Abstract. In [12, 16] we showed how to combine propositional BDI logics using Gabbay’s fibring methodology. In this paper we extend the above mentioned works by providing a tableaubased decision procedure for the combined/fibred logics. To achieve this end we first outline with an example two type ..."
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Abstract. In [12, 16] we showed how to combine propositional BDI logics using Gabbay’s fibring methodology. In this paper we extend the above mentioned works by providing a tableaubased decision procedure for the combined/fibred logics. To achieve this end we first outline with an example two types of tableau systems, (graph & path), and discuss why both are inadequate in the case of fibring. Having done that we show how to uniformly construct a tableau calculus for the combined logic using Governatori’s labelled tableau system KEM.
Preservation of interpolation by fibring
 In Carnielli et al. [2004a
"... The method of fibring for combining logics as originally proposed by Gabbay [13, 14], includes some other methods as fusion [29] as a special case. Albeit fusion is the best developed mechanism, mainly in what concerns preservation of properties as ..."
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The method of fibring for combining logics as originally proposed by Gabbay [13, 14], includes some other methods as fusion [29] as a special case. Albeit fusion is the best developed mechanism, mainly in what concerns preservation of properties as
Hierarchical Logical Consequence
"... The modern view of logical reasoning as modeled by a consequence operator (instead of simply by a set of theorems) has allowed for huge developments in the study of logic as an abstract discipline. Still, it is unable to explain why it is often the case that the same designation is used, in an ambig ..."
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The modern view of logical reasoning as modeled by a consequence operator (instead of simply by a set of theorems) has allowed for huge developments in the study of logic as an abstract discipline. Still, it is unable to explain why it is often the case that the same designation is used, in an ambiguous way, to describe several distinct modes of reasoning over the same logical language. A paradigmatic example of such a situation is ‘modal logic’, a designation which can encompass reasoning over Kripke frames, but also over Kripke models, and in any case either locally (at a fixed world) or globally (at all worlds). Herein, we adopt a novel abstract notion of logic presented as a latticestructured hierarchy of consequence operators, and explore some common prooftheoretic and modeltheoretic ways of presenting such hierarchies through a collection of meaningful examples. In order to illustrate the usefulness of the notion of hierarchical consequence operators we address a few questions in the theory of combined logics, where a suitable abstract presentation of the logics being combined is absolutely essential, and we show how to define and achieve a number of interesting preservation results for fibring, in the context of 2hierarchies.
NonTruthFunctional Fibred Semantics
"... Abstract Until recently, truthfunctionality has been considered essential to the mechanism for combining logics known as fibring. Following the first efforts towards extending fibred semantics to logics with nontruthfunctional operators, this paper aims to clarify the subject at the light of ideas ..."
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Abstract Until recently, truthfunctionality has been considered essential to the mechanism for combining logics known as fibring. Following the first efforts towards extending fibred semantics to logics with nontruthfunctional operators, this paper aims to clarify the subject at the light of ideas borrowed from the theory of general logics as institutions and the novel notion of nontruthfunctional room. Besides introducing the relevant concepts and constructions, the paper presents a detailed worked example combining classical firstorder logic with the paraconsistent propositional system C1, for which a meaningful semantics is obtained. The possibility of extending this technique to build firstorder versions of further logics of formal inconsistency is also discussed.