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26
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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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.
Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
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Cited by 8 (5 self)
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Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result
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 6 (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.
Recovering a logic from its fragments by metafibring. Logica Universalis
 In print. Preliminary version available at CLE ePrints 5(4), 2005. URL = http://www.cle.unicamp.br/eprints/vol 5,n 4,2005.html
"... In this paper we address the question of recovering a logic system by combining two or more fragments of it. We show that, in general, by fibring two or more fragments of a given logic the resulting logic is weaker than the original one, because some metaproperties of the connectives are lost after ..."
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Cited by 5 (3 self)
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In this paper we address the question of recovering a logic system by combining two or more fragments of it. We show that, in general, by fibring two or more fragments of a given logic the resulting logic is weaker than the original one, because some metaproperties of the connectives are lost after the combination process. In order to overcome this problem, the categories Mcon and Seq of multipleconclusion consequence relations and sequent calculi, respectively, are introduced. The main feature of these categories is the preservation, by morphisms, of metaproperties of the consequence relations, which allows, in several cases, to recover a logic by fibring of its fragments. The fibring in this categories is called metafibring. Several examples of wellknown logics which can be recovered by metafibring its fragments (in opposition to fibring in the usual categories) are given. Finally, a general semantics for objects in Seq (and, in particular, for objects in Mcon) is proposed, obtaining a category of logic systems
Cut elimination for a class of propositional based logics
, 2005
"... Sufficient conditions for propositional based logics to enjoy cut elimination are established. These conditions are satisfied by a wide class of logics encompassing among others classical and intuitionistic logic, modal logic S4, and classical and intuitionistic linear logic and some of their fragme ..."
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Sufficient conditions for propositional based logics to enjoy cut elimination are established. These conditions are satisfied by a wide class of logics encompassing among others classical and intuitionistic logic, modal logic S4, and classical and intuitionistic linear logic and some of their fragments. The class of logics is characterized by the type of rules and provisos used in their sequent calculi. The conditions can be checked in finite time and define relations between the rules and the provisos so that the calculus can enjoy cut elimination. A general proof of cut elimination is presented for any calculus satisfying those conditions.
Graphtheoretic fibring of logics
 Part II  Completeness preservation. Preprint, SQIG  IT and IST  TU Lisbon
, 2008
"... A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an mgraph where the nodes and the medges include the sorts and the constructors of the signatu ..."
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Cited by 4 (3 self)
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A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an mgraph where the nodes and the medges include the sorts and the constructors of the signatures at hand. Fibring of two models is an mgraph where the nodes and the medges are the values and the operations in the models, respectively. Fibring of two deductive systems is an mgraph whose nodes are language expressions and the medges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graphtheoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with nondeterministic semantics, logics with an algebraic semantics, logics with partial semantics, and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. 1
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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Cited by 3 (1 self)
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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.
Intuitionistic Logic with Classical Atoms
"... In this paper, we define a Hilbertstyle axiom system IPCCA that conservatively extends intuitionistic propositional logic (IPC) by adding new classical atoms for which the law of excluded middle (LEM) holds. We establish completeness of IPCCA with respect to an appropriate class of Kripke models. W ..."
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Cited by 2 (0 self)
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In this paper, we define a Hilbertstyle axiom system IPCCA that conservatively extends intuitionistic propositional logic (IPC) by adding new classical atoms for which the law of excluded middle (LEM) holds. We establish completeness of IPCCA with respect to an appropriate class of Kripke models. We show that IPCCA is a conservative extension of both classical propositional logic (CPC) and also IPC. We further investigate the disjunction property in IPCCA. In particular, we show that the disjunction property holds for every formula A ∨ B if either A or B does not contain classical atoms. 1 In our previous paper, we discussed the intuitionistic version of a basic Logic of Proofs [Kurokawa, 2003]. From a semantical point of view the formulas of the form x: F, denoting x is a proof of F, behave as if they were classical propositions put on Kripke models of intuitionistic propositional logic. A question appeared of how we can describe a logic in which we have propositional variables of two sorts: those that behave classically and those that behave intuitionistically. What results is a logic that combines many essential features of IPC and CPC. 1The results of this paper were obtained in November 2003. Early in 2004, the author learned from an FOM posting about A. Sakharov’s paper “Median Logic ” (submitted on February 6, 2004 to the Mathematics Preprint Server), on a firstorder intuitionistic logic with classical propositional atoms. In Sakharov’s paper, a relevant proof system with some weak form of cutelimination is given. Apart from a common design idea, this paper and “Median Logic ” do not have a significant overlap. 1 The present work is not the first attempt at such a blending of the two logics. We will examine the history of the relationship between IPC and CPC. There are numerous examples of the socalled “intermediate logics, ” which are obtained from IPC by adding axiom schemas that are weaker than those of classical logic. For instance, Dummett logic is obtained by adding the axiom schema (A → B) ∨ (B → A) to IPC. Theorems of this logic are valid in all linear Kripke models. For a survey of intermediate logics one could look at Chagrov
Interpolation via translations
"... A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties betwe ..."
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A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a KolmogorovGentzenGödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a KiriyamaOno style translation with features of a KolmogorovGentzenGödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. AMS Classification: 03C40, 03F03, 03B22
Conservativity of fibred logics without shared connectives
 SQIG  Instituto de Telecomunicações and IST  U
, 2014
"... Fibring is a general mechanism for combining logics that provides valuable insight on designing and understanding complex logical systems. Mostly, the research on fibring has focused on its model and prooftheoretic aspects, and on transference results for relevant metalogical properties. Conservati ..."
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Fibring is a general mechanism for combining logics that provides valuable insight on designing and understanding complex logical systems. Mostly, the research on fibring has focused on its model and prooftheoretic aspects, and on transference results for relevant metalogical properties. Conservativity, however, a property that lies at the very heart of the original definition of fibring, has not deserved similar attention. In this paper, we provide the first full characterization of the conservativity of fibred logics, in the special case when the logics being combined do not share connectives. Namely, under this assumption, we provide necessary and sufficient conditions for a fibred logic to be a conservative extension of the logics being combined. Our characterization relies on a key technical result that provides a complete description of what follows from a set of nonmixed hypotheses in the fibred logic, in terms of the logics being combined. With such a powerful tool in hand, we also explore a semantic application. Namely, we use our key technical result to show that finitevaluedness is not preserved by fibring.