Heterogeneous Fibring of Deductive Systems Via Abstract Proof Systems

Cruz-Filipe, Luís; Sernadas, A.; Sernadas, Cristina

doi:10.1093/jigpal/jzm057

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…SeeCruz-Filipe et al (2007), proposition 2.17, for a slightly stronger result. In particular, the juxtaposition of two nontrivial consequence relations over disjoint signatures is a strong conservative extension of each of them.…”

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confidence: 97%

“…But that would have yielded a much less interesting language. 23Cruz-Filipe et al (2007) defines a more general notion of the fibring of consequence relations that also applies in the case of nonsubstitution invariant consequence relations. By propositions 2.18 and 2.19 in that paper, the fibring of two substitution invariant consequence relations coincides with their juxtaposition.…”

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Juxtaposition: A New Way to Combine Logics

Schechter

2011

The Review of Symbolic Logic

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This paper develops a new framework for combining propositional logics, called "juxtaposition." Several general metalogical theorems are proved concerning the combination of logics by juxtaposition. In particular, it is shown that under reasonable conditions, juxtaposition preserves strong soundness. Under reasonable conditions, the juxtaposition of two consequence relations is a conservative extension of each of them. A general strong completeness result is proved. The paper then examines the philosophically important case of the combination of classical and intuitionist logics. Particular attention is paid to the phenomenon of collapse. It is shown that there are logics with two stocks of classical or intuitionist connectives that do not collapse. Finally, the paper briefly investigates the question of which rules, when added to these logics, lead to collapse. §1. Introduction. Methods of combining logics are of great interest. 1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multiple kinds of modal operators-epistemic, temporal, and deontic.There are also purely philosophical reasons to be interested in the combination of logics. One illustration of this comes from so-called collapse theorems.Suppose there is a language with two stocks of the usual logical connectives (for propositional or for first-order logic)-∧ 1 and ∧ 2 , ∨ 1 and ∨ 2 , and so on. There is a well-known result which states that given any logic for this language such that each logical constant obeys the usual natural deduction rules (for classical or even for intuitionist logic), sentences that differ only in some or all of their subscripts are intersubstitutable. 2 Corresponding connectives, such as ∧ 1 and ∧ 2 , behave as mere notational variants. In such a logic, if one stock of constants obeys the classical natural deduction rules, so does the other. This result has been used to argue for several striking philosophical theses. For instance, it has been used to argue that the logical constants in our language have unique and determinate extensions. The argument goes as follows: Suppose that one of our logical constants had an indeterminate extension. Then we could precisify the term. We could introduce two (or more) precise terms into our language with distinct extensions. These terms would presumably obey the usual natural deduction rules (understood to apply to the expanded language). But since the terms would have distinct extensions, they wouldnot be Received: February 17, 2011. 1 Methods of combining logics include fibring and its variants, as developed by Gabbay, as well as algebraic fibring and its variants, as developed by A. Sernadas and his collaborators. See Gabbay (1998) and Carnielli et al. (2008), respectively, for comprehensive overviews of this work. See Caleiro et al. (2005) for a summary of the central results concerning algebraic fibring. 2 See McG...

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Juxtaposition: A New Way to Combine Logics

Schechter

2011

The Review of Symbolic Logic

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“…The relevant material for this Chapter is the work presented in [68]. We refrain here of considering some preservation results that are there namely related to strong and weak semi-decidability.…”

Section: Heterogeneous Fibringmentioning

confidence: 99%

Analysis and Synthesis of Logics

Gabbay

Carnieli²,

Coniglio³

et al. 2008

Applied Logic Series

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SCOPE OF THE SERIESLogic is applied in an increasingly wide variety of disciplines, from the traditional subjects of philosophy and mathematics to the more recent disciplines of cognitive science, computer science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject. Springer, through its Applied Logic Series, seeks to provide a home for outstanding books and research monographs in applied logic, and in doing so demonstrates the underlying unity and applicability of logic.The titles published in this series are listed at the end of this volume. Library of Congress Control Number: 2008920294ISBN 978-1-4020-6781-5 (HB) ISBN 978-1-4020-6782-2 (e-book)Published by Springer,

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“…By considering the category of standard consequence relations (see [4,11,13]) then (ξ ∨ ¬ξ) is not a valid formula of the fibring L = C ¬∨ , of the consequence systems corresponding to classical negation L ¬ = C ¬ , 1 and classical disjunction L ∨ = C ∨ , 2 . In order to see this, consider the following matrices:…”

Section: Recovering a Logic 393mentioning

confidence: 99%

“…By considering the category Hil of Hilbert propositional calculi (see for instance [4,11,13,18,22]) which is frequently used to define fibring, then the result of Example 9 cannot be obtained. That is, if we compute the fibring of the Hilbert calculi corresponding to classical implication and to classical negation, respectively, we cannot recover classical logic.…”

Section: See Examples 4 and 6) Produces The {⇒ ¬}-Fragment Of Intuitmentioning

confidence: 99%

Recovering a Logic from Its Fragments by Meta-Fibring

Coniglio

2007

Log. univers.

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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 meta-properties of the connectives are lost after the combination process. In order to overcome this problem, the categories Mcon and Seq of multiple-conclusion consequence relations and sequent calculi, respectively, are introduced. The main feature of these categories is the preservation, by morphisms, of meta-properties 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 meta-fibring. Several examples of well-known logics which can be recovered by meta-fibring 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 called Log. A general theorem of preservation of completeness by fibring in Log is also obtained. Mathematics Subject Classification (2000). Primary 99Z99; Secondary 00A00.

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Heterogeneous Fibring of Deductive Systems Via Abstract Proof Systems

Cited by 5 publications

References 22 publications

Juxtaposition: A New Way to Combine Logics

Juxtaposition: A New Way to Combine Logics

Analysis and Synthesis of Logics

Recovering a Logic from Its Fragments by Meta-Fibring

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