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  1. Congruences and Ideals on Peirce Algebras: A Heterogeneous/Homogeneous Point of View.Sandra Marques Pinto & M. Teresa Oliveira‐Martins - 2012 - Mathematical Logic Quarterly 58 (4-5):252-262.
    For a Peirce algebra equation image, lattices equation image of all heterogenous Peirce congruences and equation image of all heterogenous Peirce ideals are presented. The notions of kernel of a Peirce congruence and the congruence induced by a Peirce ideal are introduced to describe an isomorphism between equation image and equation image. This isomorphism leads us to conclude that the class of the Peirce algebras is ideal determined. Opposed to Boolean modules case, each part of a Peirce ideal I = (...)
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  • Congruences and Ideals on Boolean Modules: A Heterogeneous Point of View.Sandra Marques Pinto & M. Teresa Oliveira-Martins - 2011 - Mathematical Logic Quarterly 57 (6):571-581.
    Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module equation image are given and the respective lattices equation image and equation image are presented. A characterization of the simple bijective Boolean modules is achieved differing from that given by Brink in a homogeneous approach. We construct the smallest and the greatest modular congruence having the same Boolean part. The same is established for modular ideals. The notions of kernel of a modular congruence and the congruence induced by a (...)
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  • 2000 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium 2000.Carol Wood - 2001 - Bulletin of Symbolic Logic 7 (1):82-163.
  • The Development of Mathematical Logic From Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press.
    The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...)
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  • Heyting Mereology as a Framework for Spatial Reasoning.Thomas Mormann - 2013 - Axiomathes 23 (1):137- 164.
    In this paper it is shown that Heyting and Co-Heyting mereological systems provide a convenient conceptual framework for spatial reasoning, in which spatial concepts such as connectedness, interior parts, (exterior) contact, and boundary can be defined in a natural and intuitively appealing way. This fact refutes the wide-spread contention that mereology cannot deal with the more advanced aspects of spatial reasoning and therefore has to be enhanced by further non-mereological concepts to overcome its congenital limitations. The allegedly unmereological concept of (...)
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  • Philosophy and Cognitive Sciences: Proceedings of the 16th International Wittgenstein Symposium (Kirchberg Am Wechsel, Austria 1993).Roberto Casati & Barry Smith (eds.) - 1994 - Vienna: Wien: Hölder-Pichler-Tempsky.
    Online collection of papers by Devitt, Dretske, Guarino, Hochberg, Jackson, Petitot, Searle, Tye, Varzi and other leading thinkers on philosophy and the foundations of cognitive Science. Topics dealt with include: Wittgenstein and Cognitive Science, Content and Object, Logic and Foundations, Language and Linguistics, and Ontology and Mereology.
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  • Topology, Connectedness, and Modal Logic.Roman Kontchakov, Ian Pratt-Hartmann, Frank Wolter & Michael Zakharyaschev - 2008 - In Carlos Areces & Robert Goldblatt (eds.), Advances in Modal Logic, Volume 7. CSLI Publications. pp. 151-176.
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  • Complete Axiomatizations for XPath Fragments.Balder ten Cate, Tadeusz Litak & Maarten Marx - 2010 - Journal of Applied Logic 8 (2):153-172.
  • Action Negation and Alternative Reductions for Dynamic Deontic Logics.Jan Broersen - 2004 - Journal of Applied Logic 2 (1):153-168.
  • Structuralism and Representation Theorems.George Weaver - 1998 - Philosophia Mathematica 6 (3):257-271.
    Much of the inspiration for structuralist approaches to mathematics can be found in the late nineteenth- and early twentieth-century program of characterizing various mathematical systems upto isomorphism. From the perspective of this program, differences between isomorphic systems are irrelevant. It is argued that a different view of the import of the differences between isomorphic systems can be obtained from the perspective of contemporary discussions of representation theorems and that from this perspective both the identification of isomorphic systems and the reduction (...)
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  • The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations.Roger D. Maddux - 1991 - Studia Logica 50 (3-4):421 - 455.
    The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schröder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarski's axiomatization led to the creation of the theory of relation algebras, and was shown to be incomplete by Roger Lyndon's discovery of nonrepresentable relation algebras. This paper introduces the calculus of relations and the (...)
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  • Relational Dual Tableau Decision Procedure for Modal Logic K.Joanna Golińska-Pilarek, Emilio Muñoz-Velasco & Angel Mora-Bonilla - 2012 - Logic Journal of the IGPL 20 (4):747-756.
    We present a dual tableau system, RLK, which is itself a deterministic decision procedure verifying validity of K-formulas. The system is constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. Furthermore, we describe an implementation of the system in Prolog, and we show some of its advantages.
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  • Mixed Algebras and Their Logics.Ivo Düntsch, Ewa Orłowska & Tinko Tinchev - 2017 - Journal of Applied Non-Classical Logics 27 (3-4):304-320.
    We investigate complex algebras of the form arising from a frame where, and exhibit their abstract algebraic and logical counterparts.
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  • Relational Proof Systems for Spatial Reasoning.Joanna Golińska-Pilarek & Ewa Orlowska - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):409-431.
    We present relational proof systems for the four groups of theories of spatial reasoning: contact relation algebras, Boolean algebras with a contact relation, lattice-based spatial theories, spatial theories based on a proximity relation.
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  • Dual Tableau-Based Decision Procedures for Relational Logics with Restricted Composition Operator.Domenico Cantone, Marianna Nicolosi Asmundo & Ewa Orlowska - 2011 - Journal of Applied Non-Classical Logics 21 (2):177-200.
    We consider fragments of the relational logic RL obtained by posing various constraints on the relational terms involving the operator of composition of relations. These fragments allow to express several non classical logics including modal and description logics. We show how relational dual tableaux can be employed to provide decision procedures for each of them.
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  • Equational Reasoning in Non-Classical Logics.Marcelo Frias & Ewa Orlowska - 1998 - Journal of Applied Non-Classical Logics 8 (1-2):27-66.
    ABSTRACT In this paper it is shown that a broad class of propositional logics can be interpreted in an equational logic based on fork algebras. This interpetability enables us to develop a fork-algebraic formalization of these logics and, as a consequence, to simulate non-classical means of reasoning with equational theories algebras.
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  • Essay Review.Alison Walsh - 1998 - History and Philosophy of Logic 19 (2):107-114.
    Nathan Houser, Don D. Roberts and James Van Evra (eds), Studies in the Logic of Charles Sanders Peirce, In:Bloomington and Indianapolis, Indiana University Press, 1997, xiii + 653 pp. £41.95. ISBN 0-253-33020-3.
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  • A Proof System for Contact Relation Algebras.Ivo Düntsch & Ewa Orłowska - 2000 - Journal of Philosophical Logic 29 (3):241-262.
    Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa and Sikorski (1963) for relation algebras generated by a contact relation.
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  • A Necessary Relation Algebra for Mereotopology.Ivo DÜntsch, Gunther Schmidt & Michael Winter - 2001 - Studia Logica 69 (3):381 - 409.
    The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T 0 topological space with an additional "contact relation" C defined by xCy x ØA (possibly) more general class of models is provided by the Region Connection Calculus (RCC) of Randell et al. We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the RCC, (...)
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  • Provability with Finitely Many Variables.Robin Hirsch, Ian Hodkinson & Roger D. Maddux - 2002 - Bulletin of Symbolic Logic 8 (3):348-379.
    For every finite n ≥ 4 there is a logically valid sentence φ n with the following properties: φ n contains only 3 variables (each of which occurs many times); φ n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): φ n has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of (...)
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  • Relevance Logic and the Calculus of Relations.Roger D. Maddux - 2010 - Review of Symbolic Logic 3 (1):41-70.
    Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.
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  • Proof Theory for Modal Logic.Sara Negri - 2011 - Philosophy Compass 6 (8):523-538.
    The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.
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  • Groups and Algebras of Binary Relations.Steven Givant & Hajnal Andréka - 2002 - Bulletin of Symbolic Logic 8 (1):38-64.
    In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras, He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jonsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one (...)
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  • Tarski's Influence on Computer Science.Solomon Feferman - unknown
    The following is the text of an invited lecture for the LICS 2005 meeting held in Chicago June 26-29, 2005.1 Except for the addition of references, footnotes, corrections of a few points and stylistic changes, the text is essentially as delivered. Subsequent to the lecture I received interesting comments from several colleagues that would have led me to expand on some of the topics as well as the list of references, had I had the time to do so.
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  • Undecidability of Representability as Binary Relations.Robin Hirsch & Marcel Jackson - 2012 - Journal of Symbolic Logic 77 (4):1211-1244.
    In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.
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  • On Eight Independent Equational Axiomatisations for Fork Algebras.Paulo As Veloso - 1998 - Bulletin of the Section of Logic 27 (3):117-129.
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  • A Finitary Relational Algebra for Classical First-Order Logic.Paulo As Veloso & Armando M. Haeberer - 1991 - Bulletin of the Section of Logic 20 (2):52-62.
  • Notions of Density That Imply Representability in Algebraic Logic.Hajnal Andréka, Steven Givant, Szabolcs Mikulás, István Németi & András Simon - 1998 - Annals of Pure and Applied Logic 91 (2-3):93-190.
    Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result (...)
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  • A Sequent Calculus for Relation Algebras.Roger Maddux - 1983 - Annals of Pure and Applied Logic 25 (1):73-101.
  • A Modal-Hamiltonian Interpretation of Quantum Mechanics.Olimpia Lombardi & Mario Castagnino - 2008 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 39 (2):380-443.
    The aim of this paper is to introduce a new member of the family of the modal interpretations of quantum mechanics. In this modal-Hamiltonian interpretation, the Hamiltonian of the quantum system plays a decisive role in the property-ascription rule that selects the definite-valued observables whose possible values become actual. We show that this interpretation is effective for solving the measurement problem, both in its ideal and its non-ideal versions, and we argue for the physical relevance of the property-ascription rule by (...)
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  • On the Consistency Problem for the INDU Calculus.Philippe Balbiani, Jean-François Condotta & Gérard Ligozat - 2006 - Journal of Applied Logic 4 (2):119-140.
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  • Dynamic Algebras: Examples, Constructions, Applications.Vaughan Pratt - 1991 - Studia Logica 50 (3-4):571 - 605.
    Dynamic algebras combine the classes of Boolean (B 0) and regular (R ; *) algebras into a single finitely axiomatized variety (B R ) resembling an R-module with scalar multiplication . The basic result is that * is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result (...)
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  • Finite, Integral, and Finite-Dimensional Relation Algebras: A Brief History.Roger D. Maddux - 2004 - Annals of Pure and Applied Logic 127 (1-3):117-130.
    Relation algebras were invented by Tarski and his collaborators in the middle of the 20th century. The concept of integrality arose naturally early in the history of the subject, as did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. Some results on these topics are presented in chronological order.
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  • A Coalgebraic Perspective on Logical Interpretations.M. A. Martins, A. Madeira & L. S. Barbosa - 2013 - Studia Logica 101 (4):783-825.
    In Computer Science stepwise refinement of algebraic specifications is a well-known formal methodology for rigorous program development. This paper illustrates how techniques from Algebraic Logic, in particular that of interpretation, understood as a multifunction that preserves and reflects logical consequence, capture a number of relevant transformations in the context of software design, reuse, and adaptation, difficult to deal with in classical approaches. Examples include data encapsulation and the decomposition of operations into atomic transactions. But if interpretations open such a new (...)
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  • Podstawy Logiki Nazw.Adelina Morawiec - 1961 - Studia Logica 12 (1):145 - 170.
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  • Dynamic Logic with Program Specifications and its Relational Proof System.Ewa Orlowska - 1993 - Journal of Applied Non-Classical Logics 3 (2):147-171.
    ABSTRACT Propositional dynamic logic with converse and test, is enriched with complement, intersection and relational operations of weakest prespecification and weakest postspecification. Relational deduction system for the logic is given based on its interpretation in the relational calculus. Relational interpretation of the operators ?repeat? and ?loop? is given.
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  • A Modal Ontology of Properties for Quantum Mechanics.Newton Costa, Olimpia Lombardi & Mariano Lastiri - 2013 - Synthese 190 (17):3671-3693.
    Our purpose in this paper is to delineate an ontology for quantum mechanics that results adequate to the formalism of the theory. We will restrict our aim to the search of an ontology that expresses the conceptual content of the recently proposed modal-Hamiltonian interpretation, according to which the domain referred to by non-relativistic quantum mechanics is an ontology of properties. The usual strategy in the literature has been to focus on only one of the interpretive problems of the theory and (...)
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  • Relational Dual Tableau Decision Procedures and Their Applications to Modal and Intuitionistic Logics.Joanna Golińska-Pilarek, Taneli Huuskonen & Emilio Muñoz-Velasco - 2014 - Annals of Pure and Applied Logic 165 (2):409-427.
    This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...)
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  • R⌝-Algebras and R⌝-Model Structures as Power Constructs.Chris Brink - 1989 - Studia Logica 48 (1):85 - 109.
    In relevance logic it has become commonplace to associate with each logic both an algebraic counterpart and a relational counterpart. The former comes from the Lindenbaum construction; the latter, called a model structure, is designed for semantical purposes. Knowing that they are related through the logic, we may enquire after the algebraic relationship between the algebra and the model structure. This paper offers a complete solution for the relevance logic R. Namely, R-algebras and R-model structures can be obtained from each (...)
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  • Peirce, Frege, the Logic of Relations, and Church's Theorem.Randall R. Dipert - 1984 - History and Philosophy of Logic 5 (1):49-66.
    In this essay, I discuss some observations by Peirce which suggest he had some idea of the substantive metalogical differences between logics which permit both quantifiers and relations, and those which do not. Peirce thus seems to have had arguments?which even De Morgan and Frege lacked?that show the superior expressiveness of relational logics.
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  • Relevance Logics and Relation Algebras.Katalin Bimbó, J. Michael Dunn & Roger D. Maddux - 2009 - Review of Symbolic Logic 2 (1):102-131.
    Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...)
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  • Reading Proofs with Understanding.George Weaver - 1988 - Theoria 54 (1):31-47.
  • A Modal-Hamiltonian Interpretation of Quantum Mechanics.Olimpia Lombardi & Mario Castagnino - 2008 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 39 (2):380-443.
    The aim of this paper is to introduce a new member of the family of the modal interpretations of quantum mechanics. In this modal-Hamiltonian interpretation, the Hamiltonian of the quantum system plays a decisive role in the property-ascription rule that selects the definite-valued observables whose possible values become actual. We show that this interpretation is effective for solving the measurement problem, both in its ideal and its non-ideal versions, and we argue for the physical relevance of the property-ascription rule by (...)
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  • Computation of Aristotle's and Gergonne's Syllogisms.S. N. Furs - 1987 - Studia Logica 46 (3):209 - 225.
    A connection between Aristotle's syllogistic and the calculus of relations is investigated. Aristotle's and Gergonne's syllogistics are considered as some algebraic structures. It is proved that Gergonne's syllogistic is isomorphic to closed elements algebra of a proper approximation relation algebra. This isomorphism permits to evaluate Gergonne's syllogisms and also Aristotle's syllogisms, laws of conversion and relations in the "square of oppositions" by means of regular computations with Boolean matrices.
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  • Weakly Associative Relation Algebras with Projections.Agi Kurucz - 2009 - Mathematical Logic Quarterly 55 (2):138-153.
    Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first-order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on (...)
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  • A Modal Ontology of Properties for Quantum Mechanics.Newton da Costa, Olimpia Lombardi & Mariano Lastiri - 2013 - Synthese 190 (17):3671-3693.
    Our purpose in this paper is to delineate an ontology for quantum mechanics that results adequate to the formalism of the theory. We will restrict our aim to the search of an ontology that expresses the conceptual content of the recently proposed modal-Hamiltonian interpretation, according to which the domain referred to by non-relativistic quantum mechanics is an ontology of properties. The usual strategy in the literature has been to focus on only one of the interpretive problems of the theory and (...)
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  • Composable Relations Induced in Networks of Aligned Ontologies: A Category Theoretic Approach.Seremeti Lambrini & Kameas Achilles - 2015 - Axiomathes 25 (3):285-311.
    A network of aligned ontologies is a distributed system, whose components are interacting and interoperating, the result of this interaction being, either the extension of local assertions, which are valid within each individual ontology, to global assertions holding between remote ontology syntactic entities through a network path, or to local assertions holding between local entities of an ontology, but induced by remote ontologies, through a cycle in the network. The mechanism for achieving this interaction is the composition of relations. In (...)
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