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115Connection structuresNotre Dame Journal of Formal Logic 32 (2): 242-247. 1991.Whitehead, in his famous book "Process and Reality", proposed a definition of point assuming the concepts of “region” and “connection relation” as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as primitive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.
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26Fuzzy logic: Mathematical tools for approximate reasoningBulletin of Symbolic Logic 9 (4): 510-511. 2003.
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574.4. Il Cervino di Varzi: similarità e oggetti vaghiRivista di Estetica 49 281-296. 2012.In the framework of fuzzy logic a definition of vague object is proposed based on the notion of fuzzy equivalence. Indeed, while a crisp object is defined as a “concrete thing” together with an equivalence, I propose to define a vague object as a “concrete thing” together with a fuzzy equivalence.
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La relazione di connessione in AN Whitehead: Aspetti matematiciEpistemologia 15 (2): 351-364. 1992.
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54An Extension Principle for Fuzzy LogicsMathematical Logic Quarterly 40 (3): 357-380. 1994.Let S be a set, P the class of all subsets of S and F the class of all fuzzy subsets of S. In this paper an “extension principle” for closure operators and, in particular, for deduction systems is proposed and examined. Namely we propose a way to extend any closure operator J defined in P into a fuzzy closure operator J* defined in F. This enables us to give the notion of canonical extension of a deduction system and to give interesting examples of fuzzy logics. In particular, the canonical exte…Read more
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71Decidability, Recursive Enumerability and Kleene Hierarchy For L‐SubsetsMathematical Logic Quarterly 35 (1): 49-62. 1989.
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48Fuzzy natural deductionZeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (1): 67-77. 1990.
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149Connecting bilattice theory with multivalued logicLogic and Logical Philosophy 23 (1): 15-45. 2014.This is an exploratory paper whose aim is to investigate the potentialities of bilattice theory for an adequate definition of the deduction apparatus for multi-valued logic. We argue that bilattice theory enables us to obtain a nice extension of the graded approach to fuzzy logic. To give an example, a completeness theorem for a logic based on Boolean algebras is proved
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98Recursively Enumerable L‐SetsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (2): 107-113. 1987.
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73Decidability, partial decidability and sharpness relation for l-subsetsStudia Logica 46 (3): 227-238. 1987.If X is set and L a lattice, then an L-subset or fuzzy subset of X is any map from X to L, [11]. In this paper we extend some notions of recursivity theory to fuzzy set theory, in particular we define and examine the concept of almost decidability for L-subsets. Moreover, we examine the relationship between imprecision and decidability. Namely, we prove that there exist infinitely indeterminate L-subsets with no more precise decidable versions and classical subsets whose unique shaded decidable …Read more
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71Grasping Infinity by Finite SetsMathematical Logic Quarterly 44 (3): 383-393. 1998.We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of wide sets
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59Decidability, Recursive Enumerability and Kleene Hierarchy ForL-SubsetsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (1): 49-62. 1989.
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74Graded consequence relations and fuzzy closure operatorJournal of Applied Non-Classical Logics 6 (4): 369-379. 1996.ABSTRACT In this work the connections between the fuzzy closure operators and the graded consequence relations are examined Namely, as it is well known, in the crisp case there is a complete equivalence between the notion of closure operator and the one of consequence relation. We extend this result by proving that the graded consequence relations are related to a particular class of fuzzy closure operators, namely the class of fuzzy closure operators that can be obtained by a chain of classical…Read more
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73Distances, diameters and verisimilitude of theoriesArchive for Mathematical Logic 31 (6): 407-414. 1992.
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153Special Issue on Point-Free Geometry and TopologyLogic and Logical Philosophy 22 (2): 139-143. 2013.In the first section we briefly describe methodological assumptions of point-free geometry and topology. We also outline history of geometrical theories based on the notion of emph{region}. The second section is devoted to concise presentation of the content of the LLP special issue on point-free theories of space.
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147Fuzzy Logic Programming and Fuzzy ControlStudia Logica 79 (2): 231-254. 2005.We show that it is possible to base fuzzy control on fuzzy logic programming. Indeed, we observe that the class of fuzzy Herbrand interpretations gives a semantics for fuzzy programs and we show that the fuzzy function associated with a fuzzy system of IF-THEN rules is the fuzzy Herbrand interpretation associated with a suitable fuzzy program.
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51Fuzzy Models of First Order LanguagesZeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24): 331-340. 1986.
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96Point-free Foundation of Geometry and Multivalued LogicNotre Dame Journal of Formal Logic 51 (3): 383-405. 2010.Whitehead, in two basic books, considers two different approaches to point-free geometry: the inclusion-based approach, whose primitive notions are regions and inclusion relation between regions, and the connection-based approach, where the connection relation is considered instead of the inclusion. We show that the latter cannot be reduced to the first one, although this can be done in the framework of multivalued logics.
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88Fuzzy logic, continuity and effectivenessArchive for Mathematical Logic 41 (7): 643-667. 2002.It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general)
