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290Whitehead's pointfree geometry and diametric posetsLogic and Logical Philosophy 19 (4): 289-308. 2010.This note is motivated by Whitehead’s researches in inclusion-based point-free geometry as exposed in An Inquiry Concerning the Principles of Natural Knowledge and in The concept of Nature. More precisely, we observe that Whitehead’s definition of point, based on the notions of abstractive class and covering, is not adequate. Indeed, if we admit such a definition it is also questionable that a point exists. On the contrary our approach, in which the diameter is a further primitive, enables us to…Read more
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142Point-Free Geometry and Verisimilitude of TheoriesJournal of Philosophical Logic 36 (6): 707-733. 2007.A metric approach to Popper's verisimilitude question is proposed which is related to point-free geometry. Indeed, we define the theory of approximate metric spaces whose primitive notions are regions, inclusion relation, minimum distance, and maximum distance between regions. Then, we show that the class of possible scientific theories has the structure of an approximate metric space. So, we can define the verisimilitude of a theory as a function of its (approximate) distance from the truth. Th…Read more
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81Fuzzy 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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66Point-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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54Special 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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51Modal logic and model theoryStudia Logica 43 (3). 1984.We propose a first order modal logic, theQS4E-logic, obtained by adding to the well-known first order modal logicQS4 arigidity axiom schemas:A → □A, whereA denotes a basic formula. In this logic, thepossibility entails the possibility of extending a given classical first order model. This allows us to express some important concepts of classical model theory, such as existential completeness and the state of being infinitely generic, that are not expressibile in classical first order logic. Sinc…Read more
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49Multivalued Logic to Transform Potential into Actual ObjectsStudia Logica 86 (1): 69-87. 2007.We define the notion of “potential existence” by starting from the fact that in multi-valued logic the existential quantifier is interpreted by the least upper bound operator. Besides, we try to define in a general way how to pass from potential into actual existence.
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47Connection 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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46Distances, diameters and verisimilitude of theoriesArchive for Mathematical Logic 31 (6): 407-414. 1992.
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40Connecting 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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39Grasping 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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37Defining Measures in a Mereological SpaceLogic and Logical Philosophy 1. forthcoming.We explore the notion of a measure in a mereological structure and we deal with the difficulties arising. We show that measure theory on connection spaces is closely related to measure theory on the class of ortholattices and we present an approach akin to Dempster’s and Shafer’s. Finally, the paper contains some suggestions for further research.
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36Recursively Enumerable L‐SetsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (2): 107-113. 1987.
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34Effectiveness and Multivalued LogicsJournal of Symbolic Logic 71 (1). 2006.Effective domain theory is applied to fuzzy logic. The aim is to give suitable notions of semi-decidable and decidable L-subset and to investigate about the effectiveness of the fuzzy deduction apparatus
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34Turing L -machines and recursive computability for L -mapsStudia Logica 48 (2). 1989.We propose the notion of partial recursiveness and strong partial recursiveness for fuzzy maps. We prove that a fuzzy map f is partial recursive if and only if it is computable by a Turing fuzzy machine and that f is strongly partial recursive and deterministic if and only if it is computable via a deterministic Turing fuzzy machine. This gives a simple and manageable tool to investigate about the properties of the fuzzy machines
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29Decidability, 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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27Fuzzy natural deductionZeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (1): 67-77. 1990.
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27Connection Structures: Grzegorczyk's and Whitehead's Definitions of PointNotre Dame Journal of Formal Logic 37 (3): 431-439. 1996.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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27Graded 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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26Pavelka's Fuzzy Logic and Free L-SubsemigroupsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 31 (7-8): 123-129. 1985.
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26Decidability, Recursive Enumerability and Kleene Hierarchy ForL-SubsetsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (1): 49-62. 1989.
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26Fuzzy logic: Mathematical tools for approximate reasoningBulletin of Symbolic Logic 9 (4): 510-511. 2003.
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26Mathematical Features of Whitehead’s Point-free GeometryIn Michel Weber (ed.), Handbook of Whiteheadian Process Thought, De Gruyter. pp. 119-130. 2008.
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24Fuzzy Models of First Order LanguagesZeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24): 331-340. 1986.
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23Fuzzy 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)
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22An 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