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15Measures in Euclidean Point-Free Geometry (an exploratory paper)Logic and Logical Philosophy 1-20. forthcoming.We face with the question of a suitable measure theory in Euclidean point-free geometry and we sketch out some possible solutions. The proposed measures, which are positive and invariant with respect to movements, are based on the notion of infinitesimal masses, i.e. masses whose associated supports form a sequence of finer and finer partitions.
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39Defining 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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15Approximate Similarities and Poincaré ParadoxNotre Dame Journal of Formal Logic 49 (2): 203-226. 2008.De Cock and Kerre, in considering Poincaré paradox, observed that the intuitive notion of "approximate similarity" cannot be adequately represented by the fuzzy equivalence relations. In this note we argue that the deduction apparatus of fuzzy logic gives adequate tools with which to face the question. Indeed, a first-order theory is proposed whose fuzzy models are plausible candidates for the notion of approximate similarity. A connection between these structures and the point-free metric space…Read more
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12Pavelka's Fuzzy Logic and Free L‐SubsemigroupsMathematical Logic Quarterly 31 (7‐8): 123-129. 1985.
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18Approximate Reasoning Based on SimilarityMathematical Logic Quarterly 46 (1): 77-86. 2000.The connection between similarity logic and the theory of closure operators is examined. Indeed one proves that the consequence relation defined in [14] can be obtained by composing two closure operators and that the resulting operator is still a closure operator. Also, we extend any similarity into a similarity which is compatible with the logical equivalence, and we prove that this gives the same consequence relation
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38Effectiveness 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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27Mathematical 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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143Point-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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49Connection 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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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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296Whitehead'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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7TuringL-machines and recursive computability forL-mapsStudia Logica 48 (2): 179-192. 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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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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28Connection 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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52Modal 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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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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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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24Fuzzy Models of First Order LanguagesZeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24): 331-340. 1986.
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67Point-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