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14Measures 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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36Defining 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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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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289Whitehead'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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5TuringL-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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25Connection 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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50Modal 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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14Approximate 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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17Approximate 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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31Effectiveness 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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33Turing 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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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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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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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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27Fuzzy natural deductionZeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (1): 67-77. 1990.
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23Connecting 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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36Recursively Enumerable L‐SetsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (2): 107-113. 1987.
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26Decidability, 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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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