-
9Mereological foundations of point-free geometry via multi-valued logicLogic and Logical Philosophy 24 (4): 535-553. 2015.We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking int…Read more
-
7Fuzziness in Italy – Traces of a scattered historyArchives for the Philosophy and History of Soft Computing 2017 (1). 2017.The history of Fuzziness in Italy is varied and scattered among a num- ber of research groups. As a matter of fact, “fuzziness” spread in Italy through a sort of spontaneous diffusion, and, also subsequently, no one felt the need to cre- ate some “national” common structure like an Association or similar things. Since a cohesive retelling would be next to impossible, a few members of the Italian fuzzy community have been asked to recount their experience and express their hopes for the future.
-
42Point-Free Geometry, Ovals, and Half-PlanesReview of Symbolic Logic 10 (2): 237-258. 2017.In this paper we develop a point-free system of geometry based on the notions ofregion,parthood, andovality, the last one being a region-based counterpart of the notion ofconvex set. In order to show that the system we propose is sufficient to reconstruct an affine geometry we make use of a theory of a Polish mathematician Aleksander Śniatycki from [15], in which the concept ofhalf-planeis assumed as basic.
-
30Measures in Euclidean Point-Free Geometry (an exploratory paper)Logic and Logical Philosophy 32 (4): 619-638. 2023.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.
-
90Defining Measures in a Mereological Space (an exploratory paper)Logic and Logical Philosophy 31 (1): 57-74. 2022.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.
-
347Whitehead'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
-
165Point-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
-
52Connection 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
-
67Vagueness and Formal Fuzzy Logic: Some CriticismsLogic and Logical Philosophy 26 (4): 431-460. 2017.In the common man reasoning the presence of vague predicates is pervasive and under the name “fuzzy logic in narrow sense” or “formal fuzzy logic” there are a series of attempts to formalize such a kind of phenomenon. This paper is devoted to discussing the limits of these attempts both from a technical point of view and with respect the original and principal task: to define a mathematical model of the vagueness. For example, one argues that, since vagueness is necessarily connected with the in…Read more
-
48Pavelka's Fuzzy Logic and Free L-SubsemigroupsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 31 (7-8): 123-129. 1985.
-
35Approximate 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
-
77TuringL-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.
-
82Multivalued 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.
-
39Approximate 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
-
78Modal 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
-
75Effectiveness 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
-
26Pavelka's Fuzzy Logic and Free L‐SubsemigroupsMathematical Logic Quarterly 31 (7‐8): 123-129. 1985.
-
50Mathematical Features of Whitehead’s Point-free GeometryIn Michel Weber and Will Desmond (ed.), Handbook of Whiteheadian Process Thought, De Gruyter. pp. 119-130. 2008.
-
88Connection 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.
-
La relazione di connessione in AN Whitehead: Aspetti matematiciEpistemologia 15 (2): 351-364. 1992.
-
33An 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