•  274
    Whitehead's pointfree geometry and diametric posets
    with Bonaventura Paolillo
    Logic 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
  •  128
    Point-Free Geometry and Verisimilitude of Theories
    Journal 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
  •  65
    Fuzzy Logic Programming and Fuzzy Control
    Studia 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.
  •  57
    Point-free Foundation of Geometry and Multivalued Logic
    with Cristina Coppola and Annamaria Miranda
    Notre 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
  •  44
    Modal logic and model theory
    with Virginia Vaccaro
    Studia 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
  •  40
    Distances, diameters and verisimilitude of theories
    Archive for Mathematical Logic 31 (6): 407-414. 1992.
  •  37
    Pointless metric spaces
    Journal of Symbolic Logic 55 (1): 207-219. 1990.
  •  37
    Connection structures
    with Loredana Biacino
    Notre 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.
  •  36
    Multivalued Logic to Transform Potential into Actual Objects
    Studia 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.
  •  34
    Grasping Infinity by Finite Sets
    with Ferrante Formato
    Mathematical 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
  •  33
    Special Issue on Point-Free Geometry and Topology
    with Cristina Coppola
    Logic 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
  •  28
    Recursively Enumerable L‐Sets
    with Loredana Biacino
    Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (2): 107-113. 1987.
  •  26
    Fuzzy logic: Mathematical tools for approximate reasoning
    Bulletin of Symbolic Logic 9 (4): 510-511. 2003.
  •  25
    Effectiveness and Multivalued Logics
    Journal 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
  •  23
    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
  •  21
    Fuzzy natural deduction
    with Roberto Tortora
    Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (1): 67-77. 1990.
  •  20
    Vagueness and Formal Fuzzy Logic: Some Criticisms
    Logic and Logical Philosophy 26 (4). 2017.
  •  20
    Fuzzy Models of First Order Languages
    with A. di Nola
    Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24): 331-340. 1986.
  •  19
    Decidability, Recursive Enumerability and Kleene Hierarchy ForL-Subsets
    with Loredana Biacino
    Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (1): 49-62. 1989.
  •  19
    Fuzzy logic, continuity and effectiveness
    with Loredana Biacino
    Archive 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)
  •  19
    Mathematical Features of Whitehead’s Point-free Geometry
    with Annamaria Miranda
    In Michel Weber (ed.), Handbook of Whiteheadian Process Thought, De Gruyter. pp. 119-130. 2008.
  •  19
    Pavelka's Fuzzy Logic and Free L-Subsemigroups
    Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 31 (7-8): 123-129. 1985.
  •  18
    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
  •  17
    Graded consequence relations and fuzzy closure operator
    Journal 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
  •  17
    An Extension Principle for Fuzzy Logics
    Mathematical 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
  •  16
    Fuzzy natural deduction
    with Roberto Tortora
    Mathematical Logic Quarterly 36 (1): 67-77. 1990.
  •  15
    Connection Structures: Grzegorczyk's and Whitehead's Definitions of Point
    with Loredana Biacino
    Notre 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
  •  14
    Approximate Reasoning Based on Similarity
    with M. Ying and L. Biacino
    Mathematical 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
  •  12
    4.4. Il Cervino di Varzi: similarità e oggetti vaghi
    Rivista di Estetica 49 281-296. 2012.
  •  11
    Approximate Similarities and Poincaré Paradox
    Notre 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