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.

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.

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.

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 versions are the L-subsets almost-everywhere indeterminate. (shrink)

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.

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 closure operators. (shrink)

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.

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.

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.

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.

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.

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 avoid (...) such a drawback. Moreover, since such a notion enables us to define a metric in the set of points, our proposal looks to be a good starting point for a foundation of the geometry metrical in nature (as proposed, for example, by L.M. Blumenthal). (shrink)

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 extension of the classical propositional calculus is defined and it is showed its connection with possibility and necessity measures. Also, the canonical extension of first order logic enables us to extend some basic notions of programming logic, namely to define the fuzzy Herbrand models of a fuzzy program. Finally, we show that the extension principle enables us to obtain fuzzy logics related to fuzzy subalgebra theory and graded consequence relation theory. (shrink)

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.

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. This avoids some of the difficulties (...) arising from the known definitions of verisimilitude. (shrink)

In the framework of fuzzy logic a definition of vague object is proposed based on the notion of fuzzy equivalence. Indeed, while a crisp object is defined as a “concrete thing” together with an equivalence, I propose to define a vague object as a “concrete thing” together with a fuzzy equivalence.

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. Since they can be expressed in (...) -logic, we are also induced to compare the expressive powers ofQS4E and . Some questions concerning the power of rigidity axiom are also examined. (shrink)

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.

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.

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 spaces is also established.

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).

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. (shrink)

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.

In this paper we present a method to reduce the decision problem of several infinite-valued propositional logics to their finite-valued counterparts. We apply our method to Łukasiewicz, Gödel and Product logics and to some of their combinations. As a byproduct we define sequent calculi for all these infinite-valued logics and we give an alternative proof that their tautology problems are in co-NP.

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.