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.

In this paper we give a representation of the free MV-algebra over one generator as a structure of functions having finite domain.

We extend de Finetti’s No-Dutch-Book Criterion to Gödel infinite-valued propositional logic.

We extend de Finetti's No-Dutch-Book Criterion to Gödel infinite-valued propositional logic.

We axiomatize the notion of state over finitely generated free NM-algebras, the Lindenbaum algebras of pure Nilpotent Minimum logic. We show that states over the free n -generated NM-algebra exactly correspond to integrals of elements of with respect to Borel probability measures.

Gödel algebras form the locally finite variety of Heyting algebras satisfying the prelinearity axiom =. In 1969, Horn proved that a Heyting algebra is a Gödel algebra if and only if its set of prime filters partially ordered by reverse inclusion–i.e. its prime spectrum–is a forest. Our main result characterizes Gödel algebras that are free over some finite distributive lattice by an intrisic property of their spectral forest.

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.

The theme of Aesthetics in Present Future concerns the new chances the arts have and the deep changes they are undergoing, due to the new media, and the digital world in which we are growingly immersed. That this world is to be understood from an aesthetic point of view, become clear if we think of how much of what we produce, and observe and study is offered through images in particular and perceptual means in general.

O objetivo deste artigo é associar a lógica de abdução de Peirce à cibernética de sistemas vivos. Sistemas vivos não podem ser entendidos através de uma epistemologia causalista, pois se comportam conforme os efeitos e não conforme as causas. A cibernética analisou o modo pelo qual máquinas não triviais se movem através de retroação, ou circuito de retorno: a cada etapa, o sistema corrige a etapa anterior, dependendo de até onde a etapa anterior pode ir em relação a um possível (...) equilíbrio. A dinâmica implica uma possível inibição da energia excessiva seguida de um possível aumento da energia insuficiente. Cada uma dessas condições é um erro que se corrige automaticamente, atingindo um estado temporário de homeostase. A auto-correção significa que é preciso adicionar à saída um impulso adicional “criativo” que seja singular e imprevisível. A lógica que é capaz de fazer sentido dessa dinâmica é a abdução de Peirce, também chamada retrodução, onde o ato inferencial retroage sobre um fato incompreensível, um “erro” epistemológico, inventando sua causa. O pensar e a auto-poiesis de sistemas vivos percorrem o mesmo caminho, substituindo estratégias ocasionais por causas universais atemporais. A abdução é a evidência de que é possível atingir uma condição de “conhecimento” sem apelar para causalidade. A possibilidade que as “leis da natureza”, possa seguir este caminho conclui o ensaio ao adicionar a noção do físico Lee Smolin do “princípio de precedência”, conforme foi inspirado pela ideia de Peirce da evolução das leis da natureza e seu caráter temporal, ou seja, contingente. (shrink)

Questo libro risulta da un’operazione filosofica molto originale: l’autrice si chiede per quali possibili motivi a Wittgenstein piacesse un giallo di Norbert Davis, Rendez-Vous with Fear, uscito negli Stati Uniti nel 1943 con il titolo The Mouse in the Mountain. Dalle lettere scambiate con l’amico Norman Malcolm sappiamo della sua passione per le hard boiled stories e per questo giallo in particolare. Con grande libertà teorica, che include introspezione e analisi epistemologica, antropologic...

Thinking through Error: The Moving Target of Knowledge argues that there is a positive view of error. Making errors does not only mean that we’ve done something wrong, but also that we —more or less unaware— are given a chance to find something new and true. Trying to avoid errors is a social request, but it is uncertainty that has a liberating function on the philosophical level, as well as on the individual, psychological level.

The first organic robots built by Tuft and Vermont University researchers pose questions to philosophy and give it a new task. The xenobots embody what phi-losophers had attempted to define as teleology. This paper addresses the way te-los can be redefined, once liberated from the suspicion of vitalism. While Darwin-ism, through a theory of evolution based on the environment, has contributed to the elimination of telos, here a new view of biology is described, which shows how evolution can be fully (...) explained through the notion of feedback, or inner re-sistance in a system, as preliminary condition for natural selection to work. (shrink)

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.

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.

Una nuova traduzione di "An Essay Concerning Toleration" di John Locke.

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)

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).

Casalini reconstructs Politics out of History by Wendy Brown by taking into particular consideration the dialogue between Brown and the works of Marx, Nietzsche and Freud. The emphasis placed on the nexus between individual and political freedom and on the distinction between “moralism” and “morality” leads the way to the exploration of the relationship between theory and politics. In the light of the contemporary crisis of the American left, the role of the theorist is that of introducing elements of discontinuity (...) in the common-sense discourse which, on the other hand, tends to be crystallized by politics. According to Brown this gap must be taken into consideration by Women Studies, since this field of study sometimes risks relying on static and institutionalized identities which can be an obstruction to its exercise of a «homeopathic function» inside democratic regimes. (shrink)

La teoria dell’oppressione di Lugones è incentrata sulla necessità per l’oppresso di coltivare una diversa logica della realtà. Le condizioni per l’emergere di una coscienza resistente poggiano su un’ontologia pluralista e sull’idea di un sé molteplice capace di viaggiare tra «mondi». Un ruolo fondamentale viene occupato dallo spazio del _limen_. Il limen costituisce un portale verso la liberazione, ma non ne garantisce l’esito. Il progetto politico di emancipazione dalle oppressioni multiple richiede la creazione di coalizioni. Un obiettivo che può essere (...) raggiunto solo attraverso una comunicazione complessa, che abbandona la logica liberale della trasparenza. (shrink)