We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. (...) Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations. (shrink)

We prove that the unification type of Łukasiewicz logic and of its equivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardiʼs algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background (...) to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra—a fundamental result that, albeit known to specialists, seems to appear in print here for the first time. (shrink)

The problem of artificial precision is a major objection to any theory of vagueness based on real numbers as degrees of truth. Suppose you are willing to admit that, under sufficiently specified circumstances, a predication of “is red” receives a unique, exact number from the real unit interval [0, 1]. You should then be committed to explain what is it that determines that value, settling for instance that my coat is red to degree 0.322 rather than 0.321. In this note (...) I revisit the problem in the important case of Łukasiewicz infinite-valued propositional logic that brings to the foreground the rôle of maximally consistent theories. I argue that the problem of artificial precision, as commonly conceived of in the literature, actually conflates two distinct problems of a very different nature. (shrink)

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.

Mundici has recently established a characterization of free finitely generated MV-algebras similar in spirit to the representation of the free Boolean algebra with a countably infinite set of free generators as any Boolean algebra that is countable and atomless. No reference to universal properties is made in either theorem. Our main result is an extension of Mundici’s theorem to the whole class of MV-algebras that are free over some finite distributive lattice.

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.

In the context of truth-functional propositional many-valued logics, Hájek’s Basic Fuzzy Logic BL [14] plays a major rôle. The completeness theorem proved in [7] shows that BL is the logic of all continuous t -norms and their residua. This result, however, does not directly yield any meaningful interpretation of the truth values in BL per se . In an attempt to address this issue, in this paper we introduce a complete temporal semantics for BL. Specifically, we show that BL formulas (...) can be interpreted as modal formulas over a flow of time, where the logic of each instant is Łukasiewicz, with a finite or infinite number of truth values. As a main result, we obtain validity with respect to all flows of times that are non-branching to the future, and completeness with respect to all finite linear flows of time, or to an appropriate single infinite linear flow of time. It may be argued that this reduces the problem of establishing a meaningful interpretation of the truth values in BL logic to the analogous problem for Łukasiewicz logic. (shrink)

We obtain an algorithm to compute finite coproducts of finitely generated Gödel algebras, i.e. Heyting algebras satisfying the prelinearity axiom =1. We achieve this result using ordered partitions of finite sets as a key tool to investigate the category opposite to finitely generated Gödel algebras . We give two applications of our main result. We prove that finitely presented Gödel algebras have free products with amalgamation; and we easily obtain a recursive formula for the cardinality of the free Gödel algebra (...) over a finite number of generators first established by A. Horn. (shrink)

We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial (...) characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper. (shrink)

The aim of Stich's book is to further the controversial thesis that the conceptual framework of ‘folk’ psychology will have no significant role to play in a mature cognitive science. Skepticism about the scientific relevance of folk psychology has been voiced by others ; but Stich's critique is both novel and more fully developed than earlier ones. The charge is not–-or not simply–-that ‘folk theory’ is a “degenerating paradigm“, or that, in general, the constructs of folk theory fail to refer (...) altogether. Stich's thesis is subtler, and rests on the claim that the individuation of folk-psychological states is irremediably vague and context-and-observer relative in a way that makes a folk-psychological taxonomy ill-suited to the requirements of scientific explanation and systematization. Stich, of course, does not reject mentalism as such, as did the behaviorists: he merely rejects the assumption–-often tacitly accepted by cognitive scientists and by their philosophical interpreters–-that the mental states posited by a mature cognitive science will essentially correspond to the intentional, propositional attitude states of folk psychology. Thus, for Stich, there is no reason to suppose that cognitive science will turn out to be a sophisticated extension of folk psychology. The “Panglossian prospect” of a conceptual reunification of the scientific image of mind with the ‘manifest’ image under the banner of cognitivism is just what this book calls into question. (shrink)

This paper discusses the traditional antagonism between the Intentionalist and the Physicalist paradigms of the nature of mind and human behaviour. After tracing the development of the concept of intentionality in contemporary analytic philosophy and delineating some of the problems it presented for the so-called Thesis of Physicalism in several of its formulations, the paper proceeds to show how a liberalized methodology of theory construction and an understanding of functional systems and artificial intelligence models may enable us to reconstruct the (...) intentional states of persons in a way compatible with the demands of physicalism. In particular, it is suggested that the intentional states which, on the intentionalist paradigm, mediate human behaviour might be understood as (theoretically postulated) functional states of a physical organism modelled on the functional states of a (probabilistic) automaton. (shrink)

Recent literature highlights that the use of smartphones during meals increases the number of calories ingested in young people. Although the distraction interferes with physiological signals of hunger and satiety, a social facilitation effect has also been suggested. Cognition is a pivotal component in regulating food intake, and activities requiring high perceptual demands should be discouraged during meals.

En este libro, que aparece en Italia en 1987, publicado muchas veces y muy comentado también en el extranjero, Vincenzo Ferrari analiza las funciones sociales del derecho en una perspectiva relativista, subjetivista y conflictualista, opuesta a la del funcionalismo sociológico. De tal manera, el derecho se revela como un medio de acción social dirigido a lograr los proyectos de seres humanos empeñados en relaciones variables de cooperación y de lucha. Así, se muestra cómo las normas jurídicas pueden orientar las (...) actitudes y desorientarlas; solucionar los conflictos y agravarlos o fomentarlos; legitimar cualquier forma de poder con argumentos tan válidos como ficticios. Solo al nivel de los ideales se redescubre el valor del derecho, su potencialidad libertadora y su naturaleza igualitaria. (shrink)

This essay addresses the methodology of philosophy of science and illustrates how formal and empirical methods can be fruitfully combined. Special emphasis is given to the application of experimental methods to confirmation theory and to recent work on the conjunction fallacy, a key topic in the rationality debate arising from research in cognitive psychology. Several other issue can be studied in this way. In the concluding section, a brief outline is provided of three further examples.

How are individuals socialized today? To answer these questions, a unique investigation has been carried out using two scales of analysis: the scale of the cosmopolitan world as well as the scale of everyday life and socialization to otherness.

Quest'opera fa parte di un più vasto lavoro di ricerca sul neo-idealismo italiano in rapporto alla politica. Il titolo - filosofia e politica - è di per sé significativo: sta a indicare chiaramente la direzione e il tipo dell'indagine, che non è empirica ma trascendentale, non riguarda cioè la dottrina politica e le scelte pratiche di Croce, staccate dall'organismo del pensiero in cui s'incorporano e analizzate alla luce di alcune categorie, magari psicologiche e sociologiche; ma riguarda i presupposti teoretici che (...) stanno a fondamento di quella dottrina e di quelle scelte. Scartato, perché fuorviante, il metodo di quelle due «false scienze» che sono la psicologia e la sociologia, l'Autore ha tentato un discorso schiettamente filosofico, che è poi discorso di fondazione e di deduzione. Per questo è stato necessario procedere a una lettura integrale di Croce, avendo in una mano l'esperienza dei «testi» e nell'altra quella categoria della «totalità», che è più facile da invocare che da impiegare. Ma è stato necessario anche dare al lavoro un «taglio» particolare, in modo da evitare le semplificazioni eccessive e le sintesi sommarie, per andare direttamente al cuore dei problemi. L'impostazione metodologica ha richiesto da una parte l'adesione all'universo spirituale dell'autore studiato, e dall'altra l'obiettivo distaccato scientifico: due condizioni fondamentali, queste, che solo in apparenza si escludono, perché in realtà si conciliano riassumendosi nel concetto di «critica interna». DAL TESTO - "A ragione Croce afferma che il liberalismo è grandemente debitore all'estetica e alla filosofia dell'economia che hanno asserito la spiritualità del corpo, l'interiorità dell'oggetto nel soggetto, sì che il suo destino è legato a quelle due scienze che unificando Dio e mondo, idea e fatto, garantiscono, contro ogni disposizione autoritaria, la spontaneità e i contrasti delle forze spirituali; e a ragione afferma che l'autoritarismo politico e morale si congiunge a tutte quelle concezioni che hanno un riferimento trascendente e oltremondano. (da Archiviostorico.info). Ristampa del libro edito da Bulzoni, nel 1976. (shrink)

The paper lists several editions of Euclid’s Elements in the Early Modern Age, giving for each of them the axioms and postulates employed to ground elementary mathematics.

In his mature works, Kuhn abandons the concept of a paradigm and becomes more interested in the analysis of the conceptual structure of scientific theories. These changes are interpreted as resulting from a ‘linguistic turn’ that Kuhn underwent sometimes around the 1980s. Much of the philosophical discussions about Kuhn’s post-‘linguistic turn’ philosophy revolves around his views on taxonomic concepts. Apart from taxonomy, however, the mature Kuhn introduces other concepts, such as conceptual networks and lexicons. This article distinguishes these three concepts (...) and shows that adopting each one of them has different impacts on Kuhn’s model of science. After distinguishing and assessing the concepts of taxonomy, network, and lexicon, it will be argued that the latter not only fits the Kuhnian model of science better, but it also helps capture the wide sense of Kuhn’s early concept of a paradigm. (shrink)