In this paper we show that the Gupta-Belnap systems S# and S* are П12. Since Kremer has independently established that they are П12-hard, this completely settles the problem of their complexity. The above-mentioned upper bound is established through a reduction to countable revision sequences that is inspired by, and makes use of a construction of McGee.

We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like “rationality at a node” and “knowledge at a node.” We distinguish between the game theorist's and the players' own “theory of the game.” The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While in general the (...) game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory. (shrink)

Quine’s “New Foundations” (NF) was first presented in Quine [1937] and later on in Quine [1963]. Ernst Specker [1958, 1962], building upon a previous result of Ehrenfeucht and Mostowski [1956], showed that NF is consistent if and only if there is a model of the Theory of Negative (and positive) Types (TNT) with full extensionality that admits of a “shifting automorphism,” but the existence of a such a model remains an open problem.

Gian Aldo Antonelli and Cristina Bicchieri. Backwards Forwards Induction.

Gian Aldo Antonelli and Cristina Bicchieri. Forward Induction.

In this paper we apply the idea of Revision Rules, originally developed within the framework of the theory of truth and later extended to a general mode of definition, to the analysis of the arithmetical hierarchy. This is also intended as an example of how ideas and tools from philosophical logic can provide a different perspective on mathematically more “respectable” entities. Revision Rules were first introduced by A. Gupta and N. Belnap as tools in the theory of truth, and they (...) have been further developed to provide the foundations for a general theory of (possibly circular) definitions. Revision Rules are non-monotonic inductive operators that are iterated into the transfinite beginning with some given “bootstrapper” or “initial guess.” Since their iteration need not give rise to an increasing sequence, Revision Rules require a particular kind of operation of “passage to the limit,” which is a variation on the idea of the inferior limit of a sequence. We then define a sequence of sets of strictly increasing arithmetical complexity, and provide a representation of these sets by means of an operator G(x, φ) whose “revision” is carried out over ω2 beginning with any total function satisfying certain relatively simple conditions. Even this relatively simple constraint is later lifted, in a theorem whose proof is due to Anil Gupta. (shrink)

In this paper we argue that Revision Rules, introduced by Anil Gupta and Nuel Belnap as a tool for the analysis of the concept of truth, also provide a useful tool for defining computable functions. This also makes good on Gupta's and Belnap's claim that Revision Rules provide a general theory of definition, a claim for which they supply only the example of truth. In particular we show how Revision Rules arise naturally from relaxing and generalizing a classical construction due (...) to Kleene, and indicate how they can be employed to reconstruct the class of the general recursive functions. We also point at how Revision Rules can be employed to access non-minimal fixed points of partially defined computing procedures. (shrink)

The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the philosophical significance (...) of each. (shrink)

Due programmi diversi si intersecano nel lavoro di Frege sui fondamenti dell’aritmetica: • Logicismo: l’aritmetica `e riducibile alla logica; • Estensionalismo: l’aritmetica `e riducibile a una teoria delle estensioni. Sia nei Fondamenti che nei Principi, Frege articola l’idea che l’aritmetica sia riducibile a una teoria logica delle estensioni.

The paper focuses on a specific point addressed in the previous article of L. Cesalli and H. Taieb The road to “ideelle Verähnlichung”, namely, the correctness of Marty’s interpretation of the early (pre-reistic) Brentanian conception of intentionality. Moving from the distinction between immanent (or intentional) object and intentional correlate, as developed by Brentano in his lectures on Descriptive Psychology, and referring to Aristotelian theory of relativa, which Brentano always remained faithful to, I show that Marty interpreted Brentano’s early conception of (...) intentionality in an immanentistic sense. Thus, the foundations for the so-called ‘ontological’ or ‘discontinuist view’ of Brentano’s intentionality thesis were put down, and through the exegesis of his pupils and Brentano’s Enkelschüler Kastil and Kraus it was then passed on, from generation to generation, until the present day. (shrink)

A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.

In this paper we introduce three methods to approach philosophical problems informationally: Minimalism, the Method of Abstraction and Constructionism. Minimalism considers the specifications of the starting problems and systems that are tractable for a philosophical analysis. The Method of Abstraction describes the process of making explicit the level of abstraction at which a system is observed and investigated. Constructionism provides a series of principles that the investigation of the problem must fulfil once it has been fully characterised by the previous (...) two methods. For each method, we also provide an application: the problem of visual perception, functionalism, and the Turing Test, respectively. (shrink)

In the paper it is argued that bridging the digital divide may cause a new ethical and social dilemma. Using Hardin's Tragedy of the Commons, we show that an improper opening and enlargement of the digital environment (Infosphere) is likely to produce a Tragedy of the Digital Commons (TDC). In the course of the analysis, we explain why Adar and Huberman's previous use of Hardin's Tragedy to interpret certain recent phenomena in the Infosphere (especially peer-to-peer communication) may not be entirely (...) satisfactory. We then seek to provide an improved version of the TDC that avoids the possible shortcomings of their model. Next, we analyse some problems encountered by the application of classical ethics in the resolution of the TDC. In the conclusion, we outline the kind of work that will be required to develop an ethical approach that may bridge the digital divide but avoid the TDC. (shrink)

While second-order quantifiers have long been known to admit nonstandard, or interpretations, first-order quantifiers (when properly viewed as predicates of predicates) also allow a kind of interpretation that does not presuppose the full power-set of that interpretationgeneral” interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.

The present paper analyzes the dialogue between Gilles Deleuze and François Jullien. We focus on three axes articulated by the idea of immanence. Firstly, we shall compare the usage of the expression “absolutization of immanence”, coined by the sinologist in order to play down transcendence in Chinese thought, which was eventually applied by Deleuze within the realm of belief in this world. Secondly, we shall examine Jullien’s critic to Deleuze on the relation between philosophy and wisdom. We sustain that even (...) if the deleuzian description presents ambivalence in the “pre-philosophical” plane of immanence, it does not entail the superiority of philosophy. Thirdly, we shall confront the divergent perspectives on immanence: Deleuze’s treatment as plan and Jullien’s conception as fonds. Finally, we shall expose the hardship to think the idea of immanence present in the work of both authors. (shrink)

El concepto de "inmanencia" ha adquirido una creciente relevancia en el campo filosófico contemporáneo. Dos destacados pensadores franceses, Gilles Deleuze y Franfois Jullien, le han otorgado un lugar fundamental en sus investigaciones. El propósito de este artículo es indagar el diálogo, poco explorado aún, entre ambos autores. Nuestra hipótesis de trabajo propone los siguientes ejes de análisis: primero, comparamos los usos de la expresión "absolutización de la inmanencia", acuñada por el sinólogo a fin de relativizar la trascendencia en el pensamiento (...) chino y luego empleada por Deleuze en el marco de la creencia en este mundo. En segundo lugar, examinamos las críticas que Jullien le dirige a Deleuze respecto a la relación entre la filosofía y la sabiduría. En tercer lugar, confrontamos sus perspectivas divergentes sobre la inmanencia en tanto "plano" (Deleuze) y como "fondo" (Jullien). Finalmente, ponemos de manifiesto la dificultad presente en ambos autores para pensar la idea de inmanencia. The concept of "immanence" has acquired substantial relevance within the contemporary philosophical field. Two influential french thinkers, Gilles Deleuze and Franfois Jullien, have conferred this concept a fundamental standing in their research. The purpose of this work is to explore the dialogue between both authors as this exploration has not been concluded. Our hypothesis puts forward the following axes of analysis. First, we shall compare the usage of the expression "absolutization of immanence" coined by the sinologist in order to put in perspective the idea of transcendence in Chinese thought. This expression was then applied by Deleuze within the frame of the belief in this world. Second, we shall examine Jullien's critique of Deleuze concerning the relationship between Philosophy and Wisdom. Third, we shall confront the divergent perspectives on immanence: as "plane" (Deleuze) and as "buttom" (Jullien). Finally, we shall show the difficulty to ponder the idea of immanence in both authors. (shrink)

Henry Leonard and Karel Lambert first introduced so-called presupposition-free (or just simply: free) logics in the 1950’s in order to provide a logical framework allowing for non-denoting singular terms (be they descriptions or constants) such as “the largest prime” or “Pegasus” (see Leonard [1956] and Lambert [1960]). Of course, ever since Russell’s paradigmatic treatment of definite descriptions (Russell [1905]), philosophers have had a way to deal with such terms. A sentence such as “the..

This article reports on the development of the managerial ethical profile (MEP) scale. The MEP scale is a multilevel, self-reporting scale measuring the perceived influence that different dimensions of common ethical frameworks have on managerial decision making. The MEP scale measures on eight subscales: economic egoism, reputational egoism, act utilitarianism, rule utilitarianism, self-virtue of self, virtue of others, act deontology, and rule deontology. Confirmatory factor analysis (CFA) was used to provide evidence of scale validity. Future research needs and the value (...) of this measure for business ethics are discussed. (shrink)

In §21 of Grundgesetze der Arithmetik asks us to consider the forms: a a2 = 4 and a a > 0 and notices that they can be obtained from a φ(a) by replacing the function-name placeholder φ(ξ) by names for the functions ξ2 = 4 and ξ > 0 (and the placeholder cannot be replaced by names of objects or of functions of 2 arguments).

Many different modes of definition have been proposed over time, but none of them allows for circular definitions, since, according to the prevalent view, the term defined would then be lacking a precise signification. I argue that although circular definitions may at times fail uniquely to pick out a concept or an object, sense still can be made of them by using a rule of revision in the style adopted by Anil Gupta and Nuel Belnap in the theory of truth.

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

With the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, this paper presents a formalization of first-order arithmetic, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.

La thèse de l’« inexistence intentionnelle » formulée par Brentano a été traditionnellement interprétée comme une théorie de la « relation intentionnelle », autrement dit de la relation entre l’acte mental et son « objet immanent » ou « intentionnel », c’est-à-dire interne à la conscience. Se fondant sur la lecture du fameux passage sur l’intentionnalité de la Psychologie du point de vue empirique , le présent article démontre que l’interprétation ontologique de la théorie de l’intentionnalité du premier Brentano est (...) insoutenable, toute dominante qu’elle est. Pour ce faire, nous partirons des sources de la pensée de Brentano, en particulier de la théorie de la perception et des relatifs d’Aristote, pour rejeter la conception de l’objet immanent ou intentionnel comme entité immanente à la conscience et mettre en évidence la continuité qui existe entre la première conception de Brentano de l’intentionnalité et la seconde, consécutive à ce qu’on définit comme le tournant réiste de sa pensée.Brentano’s thesis of “intentional inexistence” has been traditionally interpreted as a theory of “intentional relationships”, i.e., of the relation between a mental act and its “immanent” or “intentional object”, within consciousness. Starting from the famous passage on intentionality in Psychology from an Empirical Standpoint , the present paper shows that the dominant ontological interpretation of Brentano’s former theory of intentionality is untenable. Proceeding from the sources of Brentano’s thought, in particular from Aristotle’s theory of perception and of relatives, the conception of the immanent or intentional object as an immanent entity to consciousness is rejected. Instead, the continuity between Brentano’s former conception of intentionality and the subsequent one, following the so-called reistic turning-point in his thought, is highlighted. (shrink)

The purpose of this note is to acknowledge a gap in a previous paper, "The complexity of revision," and to provide a corrected version of the argument.

Husserl’s Third Logical Investigation, ostensibly dealing with the phenomenology of whole and parts, is actually meant to introduce the notion of Fundierung. This term is frequently used in the phenomenological literature, although little has been written about Fundierung itself since Husserl introduced it. Husserl himself, although he used it extensively, never again felt the need to reopen the discussion.

It has been observed that whereas painters and musicians are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work. Mathematicians, instead, are fond of passing judgment on the beauty of their favored pieces of mathematics. Even a cursory observation shows that the characteristics of mathematical beauty are at variance (...) with those of artistic beauty. For example, courses in art appreciation are fairly common; it is however unthinkable to find any mathematical beauty appreciation courses taught anywhere. The purpose of the present paper is to try to uncover the sense of the term beauty as it is currently used by mathematicians. (shrink)

This paper presents a bivalent extensional semantics for positive free logic without resorting to the philosophically questionable device of using models endowed with a separate domain of "non-existing" objects. The models here introduced have only one (possibly empty) domain, and a partial reference function for the singular terms (that might be undefined at some arguments). Such an approach provides a solution to an open problem put forward by Lambert, and can be viewed as supplying a version of parametrized truth non (...) unlike the notion of "truth at world" found in modal logic. A model theory is developed, establishing compactness, interpolation (implying a strong form of Beth definability), and completeness (with respect to a particular axiomatization). (shrink)

In this paper we introduce the notion of a set algebra S satisfying a system E of equations. After defining a notion of freeness for such algebras, we show that, for any system E of equations, set algebras that are free in the class of structures satisfying E exist and are unique up to a bisimulation. Along the way, analogues of classical set-theoretic and algebraic properties are investigated.

Offers a glimpse into the world of science and technology between 1950 and 1990 as seen through the eyes of a mathematician, and debunks various myths of scientific philosophy. Portrays some of the great scientific personalities of the period, including Stanislav Ulam, who patented the hydrogen bomb, and Jack Schwartz, one of the founders of computer science. Also discusses phenomenology of mathematics, and philosophy and computer science. Includes book reviews. For students and academics. Annotation copyright by Book News, Inc., Portland, (...) OR. (shrink)

Der Triestiner Vittorio Benussi (1878-1927), Mitglied der Grazer gegenstandstheoretischen und psychologischen Schule um Alexius Meinong, war einer der bedeutendsten Experimentalpsychologen seiner Zeit. Seine Pionierleistungen auf dem Gebiet der experimentellen Gestaltpsychologie gerieten jedoch bald durch die fortschreitende Durchsetzung der Berliner Schule der Gestalttheorie in Vergessenheit, so daß sein Werk bis heute weitgehend unbekannt geblieben ist. Benussis wissenschaftliche Tätigkeit, die sich durch eine streng experimentelle Vorgangsweise auszeichnet, erweist sich rückblickend als fruchtbarer Anknüpfungspunkt für die zeitgenössische Kognitionswissenschaft. Dies ermöglicht eine Neubewertung seiner wissenschaftlichen (...) Arbeit und jener aktpsychologischen Tradition, der er angehörte. Benussis Untersuchungen nehmen ihren Ausgangspunkt in der Voraussetzung der Intentionalität des Psychischen. Seine Arbeiten thematisieren das Problem des subjektiven Eingriffs in die Konstitution der Erfahrungsgegebenheiten, indem sie die Vorherrschaft des Subjektes über die elementaren Sinnesbedingungen betonen. In Benussis Gesamtwerk läßt sich ein einheitliches Programm erkennen: Ausgehend von der Gegenstands- und Produktionstheorie der Grazer Schule, entwickelt Benussi seinen eigenen theoretischen Standpunkt, der ihn - bewußt - der Phänomenologie Husserls nähebringt. Benussi verlagert schrittweise den Schwerpunkt seiner Forschungen von der gegenstandstheoretischen Perspektive zur Hervorhebung der latenten Subjektivität, die im Konstitutionsprozeß des Gegebenen wirksam ist. Leben und Werk Vittorio Benussis rückblickend zu betrachten, heißt sich mit einem Klassiker der Psychologie zu beschäftigen, dessen Modernität erst in der heutigen Zeit erkannt werden kann. (shrink)

Esse estudo apresenta os conceitos de documento e documentação propostos por Paul Otlet (1868-1944) presentes no âmbito da Revista do Serviço Público (RSP). Publicada desde 1937, a RSP contempla artigos nas áreas de Administração e Gestão pública, Políticas públicas e Economia do setor público, sendo considerada um importante veículo de informação no Brasil, contemporâneo ao período de efervescência do pensamento de Otlet. A pesquisa consiste numa Revisão Sistemática de Literatura (RSL) com análise documental da RSP no período entre 1951 e (...) 1975. Como procedimento de seleção dos artigos, utiliza o protocolo de pesquisa RSL elaborado por Valentim (2020). A partir do corpus de análise dos textos recuperados identifica os artigos que apresentam os conceitos de “documento” e “documentação” segundo Otlet, bem como os elementos constitutivos sobre o cenário da documentação no Brasil, na perspectiva da publicação de autores brasileiros na RSP, concluindo que os trabalhos analisados contemplam e ampliam a temática da Documentação com importantes reflexões de Otlet para a Ciência da Informação no Brasil. (shrink)

In this paper, we sketch out a simple scheme to evaluate different ways in which Western society has coped with the momentous and hidden problem of envy; afterward, we consider the consequences for the constitution of the social space that these changes entail. We will argue that envy, when considered as a primal feeling, can shed light on René Girard’s notion of metaphysical desire and on diasparagmos rituals. Then, taking into account Jean-Pierre Dupuy’s endogenous fixed point thesis—concerning the constitution of (...) autotranscendent social structures that configure themselves around an attractor, a fixed point, revealed to be a product of the process of constitution, an effect and not a cause—we will consider envy as the main feedback in the system. Starting from this theoretical scenario, we will envisage three steps—on a line sketched according to mimetic theory—marked by the particular conditions and role of the feedback. We will show what kind of balance was guaranteed to Athens (taken as an exemplar archaic society, starting from its self-representation as given by Sophocles in Oedipus the King) by the complementary rituals of pharmakos and ostrakos; what equilibrium was offered to Christian medieval social structures by the doctrine of the deadly sins; and, finally, we will take a look at our own secularized society. Our aim is to apply René Girard’s theory systematically, to show how the philosophy of history that we can infer from his works can help us to detect a curious and at the same time quite dramatic change as far as the homeostatic equilibrium of our society is concerned. Thanks to this exercise, we will be able to show how the transcendental social space— by which we mean the region of possible outcomes, in terms of constraints and possibilities, of a human life in a social context—is constituted through the victimage mechanism as defined by Girard and how this constitution is affected by the performative unveiling of the mechanism itself. (shrink)