Given two (positive) equivalence relations ∼ 1 , ∼ 2 on the set ω of natural numbers, we say that ∼ 1 is m-reducible to ∼ 2 if there exists a total recursive function h such that for every x, y ∈ ω, we have $x \sim_1 y \operatorname{iff} hx \sim_2 hy$ . We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a "uniformity property" holds). This (...) result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration. From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory. (shrink)

It is well known that, in Peano arithmetic, there exists a formula Theor (x) which numerates the set of theorems. By Gödel's and Löb's results, we have that Theor (˹p˺) ≡ p implies p is a theorem ∼Theor (˹p˺) ≡ p implies p is provably equivalent to Theor (˹0 = 1˺). Therefore, the considered "equations" admit, up to provable equivalence, only one solution. In this paper we prove (Corollary 1) that, in general, if P (x) is an arbitrary formula built (...) from Theor (x), then the fixed-point of P (x) (which exists by the diagonalization lemma) is unique up to provable equivalence. This result is settled referring to the concept of diagonalizable algebra (see Introduction). (shrink)

We generalize a well-knownSmullyan's result, by showing that any two sets of the kindC a = {x/ xa} andC b = {x/ xb} are effectively inseparable (if I b). Then we investigate logical and recursive consequences of this fact (see Introduction).

Some years ago, Yablo gave a paradox concerning an infinite sequence of sentences: if each sentence of the sequence is 'every subsequent sentence in the sequence is false', a contradiction easily follows. In this paper we suggest a formalization of Yablo's paradox in algebraic and topological terms. Our main theorem states that, under a suitable condition, any continuous function from 2N to 2N has a fixed point. This can be translated in the original framework as follows. Consider an infinite sequence (...) of sentences, where any sentence refers to the truth values of the subsequent sentences: if the corresponding function is continuous, no paradox arises. (shrink)

By an unfounded chain for a function f:X→X we mean a sequence nω of elements of X s.t. fxn+1=xn for every n. Unfounded chains can be regarded as a generalization of fixed points, but on the other hand are linked with concepts concerning non-well-founded situations, as ungrounded sentences and the hypergame. In this paper, among other things, we prove a lemma in general topology, we exhibit an extensional recursive function from the set of sentences of PA into itself without an (...) unfounded chain, and we prove that every term in a Magari algebra has an unfounded chain. (shrink)

In Zwicker (1987) the hypergame paradox is introduced and studied. In this paper we continue this investigation, comparing the hypergame argument with the diagonal one, in order to find a proof schema. In particular, in Theorems 9 and 10 we discuss the complexity of the set of founded elements in a recursively enumerable relation on the set N of natural numbers, in the framework of reduction between relations. We also find an application in the theory of diagonalizable algebras and construct (...) an undecidable formula. (shrink)

Logic and Knowledge -/- Editor: Carlo Cellucci, Emily Grosholz and Emiliano Ippoliti Date Of Publication: Aug 2011 Isbn13: 978-1-4438-3008-9 Isbn: 1-4438-3008-9 -/- The problematic relation between logic and knowledge has given rise to some of the most important works in the history of philosophy, from Books VI–VII of Plato’s Republic and Aristotle’s Prior and Posterior Analytics, to Kant’s Critique of Pure Reason and Mill’s A System of Logic, Ratiocinative and Inductive. It provides the title of an important collection of papers (...) by Bertrand Russell (Logic and Knowledge. Essays, 1901–1950). However, it has remained an underdeveloped theme in the last century, because logic has been treated as separate from knowledge. -/- This book does not hope to make up for a century-long absence of discussion. Rather, its ambition is to call attention to the theme and stimulating renewed reflection upon it. The book collects essays of leading figures in the field and it addresses the theme as a topic of current debate, or as a historical case study, or when appropriate as both. Each essay is followed by the comments of a younger discussant, in an attempt to transform what might otherwise appear as a monologue into an ongoing dialogue; each section begins with an historical essay and ends with an essay by one of the editors. -/- Carlo Cellucci is Emeritus Professor of Philosophy at the University of Rome ‘La Sapienza,’ Italy. He is currently completing a book entitled, Remaking Logic: What is Logic Really? -/- Emily Grosholz is Professor of Philosophy at the Pennsylvania State University, USA. She is the author of Representation and Productive Ambiguity in Mathematics and the Sciences (Oxford University Press, 2007). -/- Emiliano Ippoliti is a Research Fellow at the University of Rome ‘La Sapienza,’ Italy. His main interests are heuristics, the logic of discovery, and problem-solving. He is currently working on a book, Ampliating Knowledge: Data, Hypotheses and Novelty. -/- TABLE OF CONTENTS -/- Foreword .................................................................................................... ix Acknowledgements ................................................................................. xxv -/- Section I: Logic and Knowledge -/- Chapter One................................................................................................. 3 The Cognitive Importance of Sight and Hearing in Seventeenthand Eighteenth-Century Logic (Mirella Capozzi) Discussion .............................................................................................. 26 (Chiara Fabbrizi) Chapter Two .............................................................................................. 33 Nominalistic Content (Jody Azzouni) Discussion ............................................................................................... 52 (Silvia De Bianchi) Chapter Three ............................................................................................ 57 A Garden of Grounding Trees (Göran Sundholm) Discussion.......................................................................................... .. 75 (Luca Incurvati) Chapter Four .............................................................................................. 81 Logics and Metalogics (Timothy Williamson) Discussion.......................................................................................... 101 (Cesare Cozzo) Chapter Five ............................................................................................ 109 Is Knowledge the Most General Factive Stative Attitude? (Cesare Cozzo) Discussion.......................................................................................... 117 (Timothy Williamson) Chapter Six .............................................................................................. 123 Classifying and Justifying Inference Rules (Carlo Cellucci) Discussion.......................................................................................... 143 (Norma B. Goethe) -/- Section II: Logic and Science -/- Chapter Seven.......................................................................................... 151 The Universal Generalization Problem and the Epistemic Status of Ancient Medicine: Aristotle and Galen (Riccardo Chiaradonna) Discussion.......................................................................................... 168 (Diana Quarantotto) Chapter Eight........................................................................................... 175 The Empiricist View of Logic (Donald Gillies) Discussion.......................................................................................... 191 (Paolo Pecere) Chapter Nine............................................................................................ 197 Artificial Intelligence and Evolutionary Theory: Herbert Simon’s Unifying Framework (Roberto Cordeschi) Discussion.......................................................................................... 216 (Francesca Ervas) Chapter Ten ............................................................................................. 221 Evolutionary Psychology and Morality: The Renaissance of Emotivism? (Mario De Caro) Discussion.......................................................................................... 232 (Annalisa Paese) Chapter Eleven ........................................................................................ 237 Between Data and Hypotheses (Emiliano Ippoliti) Discussion.......................................................................................... 262 (Fabio Sterpetti) -/- Section III: Logic and Mathematics -/- Chapter Twelve ....................................................................................... 273 Dedekind Against Intuition: Rigor, Scope and the Motives of his Logicism (Michael Detlefsen) Discussion.......................................................................................... 290 (Marianna Antonutti) Chapter Thirteen...................................................................................... 297 Mathematical Intuition: Poincaré, Polya, Dewey (Reuben Hersh) Discussion.......................................................................................... 324 (Claudio Bernardi) Chapter Fourteen ..................................................................................... 329 On the Finite: Kant and the Paradoxes of Knowledge (Carl Posy) Discussion.......................................................................................... 358 (Silvia Di Paolo) Chapter Fifteen ........................................................................................ 363 Assimilation: Not Only Indiscernibles are Identified (Robert Thomas) Discussion.......................................................................................... 380 (Diego De Simone) Chapter Sixteen ....................................................................................... 385 Proofs and Perfect Syllogisms (Dag Prawitz) Discussion.......................................................................................... 403 (Julien Murzi) Chapter Seventeen ................................................................................... 411 Logic, Mathematics, Heterogeneity (Emily Grosholz) Discussion.......................................................................................... 427 (Valeria Giardino) -/- Contributors........................................................................................ ..... 433 Index............................................................................................... ......... 437 -/- Price Uk Gbp: 49.99 Price Us Usd: 74.99. 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One might suppose that Everettian quantum mechanics (EQM) is inhospitable to metaphysial indeterminacy (MI), given that, as A. Wilson (2020) puts it, "the central idea of EQM is to replace indeterminacy with multiplicity" (77). But as Wilson goes on to suggest, the popular decoherence-based understanding of EQM (henceforth: DEQM) appears to admit of indeterminacy in both world number and world nature, where the latter indeterminacy---our focus here---is plausibly metaphysical. After a brief presentation of DEQM (S1), we bolster the case for (...) there being MI in world nature in DEQM (S2). The remainder of the paper is devoted to a comparative assessment of the two main approaches to MI for purposes of accommodating this MI---namely, a metaphysical supervaluationist approach (as per Barnes and Williams 2011) and a determinable-based approach (as per Wilson 2013 and Calosi and Wilson 2018 and 2021). We briefly describe each approach (S3), then offer arguments in favour of a determinable-based approach to world nature MI in DEQM (S4). (shrink)

This volume is the first systematic and thorough attempt to investigate the relation and the possible applications of mereology to contemporary science. It gathers contributions from leading scholars in the field and covers a wide range of scientific theories and practices such as physics, mathematics, chemistry, biology, computer science and engineering. Throughout the volume, a variety of foundational issues are investigated both from the formal and the empirical point of view. The first section looks at the topic as it applies (...) to physics. The section addresses questions of persistence and composition within quantum and relativistic physics and concludes by scrutinizing the possibility to capture continuity of motion as described by our best physical theories within gunky space times. The second part tackles mathematics and shows how to provide a foundation for point-free geometry of space switching to fuzzy-logic. The relation between mereological sums and set-theoretic suprema is investigated and issues about different mereological perspectives such as classical and natural Mereology are thoroughly discussed. The third section in the volume looks at natural science. Several questions from biology, medicine and chemistry are investigated. From the perspective of biology, there is an attempt to provide axioms for inferring statements about part hood between two biological entities from statements about their spatial relation. From the perspective of chemistry, it is argued that classical mereological frameworks are not adequate to capture the practices of chemistry in that they consider neither temporal nor modal parameters. The final part introduces computer science and engineering. A new formal mereological framework in which an indeterminate relation of part hood is taken as a primitive notion is constructed and then applied to a wide variety of disciplines from robotics to knowledge engineering. A formal framework for discrete mereotopology and its applications is developed and finally, the importance of mereology for the relatively new science of domain engineering is also discussed. (shrink)

This paper provides the first thorough assessment of a physics-based answer, the Bound State Answer, to the Special Composition Question. According to the BSA some objects compose something if they are in a common bound state. The reasons to endorse such an answer, in particular, motivations coming from empirical adequacy and conservativeness, precision, simplicity, and parsimony, are critically addressed. I then go on to compare the BSA to other moderate answers to the SCQ and consider whether objections raised against such (...) answers can be raised against the BSA as well. I finally relate the discussion with mereological pluralism. (shrink)

Multilocation – the notion of an object being at two places – is a central notion in metaphysics. According to a widespread view, multilocation is problematic but metaphysically possible. In effect, it has been claimed that in a quantum world, multilocation is not simply possible but actual. This article provides a new argument against the latter claim: there is no quantum multilocation.

Despite recent interest in political ethnography, most of the reflection has been on the ethnographic aspect of the enterprise with much less emphasis on the question implicit in the first word of the couplet: What is actually political about political ethnography and how much should ethnographers pre-define it? The question is complicated because a central component of the definition of what is political is actually the struggle to define its jurisdiction and how it gets distinguished from what it is not. (...) In this article we aim to show how ethnography can actually lead us out of this conundrum in which the political is paradoxically both predefined and, at the same time, the open question that leads the process of inquiry. We do so by advancing a formal and relational approach that provides us with procedural tools to define the nature and specificity of the political bond not ex ante, but rather during the process of research itself. In the first part of the article we historicize the development of political ethnography as a distinct avenue for inquiry and show what have been the challenges to its normalization. This is followed by the article’s main section, which focuses on the four ways in which what is political has been conceptualized in contemporary socio-ethnographical literature. In the conclusion of the article, we advance a lowest common denominator definition proposal, with examples from other scholars as well as from our own research to illustrate how this approach would work. (shrink)

There is an interesting puzzle about the interaction between mereology, topology, and dependence. It is not only interesting in and on itself, but also reveals subtleties about the aforementioned interaction that have gone unnoticed. The puzzle has it that the following plausible claims are jointly inconsistent: wholes depend on their parts; boundaries are parts; boundaries depend on the whole they are part of. In the paper, I first argue that claims – are not as a matter of fact inconsitent insofar (...) as further assumptions are needed to get the puzzle off the ground. I consider several such assumptions, somemore plausible than others. Though I do not take any definite stance as to whether the plusibility of the assumptions considered trump that of claims –, I set forth a suggestion to replace with something similar yet interestingly different. (shrink)

There are famously two main metaphysics of persistence, namely three and four-dimensionalism. Both yield a particular solution to the so called puzzle of change. I argue that typical three-dimensionalist solutions to the puzzle face insurmountable difficulties even in the simplest relativistic setting, that of Minkowski spacetime.

En este trabajo nos proponemos acercarnos a la noción del conocimiento simbólico en la obra de Gottfried Leibniz, rastreándola en diversos textos de su autoría en que se encuentra diseminada. Comenzamos con la importante distinción entre idea y concepto, que separa el planteo de nuestro autor de las formulaciones de otros como Descartes, Malebranche y Spinoza. Seguimos con la clasificación de los tipos de conceptos que el filósofo de Leipzig lleva acabo, en la que divide a los conceptos simbólicos de (...) los intuitivos. Abordamos después las ventajas epistemológicas que presenta el pensamiento simbólico frente al pensamiento verbal, analizando sus funciones cognitivas y su rol en diferentes tipos de sistemas semióticos. Continuamos asentando el simbolismo epistemológico dentro de una teoría general de la representación, que sostiene cierto isomorfismo entre las esferas del lenguaje, el pensamiento y la realidad. Finalmente, fundamentamos ontológicamente esta analogía representativa estructural en la noción leibniziana de expresión y la vinculamos con el perspectivismo monadológico, poniendo de relieve su cercanía con algunos planteos fenomenológicos y hermenéuticos de nuestros días. (shrink)

Multilocation – the notion of an object being 18756735_00000147_text.pdfat two places – is a central notion in metaphysics. According to a widespread view, multilocation is problematic but metaphysically possible. In effect, it has been claimed that in a quantum world, multilocation is not simply possible but actual. This article provides a new argument against the latter claim: there is no quantum multilocation.

Super-substantivalism (of the type we’ll consider) roughly comprises two core tenets: (1) the physical properties which we attribute to matter (e.g. charge or mass) can be attributed to spacetime directly, with no need for matter as an extraneous carrier “on top of” spacetime; (2) spacetime is more fundamental than (ontologically prior to) matter. In the present paper, we revisit a recent argument in favour of super-substantivalism, based on General Relativity. A critique is offered that highlights the difference between (various accounts (...) of) fundamentality and (various forms of) ontological dependence. This affords a metaphysically more perspicuous view of what super-substantivalism’s tenets actually assert, and how it may be defended. We tentatively propose a re-formulation of the original argument that not only seems to apply to all classical physics, but also chimes with a standard interpretation of spacetime theories in the philosophy of physics. (shrink)

Different realistic attitudes towards wavefunctions and quantum states are as old as quantum theory itself. Recently Pusey, Barret and Rudolph on the one hand, and Auletta and Tarozzi on the other, have proposed new interesting arguments in favor of a broad realistic interpretation of quantum mechanics that can be considered the modern heir to some views held by the fathers of quantum theory. In this paper we give a new and detailed presentation of such arguments, propose a new taxonomy of (...) different realistic positions in the foundations of quantum mechanics and assess the scope, within this new taxonomy, of these realistic arguments. (shrink)

It is notoriously difficult to define Metaphysics1, its content, its method, its language, its scope. Thus I will not even try an attempt here. I will rest content to point out some widely held characterizations. A long and highly influential tradition maintains that Metaphysics is the study of being qua being. It is concerned with what there is, what kind of things are the things that there are, what properties do they have, how they are related. In this sense Metaphysics (...) deals with the more general features of reality, the most fundamental categories of being. Call this tradition General Metaphysics. (shrink)

In questo lavoro si presenta un nuovo esperimento mentale che solleva un particolare problema per quello che possiamo chiamare “fisicalismo minimale” in filosofia della mente. In particolare si argomenta che il fisicalismo minimale o i) non è in grado di fornirne un resoconto adeguato dell’esperimento mentale presentato, o ii) viene costretto a fornire un resoconto che è fortemente in contrasto con la nostra immagine scientifica del mondo. Il problema sollevato è un particolare esempio di quelli che Chalmers (1996) definisce hard (...) problems in filosofia della mente. Questa è la struttura del lavoro. Nella sezione 2. si presenta l’esperimento mentale, che chiameremo di Shem-Shaun1. Nella sezione 3. illustriamo il nostro argomento principale contro il fisicalismo minimale a partire dall’esperimento mentale della sezione precedente. Nella sezione conclusiva si discutono diversi modi di controbattere all’argomento principale e le sue conseguenze per il fisicalismo. (shrink)

Resumen: En este artículo presentamos a san Agustín como punto de partida de la reflexión de Husserl respecto del tiempo y la correlación entre memoria y Erinnerung. La investigación fenomenológica de Husserl acerca de la conciencia interna del tiempo parte de la reflexión de san Agustín por el mismo problema. En estas obras, el tiempo se puede medir porque hay una distentio animi. En Husserl, Die Erinnerung nos coloca ante una conexión infinita de “antes”, pues toda percepción se encuentra en (...) una relación de multiplicidad de percepciones. Ambos pensadores están unidos por la vocación de dar cuenta de la propia inmanencia.: This article presents Saint Agustin as a starting point for Husserl’s reflection on time and the correlation between memory and Erinnerung. Husserl’s phenomenological research on the internal consciousness of time starts from the reflection of St. Augustine on the same problem. In this works, time can be measured because there is distentio animi. In Husserl, Die Erinnerung places us in the face of an infinite “before” connection, because all perceptions are to be found on a relationship of multiplicity of perceptions. Both thinkers are united by the vocation to account for their own immanence. (shrink)

La reflexión de Friederich Schleiermacher sobre la universidad se centra en las dificultades que observa para superar el estancamiento de la educación alemana a principios del siglo XIX y establece, en ella, dos niveles: por un lado, la excesiva burocratización de la vida universitaria, que le quita lo que tiene precisamente de vital y, por otro, desde la perspectiva propiamente del profesor, la imposibilidad de cultivar el talento de los alumnos mediante clases farragosas, en las que se repite lo que (...) ya está escrito y en las que la memoria es la única medida del conocimiento. La mayoría de los profesores se encuentran en esta situación porque los conocimientos se perciben de manera fragmentaria y, por lo tanto, se transmiten sin el contenido de su unidad profunda. La Facultad de Filosofía tiene el cometido de construir el camino hacia la unidad de conocimiento y fundamentar el sentido profundo de la vida universitaria. Por ello, los cimientos de la universidad moderna descansan sobre el profesor que investiga y transmite en sus clases el pathos de aquella tarea; también descansan sobre el alumno, a quien hay que ayudar a comprender la dimensión de su talento y desde allí, emprender la tarea de la auto-educación. (shrink)

La presencia de Platón y del pensamiento platónico se ha mantenido de manera incesante a lo largo de la historia de la cultura. En esta historia, el siglo XX refleja el rechazo y la aceptación, la recepción, en suma, de un modo inusitado tal vez porque, durante este siglo, la humanidad se vio cara a cara con los infiernos que supo crear: un mundo en ruinas hacía imposible intuir la perfecta quietud de las Formas. La recepción de Platón en el (...) siglo XX bucea en la recepción de Platón en aquel mundo contradictorio: ¿cómo se integra su pensamiento fundacionalmente metafísico con la deconstrucción de la metafísica? ¿Cuáles son los alcances de la crítica que la teoría política hace, desde la época de los totalitarismos, al autor de la República? ¿Qué valor cognoscitivo y pedagógico puede tener en mito platónico para filósofos cristianos que se ocupan del pensamiento platónico? ¿Cómo recibe la poesía hispanoamericana su voz de demiurgo? Estas preguntas, de las que se ofrecen diversas respuestas en este libro, son tratadas con rigor académico y confrontadas doblemente con las fuentes: en primer término, con Platón y luego con quienes son estudiados en el presente volumen, aunque desde ángulos muy diversos: M. Heidegger, H. G. Gadamer, K. Popper, E. Cassirer, L. Marechal. La seguridad del tratamiento metodológico y lingüístico que se refleja en esta obra hace que sea altamente recomendable para alumnos de grado y de posgrado en Filosofía, Teoría Política, Literatura y Humanidades en general. (shrink)

El desplazamiento del mito de la cultura ofrece alternativas (ciencia, historia, política) que corren el riesgo de reproducir planteamientos irracionales que, en el seno de aquella forma, se encuentran superados. El pensamiento mítico ofrece soluciones novedosas para ciertos problemas actuales.

Resumen Este artículo estudia la transformación de las bases del escepticismo antiguo en el pensamiento agustiniano; para presentar el proceso se parte de la crisis que supuso la ruptura con experiencia maniquea, se distingue su preocupación y consideración sobre el modo y los límites del conocimiento y se analiza la función de la imagen mítica de Proteo en Contra Académicos; esta imagen permite ponderar la capacidad de la representación mítica como contrafuerte del carácter discursivo del saber humano. Mediante este recurso, (...) san Agustín afirma que la razón no puede quedar aislada de otros modos complementarios de comprensión, hace propio el núcleo de la actitud escéptica y amplía la noción de racionalidad para la filosofía cristiana.This paper studies the transformation of the foundations of ancient skepticism in Augustinian thought; to present the process we start from the crisis that led to the break with Manichaean experience, we distinguished their concern and consideration about the mode and limits of knowledge and analyzed the function of Proteo's mythical image in Contra academicos; this image allows us to weigh the capacity of mythical representation as a buttress of the discursive character of human knowledge. Through this resource, St. Augustine affirms that reason cannot be isolated from other complementary modes of understanding, appropriates the core of the skeptical attitude and broadens the notion of rationality for Christian philosophy. (shrink)

Kline's functional categories for the evolution of teaching blur some valuable distinctions. Moreover, her account provides no answer to the question of why direct active teaching seems to be a uniquely human phenomenon.

Develops a critique of the concept of the attention economy from the perspectives of labour, time, and power.

The relation between noise and disturbance is investigated within the general framework of Galois connections. Within this framework, we introduce the notion of leak of information, mathematically defined as one of the two closure maps arising from the observable-channel compatibility relation. We provide a physical interpretation for it, and we give a comparison with the analogous closure maps associated with joint measurability and simulability for quantum observables.