In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is ‐definable relative to the previous stages of the recursion. It is known that this principle is provable in. In the present note, we argue that a common formulation of effective transfinite recursion is too restrictive. We then propose a more liberal formulation, which appears very natural and is still (...) provable in. (shrink)

Sortal concepts are at the center of certain logical discussions and have played a significant role in solutions to particular problems in philosophy. Apart from logic and philosophy, the study of sortal concepts has found its place in specific fields of psychology, such as the theory of infant cognitive development and the theory of human perception. In this monograph, different formal logics for sortal concepts and sortal-related logical notions are characterized. Most of these logics are intensional in nature and possess, (...) in addition, a bidimensional character. That is, they simultaneously represent two different logical dimensions. In most cases, the dimensions are those of time and natural necessity, and, in other cases, those of time and epistemic necessity. Another feature of the logics in question concerns second-order quantification over sortal concepts, a logical notion that is also represented in the logics. Some of the logics adopt a constant domain interpretation, others a varying domain interpretation of such quantification. Two of the above bidimensional logics are philosophically grounded on predication sortalism, that is, on the philosophical view that predication necessarily requires sortal concepts. Another bidimensional logic constitutes a logic for complex sortal predicates. These three sorts of logics are among the important novelties of this work since logics with similar features have not been developed up to now, and they might be instrumental for the solution of philosophically significant problems regarding sortal predicates. The book assumes a modern variant of conceptualism as a philosophical background. For this reason, the approach to sortal predicates is in terms of sortal concepts. Concepts, in general, are here understood as intersubjective realizable cognitive capacities. The proper features of sortal concepts are determined by an analysis of the main features of sortal predicates. Posterior to this analysis, the sortal-related logical notions represented in the above logics are discussed. There is also a discussion on the extent to which the set-theoretic formal semantic systems of the book capture different aspects of the conceptualist approach to sortals. These different semantic frameworks are also related to realist and nominalist approaches to sortal predicates, and possible modifications to them are considered that might represent those alternative approaches. (shrink)

Modeling cognition by structural analysis of representation leads to systematic difficulties which are not resolvable. We analyse the merits and limits of a representation-based methodology to modeling cognition by treating Jackendoff's Consciousness and the Computational Mind as a good case study. We note the effects this choice of methodology has on the view of consciousness he proposes, as well as a more detailed consideration of the computational mind. The fundamental difficulty we identify is the conflict between the desire for modular (...) processors which map directly onto representations and the need for dynamically interacting control. Our analysis of this approach to modeling cognition is primarily directed at separating merits from problems and inconsistencies by a critique internal to this approach; we also step outside the framework to note the issues it ignores. (shrink)

Since the transition to democracy in the early 1990s, economic research and instruction in South Africa have become far more quantitative and technically sophisticated. In this paper, I trace and discuss reasons for these developments, and I argue that this quantification of economics should not be at the expense of exchanges with qualitative data that fail the criterion of being representative, or with other disciplines that are less quantitative. With South Africa’s complex history, persistent inequality and considerable cultural diversity, economics (...) has much to gain from interdisciplinary collaboration and mixed methods research. (shrink)

We have previously established that [Formula: see text]-comprehension is equivalent to the statement that every dilator has a well-founded Bachmann–Howard fixed point, over [Formula: see text]. In this paper, we show that the base theory can be lowered to [Formula: see text]. We also show that the minimal Bachmann–Howard fixed point of a dilator [Formula: see text] can be represented by a notation system [Formula: see text], which is computable relative to [Formula: see text]. The statement that [Formula: see text] (...) is well founded for any dilator [Formula: see text] will still be equivalent to [Formula: see text]-comprehension. Thus, the latter is split into the computable transformation [Formula: see text] and a statement about the preservation of well-foundedness, over a system of computable mathematics. (shrink)

This paper explores the perversion of reality in the context of advanced technologies, such as AI, VR, and AR, through the lens of Jean Baudrillard’s theory of hyperreality and the precession of simulacra. By examining the transformative effects of these technologies on our perception of reality, with a particular focus on the usage of digital avatars, the paper highlights the blurred distinction between the real and the simulated, where the copy becomes more ‘real’ than the original. Drawing on Baudrillard’s concept (...) of hyperreality, the paper delves into the perversion of reality as individuals seek refuge in virtual pleasure paradises and embrace artificial pleasures through their digital avatars, disconnecting from genuine human experiences. The convergence of AI, VR, and AR technologies amplifies this hyperreal condition, where digital avatars mimic or surpass the depth of human relationships, challenging our understanding of what is real. In line with Baudrillard’s theory, the paper explores the objectification and commodification of reality within digital spaces, specifically examining the digital avatars’ role in the erosion of genuine human connections. It explores the implications of these avatars in terms of consent, exploitation, and loss of authenticity, echoing Baudrillard’s concerns about the distortion of reality in contemporary society. Recognizing the implications of these technologies, the paper calls for a critical reflection on their transformative power. It emphasizes the need for a nuanced understanding of the hyperreal condition and ethical responsibility in engaging with AI, VR, and AR, particularly in relation to the usage of digital avatars. By resisting the seductive allure of digital escapism and preserving genuine human connections, we can navigate the perversion of reality and cultivate empathy, compassion, and meaningful interactions that transcend the simulated experiences offered by technology. (shrink)

In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight.

This paper explores the intersection of René Girard's mimetic theory and the algorithmic society, particularly in the context of the potential advent of Artificial General Intelligence (AGI). Girard's theory, which elucidates the dynamics of desire, rivalry, scapegoating, and the sacrificial crisis, provides a unique lens through which to examine the complexities of our relationship with AI and its role in the creation of the sacred. As individuals increasingly rely on AI recommendations, the distinction between personal choice and algorithmic manipulation becomes (...) less clear, raising concerns about the authenticity of cultural expressions and the role of algorithms in shaping cultural narratives. The triangular structure of desire, with AI as the model and individuals as the imitators, underscores the power of algorithms in this process. The sacrificial crisis, a key concept in Girard's theory, becomes a critical point of reflection in the algorithmic society. The exposure of the scapegoating mechanism reveals the destructive potential of algorithmic manipulation and calls for new forms of understanding, empathy, and non-violent solutions. This paper argues that recognizing the sacrificial crisis can prompt individuals and society to critically examine the impact of AI's influence, challenge the narratives it perpetuates, and reclaim agency in the face of algorithmic dominance. This paper further discusses the potential implications of the emergence of AGI, which could intensify the influence of algorithms on the creation of the sacred due to its advanced cognitive capabilities and deep understanding of human desires and behaviors. This could fuel a rapid evolution of the mimetic ecosystem, with profound implications for personal freedom, independent decision-making, and the formation and preservation of individual identity. This paper concludes by emphasizing the need for responsible algorithmic practices and ethical considerations to ensure that the creation of the sacred serves the common good in the algorithmic society. (shrink)

In previous work, the author has shown that $\Pi ^1_1$ -induction along $\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of Girard’s dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called 2-ptykes. We show that $\Pi (...) ^1_2$ -induction along $\mathbb N$ is equivalent to the existence of fixed points for all 2-ptykes that satisfy a certain normality condition. Beyond this specific result, the paper paves the way for the analysis of further $\Pi ^1_4$ -statements in terms of well ordering principles. (shrink)

Michael Rathjen and the present author have shown that ‐bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in. In this note we show that the base theory can be weakened to. Our argument makes crucial use of a normal function f with and. We shall also exhibit a normal function g with and.

In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$, for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

We shall distinguish between sortal predication and standard predication. The former kind of predication necessarily involves sortal concepts but the latter, as it is customarily viewed, does not. It is generally thought that the only essential occurrence of a concept in a standard predication is the concept being predicated. In this paper, we shall put forward an alternative view. We shall propose to understand standard predication as a cognitive act essentially requiring sortal concepts. We shall call this view conceptual predication (...) sortalism and ground it on the basis of epistemic-semantic reasons. Concepts are understood as intersubjectively realizable capacities or abilities that fulfill particular cognitive roles, such as those of classification, categorization, individuation, and referring. (shrink)

Cognition is the flexible coupling of perception and action. Whether direct or complex, this coupling depends on representing information and operating upon it. Thus, representation and its partner, processing, are the most fundamental of ideas in cognitive science. Representations are the bundles of information on which processes operate. Cognitive processes such as perception and attention encode information from the world, thus creating or changing our representations. Processes of reasoning and decision making operate on representations to form new beliefs and to (...) specify particular actions. Process refers to the dynamic use of information. Representation refers to the information available for use. Loosely speaking, representations include the ideas, sights, images, and beliefs that fill our thoughts and also the sensations and dispositions which may fall outside our awareness. Because representation is such a central concept in cognitive science, the term is used in a number of related senses, and I will note a couple of these more specialized uses as the need arises. (shrink)

Logic and philosophy of logic have increasingly become areas of research and great interest in Latin America and Spain, where significant work has been done and continues to be done in both of these fields. The goal of this volume is to draw attention to this work through a collection of original and unpublished papers by specialists from Latin America and Spain. Some of the papers are of importance for set-theory and model theory. They cover topics such as the foundations (...) of paraconsistency, the use of paraconsistent logic as a basis for set-theory, and the methodological aspects in both the justification of new axioms in set theory and the formalization of pre-theoretic notions. Other papers are related to epistemic logic. They deal with the issues of abduction and the choice of the simplest hypothesis, the definition of group probability, and the nature of explanation and understanding in such logic. There are also papers on logical paradoxes, the semantics of names, and the nature of relations. Max A. Freund is Professor of Logic and Philosophy at the University of Costa Rica and the National University of Costa Rica. He is co-author of the book Modal Logic its syntax and semantics, as well as the author of Judicial Logic, and of a forthcoming book The Logic of Sortals. Max Fernandez de Castro is Professor of Logic and Philosophy at the Autonomous Metropolitan University of Mexico. He is the author of the book Quine y la Ontologíaa and co-author of the books Lógica Matemática I: lógica proposicional, intuicionista y modal, Lógica Matemática II: clásica, intuicionista y modal, Teoría de Conjuntos, Lógica y Temas Afines I. Marco Ruffino is Professor of Philosophy at the University of Campinas and the editor of Manuscrito, the Brazilian international journal for Analytic Philosophy. He has many publications, in important philosophical journals, in the philosophy of language, of logic, and of mathematics; in the history of analytic philosophy, and on Frege and Wittgenstein. (shrink)

"À son décès, Julien Freund laissait parmi ses papiers un recueil de textes inédits, sous forme épistolaire, adressés à ses amis et collègues : les Lettres de la vallée. Dans ces méditations, rédigées en sa retraite vosgienne - qui n'est pas sans évoquer la Hütte de Heidegger, de l'autre côté du Rhin - et loin des servitudes propres aux écrits universitaires, l'auteur de L'Essence du politique s'exprime en toute liberté, à bâtons rompus, sur ses sujets de prédilection (le politique, (...) l'idéologie, la guerre, les totalitarismes, l'utopie, la décadence), mais également l'enseignement, le journalisme, la connaissance, le droit, la morale, la métaphysique, les limites... Ce livre forme la clef de voûte où se rejoignent les grands thèmes de sa pensée."--Page 4 de la couverture. (shrink)

A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations (...) is inspired by the results of [12] on nonmonotonic inference relations, and relies on some of the definitions found there. (shrink)

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ( $\mathbf {PA}$ ) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$ -sentences. These sentences assert that each computably enumerable ( $\Sigma ^0_1$ -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is (...) relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$ -independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$ -sentences. (shrink)

We develop a set-theoretic semantics for Cocchiarella's second-order logical system . Such a semantics is a modification of the nonstandard sort of second-order semantics described, firstly, by Simms and later extended by Cocchiarella. We formulate a new second order logical system and prove its relative consistency. We call such a system and construct its set-theoretic semantics. Finally, we prove completeness theorems for proper normal extensions of the two systems with respect to certain notions of validity provided by the semantics.

There is widespread and growing concern among U.S. consumers about the treatment of farmed animals, and consumers are consequently paying attention to food product labels that indicate humane production practices. However, labels vary in their standards for animal welfare, and prior research suggests that consumers are confused by welfare-related labels: many shoppers cannot differentiate between labels that indicate changes in the way animals are raised and those that do not. We administered a survey to 1,000 American grocery shoppers to better (...) understand the extent to which consumers purchase and pay more for food with certain labels based on an assumption of welfare improvement. Results showed that 86% of shoppers reported purchasing at least one product with the following labels in the last year: “cage or crate-free”, “free-range”, “pasture-raised”, “natural”, “organic”, “no hormone”, “no antibiotic”, “no rBST”, “humane”, “vegetarian-fed”, “grass-fed”, “farm-raised”. Of those who purchased one of the aforementioned labels, 89% did so because they thought the label indicated higher-welfare production practices, and 79% consciously paid more for the product with the label because they thought that the label indicated better-than-standard animal welfare. However, many of these labels lack uniform standards for the production practices they represent, and some labels represent production practices that do not influence animal welfare, thus the degree of the animal welfare impact of a given label is highly variable. These results indicate that labels need to clearly and accurately specify their animal welfare benefits to improve the consumers’ ability to purchase products that align with their expectations. (shrink)

A formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. The absolute consistency of the system is proved. A completeness proof for the system is also constructed. This proof is relative to a concept of logical validity provided by a semantics, which assumes as its philosophical background an approach to sortals from a modern form of conceptualism.

This paper investigates the responses of GPT-4, a state-of-the-art AI language model, to ten prominent philosophical paradoxes, and evaluates its capacity to reason and make decisions in complex and uncertain situations. In addition to analyzing GPT-4's solutions to the paradoxes, this paper assesses the model's Theory of Mind (ToM) capabilities by testing its understanding of mental states, intentions, and beliefs in scenarios ranging from classic ToM tests to complex, real-world simulations. Through these tests, we gain insight into AI's potential for (...) social reasoning and its capacity for more sophisticated forms of human-AI interaction. The paper also explores the limitations and biases of AI-generated reasoning and its implications for our comprehension of complex philosophical problems. (shrink)

It is generally accepted that H. Friedman’s gap condition is closely related to iterated collapsing functions from ordinal analysis. But what precisely is the connection? We offer the following answer: In a previous paper we have shown that the gap condition arises from an iterative construction on transformations of partial orders. Here we show that the parallel construction for linear orders yields familiar collapsing functions. The iteration step in the linear case is an instance of a general construction that we (...) call ‘Bachmann–Howard derivative’. In the present paper, we focus on the unary case, i.e., on the gap condition for sequences rather than trees and, correspondingly, on addition-free ordinal notation systems. This is partly for convenience, but it also allows us to clarify a phenomenon that is specific to the unary setting: As shown by van der Meeren, Rathjen and Weiermann, the gap condition on sequences admits two linearizations with rather different properties. We will see that these correspond to different recursive constructions of sequences. (shrink)

Grandiose narcissistic traits refer to exploitative and arrogant attitudes, while vulnerable narcissistic traits entail hypersensitivity to judgment and low self-esteem. Little is known about how individuals with narcissistic traits can improve their attitudes toward themselves and others. The current research puts self- and other compassion forward as possible targets to alleviate some of destructive patterns of narcissism. Generally, self-compassion has previously been associated with beneficial effects on psychological wellbeing, while other compassion is advantageous for interpersonal relationships. This study explored the (...) relationship between narcissistic traits and the efficacy of experimental compassion inductions. Student and community participants completed grandiose and vulnerable narcissistic trait, SC and OC state questionnaires, and either an SC or OC induction. It was expected that individuals with higher narcissistic traits would benefit from the inductions and show higher SC after but would have greater difficulty showing meaningful increases in OC. The results indicated that individual differences in grandiose and vulnerable narcissistic traits are related to the magnitude of improvements following the inductions: the theorized lack of SC in individuals with vulnerable oversensitivity to judgment traits seems possible to be counteracted through different types of compassion exercises. Moreover, higher grandiose exploitativeness–entitlement and global vulnerable narcissistic traits related to less increases than others. However, directly inducing OC in individuals with these traits was linked to greater OC improvements than improvements after inducing SC. Overall, the present findings suggest that self-compassionate behavior can be improved in individuals with high oversensitivity and that other compassionate behavior could potentially be increased if, specifically, other compassion exercises are utilized when higher levels of certain narcissistic traits are present. (shrink)

Lange Zeit hat die Idee eines musikalischen Fortschritts gleichermassen fasziniert und polarisiert. Als zentraler Bestandteil der Diskurse um die Neue Musik verlangt sie nach einer differenzierten historischen Betrachtung. Anhand von reichhaltigem Textmaterial analysiert Julia Freund die zentralen Konzepte und Argumentationslinien und entwirft ein vielfältiges Panorama der Debatten der 1950er Jahre. Ausgangspunkt ist ein close reading der Schriften und Vorlesungen Theodor W. Adornos, dessen Fortschrittsbegriff im Rahmen seines philosophischen Projekts der Aufklärungskritik greifbar wird. In einem zweiten und dritten Schritt nimmt (...) die Autorin die Denkfiguren und Narrative der seriellen Komponisten (darunter K. Stockhausen und P. Boulez) sowie die Gegenentwürfe ihrer Kritiker (wie P. Hindemith oder Fr. Blume) in den Blick"--Back cover. (shrink)

Within legal contexts, a claim to knowledge requires that the evidence appealed to fulfills conditions established by legal norms that belong to the so-called Evidence Law. In addition, the epistemic justification presupposed by this kind of knowledge is temporally affected by the changing character of the evidence and the judicial discretion usually exercised in the applications of the Evidence Law. As such, these two scenarios involve three modalities: temporal, epistemic, and deontic. The present paper philosophically discusses the links between these (...) modal dimensions in the above legal context. Based on the discussion, a formal tense-epistemic-deontic logic is developed. That is, the paper characterizes a formal language formally representing the three dimensions through specific logical operators, constructs a formal semantics for the language, and specifies a formal logical system. Finally, the soundness and completeness of the system with respect to the semantics is proved. (shrink)

El artículo formula una interpretación de la computabilidad desde la perspectiva del conceptualismo realista. En esta interpretación, la noción central es la de concepto computable, el cual se entiende como cierto tipo de capacidad cognitiva. Aquí se muestra cómo difiere esa interpretación conceptualista de la clásica, denominada teoría de la computabilidad efectiva, en la cual el concepto fundamental es el de algoritmo; también se discute la relación entre estas dos interpretaciones. La discusión explora las consecuencias de la idea de que (...) los contenidos de los conceptos computables se pueden expresar en algoritmos. En un apéndice se plantean, muy brevemente, las diferencias entre el enfoque conceptualista de la computabilidad y otros dos enfoques no clásicos: el intuicionista y el representacionista./// This paper formulates an interpretaron of computability from the perspective of realist conceptualism. The key notion within this theory is that of a computable concept, where a concept is a cognitive capacity of a certain sort. Differences and connections between the conceptualist interpretation and the classical one, customarily referred to as the theory of effective computability, for which the key concept is that of an algoríthm, are shown in the text. Consequences of the idea that the content of computable concepts can be expressed in algoríthms are explored. An appendix briefly outlines how the present conceptualist approach differs from two other non-classical interpretations of computability, the intuitionist and the representationist. (shrink)