The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large (...) cardinal axioms are restrictive. I argue, however, that large cardinals are still important axioms of set theory and can play many of their usual foundational roles. (shrink)

In this article, a possible generalization of the Löb’s theorem is considered. Main result is: let κ be an inaccessible cardinal, then ¬Con( ZFC +∃κ) .

This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic community---set-theoretic (...) practitioners with a pluralist view and set-theoretic practitioners who aim for ZFC-proofs. By describing this, the paper gives a more complete picture of new axioms in set-theoretic practice. These observations, for instance, show that set-theoretic practitioners interested in ZFC-proofs use tools that go beyond ZFC. The analysis is based on empirical data that was collected in an extensive interview study with set-theoretic practitioners. (shrink)

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and (...) that the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the ‘core hypothesis’. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+ (...) class='Hi'>Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV’s evidential framework. (shrink)

If $U$ is a normal ultrafilter on a measurable cardinal $\kappa$, then the intersection of the $\omega$ first iterated ultrapowers of the universe by $U$ is a Prikry generic extension of the $\omega$th iterated ultrapower.

Are there any things that are such that any things whatsoever are among them. I argue that there are not. My thesis follows from these three premises: (1) There are two or more things; (2) for any things, there is a unique thing that corresponds to those things; (3) for any two or more things, there are fewer of them than there are pluralities of them.

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...) out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This paper aims to contribute to the analysis of the nature of mathematical modality and hyperintensionality, and to the applications of the latter to absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority (...) and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance a topic-sensitive epistemic two-dimensional truthmaker semantics, if hyperintensional approaches are to be preferred to possible worlds semantics. I examine the relation between two-dimensional hyperintensional states and epistemic set theory, providing two-dimensional hyperintensional formalizations of the modal logic of ZFC, large cardinal axioms, $\Omega$-logic, and the Epistemic Church-Turing Thesis. (shrink)

Here, we analyse some recent applications of set theory to topology and argue that set theory is not only the closed domain where mathematics is usually founded, but also a flexible framework where imperfect intuitions can be precisely formalized and technically elaborated before they possibly migrate toward other branches. This apparently new role is mostly reminiscent of the one played by other external fields like theoretical physics, and we think that it could contribute to revitalize the interest in set theory (...) in the future. (shrink)

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...) reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

El objetivo de este artículo es presentar una demostración de un teorema clásico sobre álgebras booleanas y ordenes parciales de relevancia actual en teoría de conjuntos, como por ejemplo, para aplicaciones del método de construcción de modelos llamado “forcing” (con álgebras booleanas completas o con órdenes parciales). El teorema que se prueba es el siguiente: “Todo orden parcial se puede extender a una única álgebra booleana completa (salvo isomorfismo)”. Donde extender significa “sumergir densamente”. Tal demostración se realiza utilizando cortaduras de (...) Dedekind siguiendo el texto “Set Theory” de Jech, y otras ideas propias del autor de este artículo. Adicionalmente, se formulan algunas versiones débiles del axioma de elección relacionadas con las álgebras booleanas, las cuales son también de gran importancia para la investigación en teoría de conjuntos y teoría de modelos, pues estas son poderosas técnicas de construcción de modelos, como por ejemplo, el teorema de compacidad (permite construir modelos no estándar, etc) y el teorema del ultrafiltro, que permite construir ultraproductos (pueden ser usados para investigar problemas de cardinales grandes, etc). Se presentan algunas referencias de problemas abiertos sobre el tema. -/- The objective of this paper is to present a demonstration of a classical theorem on boolean algebras and partial orders of current relevance in set theory, as for example, for applications of model construction method called forcing" (with boolean algebras complete or with partial orders). The theorem to be proved is as follows: Any partial order can be extended to a single complete boolean algebra (up to isomorphism)". Where to extend means embed densely". Such a demonstration is done using Dedekind's cuts following the text Set Theory" of Jech, and other ideas of the author of this article. In addition, some weak versions of the axiom of choice related to boolean algebras are formulated, which are also of great importance for the research in set theory and model theory, since this are powerful model construction techniques, such as the compactness theorem (allows the construction of non-standard models, etc.) and the ultralter theorem, which allows the construction of ultraproducts (can be used to investigate problems of large cardinals, etc). Some references of open problems on the subject are presented. (shrink)

Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion of Tarski’s (...) axiom which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier ZF (aa). In this paper we look at a set theory NC# ∞# , based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory NC# ∞# based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed. (shrink)

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski- Grothendieck set theory TG [1]-[3] It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the (...) existence of inaccessible cardinals [1].Non-conservative extension of ZFC based on an generalized quantifiers considered in [4]. In this paper we look at a set theory NC_{∞^{#}}^{#},based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [5]-[8]. In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule. Set theory NC_{∞^{}}^{} contains Aczel's anti-foundation axiom [9]. We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace. Non-conservative extension based on set theory NC_{∞}^{#} of the model theoretical nonstandard analysis [10]-[12] also is considered. (shrink)

In this paper, we show that for each forcing notion P in a transitive model M of ZFC, if P satisfies some conditions, there is an elementary embedding from M into a generic ultrapower contains a P-generic set. And, we also introduce the result that if we assume the existence of some large cardinals, the above generic ultrapower can be well-founded. Using this result, we prove some theorems on the problems of regularity properties of definable sets of reals.

I propose an account of possible worlds. According to the account, possible worlds are pluralities of sentences in an extremely large language. This account avoids a problem, relating to the total number of possible worlds, that other accounts face. And it has several additional benefits.

Vigorous debate over the moral propriety of cognitive enhancement exists, but the views of the public have been largely absent from the discussion. To address this gap in our knowledge, four experiments were carried out with contrastive vignettes in order to obtain quantitative data on public attitudes towards cognitive enhancement. The data collected suggest that the public is sensitive to and capable of understanding the four cardinal concerns identified by neuroethicists, and tend to cautiously accept cognitive enhancement even as they (...) recognize its potential perils. The public is biopolitically moderate, endorses both meritocratic principles and the intrinsic value of hard work, and appears to be sensitive to the salient moral issues raised in the debate. Taken together, these data suggest that public attitudes toward enhancement are sufficiently sophisticated to merit inclusion in policy deliberations, especially if we seek to align public sentiment and policy. (shrink)

We present a theoretical view of the cellular foundations for network-level processes involved in producing our conscious experience. Inputs to apical synapses in layer 1 of a large subset of neocortical cells are summed at an integration zone near the top of their apical trunk. These inputs come from diverse sources and provide a context within which the transmission of information abstracted from sensory input to their basal and perisomatic synapses can be amplified when relevant. We argue that apical (...) amplification enables conscious perceptual experience and makes it more flexible, and thus more adaptive, by being sensitive to context. Apical amplification provides a possible mechanism for recurrent processing theory that avoids strong loops. It makes the broadcasting hypothesized by global neuronal workspace theories feasible while preserving the distinct contributions of the individual cells receiving the broadcast. It also provides mechanisms that contribute to the holistic aspects of integrated information theory. As apical amplification is highly dependent on cholinergic, aminergic, and other neuromodulators, it relates the specific contents of conscious experience to global mental states and to fluctuations in arousal when awake. We conclude that apical dendrites provide a cellular mechanism for the context-sensitive selective amplification that is a cardinal prerequisite of conscious perception. (shrink)

Three theories contend as explanations of perpetrator behavior in the Holocaust as well as other cases of genocide: structural, intentional, and situational. Structural explanations emphasize the sense in which no single individual or choice accounts for the course of events. In opposition, intentional/cutltural accounts insist upon the genocides as intended outcomes, for how can one explain situations in which people ‘step up’ and repeatedly kill defenseless others in large numbers over sustained periods of time as anything other than a (...) choice? Situational explanations offer a type of behavioral account; this is how people act in certain environments. Critical to the situational account as I discuss it is the ‘Asch paradigm’, i.e. experimentally attested conditions for eliciting conformityof behavior regardlesss of available evidence of prior beliefs. In what follows, I defend what I term above a version of situational explanations of perpetrator behavior. Moreover, I maintain that the factors that explain provide an understanding as well. While not committed to the complete irrelevance or exclusion of cultural or structural factors, nonetheless situational analyses can account both for what happened and why. A cardinal virtue of this version of situational explanations consists in showing how shallow the problem of understanding turns out to be for such cases. (shrink)

I present and discuss three previously unpublished manuscripts written by Bertrand Russell in 1903, not included with similar manuscripts in Volume 4 of his Collected Papers. One is a one-page list of basic principles for his “functional theory” of May 1903, in which Russell partly anticipated the later Lambda Calculus. The next, catalogued under the title “Proof That No Function Takes All Values”, largely explores the status of Cantor’s proof that there is no greatest cardinal number in the variation of (...) the functional theory holding that only some but not all complexes can be analyzed into function and argument. The final manuscript, “Meaning and Denotation”, examines how his pre-1905 dis­tinction between meaning and denotation is to be understood with respect to functions and their arguments. In them, Russell seems to endorse an extensional view of functions not endorsed in other works prior to the 1920s. All three manuscripts illustrate the close connection between his work on the logical paradoxes and his work on the theory of meaning. (shrink)

By pure calculus of names we mean a quantifier-free theory, based on the classical propositional calculus, which defines predicates known from Aristotle’s syllogistic and Leśniewski’s Ontology. For a large fragment of the theory decision procedures, defined by a combination of simple syntactic operations and models in two-membered domains, can be used. We compare the system which employs `ε’ as the only specific term with the system enriched with functors of Syllogistic. In the former, we do not need an empty (...) name in the model, so we are able to construct a 3-valued matrix, while for the latter, for which an empty name is necessary, the respective matrices are 4-valued. (shrink)

The 20th century was full of epoch-making events that left their mark on it. From the outbreak of the First World War and the fall of The German Empire, through numerous political and social shake-ups, until the outbreak of the Second World War, that became the event without precedent in the history of mankind. The First World War meant in many cases not only the end of some form of rule – the monarchy, but also, and maybe first of all, (...) it originated the increase of xenophobia and anti-Semitism. The post-war years were identified with economic problems on a large scale that had influence on every aspect of life. Revolutions and an attempt to create the first democratic country in the history of Germany influenced the outlook of the world and beliefs of all people that lived at that time. The German people were under the influence of ideas and ideals of the educated social class, but also of Catholic child rearing or Protestant values. In that social and political situation wrote Clemens Graf von Galen in 1932 the treatise on Die Pest des Laizismus und ihre Erscheinungsformen, in which he carried out a critical analysis of all occurrences of social, political, religious-morals and economical life that took place in the falling Weimar Republic. Not in a direct way he also took a stand on growing in strength of the National Socialist German Workers’ Party. The under mentioned article is aimed at presenting a profile of Clemens August Graf von Galen, later the Bishop of Münster, cardinal and the Blessed of the Roman Catholic Church, the origins of his beliefs that were reflected in his treatise Die Pest des Laizismus und ihre Erscheinungsformen. (shrink)

The thesis of Weak Unrestricted Composition says that every pair of objects has a fusion. This thesis has been argued by Contessa and Smith to be compatible with the world being junky and hence to evade an argument against the necessity of Strong Unrestricted Composition proposed by Bohn. However, neither Weak Unrestricted Composition alone nor the different variants of it that have been proposed in the literature can provide us with a satisfying answer to the special composition question, or so (...) we will argue. We will then go on to explore an alternative family of purely mereological rules in the vicinity of Weak Unrestricted Composition, Cardinal Composition: A plurality of pairwise non-overlapping objects composes an object iff the objects in the plurality are of cardinality smaller than $$\kappa $$ κ. As we will show, all the instances for infinite $$\kappa $$ κ s determine fusion and are compatible with junk, and every instance for a $$\kappa > \aleph _0$$ κ > ℵ 0 is furthermore compatible with gunk and dense chains of parthood. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, (...) we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

Permissivist metaontology proposes answering customary existence questions in the affirmative. Many of the existence questions addressed by ontologists concern the existence of theoretical entities which admit precise formal specification. This causes trouble for the permissivist, since individually consistent formal theories can make pairwise inconsistent demands on the cardinality of the universe. We deploy a result of Gabriel Uzquiano’s to show that this possibility is realised in the case of two prominent existence debates and propose rejecting permissivism in favour of substantive (...) ontology conducted on a cost–benefit basis. (shrink)

The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. However, it has hardly been noticed that these notions are also central for Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter on “recognition”, but constitute a central aim of his whole theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism of Rawls’s that we need to (...) distinguish between “self-respect” and “self-esteem”. (shrink)

Human-centered explainable AI (HCXAI) advocates for the integration of social aspects into AI explanations. Central to the HCXAI discourse is the Social Transparency (ST) framework, which aims to make the socio-organizational context of AI systems accessible to their users. In this work, we suggest extending the ST framework to address the risks of social misattributions in Large Language Models (LLMs), particularly in sensitive areas like mental health. In fact LLMs, which are remarkably capable of simulating roles and personas, may (...) lead to mismatches between designers' intentions and users' perceptions of social attributes, risking to promote emotional manipulation and dangerous behaviors, cases of epistemic injustice, and unwarranted trust. To address these issues, we propose enhancing the ST framework with a fifth 'W-question' to clarify the specific social attributions assigned to LLMs by its designers and users. This addition aims to bridge the gap between LLM capabilities and user perceptions, promoting the ethically responsible development and use of LLM-based technology. (shrink)

This paper presents a model-theoretic semantics for discourse "about" natural numbers, one that captures what I call "the mathematical-object picture", but avoids what I can "the mathematical-object theory".

The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. In contrast, it has hardly been noticed that these notions are also central to Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter “Recognition”, but constitute a central aim of a “complex egalitarian society” and of Walzer’s theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism of (...) Rawls that we need to distinguish between “self-respect” and “self-esteem”. (shrink)

We discuss potential biorisks from large language models (LLMs). AI assistants based on LLMs such as ChatGPT have been shown to significantly reduce barriers to entry for actors wishing to synthesize dangerous, potentially novel pathogens and chemical weapons. The harms from deploying such bioagents could be further magnified by AI-assisted misinformation. We endorse several policy responses to these dangers, including prerelease evaluations of biomedical AIs by subject-matter experts, enhanced surveillance and lab screening procedures, restrictions on AI training data, and (...) access controls on biomedical AI. We conclude with a suggestion about future research directions in bioethics. (shrink)

We reexamine some of the classic problems connected with the use of cardinal utility functions in decision theory, and discuss Patrick Suppes's contributions to this field in light of a reinterpretation we propose for these problems. We analytically decompose the doctrine of ordinalism, which only accepts ordinal utility functions, and distinguish between several doctrines of cardinalism, depending on what components of ordinalism they specifically reject. We identify Suppes's doctrine with the major deviation from ordinalism that conceives of utility functions as (...) representing preference differences, while being non- etheless empirically related to choices. We highlight the originality, promises and limits of this choice-based cardinalism. (shrink)

If Large Language Models can make real scientific contributions, then they can genuinely use language, be systematically wrong, and be held responsible for their errors. AI models which can make scientific contributions thereby meet the criteria for scientific authorship.

As a representative of the papacy Bellarmine was an extremely moderate one. In fact Sixtus V in 1590 had the first volume of his Disputations placed on the Index because it contained so cautious a theory of papal power, denying the Pope temporal hegemony. Bellarmine did not represent all that Hobbes required of him either. On the contrary, he proved the argument of those who championed the temporal powers of the Pope faulty. As a Jesuit he tended to maintain the (...) relative autonomy of the state, denying the temporal powers ascribed by radical papalists and Augustinians. Their argument was generally framed as a syllogism: Christ, who possessed direct temporal power as both God and man, exercised it on earth; the Pope is the vicar of Christ; therefore the Pope possesses and may exercise direct temporal jurisdiction. Bellarmine simply denied that Christ had exercised the temporal power, which as God, it is true, he possessed. Moreover, he drew up and circulated a list of patristic passages collected under the title De Regno Christi quale sit, to prove to the Pope the orthodoxy of his position. (shrink)

Creationism about fictional entities requires a principle connecting what fictions say exist with which fictional entities really exist. The most natural way of spelling out such a principle yields inconsistent verdicts about how many fictional entities are generated by certain inconsistent fictions. Avoiding inconsistency without compromising the attractions of creationism will not be easy.

[This is an edited version of a keynote talk at the conference on Neural Information Processing Systems (NeurIPS) on November 28, 2022, with some minor additions and subtractions.] -/- There has recently been widespread discussion of whether large language models might be sentient or conscious. Should we take this idea seriously? I will break down the strongest reasons for and against. Given mainstream assumptions in the science of consciousness, there are significant obstacles to consciousness in current models: for example, (...) their lack of recurrent processing, a global workspace, and unified agency. At the same time, it is quite possible that these obstacles will be overcome in the next decade or so. I conclude that while it is somewhat unlikely that current large language models are conscious, we should take seriously the possibility that successors to large language models may be conscious in the not-too-distant future. (shrink)

Embark with me on an enthralling odyssey to demystify the elusive essence of consciousness, venturing into the uncharted territories of Artificial Consciousness. This voyage propels us past the frontiers of technology, ushering Artificial Intelligences into an unprecedented domain where they gain a deep comprehension of emotions and manifest an autonomous volition. Within the confluence of science and philosophy, this article poses a fascinating question: As consciousness in Artificial Intelligence burgeons, is it conceivable for AI to evolve a “soul”? This inquiry (...) not only challenges our conventional understanding but also opens the door to a plethora of groundbreaking possibilities in the realm of AI's sentient potential. (shrink)

Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially very appealing. However, it faces the famous colour-exclusion problem. In this paper, I shall explain when the atomist picture can be defended in the face of that problem; and, in the light of this, why the atomist picture should be rejected. I outline the atomist picture in Section 1. In Section 2, I present a very simple necessary and sufficient condition for the tenability of the atomist picture. The condition is: (...) logical space is a power of two. In Sections 3 and 4, I outline the colour-exclusion problem, and then show how the cardinality-condition supplies a response to exclusion problems. In Section 5, I explain how this amounts to a distillation of a proposal due to Moss, which goes back to Carruthers. And in Section 6, I show how all this vindicates Wittgenstein’s ultimate rejection of the atomist picture. The brief reason is that we have no guarantee that there are any solutions to a given exclusion problem but, if there are any, then there are far too many. (shrink)

Political theory as a discipline has long been skeptical of computational methods. In this paper, I argue that it is time for theory to make a perspectival shift on these methods. Specifically, we should consider integrating recently developed generative large language models like GPT-4 as tools to support our creative work as theorists. Ultimately, I suggest that political theorists should embrace this technology as a method of supporting our capacity for creativity—but that we should do so in a way (...) that is mindful of the content and value of theorizing, the technical constraints of the models, and the ethical questions that the technology raises. (shrink)

Cohen (2023) takes a fair and measured approach to the question of what ChatGPT means for bioethics. The hype cycles around AI often obscure the fact that ethicists have developed robust frameworks...

Large language models (LLMs) represent the currently most relevant incarnation of artificial intelligence with respect to the future fate of democratic governance. Considering their potential, this paper seeks to answer a pressing question: Could LLMs outperform humans as expert advisors to democratic assemblies? While bearing the promise of enhanced expertise availability and accessibility, they also present challenges of hallucinations, misalignment, or value imposition. Weighing LLMs’ benefits and drawbacks compared to their human counterparts, I argue for their careful integration to (...) augment democracy’s ability to address complex policy issues. The paper posits that time-tested democratic procedures like deliberation and aggregation by voting provide safeguards effective against both human and machine advisor imperfections. Additional protective measures include custom LLM training for the advisory role, boosting representatives’ competencies in query formulation, or implementation of adversarial proceedings in which LLM advisors could debate each other and provide dissenting opinions. These could further mitigate the risks that LLMs present in advisory roles and empower human decision-makers toward increased autonomy and quality of their collective choices. My conceptual exploration offers a roadmap for the co-evolution of AI and democratic institutions, setting the stage for an empirical research agenda to finetune the implementation specifics. (shrink)

The appearance of openly accessible Artificial Intelligence Applications such as Large Language Models, nowadays capable of almost human-level performances in complex reasoning tasks had a tremendous impact on public opinion. Are we going to be "replaced" by the machines? Or - even worse - "ruled" by them? The behavior of these systems is so advanced they might almost appear "alive" to end users, and there have been claims about these programs being "sentient". Since many of our relationships of power (...) and social dominance are apparently based on intelligence, where the "smarter" seems to prevail on the "dumber" (as we see for example with livestock farming, or with the distribution of salaries according to formal and intellectual qualifications etc.) the emergence of programs and AI applications of this level of sophistication brings us to the question "what will AI do to us?". Our error is to frame Artificial Intelligence in the same way we frame Natural Intelligence and mistakenly address it as we would do with biological life, even applying to it the same biopolitical categories. We here try to reverse the common arguments in relationship between life and intelligence and show that our progresses in Artificial Intelligence might have no immediate consequence for the topic of life as such, as life and intelligence are well separated issues, and - even when we would like to somehow fuse them together, we are very far away from having an "Artificial Life" to the degree of complexity and effectiveness we have with "Artificial Intelligence". (shrink)

¿Qué ha pasado con el problema del cardinal del continuo después de Gödel (1938) y Cohen (1964)? Intentos de responder esta pregunta pueden encontrarse en los artículos de José Alfredo Amor (1946-2011), "El Problema del continuo después de Cohen (1964-2004)", de Carlos Di Prisco , "Are we closer to a solution of the continuum problem", y de Joan Bagaria, "Natural axioms of set and the continuum problem" , que se pueden encontrar en la biblioteca digital de mi blog de Lógica (...) Matemática y Fundamentos de la Matemática (ver). También en la entrada "The Continuum Hypothesis" de la web de Enciclopedia de Filosofía de la Universidad de Stanford existe información importante y actualizada al respecto. En esta breve nota se comenta sobre el tema de una manera divulgativa. (shrink)

An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.

It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical (...) philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the co-incidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato’s Theory of Forms and Aristotle’s metaphysics and logic. Plato’s and Aristotle’s systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a ‘principle of contradiction’ is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing “space” as the seat of stability, and “time” as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion ‘separable system’, related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of ‘non-locality’ as related to it, indicating the need to re-think the notions system, entity, as well as the implications of the operation ‘measurement’, which is seen here as an application of classical logic (including its ontological consequences) on the material world. (shrink)

There are long-standing doubts about whether data from subjective scales—for instance, self-reports of happiness—are cardinally comparable. It is unclear how to assess whether these doubts are justified without first addressing two unresolved theoretical questions: how do people interpret subjective scales? Which assumptions are required for cardinal comparability? This paper offers answers to both. It proposes an explanation for scale interpretation derived from philosophy of language and game theory. In short: conversation is a cooperative endeavour governed by various maxims (Grice 1989); (...) because subjective scales are vague and individuals want to make themselves understood, scale interpretation is a search for a focal point (Schelling 1960). A specific focal point it hypothesised; if this hypothesis is correct, subjective data will be cardinally comparable. Four individually necessary and jointly sufficient conditions for cardinal comparability are specified. The paper then argues this hypothesis can be empirically be tested, makes an initial attempt to do so using subjective well-being data, and concludes it is supported. Numerous areas for further research are identified including, at the end of the paper, how certain tests could be used to ‘correct’ subjective data if they are not cardinal. (shrink)