We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for +, then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1-Erdös cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω1-Erdös (...) cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L[E]. The least cardinal λ with the Chang property ↠ is equal to the least ω1-Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non-stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j is the least ω1-Erdös cardinal. (shrink)

We present a question about Suslin trees and the weak square hierarchy which was contributed to the list of open problems of the BIRS workshop.

We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form . We generalize to some combinatorial principles that were shown by Jensen to hold in L. We show that satisfies the statement: “□κ holds whenever κ the least measurable cardinal λ of order λ++”. We introduce a hierarchy of combinatorial principles □κ, λ for 1 λ κ such that □κ□κ, 1 □κ, λ □κ, (...) κ□κ*. We prove that if holds in V. As an application, we show that ZFC + PFA Con. We also obtain one Woodin cardinal as a lower bound on the consistency strength of stationary reflection at κ+ for a singular, countably closed limit cardinal κ such that # exists; likewise for the failure of □κ* at such a κ. (shrink)

We prove that in all Mitchell-Steel core models, □ κ holds for all κ. (See Theorem 2.). From this we obtain new consistency strength lower bounds for the failure of □ κ if κ is either singular and countably closed, weakly compact, or measurable. (Corallaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □ κ holds (...) iff κ is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.). (shrink)

We present a general construction of a □κ-sequence in Jensen's fine structural extender models. This construction yields a local definition of a canonical □κ-sequence as well as a characterization of those cardinals κ, for which the principle □κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.

We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal k ≥ N₃ such that □k and □(k) fail. 2) There is a cardinal k such that k is weakly compact in the generic extension by Col(k, k⁺). Of special interest is 1) with k = N₃ since it follows from PFA by theorems of (...) Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over $K^c ||k.$. (shrink)

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof carries a partial square.

It is obvious that ♦ implies the existence of an antichain of stationary sets of cardinality which is the largest possible cardinality. We show that the obvious antichain is not maximal and find a less obvious extension of it by ℵ2 more stationary sets.

We prove new upper bound theorems on the consistency strengths of SPFA (θ), SPFA(θ-linked) and SPFA(θ⁺-cc). Our results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(c-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(c-linked) and PFA(c-linked) are each equiconsistent with the existence of a $\Sigma _{1}^{2}$ -indescribable cardinal. Our (...) upper bound for SPFA(c-c.c.) is a $\Sigma _{2}^{2}$ -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(c⁺-linked) is a cardinal κ that is $(\kappa ^{+},\Sigma _{1}^{2})$ -subcompact, which is strictly weaker than κ⁺-supercompact. The axiom MM(c) is a consequence of SPFA(c⁺-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(c⁺⁺-c.c.) is a cardinal κ that is $(\kappa ^{+},\Sigma _{2}^{2})$ -subcompact, which is also strictly weaker than κ⁺-supercompact. (shrink)

We introduce the weak partial square principles View the MathML source and View the MathML source, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial square principles fail. The main result of the paper is that □λ,κ does not imply View the MathML source.

We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, assuming some restriction of the Generalized Continuum Hypothesis, there are many outer automorphisms.

We prove that if image is a Jensen extender model, then image satisfies the Gap-1 morass principle. As a corollary to this and a theorem of Jensen, the model image satisfies the Gap-2 Cardinal Transfer Property → for all infinite cardinals κ and λ.

Assume that there is no transitive class model of [Formula: see text] with a Woodin cardinal. Let [Formula: see text] be a singular ordinal such that [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] is a regular cardinal in K. Then [Formula: see text] is a measurable cardinal in K. Moreover, if [Formula: see text], then [Formula: see text].

Durkheim écrit ce livre avec un but double : d'abord il voulait expliquer ce qui crée une société, ce qui la tient ensemble ; ensuite il voulait éclaircir l'influence qu'a la société sur la pensée logique. Pour Durkheim, la religion est la clé utilisée pour déverrouiller ces deux problématiques.Dans ce livre, Durkheim argumente que les représentations religieuses sont en fait des représentations collectives : l'essence du religieux ne peut être que le sacré. Il est une caractéristique qui se trouve universellement (...) dans toutes les religions. Le sacré, être collectif et impersonnel, représente ainsi la société elle-même. Ensuite il argumente que la société a un impact majeur dans la genèse de la pensée logique. C'est la société qui fournit à l'individu les catégories (le temps, l'espace, le nombre etc.) et les concepts nécessaires pour la pensée logique. (shrink)

There is a long tradition in philosophy and literary theory defending the view that engagement with literature promotes readers’ empathy. Until the last century, few of the empirical claims adduced in that tradition were investigated experimentally. Recent work in psychology and neuropsychology has now shed new light on the interplay of empathy and literature. This article surveys the experimental findings, addressing three central questions: What is it to read empathically? Does reading make us more empathic? What characteristics of literature, if (...) any, affect readers’ empathy? While experimental studies have delivered no conclusive answers to these questions, it has exposed their psychological complexity and constructed a more nuanced picture of the diverse ways in which literature interacts with our empathic capacities. (shrink)

Издание содержит: хрестоматия; философы; психологи; литературоведы, искусствоведы; мастер изобразительного искусства и др.

When we perceive an object, we perceive the object from a perspective. As a consequence of the perspectival nature of perception, when we perceive, say, a circular coin from different angles, there is a respect in which the coin looks circular throughout, but also a respect in which the coin's appearance changes. More generally, perception of shape and size properties has both a constant aspect—an aspect that remains stable across changes in perspective—and a perspectival aspect—an aspect that changes depending on (...) one's perspective on the object. How should we account for the perspectival aspect of spatial perception? We present a framework within which to discuss the perspectival aspect of perception and put forward three desiderata that any account of the perspectival aspect of perception should satisfy. We discuss views on which the perspectival aspect of perception is analyzed in terms of constitutively mind-dependent appearance properties as well as views on which the perspectival aspect of perception is analyzed in terms of representations of mind-independent perspectival properties. (shrink)

With special emphasis on Foucault's many recently published lecture series this book provides an updated, comprehensive presentation of his most important diagnoses, his many ground-breaking analytical concepts as well as a systematic account of his unique conception of philosophy.

We argue that the trend toward providing users with feasible and actionable explanations of AI decisions—known as recourse explanations—comes with ethical downsides. Specifically, we argue that recourse explanations face several conceptual pitfalls and can lead to problematic explanation hacking, which undermines their ethical status. As an alternative, we advocate that explanations of AI decisions should aim at understanding.

In his Treatise, Hume confronted the tensions between his project of uncovering the causal operations of the human mind and the extreme skeptical tendencies of his system. Louis Loeb argues that Hume overreaches, and he advances a controversial interpretation of Hume's epistemological framework that shows how Hume could have avoided the more destructive positions in his work.

Hume's claim that a state is a belief is often intertwined—though without his remarking on this fact—with epistemic approval of the state. This requires explanation. Beliefs, in Hume's view, are steady dispositions (not lively ideas), nature's provision for a steady influence on the will and action. Hume's epistemic distinctions call attention to circumstances in which the presence of conflicting beliefs undermine a belief's influence and thereby its natural function. On one version of this interpretation, to say that a belief is (...) justified, ceteris paribus, is to say that for all that has been shown the belief would be steady in its influence under suitable reflection. On a second version, it is to say that prima facie justification is an intrinsic property of the state, in virtue of its steadiness. These versions generate different understandings of the relationship between Parts iii and iv of Book I of the Treatise. (shrink)