We study the Katětov order on Borel ideals. We prove two structural theorems, one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals.

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martinʼs number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.

We investigate the pair-splitting number $\germ{s}_{pair}$ which is a variation of splitting number, pair-reaping number $\germ{r}_{pair}$ which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size.

The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer–Specker group Zω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}^{\omega}}}$$\end{document}. The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and Steprāns :52–59, 2006).

In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers o and a.

A metric space (X, d) is monotone if there is a linear order < on X and a constant c such that d(x, y) ≤ c d(x, z) for all x < y < z in X. We investigate cardinal invariants of the σ-ideal Mon generated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are (...) also investigated. In particular, we show that non(Mon) ≥ ������ σ-linked , but non(Mon) < ������ σ-centered is consistent. Also cov(Mon) < c and cof(������) < cov(Mon) are consistent. (shrink)

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ <ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ (...) is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A Čech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals. (shrink)

It is relatively consistent with ZFC that every countable $FU_{fin} $ space of weight N₁ is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

Let ${\cal I}$ be an ideal on ω. By cov${}_{}^{\rm{*}}$ we denote the least size of a family ${\cal B} \subseteq {\cal I}$ such that for every infinite $X \in {\cal I}$ there is $B \in {\cal B}$ for which $B\mathop \cap \nolimits X$ is infinite. We say that an AD family ${\cal A} \subseteq {\cal I}$ is a MAD family restricted to${\cal I}$ if for every infinite $X \in {\cal I}$ there is $A \in {\cal A}$ such that $|X\mathop (...) \cap \nolimits A| = \omega$. Let a$\left$ be the least size of an infinite MAD family restricted to ${\cal I}$. We prove that If $max${a,cov${}_{}^{\rm{*}}\}$ then a$\left = {\omega _1}$, and consequently, if ${\cal I}$ is tall and $\le {\omega _2}$ then a$\left = max$ {a,cov${}_{}^{\rm{*}}\}$. We use these results to prove that if c$\le {\omega _2}$ then o$= \overline o$ and that as$= max${a,non$\}$. We also analyze the problem whether it is consistent with the negation of CH that every AD family of size ω1 can be extended to a MAD family of size ω1. (shrink)

A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent formulation (...) of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of [Formula: see text] for all [Formula: see text]. (shrink)

If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma}$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models (...) of $\mathsf{ZFC}$ there are $\mathsf{MAD}$ families whose Mathias forcing does not add a dominating real. We also study ideals generated by branches, and we uncover a close relation between Canjar ideals and the selection principle $S_{\mathrm{fin}}$ on subsets of the Cantor space. (shrink)

We consider several game versions of the cardinal invariants \, \ and \. We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \ and \ are both relatively consistent with ZFC, where \ and \ are the principal game versions of \ and \, respectively. The corresponding question for \ remains open.

We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a Gδ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we (...) show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets. (shrink)

This book defends a form of ethical intuitionism, according to which (i) there are objective moral truths; (ii) we know some of these truths through a kind of immediate, intellectual awareness, or "intuition"; and (iii) our knowledge of moral truths gives us reasons for action independent of our desires. The author rebuts all the major objections to this theory and shows that the alternative theories about the nature of ethics all face grave difficulties.

The first edition of The Scientific Background to Modern Philosophy took the dialogue of science and philosophy from Aristotle through to Newton. This second edition adds eight chapters, taking the dialogue through the Enlightenment and up to Darwin. This anthology is an attempt to help bridge the gap between the history of science and the history of philosophy.

Part I: The representation of life -- Can life be given a real definition? -- The representation of the living individual -- The representation of the life-form itself -- Part II: Naive action theory -- Types of practical explanation -- Naive explanation of action -- Action and time -- Part III: Practical generality -- Two tendencies in practical philosophy -- Practices and dispositions as sources of the goodness of individual actions -- Practice and disposition as sources of individual action.

Virtually all philosophers agree that for a belief to be epistemically justified, it must satisfy certain conditions. Perhaps it must be supported by evidence. Or perhaps it must be reliably formed. Or perhaps there are some other "good-making" features it must have. But does a belief's justification also require some sort of awareness of its good-making features? The answer to this question has been hotly contested in contemporary epistemology, creating a deep divide among its practitioners. Internalists, who tend to focus (...) on scientific or theoretical beliefs as the ideal, insist that such awareness is required for justification. Externalists, who think children's ordinary beliefs in obvious facts are paradigm cases of justified belief, say it isn't required. Michael Bergmann's book offers a decisive refutation of internalism and a sustained defense of externalism. (shrink)

We study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number ${\mathfrak{b}}$ , the intersection number of the class of all meager ideals is equal to ${\mathfrak{h}}$ and the intersection number of the class of all F σ ideals is between ${\mathfrak{h}}$ and ${\mathfrak{b}}$ , consistently different from both.

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note (...) we construct, in ZFC, ${2^{\mathfrak{c}}}$ non-metrizable infinite pairwise non-homeomorphic minimal sets on compact connected linearly ordered spaces. (shrink)

In this article we, given a free ultrafilter p on ω, consider the following classes of ultrafilters:(1) T(p) - the set of ultrafilters Rudin-Keisler equivalent to p,(2) S(p)={q ∈ ω*:∃ f ∈ ω ω , strictly increasing, such that q=f β (p)},(3) I(p) - the set of strong Rudin-Blass predecessors of p,(4) R(p) - the set of ultrafilters equivalent to p in the strong Rudin-Blass order,(5) P RB (p) - the set of Rudin-Blass predecessors of p, and(6) P RK (p) (...) - the set of Rudin-Keisler predecessors of p,and analyze relationships between them. We introduce the semi-P-points as those ultrafilters p ∈ ω* for which P RB (p)=P RK (p), and investigate their relations with P-points, weak-P-points and Q-points. In particular, we prove that for every semi-P-point p its α-th left power α p is a semi-P-point, and we prove that non-semi-P-points exist in ZFC. Further, we define an order ⊴ in T(p) by r⊴q if and only if r ∈ S(q). We prove that (S(p),⊴) is always downwards directed, (R(p),⊴) is always downwards and upwards directed, and (T(p),⊴) is linear if and only if p is selective.We also characterize rapid ultrafilters as those ultrafilters p ∈ ω* for which R(p)∖S(p) is a dense subset of ω*.A space X is M-pseudocompact (for ) if for every sequence (U n ) n < ω of disjoint open subsets of X, there are q ∈ M and x ∈ X such that x=q-lim (U n ); that is, for every neighborhood V of x. The P RK (p)-pseudocompact spaces were studied in [ST].In this article we analyze M-pseudocompactness when M is one of the classes S(p), R(p), T(p), I(p), P RB (p) and P RK (p). We prove that every Frolik space is S(p)-pseudocompact for every p ∈ ω*, and determine when a subspace with is M-pseudocompact. (shrink)

Externalist theories of justification create the possibility of cases in which everything appears to one relevantly similar with respect to two propositions, yet one proposition is justified while the other is not. Internalists find this difficult to accept, because it seems irrational in such a case to affirm one proposition and not the other. The underlying internalist intuition supports a specific internalist theory, Phenomenal Conservatism, on which epistemic justification is conferred by appearances.

At the University of Sheffield during 2011 and 2012, a leading group of philosophers, psychologists, and others gathered to explore the nature and significance of implicit bias. The two volumes of Implicit Bias and Philosophy emerge from these workshops. Each volume philosophically examines core areas of psychological research on implicit bias as well as the ramifications of implicit bias for core areas of philosophy. Volume I: Metaphysics and Epistemology is comprised of two parts: “The Nature of Implicit Attitudes, Implicit Bias, (...) and Stereotype Threat,” and “Skepticism, Social Knowledge, and Rationality.” The first part contains chapters examining the relationship between implicit attitudes and “dual process” models of the mind; the role of affect in the formation and change of implicit associations; the unity (or disunity) of implicit attitudes; whether implicit biases are mental states at all; and whether performances on stereotype-relevant tasks are automatic and unconscious or intentional and strategic. The second part contains chapters examining implicit bias and skepticism; the effects of implicit bias on scientific research; the accessibility of social stereotypes in epistemic environments; the effects of implicit bias on the self-perception of members of stigmatized social groups as rational agents; the role of gender stereotypes in philosophy; and the role of heuristics in biased reasoning. (shrink)

Causation: A Realist Approach Traditional empiricist accounts of causation and laws of nature have been reductionist in the sense of entailing that given a complete specification of the non-causal properties of and relations among particulars, it is therefore logically determined both what laws there are and what events are causally related. It is argued here, however, that reductionist accounts of causation and of laws of nature are exposed to decisive objections, and thus that the time has come for empiricists to (...) break with that tradition. -/- The basic goal of this book, therefore, is to set out and defend realistic accounts of those concepts. In the case of causal relations, for example, Tooley maintains that causation is basically a matter of theoretical relations between states of affairs that underlie and explain relative frequencies. He also argues that such an approach avoids the objections that tell against reductionist accounts and that it does so without making causal relations epistemologically inaccessible. (shrink)

In this engaging and spirited text, Michael Lynch argues that truth does matter, in both our personal and political lives. He explains that the growing cynicism over truth stems in large part from our confusion over what truth is.

Dignity plays a central role in current thinking about law and human rights, but there is sharp disagreement about its meaning. Combining conceptual precision with a broad historical background, Michael Rosen puts these controversies in context and offers a novel, constructive proposal. “Penetrating and sprightly...Rosen rightly emphasizes the centrality of Catholicism in the modern history of human dignity. His command of the history is impressive...Rosen is a wonderful guide to the recent German constitutional thinking about human dignity...[Rosen] is in (...) general an urbane and witty companion, achieving his aim of accessibly written philosophy.” —Samuel Moyn, The Nation “[An] elegant, interesting and lucid exploration of the concept of dignity...Drawing on classical, liberal and Catholic traditions, Rosen hopes to rehabilitate dignity to its rightful place near the centre of moral thought...Rosen's admirable book deserves wide attention from political theorists, jurisprudes and political philosophers.” —Simon Blackburn, Times Higher Education “Dignity deserves to be widely read, not only for its intrinsic interest, but also as a corrective to the habit of discussing such topics in abstraction from their social context. Whether or not one agrees with Rosen's arguments, there can be no doubt he has widened our horizons.” —Rae Langton, Times Literary Supplement. (shrink)

edition (unseen), $12.95. traditions, bringing into being new modes of understanding. Paper Anthropology, and particularly ethnography, is torn between two quests, one to capture the diversity of social life and the other to discover universal principles structuring that diversity. Jackson examines these quests within the context of ethnographic fieldwork, focusing on the relationship between ethnographers and the people they study. He is concerned with defining the anthropological project as something more than the projection of the anthropologist's traditions and concerns onto (...) an alien culture. Rather, he would have the project open a genuine dialogue between people from different cultures or Annotation copyrighted by Book News, Inc., Portland, OR. (shrink)

Michael G. Titelbaum presents a new Bayesian framework for modeling rational degrees of belief—the first of its kind to represent rational requirements on agents who undergo certainty loss.

In this paper, I propose a new way of defining sport that I call a ‘core-periphery’ model. According to a core-periphery model, sport comes in degrees – what I refer to as ‘sport-likeness’ – and the aim of the philosopher of sport is to chart those dimensions along which an activity can be more or less a sport. By introducing the concept of sport-likeness, the core-periphery model complicates the picture of what is or is not a sport and encourages philosophers (...) interested in defining sport to engage with the social sciences in exploring the extension of the term sport in common usage. In this paper I present the results of a small survey about attitudes to sport, and use it to illustrate how a core-periphery definition of sport would proceed. (shrink)

While philosophers hold that it is patently absurd to blame robots or hold them morally responsible [1], a series of recent empirical studies suggest that people do ascribe blame to AI systems and robots in certain contexts [2]. This is disconcerting: Blame might be shifted from the owners, users or designers of AI systems to the systems themselves, leading to the diminished accountability of the responsible human agents [3]. In this paper, we explore one of the potential underlying reasons for (...) robot blame, namely the folk's willingness to ascribe inculpating mental states or "mens rea" to robots. In a vignette-based experiment (N=513), we presented participants with a situation in which an agent knowingly runs the risk of bringing about substantial harm. We manipulated agent type (human v. group agent v. AI-driven robot) and outcome (neutral v. bad), and measured both moral judgment (wrongness of the action and blameworthiness of the agent) and mental states attributed to the agent (recklessness and the desire to inflict harm). We found that (i) judgments of wrongness and blame were relatively similar across agent types, possibly because (ii) attributions of mental states were, as suspected, similar across agent types. This raised the question - also explored in the experiment - whether people attribute knowledge and desire to robots in a merely metaphorical way (e.g., the robot "knew" rather than really knew). However, (iii), according to our data people were unwilling to downgrade to mens rea in a merely metaphorical sense when given the chance. Finally, (iv), we report a surprising and novel finding, which we call the inverse outcome effect on robot blame: People were less willing to blame artificial agents for bad outcomes than for neutral outcomes. This suggests that they are implicitly aware of the dangers of overattributing blame to robots when harm comes to pass, such as inappropriately letting the responsible human agent off the moral hook. (shrink)

In previous work I have argued against internalism by means of a dilemma intended to force all internalists to accept one of two undesirable options: either their internalism is unmotivated or it is saddled with vicious regress problems. Recently it has been argued that Phenomenal Conservatism—a theory of justification according to which justification depends on seemings—is a kind of internalism that can escape this dilemma. In this paper, I argue that Phenomenal Conservatism cannot escape my dilemma for internalism. In order (...) to do this, it is helpful to get clear on what Phenomenal Conservatism is saying, and for that reason I begin, in section I, by examining in some detail what seemings are. I then argue, in the remainder of the paper, that, whichever horn of the dilemma for internalism they take, Phenomenal Conservatives are in trouble. (shrink)

This volume presents a selection of the most influential recent discussions of the crucial metaphysical questions: what is it for one event to cause another? The subject of causation bears on many topics, such as time, explanation, mental states, the laws of nature, and the philosphy of science.

Can an ugly experiment be a good experiment? Philosophers have identified many beautiful experiments and explored ways in which their beauty might be connected to their epistemic value. In contrast, the present chapter seeks out (and celebrates) ugly experiments. Among the ugliest are those being designed by AI algorithms. Interestingly, in the contexts where such experiments tend to be deployed, low aesthetic value correlates with high epistemic value. In other words, ugly experiments can be good. Given this, we should conclude (...) that beauty is not generally necessary or sufficient for epistemic value, and increasing beauty will not generally tend to increase epistemic value. (shrink)

Radical skepticism endorses the extreme claim that large swaths of our ordinary beliefs, such as those produced by perception or memory, are irrational. The best arguments for such skepticism are, in their essentials, as familiar as a popular science fiction movie and yet even seasoned epistemologists continue to find them strangely seductive. Moreover, although most contemporary philosophers dismiss radical skepticism, they cannot agree on how best to respond to the challenge it presents. In the tradition of the 18th century Scottish (...) philosopher, Thomas Reid, Radical Skepticism and Epistemic Intuition joins this discussion by taking up four main tasks. First, it identifies the strongest arguments for radical skepticism, namely, underdetermination arguments, which emphasize the gap between our evidence and our ordinary beliefs based on that evidence. Second, it rejects all inferential or argument-based responses to radical skepticism, which aim to lay out good noncircular reasoning from the evidence on which we base our ordinary beliefs to the conclusion that those beliefs are probably true. Third, it develops a commonsense noninferential response to radical skepticism with two distinctive features: it consciously and extensively relies on epistemic intuitions, which are seemings about epistemic goods, such as knowledge and rationality, and it can be endorsed without difficulty by both internalists and externalists in epistemology. Fourth, and finally, it defends this commonsense epistemic-intuition-based response to radical skepticism against a variety of objections, including those connected with underdetermination worries, epistemic circularity, disagreement problems, experimental philosophy, and concerns about whether it engages skepticism in a sufficiently serious way. (shrink)