We prove that there is a -ideal of compact sets which is strictly above in the Tukey order. Here is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličković asked in [Alain Louveau, Boban Veličković, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 171–195].

Given a compact Polish space E and the hyperspace of its compact subsets , we consider Gδσ-ideals of compact subsets of E. Solecki has shown that any σ-ideal in a broad natural class of Gδ ideals can be represented via a compact subset of ; in this article we examine the behaviour of Gδ subsets of E with respect to the representing set. Given an ideal I in this class, we construct a representing set (...) that recognises a compact subset of E as being “small” precisely when it is in I, and recognises a Gδ subset of E as being “small” precisely when it is covered by countably many compact sets from I. (shrink)

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

We prove several trichotomy results for ideals of compact sets. Typically, we show that a "sufficiently rich" universally Baire ideal is either $\Pi _{3}^{0}$-hard, or $\Sigma _{3}^{0}$-hard, or else a σ-ideal.

We extract some properties of Mahlo’s operation and show that some other very natural operations share these properties. The weakly compact sets form a similar hierarchy as the stationary sets. The height of this hierarchy is a large cardinal property connected to saturation properties of the weakly compact ideal.

We study the structure of analytic ideals of subsets of the natural numbers. For example, we prove that for an analytic ideal I, either the ideal {X (Ω × Ω: En X ({0, 1,…,n} × Ω } is Rudin-Keisler below I, or I is very simply induced by a lower semicontinuous submeasure. Also, we show that the class of ideals induced in this manner by lsc submeasures coincides with Polishable ideals as well as analytic P-ideals. We (...) study this class of ideals and characterize, for example, when the ideals in it are Fσ or when they carry a locally compact group topology. We apply these results to Borel partial orders to rederive a theorem of Todorcevic and to Borel equivalence relations to answer a question of Kechris and Louveau. As another application we give a characterization of σ-ideals of μ-zero sets for Maharam submeasures μ on the Cantor set which is to a large extent analogous to a characterization of the meager ideal due to Kechris and the author. (shrink)

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study (...) the 1—1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

This paper develops a moderate pessimist account of moral deference. I argue that while some pessimist explanations of the puzzle of moral deference have been misguided in matters of detail, they nevertheless share an important insight, namely that there is a justified moral agency ideal grounded in pro tanto reasons against moral deference. This thought is unpacked in terms of a set of values associated with the practice of morality. I conclude by suggesting that the solution to the puzzle of (...) moral deference developed here gives us a plausible recipe for generalizing to certain cases of nonmoral deference as well. (shrink)

We generalize a result on True Arithmetic by Lachlan and Soare to certain other completions of Peano Arithmetic. If $\mathscr{T}$ is a completion of $\mathscr{PA}$, then Rep denotes the family of sets $X \subseteq \omega$ for which there exists a formula $\varphi$ such that for all $n \in \omega$, if $n \in X$, then $\mathscr{T} \vdash \varphi})$ ) and if $n \not\in X$, then $\mathscr{T} \vdash \neg\varphi})$. We show that if $\mathscr{S,J} \subseteq \mathscr{P}$ such that $\mathscr{S}$ is a Scott (...) set, $\mathscr{J}$ is a jump ideal, $\mathscr{S} \subset \mathscr{J}$ and for all $X \in \mathscr{J}$, there exists $C \in \mathscr{S}$ such that C is a "coding" set for the family of subtrees of 2$^{<\omega}$ computable in X, and if $\mathscr{T}$ is a completion of $\mathscr{PA}$ such that Rep = $\mathscr{S}$, then there exists a model $\mathscr{A}$ of $\mathscr{T}$ such that $\mathscr{J}$ is the Scott set of $\mathscr{A}$ and no enumeration of Rep is computable in $\mathscr{A}$. The model $\mathscr{A}$ of $\mathscr{T}$ is obtained via a new notion of forcing. Before proving our main result, we demonstrate the existence of uncountably many different pairs satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2$^{<\omega}$. In particular, $C \subseteq \omega$ is a coding set for the family of subtrees of 2$^{<\omega}$ computable in X if and only if for all trees T $\subseteq 2^{<\omega}$ computable in X, if $\chi_C$ is a path through T, then there exists $\sigma \in T$ such that $\sigma \subset \chi_C$ and every extension of $\sigma$ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2$^{<\omega}$. We show they are identical. (shrink)

We show that a first category homogeneous zero-dimensional Borel set X can be embedded in as an ideal on ω if and only if X is homeomorphic to X × X if and only if X is Wadge-equivalent to X × X. Furthermore, we determine the Wadge classes of such X, thus giving a complete picture of the possible descriptive complexity of Borel ideals on ω. We also discuss the connection with ideals of compact sets.

This book offers an ideal first reader in ancient Greek. It presents a selection of extracts from a comprehensive range of Greek authors, from Homer to Plutarch, together with generous help with vocabulary and grammar. The passages have been chosen for their intrinsic interest and variety, and brief introductions set them in context. All but the commonest Greek words are glossed as they occur and a general vocabulary is included at the back. Although the book is designed to be used (...) by students who have only completed a beginner's course in Greek, students of all ages will find that it helps them to improve their ability and confidence in reading a wide range of prose and verse texts, while at the same time providing a fascinating and copiously illustrated introduction to the riches and variety of ancient Greek literature over a period of almost a thousand years. (shrink)

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters. There is also a substantial interest in nicely definable ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic (...) class='Hi'>ideals occur naturally in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space, see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P: X ≥I Y iff X \ Y ϵ I and the partial order.2. Boolean algebras of the form P/I and their automorphisms.3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I.In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3. (shrink)

Recent decades have seen attacks on philosophy as an irrelevant field of inquiry when compared with science. In this book, Graham McFee defends the claims of philosophy against attempts to minimize either philosophy’s possibility or its importance by deploying a contrast with what Wittgenstein characterized as the “dazzling ideal” of science. This ‘dazzling ideal’ incorporates both the imagined completeness of scientific explanation—whereby completing its project would leave nothing unexplained—and the exceptionless character of the associated conception of causality. On such a (...) scientistic world-view, what need is there for philosophy? In his defense of philosophy, McFee shows that rejecting such scientism is not automatically anti-scientific, and that it permits granting to natural science its own truth-generating power. Further, McFee argues for contextualism in the project of philosophy, and sets aside the pervasive requirement for exceptionless generalizations while relating his account to interconnections between the concepts of person, substance, agency, and causation. (shrink)

Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω (...) 1 such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory. (shrink)

McKenna’s critique (rather than criticisms) of idealized approaches to epistemology is an important contribution to the literature. In this brief discussion, I set out his main concerns about more idealized approaches, within and beyond social epistemology, before turning to some issues I think he neglects. I suggest that it’s important to pay attention to the prestige hierarchy in philosophy, and to how that hierarchy can serve ideological purposes. The greater prestige of more abstract approaches plays a role in determining what (...) issues and what voices we attend to. (shrink)

We improve results of Marczewski, Frankiewicz, Brown, and others comparing the σ-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis of σ-ideals (...) to include the completely Ramsey null sets relative to a Ramsey ultrafilter and obtain all 32 possible examples of sets in some ideals and not others, some under the assumption of MA, but most in ZFC alone. We also improve upon the known constructions of a Marczewski measure zero set which is not Ramsey by using a set theoretic hypothesis which is weaker than those used by other authors. We give several consistency proofs: one concerning the relative sizes of the covering numbers for the meager sets and the completely Ramsey null sets; another concerning the size of non(CRU 0); and a third concerning the size of add(CRU 0). We also study those classes of perfect sets which are bases for the class of always first category sets. (shrink)

We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the axiom (...) of Dependent Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF. (shrink)

This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L$, JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of (...) $\omega $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model.In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property.Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show that there are no infinite mad families in $L$ under large cardinals and that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$, respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L$. Part I concludes with Chapter 5, a short list of open questions.In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s.The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T, then $L$ -absoluteness for ccc posets must fail.We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L.In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L$ -absoluteness for $\lambda $ -linked proper posets.Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L.prepared by Zach Norwood.E-mail: [email protected]. (shrink)

The United States Constitution embodies both the real and the ideal. It is a concrete written text that uses particular words, has a history, and possesses certain limits; it is also a statement of the aspirations and dreams of a society. This dual identity requires that the Constitution be understood both as written positive law, and as an expression of a national vision and set of ideals. ;I argue for a conceptual theory of law that is positivistic in the (...) sense that it relies for the identification of law on social conventions that form the foundations for determining what counts as law. My view is drawn from H. L. A. Hart's legal positivist view, but it departs from this view to argue that judicial custom can be, along with Hart's rule of recognition, an additional social custom that provides criteria for identifying and creating law. Judicial custom involves the acceptance over time by judges of certain principles as authoritative in the context of adjudicative decisionmaking. I contend that both rules and principles can be part of law, and that principles attain the status of law through judicial custom. ;I rely on this understanding of law to argue for a theory of constitutional adjudication that is an alternative to both originalism and traditional non-originalist theories. Originalism is the view that judges can decide constitutional cases exclusively on the basis of two grounds: the text or the original intentions or understanding of its framers. The theory of law that I defend accepts that these grounds are important for any constitutional decision, but holds that constitutional decisions can also legitimately be made on the basis of constitutional theories. Such theories explain the often broad and abstract language of constitutional provisions by articulating hypotheses about the principles that underlie the text. I argue that the central constitutional commitment is to each individual's right to self-determination. This requires protection of an aspect of personal liberty that involves the deeply personal and unique nature of each person's aspirations, ideas, and initiatives. (shrink)

Carnap's ideal of explication has become a key concept in analytic philosophy and the basis of a method of analysis which may be considered as an alternative to various forms of naturalism, including Quine's conception of a naturalized epistemology. More recently, new light has been shed on this aspect of the classical Carnap-Quine debate by contemporary philosophers. Whereas Michael Friedman articulated a notion of relativized a priori which owes much to Carnap's internal/external distinction, André Carus attempted to restate Carnap's ideal (...) of explication in a way that bridges the gap between conceptual engineering and naturalism. On the other hand, Mark Wilson argued that concepts develop in unpredictable ways and cannot be planned in the way classical analytic philosophy has assumed from Frege and Russell onwards. This book consists of a series of defences as well as critiques of Carnap's programme, setting it in its historical context, discussing specific cases of explications, and enriching the on-going debate on conceptual engineering and naturalism in analytic philosophy.Carnap's ideal of explication has become a key concept in analytic philosophy and the basis of a method of analysis which may be considered as an alternative to various forms of naturalism, including Quine's conception of a naturalized epistemology. More recently, new light has been shed on this aspect of the classical Carnap-Quine debate by contemporary philosophers. Whereas Michael Friedman articulated a notion of relativized a priori which owes much to Carnap's internal/external distinction, André Carus attempted to restate Carnap's ideal of explication in a way that bridges the gap between conceptual engineering and naturalism. On the other hand, Mark Wilson argued that concepts develop in unpredictable ways and cannot be planned in the way classical analytic philosophy has assumed from Frege and Russell onwards. This book consists of a series of defences as well as critiques of Carnap's programme, setting it in its historical context, discussing specific cases of explications, and enriching the on-going debate on conceptual engineering and naturalism in analytic philosophy. (shrink)

We investigate the cofinality of the partial order N κ of functions from a regular cardinal κ into the ideal N of Lebesgue measure zero subsets of R. We show that when add(N) = κ and the covering lemma holds with respect to an inner model of GCH, then cf(N κ ) = max {cf(κ κ ), cf([ cf(N)] κ )}. We also give an example to show that the covering assumption cannot be removed.

This article offers a study of the early formation and development of the ideal of harmony in ancient Chinese philosophy and ancient Greek philosophy. It shows that, unlike the Pythagorean notion of harmony, which is primarily based on a linear progressive model with a pre-set order, the ancient Chinese concept of harmony is best understood as a comprehensive process of harmonization. It encompasses spatial as well as temporal dimensions, metaphysical as well as moral and aesthetical dimensions. It is a fundamentally (...) open notion in the sense that it does not aim to conform to any pre-set order. This broader, richer, and more liberal understanding of harmony has had a profound influence on Chinese culture as whole in its long history. (shrink)

In this survey paper we collect several known results on destroying tall ideals on countable sets and maximal almost disjoint families with forcing. In most cases we provide streamlined proofs of the presented results. The paper contains results of many authors as well as a preview of results of a forthcoming paper of Brendle, Guzmán, Hrušák, and Raghavan.

The Clinical ethics centre in Paris offers its services equally to doctors and patients/proxies. Its primary goal is to re-equilibrate doctor–patient roles through giving greater voice to patients individually in medical decisions. Patients are present at virtually all levels, initiating consults, providing their point of view and receiving feedback. The implications of patients' involvement are threefold. At an operational level, decision-making is facilitated by repositioning the debate on ethical grounds and introducing a dynamic of decisional partnership, although contact with patients (...) can make it difficult to deny their demands and set the limits of our role. Ethically, it reinforces patients' autonomy and grants them a place of veritable ethics ‘actors’, with the danger that this may become excessively autonomy oriented. Finally, at a collective level, the programme fulfils its political purpose in promoting patients' rights and the ideal of démocratie sanitaire, but complicates balancing individual demands with collective values. (shrink)

Are societies required to pursue continual economic growth as a matter of justice? In “The Value of Economic Growth”, Julie Rose considers three arguments in favor of the need for continual economic growth, each of which revolves around the instrumental value of economic growth for promoting an important good that is needed for a just society. In each case, Rose argues that there are mechanisms other than economic growth that could allow a society to deliver the relevant goods, and thus (...) meet the demands of justice with respect to those goods. I raise a set of issues for Rose’s argument that put pressure on the normative significance of her discussion. At the heart of these issues are ones about which possibilities Rose considers and which idealizations she makes. These issues tie into more general questions about the aims and methodology of normative work in social/political philosophy. Thus, in addition to being a contribution to the debate over the politics of economic growth, this paper can be understood partly as a case study in how reflection on these kinds of issues – ones about modality, idealization, and methodology – can matter to how we evaluate specific arguments in social/political philosophy. (shrink)

For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln if and only if κ is a Πn1-indescribable cardinal and every Πn1-indescribable subset of κ has a Πn1-indescribable proper initial segment. The Πn1-reflection principle Refln generalizes a certain stationary reflection principle and implies that κ is Πn1-indescribable of order ω. We define a forcing which shows that the converse of this implication can be false in the case n=1; that is, we show that κ being Π11-indescribable of (...) order ω need not imply Refl1. Moreover, we prove that if κ is -weakly compact where α<κ+, then there is a forcing extension in which there is a weakly compact set W⊆κ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and κ remains -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down. (shrink)

In the past decade, scholars such as Samuel Freeman, Martin O’Neill, Alan Thomas and others have argued that no matter how widely Rawls’s theory of justice (TJ) was understood as a defence of the welfare state (WS), the socio-economic system Rawls defends and always defended is property-owing democracy (POD). In this article I present the argument that Rawls did not defend POD in TJ. However, while the claim that it was POD the socio-economic system implied by the principle of difference (...) generated heated reactions by defenders of the WS, the argument I raise differs from most of them. In the article I maintain that, for Rawls, principles of justice underdetermine the choice of social systems, and that although this is true of both principles, it is especially true of the difference principle, whose ability to arbitrate between different social and economic policies is limited. I show how the attempt to uniquely determine the set of social and economic policies authorized by the difference principle clashes with what Rawls explicitly states in TJ and contrasts with central aspects of his sociological and political conception of socio-economic decision-making. Although my aim is to show that the difference principle itself does not discriminate among socio-economic systems, this does not preclude that additional reasons can help in narrowing the range of socio-economic systems allowed. With reference to the ideal of social union that Rawls expounds in the third part of TJ, I argue that TJ does, in fact, offer additional reasons to appraise the WS as preferable to POD. I maintain that the centrality Rawls attributes to the social unity of a just society provides the basis for an understanding of Rawls’s position towards POD after TJ. (shrink)

ArgumentThis paper analyses a process of co-construction of knowledge and its multiple forms of communication in a country of the European periphery in the early twentieth century. It focuses on Lieutenant Manuel Soares de Melo e Simas, a politically engaged Portuguese astronomer, who moved from amateur to professional during the political transition from the monarchy to the republic. Melo e Simas paralleled his professional career in continuous activity of communicating science to the public in the context of republicanism in a (...) double way, by responding to the agenda of republicanism and by playing an active role in shaping it. He aimed at educating lay audiences in the various ways of astronomy, and he reached out to as many people as possible by exploring a multitude of communication channels, from lectures to articles in newspapers and journals. Voiced often within newly created republican institutions, the praxis and the ideas of Melo e Simas helped to mold the new republican scientific ethos. By going beyond mere emphasis on scientism and positivism, usually taken to be the defining characteristics of the new republican ethos, this paper argues that science and the specificities of its multiple forms of communication were central to the way Melo e Simas shaped the republican ideology. Furthermore, popularization of science was used to legitimize the status of professional scientists at the same time that it helped reinforce their institutional setting, still to be negotiated in the forthcoming decades through a complex process which deserves further historical analysis. (shrink)

Let A and B be subsets of the space 2 N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2 N into 2 N such that A = Φ -1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, $a \subset b$ implies $\Phi (a) \subset \Phi(b)$ . The set A is said to be (...) monotone if a ∈ A and $a \subset b$ imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The ▵ 0 1 sets are all reducible to the (Σ 0 1 but not ▵ 0 1 ) sets, which are in turn all reducible to the strictly ▵ 0 2 sets, which are all in turn reducible to the strictly Σ 0 2 sets. In addition, the nontrivial Σ 0 n sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly Π 0 2 monotone sets which have different monotone degrees. We show that every Σ 0 2 monotone set is actually Σ 0 2 positive. We also consider reducibility for subsets of the space of compact subsets of 2 N . This leads to the result that the finitely iterated Cantor-Bendixson derivative D n is a Borel map of class exactly 2n, which answers a question of Kuratowski. (shrink)

There is a growing need to implement anticipatory climate change adaptation measures, particularly in vulnerable sectors, such as in agriculture. However, setting goals to adapt is wrought with several challenges. This paper discusses two sets of challenges to goals of anticipatory adaptation, of empirical and normative character. The first set of challenges concern issues such as the extent to which the climate will change, the local impacts of such changes, and available adaptive responses. In the second set of uncertainties (...) are issues such as the distribution of burdens to enhance adaptive capacities in vulnerable agents with a legitimate claim to such resources, and what anticipatory adaptation ideally should result in. While previous discussions have been limited to either discuss the first or second set of uncertainties, this paper suggests that both dimensions should be considered when setting goals in social planning with long time frames. A taxonomy will be suggested that combines both dimensions. Furthermore, strategies for managing situations in which there are either empirical, or normative, uncertainties will be proposed which could be used in social decision-making with long planning time-frames in which goals must be set. (shrink)

In this book the authors introduce and study the following notions: Neutrosophic Crisp Points, Neutrosophic Crisp Relations, Neutrosophic Crisp Sets, Neutrosophic Set Generated by (Characteristic Function), alpha-cut Level for Neutrosophic Sets, Neutrosophic Crisp Continuous Function, Neutrosophic Crisp Compact Spaces, Neutrosophic Crisp Nearly Open Sets, Neutrosophic Crisp Ideals, Neutrosophic Crisp Filter, Neutrosophic Crisp Local Functions, Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals, Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity, Neutrosophic Topological Region, Neutrosophic Closed Set (...) and Neutrosophic Continuous Function, etc. They compute the distances between neutrosophic sets and extend it to Neutrosophic Hesitancy Degree. The authors also generalize the Crisp Topological Space and Intuitionistic Topological Space to the notion of Neutrosophic Crisp Topological Space. At the end, they present applications to Neutrosophic Database, and show a security scheme based on Public Key Infrastructure (PKI) using Neutrosophic Logic Manipulation. The authors utilize neutrosophic sets in order to analyze social networks data conducted through learning activities, and for the Geographical Information Systems (GIS) they employ fundamental concepts and properties of a Neutrosophic Spatial Region. Keywords: Neutrosophic Crisp Points; Neutrosophic Crisp Relations; Neutrosophic Crisp Sets; Neutrosophic Crisp Continuous Function; Neutrosophic Crisp Compact Spaces; Neutrosophic Crisp Nearly Open Sets; Neutrosophic Crisp Ideals; Neutrosophic Crisp Filter; Neutrosophic Crisp Local Functions; Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals; Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity; Neutrosophic Topological Region; Neutrosophic Closed Set and Neutrosophic Continuous Function; Neutrosophic Hesitancy Degree; Neutrosophic Crisp Topological Space; Neutrosophic Database; Neutrosophic Logic Manipulation; Neutrosophic Spatial Region. (shrink)

It is tempting and not at all uncommon to find the striking—even noble—visage of an Ideal Observer staring out from the center of Hume's moral theory. When Hume claims, for instance, that virtue is “ whatever mental action or quality gives to a spectator the pleasing sentiment of approbation ,” it is only natural to think that he must have in mind not just any spectator but a spectator who is fully informed and unsullied by prejudice. And when Hume writes (...) that “the true standard of taste and beauty” is set by those who exhibit “[s]trong sense, united to delicate sentiment, improved by practice, perfected by comparison, and cleared of all prejudice,” he appears to describe a character no ordinary human could actually possess. Indeed, Hume's frequent appeals to the moral sentiments of spectators, his insistence that those sentiments depend upon taking “the general survey,” and his persistent invocation of the general point of view , together make the temptation almost irresistible. (shrink)

Relationships between self-ratings and expectations of an ideal U.S. president, were studied in 43 men drawn from a university setting in the eastern coast of the U.S.A. The men first rated themselves on personality variables, life choices (agentic and communal), peacefulness, spirituality, and morality. Then they were presented with a vignette requesting that they describe an ideal U.S. president on inventories measuring personality variables, life choices, peacefulness, spirituality, and morality. For the rating of the ideal U.S. president, they also were (...) asked to respond to a 20 item questionnaire that was a composite of several factors on organization and leadership, morality, spirituality, and peacefulness. The respondents belonged to one of seven different political persuasions, similar in some ways to different cultures. Self-ratings of the men and expectations of the president were highly correlated for extraversion, openness, trait morality, agentic and communal life choices. However, no significant correlations were found between the self-ratings and expectations of the president for neuroticism, agreeableness, conscientiousness, peacefulness, nor state morality. The men were also presented with vignettes for the ideal physician and ideal automechanic and asked to rate each of them on the inventory items. Overall, the U.S. President was rated as more neurotic and immoral in terms of ingrained ideas of right and wrong, but also as more caring for others, transcendent, seeking goodness and truth, forgiving, cooperative, and most concerned with matters of justice and mercy, and more concerned with both agentic (power-seeking) and communal (community-minded) life choices than were either the ideal physician or ideal automechanic. The ideal physician was rated as highest in extra-version, openness, agreeableness, conscientiousness, and overall peacefulness, and lowest in neuroticism. The ideal automechanic was rated as highest in state or situational immorality, and lowest in both agentic (power-seeking, business-mindedness) and communal (community-mindedness) life choices, and also lowest in caring for others well-being, transcendence, seeking goodness and truth, forgiveness and cooperation, being concerned with justice and mercy, overall expectations, overall spirituality, and overall organization and leadership. In general, the self-ratings were significantly related to ratings/expectations, of the U.S. President, ideal physician, and ideal automechanic. The men seemed to identify more with the automechanic than with the present or physician. (shrink)

On Exceeding Determination and the Ideal of Reason: Immanuel Kant, William Desmond, and the Noumenological Principle examines the critical philosophy of Immanuel Kant, as it bears on theological principles. Focusing on the foundational ideas (of self, world, and God) that constitute Kant's metaphysical system, Shaw argues that these ideal projections of the rational structures of the thinking subject only conceal and obfuscate the more robust sense of the real that exists behind all phenomenal appearances. This book aims to critically assess (...) the theological content presented in the philosophy of Kant whilst reconstructing and developing some of Kant's more obscure points on the epistemological limitations of pure reason and the borders of knowledge itself. Shaw draws into dialogue with the writings of the contemporary philosopher William Desmond to demonstrate some of the problems of Kantian thought when it comes to the deeper mysteries of being. Desmondian philosophy proves to be a powerful influence over much of this work, as Shaw advances upon the delicate nexus where philosophy and theology convene. As a bold addition to this work, Shaw lays the groundwork for a new discourse in theology and the philosophy of religion: "Noumenology". Noumenology, which is Shaw's response to the deficiencies in methods concerning many of the modern approaches to phenomenology, is a call for the recalibration of the starting point for eidetic inquiry through the proposal of a non-discursive approach to the sensible, emotional, and intellectual - or perhaps, the all together spiritual. To accomplish this, Shaw reaches back to Kant for a heterodox approach to the language and content surrounding the noumenonal to set forth his theological interpretation of the "Noumenological Principle". (shrink)

Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well. In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of entailment, hence (...) the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, GödelΔ, and Gödel∼ logics. (shrink)

It is shown how the development of physics has involved making explicit what were homocentric projections which had heretofore been implicit, indeed inexpressible in theory. This is shown to support a particular notion of the invariant as the real. On this basis the divergence in ideals of physical intelligibility between Bohr and Einstein is set out. This in turn leads to divergent, but explicit, conceptions of objectivity and completeness for physical theory. *I am indebted to Dr. G. McLelland. Professor (...) F. Rohrlich and an anonymous referee of this journal for several improvements in the formulation of the paper. (shrink)

This book tells a single story, in many voices, about a serious and sustained set of changes in mathematics teaching practice in a high school and how those efforts influenced and were influenced by a local university. It includes the writings and perspectives of high school students, high school teachers, preservice teacher candidates, doctoral students in mathematics education and other fields, mathematics teacher educators, and other education faculty. As a whole, this case study provides an opportunity to reflect on reform (...) visions of mathematics for all students and the challenges inherent in the implementation of these visions in US schools. It challenges us to rethink boundaries between theory and practice and the relative roles of teachers and university faculty in educational endeavors. (shrink)

The Ideal of a Rational Morality collects the most important essays by the distinguished moral philosopher Marcus G. Singer. Its guiding theme is the concept of a morality based in reason, which is presupposed in ordinary moral contexts and provides an ideal for improving ordinary morality and correcting moral judgements. Singer makes compelling claims that certain fundamental presuppositions are inescapable in moral thought, that fundamental moral principles can be proved, and that the concepts of truth and 'common sense' are essential (...) to ethics.Subsequent essays proceed to analyse the nature of value judgements and of moral judgements, emphasising the vital importance of certain basic distinctions. There is a discussion of race and racism, presenting new ideas about the nature of both. The relation of law to morality is considered, as are the relations between moral judgements of individual persons and the actions and moral judgements of institutions, leading to an examination of the relations between moral issues and social problems. A particularly well-known essay on the Golden Rule is reproduced, where Singer's analysis shows it to be applicable and defensible as it stands, and finally the nature of happiness is explored through a discussion of John Stuart Mill's moral philosophy.Each essay is supplemented by a set of Additional Notes and Comments, with supplementary pieces and response to criticisms. The Ideal of a Rational Morality will be fascinating reading for anybody seeking rigour and clarity in ethical issues. (shrink)

Pascal described human beings as ‘thinking reeds’, weak in flesh but magnificent in mind. While it is a poetic image, it is also an ambivalent one and may suggest an inappropriately dualist view of human nature. It is important to realise that not only are we thinking reeds but that we are thinking because we are reeds. In fact, rationality is reed-like itself, very much of a kind with the rest of human nature. It is now more than two and (...) half centuries since David Hume first pointed out the lack of an argument that would fully justify claims about matters of fact. Being neither made evident by our observations nor arising out of the mere considera-tion of relations of ideas, claims such as that the turkey will be fed dinner tomo-rrow – rather than being had for dinner have remained problematic ever since. Many attempts have been made to show that something of the beauty and certainty of reasoning about relations of ideas could be recaptured in our dealings with matters of fact, but all attempts have remained mere shadows of what we tried to grasp. Hume’s argument stands. An infinite being might watch countless sun-sets and yet should witness each new sun-rise with surprise, always withholding its judgement regarding what will follow. (shrink)