This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions.

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)

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are : Every product of a concentrated space with a Hurewicz S1S1 space satisfies S1S1. On the other hand, assuming the (...) Continuum Hypothesis, for each Sierpiński set S there is a Luzin set L such that L×SL×S can be mapped onto the real line by a Borel function. Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy. Assuming d=ℵ1d=ℵ1, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers.A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than addadd, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy >ℵ1.Our results improve upon and unify a number of results, established earlier by many authors. (shrink)

In this paper, we are interested in parallels to the classical notions of special subsets in defined in the generalized Cantor and Baire spaces (2κ and ). We consider generalizations of the well‐known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ‐sets, γ‐sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less‐know classes like X‐small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and (...) Laver‐null sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems. (shrink)

We prove the following theorems: (1) If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s 0 . (2) If X has Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. (3) If X has strong measure zero and Hurewicz's covering property then its algebraic sum with any set (...) in APC ' is a set in APC '. (APC ' is included in the class of sets always of first category, and includes the class of strong first category sets.) These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the γ-property and of a first category set is a first category set, and Bartoszynski and Judah's characterization of SR M -sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts. (shrink)

Using a dictionary translating a variety of classical and modern covering properties into combinatorial properties of continuous images, we get a simple way to understand the interrelations between these properties in ZFC and in the realm of the trichotomy axiom for upward closed families of sets of natural numbers. While it is now known that the answer to the Hurewicz 1927 problem is positive, it is shown here that semifilter trichotomy implies a negative answer to a slightly stronger form (...) of this problem. (shrink)

CH implies that selective separability is not preserved by finite powers . In ZFC, selective separability does not imply H-separability.

All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q \subseteq X}$$\end{document} there exists a crowded Q′⊆Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q'\subseteq Q}$$\end{document} with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^\omega}$$\end{document}.Every closed subspace of X (...) is a Baire space. While is the well-known property of being completely Baire, properties and have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications →→→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \rightarrow \rightarrow \rightarrow }$$\end{document} hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a ZFC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf ZFC}$$\end{document} counterexample and a consistent definable counterexample of lowest possible complexity to the implication ←\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \leftarrow }$$\end{document} for i=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i = 1, 2, 3}$$\end{document}. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. (shrink)

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allow us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally (...) definable covering maps; monodromy equivalence for locally constant o-minimal sheaves – from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert–van Kampen theorems. (shrink)

In a previous paper-[17]-we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove: 1. If a metric space has a covering property (...) of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3). 2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9). 3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12). Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems. (shrink)

We introduce a class of proper posets which is preserved under countable support iterations, includes $\omega ^\omega$ -bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.

What is property, and why does our species happen to have it? In The Property Species, the economist Bart Wilson explores how we acquire, perceive, and know the custom of property, and why this might be relevant to social scientists, philosophers, and legal scholars for understanding how property works in the twenty-first century.

In this eye-opening study at the intersection of psychoanalytic theory and political organization and thought, Elliott Schwebach explores why property can be understood to be oppressive and how political theory overlooks its unique significance as a pillar of social violence. Synthesizing insights from Pierre-Joseph Proudhon, Sigmund Freud, Ives Hendrick and Frantz Fanon, Schwebach investigates human activity as shaped by the effects of property regimes and traces broader implications for understanding the legacies of colonial domination. He then shifts focus (...) to contemporary eco-theory, challenging the Lockeanism that continues to characterize premodern Indigenous environmental engagements and presenting novel frameworks for understanding healthy ecopolitical activity based upon the trajectories of psychological drives. This unique perspective validates creative expressions of decolonial resistance and offers fruitful alternatives to customary positions in psychoanalytic and environmental political philosophy. The book will be an indispensable resource for scholars of property, Freudian psychology, political ecology, and the visionary thought of Frantz Fanon. (shrink)

Many predicates are dispositional. Some show this by a suffix like "-ible", -uble", or "-able": sugar is soluble in water, gasoline is flammable. Others have no such suffix and don't wear their dispositionality on their sleeves. Yet part of what it is to be solid is to be disposed to resist deformation, and part of what it is to be red is to appear red to normal human observers in normal lighting conditions. However, there is no agreement as to whether (...) dispositional predicates may be given a realist interpretation. For many authors, propositions containing them are made true by states of affairs (or facts) containing categorical, rather than dispositional properties. Many also claim that the states of affairs that make true attributions of dispositions to macroscopic objects are microscopic states of affairs concerning their parts. For example, what makes a vase fragile is the microscopic structure of its molecular constituents, which is what makes the vase break when it falls. Against these claims, I will argue that what makes a dispositional predicate apply to an object, whether macroscopic or microscopic, whether in common sense or science, is the object's having what I will call a powerful property. If the object is macroscopic, it is another matter whether the property is microreducible. My reasons for supposing that these powerful properties exist are those for postulating theoretical properties generally: they unify existing explanations and suggest new ones. (shrink)

SHOEMAKER IS WRONG TO CLAIM THAT ALL THE GENUINE PROPERTIES OF THINGS ARE NOTHING BUT POTENTIALITIES FOR CONTRIBUTING TO THE CAUSAL POWERS OF THINGS. FOR THE ONLY GROUNDS FOR ATTRIBUTING CAUSAL POWERS TO THINGS ARE IN TERMS OF THE EFFECTS WHICH THOSE THINGS TYPICALLY PRODUCE. BUT ALL EFFECTS ARE ULTIMATELY INSTANTIATIONS OF PROPERTIES, AND IF THESE WERE NOTHING BUT POTENTIALITIES TO PRODUCE EFFECTS, THERE WOULD BE A VICIOUS INFINITE REGRESS, AND NO ONE WOULD EVER BE JUSTIFIED IN ATTRIBUTING PROPERTIES TO (...) OBJECTS. (shrink)

Hegel is aware that it is only in the modern world, with the emergence of civil society, that ‘the freedom of property has been recognized here and there as a principle’. Nonetheless, he contends, property is made necessary by the very idea of freedom itself. The purpose of this essay is to explain why this is the case by tracing the logic that leads in Hegel's Philosophy of Right from freedom, through right, to property and its use. (...) I conclude by briefly comparing Hegel and Marx on the topic of ‘value’. (shrink)

Orthodox ideas of ownership tend to depict property as a private domain that expresses the owner?s formal rights. Yet equity does much to resist this outlook, deploying ethically-loaded ideas such as conscience and articulating an interpersonal and distinctly duty-driven character to property relations. Focusing on English case law, this article suggests that we can gather various strands of equitable property norms, particularly those derived from the constructive trust, around relationships of responsibility and vulnerability. Furthermore, the article asks (...) what such equitable ideas about property might therefore tell us about the purportedly possessive character of human subjectivity. Rather than constructing people as sovereign, autonomous proprietors, we might read equity as tracing an ethical quality to the way we perceive ourselves and others through property-related ideas. (shrink)

The case of highland Yemen up to around the middle of the twentieth century involves a history different from most Muslim societies in that, from 1919, the Yemeni state was independent. The problem I address concerns the utility of thinking about the highland property regime in this era in relation to the categories of "private" and "public." What sort of antecedents existed, at the level of property relations, for later commercial transformations that would culminate in such things as (...) Pizza Hut franchises? In Yemen, the polity of the period was a type of Islamic state, one based on the shari`a in ideology and in application and headed by a classical form of jurist-leader, a figure known as an imam. Prior to the twentieth century, this Yemeni form of a shari`a-based Islamic polity had a thousand year history in the highlands. Shari`a courts had exclusive competence as state tribunals, and judges, trained on old-style law books, heard the full gamut of litigation. By contrast, colonial changes in other Muslim societies typically entailed a sharp alteration and restriction of the sphere of the shari`a, often limiting application to the domain of "family law" alone. At the same time, the highlands had yet to commence the other great and, elsewhere, ongoing modern transformation of the shari`a, codification and legislation, which would not begin in Yemen until after the Revolution of 1962.Highland society at mid-century was agrarian, based primarily on settled plow cultivation, and the associated property regime was almost exclusively "private." In Yemen, the "private" property regime centered conceptually on milk, a category of individual ownership of immoveable property, and on the concept of mal, a commercial commodity. As a form of private, landed property, milk involved rights that could be acquired, alienated and inherited, and the associated agricultural production was based on lease contracts between landlords and sharecroppers. Pious endowments were the basis for one of the great Muslim "public" institutions, which supported mosques, schools, water systems, etc. Another fundamental "public" institution was an extension of the "private" commercial notion of mal into the state institution of the 'House of Mal,' or Treasury. Other key "public" institutions were the imamate itself, a form of Islamic state, and the shari`a court. This shari`a system can be thought of in terms of three levels of legal texts. At the highest level, shari`a doctrine, the jurisprudence of the period, constituted an ideology of the property regime. The doctrine provided models for both the range of substantive undertakings and for court processes, but the relationship of these models to the property relations and to litigation on the ground is a key question. At the lowest level, the routine documents of ongoing, uncontested practice proliferated. These included ordinary sale documents, leases, marriage contracts, endowment instruments, wills and estate inheritance instruments. Between high doctrine and low instruments were the records of shari`a court judgments, records in which the conflicts and contradictions of the property system were expressed, argued and ruled upon. In certain problem areas of the law, the ruling imams's personal doctrinal "choices" were designed to guide court judges in their rulings. (shrink)

Revised and reprinted in Handbook of Philosophical Logic, volume 10, Dov Gabbay and Frans Guenthner (eds.), Dordrecht: Kluwer, (2003). -- Two sorts of property theory are distinguished, those dealing with intensional contexts property abstracts (infinitive and gerundive phrases) and proposition abstracts (‘that’-clauses) and those dealing with predication (or instantiation) relations. The first is deemed to be epistemologically more primary, for “the argument from intensional logic” is perhaps the best argument for the existence of properties. This argument is presented (...) in the course of discussing generality, quantifying-in, learnability, referential semantics, nominalism, conceptualism, realism, type-freedom, the first-order/higher-order controversy, names, indexicals, descriptions, Mates’ puzzle, and the paradox of analysis. Two first-order intensional logics are then formulated. Finally, fixed-point type-free theories of predication are discussed, especially their relation to the question whether properties may be identified with propositional functions. (shrink)

This paper initiates the study of sets in Euclidean spaces ℝn that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, n ], we are interested in the connectivity properties of the set DIMI, consisting of all points in ℝn whose dimensions lie in I, and of its dual DIMIstr, consisting of all points whose strong dimensions lie in I. If I is [0, 1) or.

_Property Rights: Philosophic Foundations,_ first published in 1977, comprehensively examines the general justifications for systems of private property rights, and discusses with great clarity the major arguments as to the rights and responsibilities of property ownership. In particular, the arguments that hold that there are natural rights derived from first occupancy, labour, utility, liberty and virtue are considered, as are the standard anti-property arguments based on disutility, virtue and inequality, and the belief that justice in distribution must (...) take precedence over private ownership. Lawrence Becker goes on to contend that there are four sound lines of argument for private property that, together with what is sound in the anti-property arguments, must be co-ordinated to form the foundations of a new theory. He therefore expounds a concise but sophisticated theory of property that is relevant to the modern world, and concludes by indicating some of the implications of his theory. (shrink)

The main claim of this paper is that, contrary to the received view, Ross’ doctrine of resultance does not provide a premise in favour of non-naturalism, but rather makes possible a viable form of non-reductionist naturalism. This is argued mainly by viewing resultance as constitution, where resultant properties are constituted by those natural properties from which they result. Accordingly, resultant properties and their constitutive properties can be viewed as placed in the same ontological realm. However, to rule out reductionism, constitution (...) is to be considered as not implying identity. Some arguments in favour of this view of constitution, above all in the moral realm, are presented in the last sections. (shrink)

Aesthetic properties figure prominently in our daily lives, our conversations and many actions we take. Yet theoretical disagreement prevails over their nature, their variety, their epistemic and metaphysical status. This overview highlights the heterogeneity of aesthetic properties and examines repercussions for explanation. Aesthetic properties belong to natural objects or scenes, to artworks in any medium, to artefacts and built environments across historical eras; and they draw a wide variety of responses such as our perceptions, emotions or imaginative thought. Historicism about (...) artworks carries over to at least some aesthetic properties that depend on facts of their broader historical context just as works do. One issue is whether aesthetic properties can be demarcated solely in terms of non-aesthetic properties. If not, explanations of aesthetic properties need to draw on other aesthetic notions such as aesthetic experience, attention, judgement or value. The paper examines several accounts of aesthetic value to show how they extend to aesthetic properties. The paper concludes with an overview of the current metaphysical debate between realists and non-realists about aesthetic properties. (shrink)

Powers are popularly assumed to be distinct from, and dependent upon, inert qualities, mainly because it is believed that qualities have their nature independently of other properties while powers have their nature in virtue of a relation to distinct manifestation property. George Molnar and Alexander Bird, on the other hand, characterize powers as intrinsic and relational. The difficulties of reconciling the characteristics of being intrinsic and at the same time essentially related are illustrated in this paper and it is (...) argued that the reasons for thinking of powers as essentially relational are based on misguided epistemological consideration. Finally, I present a way of thinking of fundamental properties as primitive natures that we can only understand in virtue of what they do but which we should not think of as being ontologically constituted by these doings. According to this view, properties are both qualities and powers. (shrink)

A conception of the person as a subjective being plays a crucial, though frequently overlooked, role in the justification of unequal property in liberal theories. Unger's ascription of individualism to general liberal legal theory can be concretely defended with respect to liberal theories of property. Identifying a common fundamental structure calls in question the conventional view that the liberal legal theories rest on an ensemble of different moral foundations. So important is subjective personality to the moral basis for (...) highly unequal property that such property looks to be morally untenable if this concept of the person is invalid. The analysis points to social theory of personality as a major direction for research into an alternative theory of egalitarian property right. (shrink)

Our days are a vast, intricate, evolving dance of mutual understandings. We stop at a traffic light, offer a plastic card as payment for a meal, leave our weapons at home, or enter a voting booth. We live and work in close proximity, at high speed, with few collisions: on our roads and in our neighborhoods, places of worship, and places of business. Somehow, having all those people around is more liberating than stifling. The secret is that we know roughly (...) what to expect from each other. Knowing what to expect enables us to adapt to each other. Not being obliged to conform to expectations—being free to test the previously untested— is likewise a great benefit. The two benefits seem mutually exclusive, yet property rights, combined with freedom of contract, enable us to reap both at once. We can rely on being able to go to market and find someone selling cauliflower at an affordable price. We can also rely on being able to go to market and find someone rendering obsolete what a few years ago had been cutting-edge technology. We make progress by testing what has not previously been tested. We experiment. One problem with experiments is: many of them don’t work.1 Or, the ideas being tested turn out to be bad ideas. Thus a successful society encourages people not only to experiment, but also to shut down experiments whose inspiration proves unsound. What kind of framework encourages experimentation without at the same time perpetuating bad ideas? Here is one hypothesis: in societies that sustain progress over long periods, people are free to experiment at their own expense and free from having to pay for other people’s bad ideas. This is the true test of a system of property. It is natural to assume instead that the true test of a system of property is a question of whether the system is just. That is, philosophers should theorize about justice first, and only then begin to theorize about what can legitimately become a person’s property. I have become skeptical about this. (shrink)