Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices (...) of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (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)

We prove that Martin’s Axiom implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while \ is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel’s conjecture.

The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.

In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets.

We establish a surprising connection between Menger's classical covering property and Blass-Laflamme's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.

In this paper we review Shelah's strong covering property and its applications. We also extend some of the results of Shelah and Woodin on the failure of equation image by adding a real.

We give topological characterizations of filters${\cal F}$onωsuch that the Mathias forcing${M_{\cal F}}$adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is (...) not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties. (shrink)

$\underline{\text{Saturation is} (\mu, \kappa)-\text{transferable in} T}$ if and only if there is an expansion T 1 of T with ∣ T 1 ∣ = ∣ T ∣ such that if M is a μ-saturated model of T 1 and ∣ M ∣ ≥ κ then the reduct M ∣ L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ 0 , λ)- transferable or (κ (T), λ)-transferable for all λ. (...) Further if for some $\mu \geq \mid T \mid, 2^\mu > \mu^+$ , stability is equivalent to for all μ ≥ ∣ T ∣, saturation is (μ, 2 μ )- transferable. (shrink)

The weak non-finite cover property (wnfcp) was introduced in [1] in connection with "axiomatizability" of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory TP of lovely pairs of (...) models of T (true in the stable case). While the question remains open, we show, among other things, that if (for a T with the wnfcp) TP is low, then TP has the wnfcp. To study this question, we describe "double lovely pairs", and, along the way, we develop the notion of a "lovely n-tuple" of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models of stable theories [2]. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Leibniz & Clarke: A Study of Their Correspondence by Ezio VailatiJan A. CoverEzio Vailati. Leibniz & Clarke: A Study of Their Correspondence. New York and Oxford: Oxford University Press, 1997. Pp. xii + 250. Cloth, $45.00.When Leibniz received the 1710 issue of the Royal Society’s Philosophical Transactions in early 1711, he read John Keill’s public charge that he had stolen the calculus from Newton. Leibniz twice sought amends (...) from the Society in the form of a retraction, in response to which a committee was formed to investigate and issue a report. The resulting Commercium epistolicum, written by the good Isaac Newton himself and condemning Leibniz, was published in early 1713. In Vienna at the time, Leibniz learned of its contents in a June letter from Johann Bernoulli, who complained about this “scarcely civilized way” of holding what was in effect a tribunal from which the chief witness was absent. By year’s end Leibniz had enlisted Wolff to see into print Leibniz’s anonymous statement—overstatement, in fact—of his case, and nine months later Keill replied with an equally overstated rejoinder. Late autumn of 1714 found Leibniz back in Hannover, suffering from gout and the tiresome pressure to advance his work on the history of the house of Brunswick-Lüneburg. If only he felt better and had more free time, he could collect his mathematical papers and set an end to the dispute by writing his own Commercium epistolicum: as it was (Leibniz told Wolff in the spring of 1715), he had no intention of giving Newton’s cronies the luxury of his time.The Leibniz-Clarke exchange began in the shadows of this dispute, contributing to Leibniz’s already established distaste for parts of British philosophy. Already in 1690 he was describing (to Huygens and others) Newton’s gravitational attraction as “inexplicable.” Soon after Leibniz was surprised to discover that Locke had accepted Newtonian attraction in the second edition of the Essay—surprised, at least, until he came to see it as part and parcel of the dangerous idea that God could attach any old property whatever to matter if He chose, including thought. Materialism and frequent miracles [End Page 533] were just around the corner, and from there it was a short distance to the mortality of the soul, and a mean view of God’s power; the decay of natural theology itself could only follow. Leibniz’s misgivings didn’t see the light of day in the early 1700s, when he penned them in the Preface to the Nouveaux Essais; but he went out of his way to work an explicit complaint about Mr. Newton and Mr. Locke into the Théodicée of 1710 (§19 of the Preliminary Discourse). The calculus dispute followed. By the time Leibniz got round to writing Princess Caroline about the dispute in a letter of May 1715, he was insisting that the Newtonians deserved public censure, and that the Britons deserved Leibniz as historiographer of England. For her own part, Caroline now wondered if Samuel Clarke, who had translated Newton’s Opticks into Latin, would in the end be unbiased enough to translate Leibniz’s Théodicée into English. Moreover Caroline doubted that Clarke’s notion of the soul was theologically innocent. Replying in November, Leibniz summarized his view that natural religion was in great decline in England, and the famous exchange began.Despite there being no monograph devoted to it until now, the Correspondence ıs (and was) famous. If a book-length study is overdue, so perhaps are three reminders—that a significant portion of the Correspondence is devoted to issues other than space and time and gravitation, that the depth to which its issues are pursued was compromised by impatience on the side of both parties, and that Clarke’s philosophical views aren’t so well-understood as Leibniz’s own. Nowhere does Vailati set out to make these points explicitly, but they emerge over the course of his book, and together with exegetical soundness are its chief contributions. While this volume makes no historical or philosophical waves, it indirectly (perhaps unwittingly) explains why none are readily forthcoming in a book... (shrink)

We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set R of real numbers have the covering property of Menger if, and only if, each filter on (...) ω associated with its countable ω-cover is a P + filter. (shrink)

We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacín and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive cover of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis and Moosa-Pillay and show that they do not hold in the counterexample. In order to do so, a (...) study of imaginaries in additive covers is developed. As a by-product of the presentation, we observe that a pure binding-group-theoretic account of the CBP is unlikely. (shrink)

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if $\delta$ has the tree property then $\delta^{+K} = \delta^+$ and $\delta$ is weakly compact in K.

We study the Vapnik–Chervonenkis density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm {U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

Our investigation is concerned with the finite model property with respect to admissible rules. We establish general sufficient conditions for absence of fmp w. r. t. admissibility which are applicable to modal logics containing K4: Theorem 3.1 says that no logic λ containing K4 with the co-cover property and of width > 2 has fmp w. r. t. admissibility. Surprisingly many, if not to say all, important modal logics of width > 2 are within the scope of this theorem–K4 itself, (...) S4, GL, K4.1, K4.2, S4.1, S4.2, GL.2, etc. Thus the situation is completely opposite to the case of the ordinary fmp–the absolute majority of important logics have fmp, but not with respect to admissibility. As regards logics of width ≤ 2, there exists a zone for fmp w. r. t. admissibility. It is shown that all modal logics A of width ≤ 2 extending S4 which are not sub-logics of three special tabular logics have fmp w.r.t. admissibility. (shrink)

Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, the Kolmogorov sum (...) rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms of quantum covers. (shrink)

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.

Results of Sierpiski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is ‘narrow’ in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set ω × ω is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. (...) Such partitions or coverings can exist only when the sets forming the product are of limited size.This paper considers such coverings for products of infinitely many sets . In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering , then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on.One result given here is a relative consistency proof for a hypothesis used by S. Mrówka to produce a counterexample in the dimension theory of metric spaces. (shrink)

We use tools of mathematical logic to analyse the notion of a path on a complex algebraic variety, and are led to formulate a “rigidity” property of fundamental groups specific to algebraic varieties, as well as to define a bona fide topology closely related to etale topology. These appear as criteria for ℵ1-categoricity, or rather stability and homogeneity, of the formal countable language we propose to describe homotopy classes of paths on a variety, or equivalently, its universal covering space.Technically, (...) for a variety A defined over a finite field extension of the field Q of rational numbers, we introduce a countable language L describing the universal covering space of A, or, equivalently, homotopy classes of paths in A. Under some assumptions on A source we show that the universal covering space of A is an analytic Zariski structure [26]), and present an Lω1ω)-sentence axiomatising the class containing the structure and that is stable and homogeneous over elementary submodels. The “rigidity” condition on fundamental groups says that projection of the fundamental group of a variety is the fundamental group of the projection, up to finite index and under some irreducibility assumptions, and is used to prove that the projection of an irreducible closed set is closed in the analytic Zariski structure.In particular, we define an analytic Zariski structure on the universal covering space of an Abelian variety defined over a finite extension of the field Q of rational numbers. (shrink)

We provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques-thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a finite conformal hypergraph. In terms (...) of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of first-order logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF. (shrink)

Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.

While the phrase "metaphysics of science" has been used from time to time, it has only recently begun to denote a specific research area where metaphysics meets philosophy of science—and the sciences themselves. The essays in this volume demonstrate that metaphysics of science is an innovative field of research in its own right. The principle areas covered are: The modal metaphysics of properties: What is the essential nature of natural properties? Are all properties essentially categorical? Are they (...) all essentially dispositions, or are some categorical and others dispositional? Realism in mathematics and its relation to science: What does a naturalistic commitment of scientific realism tell us about our commitments to mathematical entities? Can this question be framed in something other than a Quinean philosophy? Dispositions and their relation to causation: Can we generate an account of causation that takes dispositionality as fundamental? And if we take dispositions as fundamental, what is the ontological ground of dispositions? Pandispositionalism: Could all properties be dispositional in nature? Natural kinds: Are there natural kinds, and if so what account of their nature should we give? For example, do they have essences? Here we consider how these issues may be illuminated by considering examples from reals science, in particular biochemistry and neurobiology. (shrink)

We study the embedding property in the category of sorted profinite groups. We introduce a notion of the sorted embedding property (SEP), analogous to the embedding property for profinite groups. We show that any sorted profinite group has a universal SEP-cover. Our proof gives an alternative proof for the existence of a universal embedding cover of a profinite group. Also our proof works for any full subcategory of the sorted profinite groups, which is closed under taking finite quotients, fibre products, (...) and inverse limits. We also show that any sorted profinite group having SEP has a sorted complete system whose theory is $\omega $ -categorical and $\omega $ -stable under the assumption that the set of sorts is countable. (shrink)

Changes in attitude toward intellectual property are covered here.

This book surveys the leading modern theories of property - Lockean, libertarian, utilitarian/law-and-economics, personhood, Kantian and human flourishing - and then applies those theories to concrete contexts in which property issues have been especially controversial. These include redistribution, the right to exclude, regulatory takings, eminent domain and intellectual property. The book highlights the Aristotelian human flourishing theory of property, providing the most comprehensive and accessible introduction to that theory to date. The book's goal is neither to cover every conceivable theory (...) nor to discuss every possible facet of the theories covered. Instead, it aims to make the major property theories comprehensible to beginners, without sacrificing accuracy or sophistication. The book will be of particular interest to students seeking an accessible introduction to contemporary theories of property, but even specialists will benefit from the book's lucid descriptions of contemporary debates. (shrink)

Djordjević [Dj 1] proved that under natural technical assumptions, if a complete L n -theory is stable and has amalgamation over sets, then it has arbitrarily large finite models. We extend his study and prove the existence of arbitrarily large finite models for classes of models of L n -theories (maybe omitting types) under weaker amalgamation properties. In particular our analysis covers the case of vector spaces.

Using the concept of adjunction, for the comprehension of the structure of a complex system, developed in Part I, we introduce the notion of covering systems consisting of partially or locally defined adequately understood objects. This notion incorporates the necessary and sufficient conditions for a sheaf theoretical representation of the informational content included in the structure of a complex system in terms of localization systems. Furthermore, it accommodates a formulation of an invariance property of information communication concerning the analysis (...) of a complex system. (shrink)

This article focuses on the analysis of the main positions of the Constitutional Court of the Republic of Lithuania in the cases of intellectual property law. In the article three judgments and the positions of the Constitutional Court extracted therefrom are analysed. The Constitutional Court has formed several important positions with reference to intellectual property law regarding usage of property protection norms for the protection of intellectual property, requirements of application of compensation as an alternative to damages compensation and the (...) calculation of criterions thereof as well as functional legal protection of trademarks. It should be noted that Article 23(1) of the Constitution of the Republic of Lithuania has extended protection to property and, as the Constitutional Court of the Republic of Lithuania has noted, the constitutional protection of property covers not only the protection of tangible, but also of intellectual property. In the same judgment, the Constitutional Court stated another important intellectual property related rule, namely that constitutionally established property protection rights have to be implemented regardless of whether a corresponding international treaty has not been ratified, as protection of authors’ rights is granted under Articles 23 and 42(3) of the Constitution of the Republic of Lithuania. However, the second position stating that the rights of foreign subjects are to be implemented even though a corresponding international treaty has not been signed is to be criticised as by coming to such a conclusion the Constitutional Court has ignored the otherwise generally recognised principle of territoriality of intellectual property rights. In two other cases the Constitutional Court has made important statements relevant to intellectual property law. First of all, the Court stated that compensation instead of claiming actual damages is constitutional. By analysing the positions formulated by the Court, an indirect conclusion can be made to the effect that compensation instead of claiming actual damages is necessary in order to ensure effective protection of intellectual property rights. In other words, the purposes of compensation can be recognised: restoring (compensating) the infringed interests of the injured party; simplifying judicial proceedings, whereby faster proceedings (litigation) and easier substantiation is sought; and the preventive purpose (strong preventive effect). However, the Court’s argument regarding the criterion of compensation calculation, saying that the damages incurred by the intellectual property rights owner depend on the sale value of the product concerned is criticisable, as it is rarely the case that the licence fee or profit acquired in the result of unauthorised use of intellectual property objects is calculated with reference to the final sale price of the product. Notwithstanding the arguably weak argument of the Court, the Court’s position regarding the criterion of final sale price itself does not preclude the possibility of reaching a fair final decision. Finally, the Court acknowledged that trademark protection was limited to the economic functions of the trademark and the full protection thereof. (shrink)

Let T be a strongly minimal theory with quantifier elimination. We show that the class of existentially closed models of T{“σ is an automorphism”} is an elementary class if and only if T has the definable multiplicity property, as long as T is a finite cover of a strongly minimal theory which does have the definable multiplicity property. We obtain cleaner results working with several automorphisms, and prove: the class of existentially closed models of T{“σi is an automorphism”: i=1,2} is (...) an elementary class if and only if T has the definable multiplicity property. (shrink)

It is an honor to have been given the opportunity by the editor to reply to J.A. Cover’s review of Leibniz’s Metaphysics, and to have a chance to revisit, five years after the book’s publication, the still-active battleground of intrinsic and extrinsic properties, the extensionality and intensionality of perception, and the reality of aggregates and to say more, a little informally perhaps, about about some methodological questions in Leibniz scholarship. Cover’s review when it appeared gave me a great deal (...) of pleasure because he addressed sharply and directly exactly the central issues, in a way which was not in the least distanziert, and defended several alternative critical positions concisely and eloquently. (shrink)

Measuring and assessing the different aspects of gambling behavior and its consequences is crucial for planning prevention, treatment, and understanding the development of at-risk and problem gambling. Studies indicate that instruments measuring problem gambling produce different results based on the characteristics of the population assessed. To accurately measure at-risk and problem gambling behavior, especially in a low-risk population, measures must cover a wider set of dimensions than the negative consequences already manifest. The Jonsson-Abbott Scale includes items that cover overconsumption, actions (...) that reinforce gambling behavior, and belief in gambling fallacies, based on a three-factor structure and has previously demonstrated good psychometric properties. However, there is a need to investigate how the instrument also functions in low-risk populations. This study aims to do so using both confirmatory factor and Rasch analysis; this research included 1,413 Swedish participants who endorsed at least one JAS item. The results replicated the previous three-factor solution and indicated that the instrument had good reliability. In addition, the results demonstrated that the three factors are independent, and the overall score per factor needs to be analyzed. In summary, the JAS appears suitable for use in low-risk populations to measure various aspects of gambling behavior. (shrink)

It is an honor to have been given the opportunity by the editor to reply to J.A. Cover’s review of Leibniz’s Metaphysics, and to have a chance to revisit, five years after the book’s publication, the still-active battleground of intrinsic and extrinsic properties, the extensionality and intensionality of perception, and the reality of aggregates and to say more, a little informally perhaps, about about some methodological questions in Leibniz scholarship. Cover’s review when it appeared gave me a great deal (...) of pleasure because he addressed sharply and directly exactly the central issues, in a way which was not in the least distanziert, and defended several alternative critical positions concisely and eloquently. (shrink)

The purpose of this paper is to argue that the tactic of granting a fetus the legal status of a person will not, contrary to the expectations of opponents of abortion, provide grounds for a general prohibition on abortions. I begin by examining two arguments, one moral (J. J. Thomson's A Defense of Abortion) and the other legal (D. Regan's Rewriting Roe v. Wade), which grant the assumption that a fetus is a person and yet argue to the conclusion that (...) abortion is permissible. However, both Thomson and Regan rely on the so-called bad samaritan principle. This principle states that a person has a right to refuse to give aid. Their reliance on this principle creates problems, both in the moral and the legal contexts, since the bad samaritan principle is intended to apply to passive refusals to aid; abortion, however, does not look like any such passive denial of aid, and so it does not seem like the sort of action covered by the bad samaritan principle. In defense of the positions outlined by Thomson and Regan, I argue that the apparent asymmetry between abortion and the usual type of case covered by the bad samaritan principle is only apparent and not a genuine problem for their analyses. I conclude with a defense of the morality of the bad samaritan principle. (shrink)

On pp. 55–58 of Philosophical Problems of Statistical Inference, I argue that in light of unsatisfactory after-trial properties of “best” Neyman-Pearson confidence intervals, we can strengthen a traditional criticism of the orthodox N-P theory. The criticism is that, once particular data become available, we see that the pre-trial concern for tests of maximum power may then misrepresent the conclusion of such a test. Specifically, I offer a statistical example where there exists a Uniformly Most Powerful test, a test of (...) highest N-P credentials, which generates a system of “best” confidence intervals with exact confidence coefficients. But the [CIλ] intervals have the unsatisfactory feature that, for a recognizable set of outcomes, the interval estimates cover all parameter values consistent with the data, at strictly less than 100% confidence. (shrink)

As a generalization of Zhan’s method, the present paper aims to define the family of complementary fuzzy β -neighborhoods and thus three kinds of covering-based multigranulation -fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation -fuzzy rough sets are defined and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations (...) and illustrates with a numerical example to describe its performance. (shrink)

My focus in this essay is on those singular causal explanations which purport to explain the occurrence of some particular event by means of a claim of the following general sort The occurrence of event caused the occurrence of event.Examples include sentences like The short circuit caused the fire’ and The impact of the hammer caused the shattering of the glass,’ Many philosophers hold that there is a sharp distinction to be drawn between singular causal explanations and those sentences which (...) simply report causal connections. While singular causal sentences are said to typically relate events, and to be extensional, singular causal explanations are held to relate items which are sentential or fact-like in structure and to be non-extensional.1 Singular causal explanations are held on this view to always be explanations of why events or other objects belong to certain kinds or possess certain properties. (shrink)

We define a class of productive σ-ideals of subsets of the Cantor space 2 ω and observe that both σ-ideals of meagre sets and of null sets are in this class. From every productive σ-ideal I we produce a σ-ideal I κ , of subsets of the generalized Cantor space 2 κ . In particular, starting from meagre sets and null sets in 2 ω we obtain meagre sets and null sets in 2 κ , respectively. Then we investigate additivity, (...) covering number, uniformity and cofinality of I κ . For example, we show that non(I = non(I ω 1 ) = non(I ω 2 ). Our results generalizes those from [5]. (shrink)

We prove that every monadic second-order property of the unfolding of a transition system is a monadic second-order property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.

In cognitive science, the dynamical systems theory (DST) has recently been advocated as an approach to cognitive modeling that is better suited to the dynamics of cognitive processes than the symbolic/computational approaches are. Often, the differences between DST and the symbolic/computational approach are emphasized. However, alternatively their commonalities can be analyzed and a unifying framework can be sought. In this paper, the possibility of such a unifying perspective on dynamics is analyzed. The analysis covers dynamics in cognitive disciplines, as well (...) as in physics, mathematics and computer science. The unifying perspective warrants the development of integrated approaches covering both DST aspects and symbolic/computational aspects. The concept of a state-determined system, which is based on the assumption that properties of a given state fully determine the properties of future states, lies at the heart of DST. Taking this assumption as a premise, the explanatory problem of dynamics is analyzed in more detail. The analysis of four cases within different disciplines (cognitive science, physics, mathematics, computer science) shows how in history this perspective led to numerous often used concepts within them. In cognitive science, the concepts desire and intention were introduced, and in classical mechanics the concepts momentum, energy and force. Similarly, in mathematics a number of concepts have been developed to formalize the state-determined system assumption [e.g. derivatives (of different orders) of a function, Taylor approximations]. Furthermore, transition systems - a currently popular format for specification of dynamical systems within computer science - can also be interpreted from this perspective. One of the main contributions of the paper is that the case studies provide a unified view on the explanation of dynamics across the chosen disciplines. All approaches to dynamics analyzed in this paper share the state-determined system assumption and the (explicit or implicit) use of anticipatory state properties. Within cognitive science, realism is one of the problems identified for the symbolic/computational approach - i.e. how do internal states described by symbols relate to the real world in a natural manner. As DST is proposed as an alternative to the symbolic/computational approach, a natural question is whether, for DST, realism of the states can be better guaranteed. As a second main contribution, the paper provides an evaluation of DST compared to the symbolic/computational approach, which shows that, in this respect (i.e. for the realism problem), DST does not provide a better solution than the other approaches. This shows that DST and the symbolic/computational approach not only have the state-determined system assumption and the use of anticipatory state properties in common, but also the realism problem. (shrink)

In this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω 1 . Define $\operatorname{cof}(H)$ (...) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that $\operatorname{card}(\lfloor\log_2(H)\rfloor) \leq \operatorname{cof}(H) \leq \operatorname{card}(2^H)$ , where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes $M \preccurlyeq N$ and hyperfinite integer H ∈ M such that H is not enlarged by N, 2 H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe. (shrink)