It is shown that $\operatorname{Con}(ZF)$ implies $\operatorname{Con}(ZFC$ + there exists a Dowker filter on ω2). The problem is also considered on higher cardinals.

Wales uses languages with both regular (Welsh) and irregular (English) counting systems. Three groups of 6- and 8-year-old Welsh children with varying degrees of exposure to the Welsh language—those who spoke Welsh at both home and school; those who spoke Welsh only at home; and those who spoke only English—were given standardized tests of arithmetic and a test of understanding representations of two-digit numbers. Groups did not differ on the arithmetic tests, but both groups of Welsh speakers read and compared (...) 2-digit numbers more accurately than monolingual English children. A similar study was carried out with Tamil/English bilingual children in England. The Tamil counting system is more transparent than English but less so than Welsh or Chinese. Tamil-speaking children performed better than monolingual English-speaking children on one of the standardized arithmetic tests but did not differ in their comparison of two-digit numbers. Reasons for the findings are discussed. (shrink)

This commentary focuses on one of the many issues raised in Geary's target article: the importance of gender differences in spatial ability to gender differences in mathematics. I argue that the evidence for the central role of spatial ability in mathematical ability, or in gender differences in it, is tenuous at best.

Many studies have indicated that mathematics anxiety, and other negative attitudes and emotions toward mathematics, are pervasive and are associated with lower mathematical performance. Some previous research has suggested that working memory is related to both mathematics anxiety and mathematics. Moreover, both gender and chosen course of study appeared likely to influence students’ attitudes to mathematics. In the present study, 40 university undergraduates completed a battery of assessments investigating working memory, attitude to mathematics, test anxiety. and mental and written arithmetic. (...) Attitudes to mathematics were significantly associated with the other variables: working memory, test anxiety, and both measures of mathematical performance. The other variables were not strongly associated with one another. There were no gender differences in mathematical performance, but females exhibited more negative attitudes to mathematics and higher test anxiety than males. After controlling for test anxiety, there ceased to be significant gender differences in attitudes to mathematics. Science students had more positive attitudes to mathematics than humanities students, but the groups did not differ in test anxiety, Science students were better at written but not mental arithmetic. They were also better at working memory, but this was not a significant covariate when the groups were compared on mathematical performance and attitudes to mathematics The results are discussed, with particular focus on implications for future research on influences on mathematics anxiety. (shrink)

Numbers are vital to so many areas of life: in science, economics, sports, education, and many aspects of everyday life from infancy onwards. This handbook brings together the different research areas that make up the vibrant field of numerical cognition in one comprehensive and authoritative volume.

his is a book to broaden the mind. It brings the reader into a world only vaguely imaginable and richly enlightens it with extraordinary attention to interesting historical details. It is beautifully written and endlessly interesting.

The Filter Dichotomy says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A medial limit is a universally measurable function from [Formula: see text] to the unit interval [0, 1] which is finitely additive for disjoint sets, and maps singletons to 0 and ω to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of (...) medial limits. We show that the Filter Dichotomy implies that there are no medial limits. (shrink)

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

With the use of a suitable assumption about the structure of the class of experimental filters, it is shown that the sequence of alternating replicas of two filters is their greatest lower bound, as Jauch suggests. A generalization of his suggestion yields the greatest lower bound of a denumerable set of filters. The criteria of admissibility of filters are briefly discussed.

A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ (...) and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics. (shrink)

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

The present paper is thought as a formal study of distributive closure systems which arise in the domain of sentential logics. Special stress is laid on the notion of a C-filter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of C-filters in every algebra similar to the language of C is distributive. Theorem IV.2 in Section IV gives a method of axiomatization of those (...) filter distributive logics for which the class Matr (C) prime of C-prime matrices (models) is axiomatizable. In Section V, the attention is focused on axiomatic strengthenings of filter distributive logics. The theorems placed there may be regarded, to some extent, as the matrix counterparts of Baker's well-known theorem from universal algebra [9, § 62, Theorem 2]. (shrink)

We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^0_1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse (...) mathematics result that the principle "every countable poset has a maximal filter" is equivalent to ACA₀ over RCA₀. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a \Sigma^0_1 unbounded filter, and there is a computable poset with no \Pi^0_1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA₀ over RCA₀. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets. (shrink)

We propose a model of uncertain belief. This models coherent beliefs by a filter, ????, on the set of probabilities. That is, it is given by a collection of sets of probabilities which are closed under supersets and finite intersections. This can naturally capture your probabilistic judgements. When you think that it is more likely to be sunny than rainy, we have{????|????(????????????????????)>????(????????????????????)}∈????. When you think that a gamble ???? is desirable, we have {????|Exp????[????]>0}∈????. It naturally extends the model of (...) credal sets; and we will show it captures all the expressive power of the desirable gambles model. It also captures the expressive power of sets of desirable gamble sets (with a mixing axiom, but no Archimadean axiom). (shrink)

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.

In this paper, we study (open) filters and deductive systems of self-distributive weak Heyting algebras (SDWH-algebras) and obtain some results which determine the relationship between them. We show that the variety of SDWH-algebras is not weakly regular and every open filter is the kernel of at least one congruence relation. Then it presents necessary and sufficient conditions for the existence of a one to one correspondence between open filters and congruence relations.

It is often suggested that we are equipped with a set of cognitive tools that help us to filter out unreliable testimony. But are these tools effective? I answer this question in two steps. Firstly, I argue that they are not real-time effective. The process of filtering, which takes place simultaneously with or right after language comprehension, does not prevent a particular hearer on a particular occasion from forming beliefs based on false testimony. Secondly, I argue that they are (...) long-term effective. Some hearers sometimes detect false testimony, which increases speakers’ incentives for honesty and stabilizes the practice of human communication in which deception is risky and costly. In short, filtering prevents us from forming a large number of beliefs based on false testimony, not by turning each of us into a high-functioning polygraph but by turning the social environment of human communication into one in which such polygraphs are not required. Finally, I argue that these considerations support strong anti-reductionism about testimonial entitlement. (shrink)

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it holds on a (...) stationary subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $$\kappa $$, where $$\kappa $$ is a regular cardinal. The corresponding new notion is called $$\kappa $$ -filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different $$\kappa $$ -filter pairs give rise to a fixed logic of cardinality $$\kappa (...) $$. To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality $$\kappa $$. Along the way we use $$\kappa $$ -filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair. (shrink)

Assuming the consistency of the theory “ZFC + there exists a measurable cardinal”, we construct 1. a model in which the first cardinal κ, such that 2κ > κ+, bears a normal filter F whose associated boolean algebra is κ+-distributive ,2. a model where there is a measurable cardinal κ such that, for every regular cardinal ρ < κ, 2ρ = ρ++ holds,3. a model of “ZFC + GCH” where there exists a non-measurable cardinal κ bearing a normal (...) class='Hi'>filter F whose associated boolean algebra is κ+-distributive. (shrink)

Advances in artificial intelligence, as well as its increased presence in everyday life, have brought the emergence of many new phenomena, including an intriguing appearance of what seems to be a variant of body dysmorphic disorder, coined “Snapchat dysmorphia”. Body dysmorphic disorder is a DSM-5 psychiatric disorder defined as a preoccupation with one or more perceived defects or flaws in physical appearance that are not observable or appear slight to others. Snapchat dysmorphia is fueled by automated selfie filters that reflect (...) unrealistic sociocultural standard. In this paper, we discuss how body dysmorphic disorder and related body image distortions could arise, using the conceptual resources provided by the active inference framework. We suggest that these disorders involve dysfunctional self-modelling which entails maladaptive internalization of sociocultural preferences during adolescent identity formation. Identity formation is hereby described as cycles of interpersonal active inference that arbitrate between identity exploration and commitment. We propose that impaired self-modelling is unable to reduce interpersonal uncertainty during identity exploration, which, over time, degenerates into uncontrollable epistemic habits that isolate the body image from corrective sensory evidence. In light of these insights, we subsequently explore some of the consequences of image-centered social media platforms on the identity formation process. We conclude that heightened interpersonal uncertainty in this novel context could precipitate the onset of body dysmorphic disorder and related body image distortions, particularly when selfie filters are involved. (shrink)

We find some characterizations of the Axiom of Choice in terms of certain families of open sets in T1 spaces.

Decision-making has a high practical relevance for daily performance. Its relation to other cognitive abilities such as executive control and memory is not fully understood. Here we asked whether training of either attentional filtering or memory storage would influence decision-making as indexed by repetitive assessments of the Iowa Gambling Task (IGT). The IGT was developed to assess and simulate real-life decision-making (Bechara et al., 2005). In this task, participants gain or lose money by developing advantageous or disadvantageous decision strategies. On (...) five consecutive days we trained 29 healthy young adults (20-30 years) either in working memory storage or attentional filtering and measured their IGT scores after each training session. During memory training (MT) subjects performed a computerized delayed match-to-sample task where two displays of bars were presented in succession. During filter training (FT) participants had to indicate whether two simultaneously presented displays of bars matched or not. Whereas in MT the relevant target stimuli stood alone, in FT the targets were embedded within irrelevant distractors (bars in a different color). All subjects within each group improved their performance in the trained cognitive task. For the IGT, we observed an increase over time in the amount of money gained in the FT group only. Decision-making seems to be influenced more by training to filter out irrelevant distractors than by training to store items in working memory. Selective attention could be responsible for the previously noted relationship between IGT performance and working memory and is therefore more important for enhancing efficiency in decision-making. (shrink)

The notions of soju filters in a hoop algebra are introduced, and related properties are investigated. Relations between a soju sub-hoop, a soju filter and an implicative soju filter are discussed. Conditions for a soju filter to be implicative are displayed, and characterizations of an implicative soju filters are considered. The extension property of an implicative soju filter is established.

Leibniz filters play a prominent role in the theory of protoalgebraic logics. In [3] the problem of the definability of Leibniz filters is considered. Here we study the definability of Leibniz filters with parameters. The main result of the paper says that a protoalgebraic logic S has its strong version weakly algebraizable iff it has its Leibniz filters explicitly definable with parameters.

The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not (...) refrain from a philosophical interpretation of the results from the paper. (shrink)

In an earlier paper [Rational choice and AGM belief revision, _Artificial Intelligence_, 2009] a correspondence was established between the set-theoretic structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of those structures in terms of one-shot belief revision by relating them to the trichotomous attitude towards information studied in Garapa (Rev Symb Logic, 1–21, 2020) where information may be either (...) (1) fully accepted or (2) rejected or (3) taken seriously but not fully accepted. We begin by introducing the syntactic notion of _filtered belief revision_ and providing a characterization of it in terms of a mixture of both AGM revision and contraction. We then establish a correspondence between the proposed notion of filtered belief revision and the above-mentioned set-theoretic structures, interpreted as semantic _partial_ belief revision structures. We also provide an interpretation of the trichotomous attitude towards information in terms of the degree of implausibility of the information. (shrink)

Schemas contribute to adaptation, filtering novelty though knowledge-expectancy structures, the residue of past contingencies and their consequences. Adaptation requires a balance between flexible, dynamic context-sensitivity and the cognitive efficiency that schemas afford in promoting persistent goal pursuit despite distraction. Affects can form and disrupt schemas. Transient affective experiences systematically alter selectivity of attentiveness to the directly experienced present environment, the internal environment, and to the stored experiences of memory. Enduring personal stylistic predispositions, like implicit motives and affective schemas, influence how (...) experience is perceived, responded to, and integrated; they shape memory and influence present experiential patterns, individually and intersubjectively. Such systematic influences are potential sources of error in the study of memory if not mapped; so far, individual personality differences have just been a source of complication in the literature on emotion-congruent perception and memory. I synthesize what findings there are about how personality differences, emotions, and affects contribute to the structuring and integration of perceptions and memories both directly and by way of hot, affectively-anchored schemas. Case studies from experimental and personality psychology highlight a conception of personality and affective experience relevant to memory research and cognitive science. (shrink)

This article develops a theoretical framework for the analysis of corporate political involvement, as mediated not only by environmental factors but also by characteristics of the firm itself. Consistent with previous research, the authors examine how policy salience and the political, macroeconomic, and industry environments influence the level of a firm's involvement in politics. In addition, the authors' model takes into account such influences as corporate structure, resources, political experience, and stakeholder dependence.

An adequate moral theory must – or so many philosophers have argued – be compatible with the attitudes and practical requirements of deep friendship. Bernard Williams suggested that the decision procedure required by both deontology and consequentialism inserts a fetishising filter between the natural moral motivation of any normal person to prioritise friends and the decision to act on it. But this injects ‘one thought too many’ into the moral reaction mechanism. It is standardly assumed that virtue ethics is (...) somehow immune to this objection. The present article explores this assumption and finds it wanting in various respects. Virtue ethics filters friendship throughphronesisand thus inserts an extra thought into the mechanism in question. To escape Williams's curse, the only way is to argue that the extra thought required by virtue ethics is not ‘one thought too many’. The article closes with an attempt to show that, contra deontology, the friendship motivation in virtue ethics is derived from the moral virtue, not the intellectual filter, and, contra consequentialism,phronesisdoes not require the maximisation of value. The presumed advantage of virtue ethics must lie in the content of its filter rather than the filter's non-existence. (shrink)

This article explores what John Dewey’s political philosophy can offer in regard to the current crisis in digital democracy. It focuses on two digital mechanisms, the “filter bubble” and the “echo chamber”. While there is a prominent, Dewey-inspired debate on “digital publics” in the literature, a reconstruction of the Deweyan concepts of the public and of shared experience shows that it does not adequately reflect the aspect of situated and embodied experience. Based on this, it is shown that digital (...) media offerings must also be rooted in local contexts of experience in order to answer the challenge of those two problematic mechanisms. (shrink)

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider (...) examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations. (shrink)

Online information intermediaries such as Facebook and Google are slowly replacing traditional media channels thereby partly becoming the gatekeepers of our society. To deal with the growing amount of information on the social web and the burden it brings on the average user, these gatekeepers recently started to introduce personalization features, algorithms that filter information per individual. In this paper we show that these online services that filter information are not merely algorithms. Humans not only affect the design (...) of the algorithms, but they also can manually influence the filtering process even when the algorithm is operational. We further analyze filtering processes in detail, show how personalization connects to other filtering techniques, and show that both human and technical biases are present in today’s emergent gatekeepers. We use the existing literature on gatekeeping and search engine bias and provide a model of algorithmic gatekeeping. (shrink)

In this paper, the notion of the radical of a filter in BL-algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV-algebra if and only if Ds ⊆ F. After that we define the notion of semi maximal filter in BL-algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL-algebra. Moreover, we (...) prove that A/F is a semi simple BL-algebra if and only if F is a semi maximal filter of A. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

To the French critics who originally labelled certain films noir it seemed that a class of Hollywood products had gone darker during the war years – as though a dark filter had been placed over the lens. Films were not designed or marketed as noir, and retrospectively noir's status as a genre is still unsettled. Yet there is widespread interest today in experiencing diverse films as noir, and even in using a Noir Filter in Instagram and video games. (...) Pursuing the filter clue, the noir experience can be thought of as subjection to a dark filtering of narrative. Image filtering styles can help to resolve not only the puzzle of noir's quasi-genre status but also an issue of general interest in aesthetic experience. The use of image filters makes a distinctively powerful contribution to the experience of a work or genre, or to the cultural dominance of an aesthetic regime, due to the unobtrusive formative role it plays in the economy of experience. (shrink)

This research work proposes an implementation of adept content-based music retrieval technique that attempts to address the demands of the rising availability of digital music. The primary objective of this research work is to balance the perilous impact of non-relevant songs through progressive filtering for query by humming music information retrieval system. The PF is a technique of searching in manifolds for problem solving through reduced search space. A new strategy for empirical mode decomposition analysis is adopted, and outcomes are (...) propelled as a significant source of information for multiresolution histogram representations. Finally, the concept of PF is realized through MRH to retrieve the desired song by humming. The proposed multiresolution QBH system is evaluated on a music database of 1200 fragments, and the experimental results effectively locate the humming patterns and recite the eminence of EMD analysis for retrieving the desired song through humming. (shrink)