We have a variety of different ways of dividing up, classifying, mapping, sorting and listing the objects in reality. The theory of granular partitions presented here seeks to provide a general and unified basis for understanding such phenomena in formal terms that is more realistic than existing alternatives. Our theory has two orthogonal parts: the first is a theory of classification; it provides an account of partitions as cells and subcells; the second is a theory of reference (...) or intentionality; it provides an account of how cells and subcells relate to objects in reality. We define a notion of well-formedness for partitions, and we give an account of what it means for a partition to project onto objects in reality. We continue by classifying partitions along three axes: (a) in terms of the degree of correspondence between partition cells and objects in reality; (b) in terms of the degree to which a partition represents the mereological structure of the domain it is projected onto; and (c) in terms of the degree of completeness with which a partition represents this domain. (shrink)

There are some who defend a view of vagueness according to which there are intrinsically vague objects or attributes in reality. Here, in contrast, we defend a view of vagueness as a semantic property of names and predicates. All entities are crisp, on this view, but there are, for each vague name, multiple portions of reality that are equally good candidates for being its referent, and, for each vague predicate, multiple classes of objects that are equally good candidates for being (...) its extension. We provide a new formulation of these ideas in terms of a theory of granular partitions. We show that this theory provides a general framework within which we can understand the relation between vague terms and concepts on the one hand and correlated portions of reality on the other. We also sketch how it might be possible to formulate within this framework a theory of vagueness which dispenses with the notion of truth-value gaps and other artifacts of more familiar approaches. (shrink)

In this paper we propose a formal theory of partitions (ways of dividing up or sorting or mapping reality) and we show how the theory can be applied in the geospatial domain. We characterize partitions at two levels: as systems of cells (theory A), and in terms of their projective relation to reality (theory B). We lay down conditions of well-formedness for partitions and we define what it means for partitions to project truly onto reality. We (...) continue by classifying well-formed partitions along three axes: (a) degree of correspondence between partition cells and objects in reality; (b) degree to which a partition represents the mereological structure of the domain it is projected onto; and (c) degree of completeness and exhaustiveness with which a partition represents reality. This classification is used to characterize three types of partitions that play an important role in spatial information science: cadastral partitions, categorical coverages, and the partitions involved in folk categorizations of the geospatial domain. (shrink)

The theory of granular partitions is designed to capture in a formal framework important aspects of the selective character of common-sense views of reality. It comprehends not merely the ways in which we can view reality by conceiving its objects as gathered together not merely into sets, but also into wholes of various kinds, partitioned into parts at various levels of granularity. We here represent granular partitions as triples consisting of a rooted tree structure as first (...) component, a domain satisfying the axioms of Extensional Mereology as second component, and a mapping (called ’projection’) of the first into the second as a third component. We define ordering relations among granular partitions the resulting structures are called partition frames. We then introduce an axiomatic theory which sentences are interpreted in partition frames. (shrink)

The theory of granular partitions (TGP) is a new approach to the understanding of ontologies and other classificatory systems. The paper explores the use of this new theory in the treatment of task-based clinical guidelines as a means for better understanding the relations between different clinical tasks, both within the framework of a single guideline and between related guidelines. We used as our starting point a DAML+OIL-based ontology for the WHO guideline for hypertension management, comparing this with related (...) guidelines and attempting to show that TGP provides a flexible and highly expressive basis for the manipulation of ontologies of a sort which might be useful in providing more adequate Computer Interpretable Guideline Models (CIGMs) in the future. (shrink)

Abstract: We propose a view of vagueness as a semantic property of names and predicates. All entities are crisp, on this semantic view, but there are, for each vague name, multiple portions of reality that are equally good candidates for being its referent, and, for each vague predicate, multiple classes of objects that are equally good candidates for being its extension. We provide a new formulation of these ideas in terms of a theory of granular partitions. We show (...) that this theory provides a general framework within which we can understand the relation between vague terms and concepts and the corresponding crisp portions of reality. We also sketch how it might be possible to formulate within this framework a theory of vagueness which dispenses with the notion of truth-value gaps and other artifacts of more familiar approaches. Central to our approach is the idea that judgments about reality involve in every case (1) a separation of reality into foreground and background of attention and (2) the feature of granularity. On this basis we attempt to show that even vague judgments made in naturally occurring contexts are not marked by truth-value indeterminacy. We distinguish, in addition to crisp granular partitions, also vague partitions, and reference partitions, and we explain the role of the latter in the context of judgments that involve vagueness. We conclude by showing how reference partitions provide an effective means by which judging subjects are able to temper the vagueness of their judgments by means of approximations. (shrink)

In this paper I consider one kind of vague linguistic expression: adjec- tives like tall, big, expensive. These are called gradable adjectives. The most well-known linguistic theories that account for them are the so-called degree-based theories. In this paper I present a formal model that accounts for vague gradable adjectives as an alternative to degree-based theories. The model is built on two basic ingredients: (i) comparison classes and (ii) gran- ular partitions. (i) Comparison classes are introduced to account for (...) the context-sensitivity of vague adjectives. The extension of the predicate being tall in the comparison class of men is different from its extension in the com- parison class of children. (ii) We can look at the elements of a context under different standards of precision, each of them corresponding to a granular level of observation. The finer the level is, the more differences between the individuals are detected. Granular partitions are used to represent in- distinguishability relations between objects with respect to the properties expressed by vague adjectives. (shrink)

Published in extended form as "Endurants and Perdurants in Directly Depicting Ontologies", -/- We propose an ontological theory that is powerful enough to describe both complex spatio-temporal processes and the enduring entities that participate in such processes. For this purpose we distinguish between ontologies and metaontology. Ontologies are based on very simple directly depicting languages and fall into two major categories: ontologies of type SPAN and ontologies of type SNAP. These represent two complementary perspectives on reality and result in distinct (...) though compatible systems of categories. In a SNAP (snapshot) ontology we have the enduring entities in a given domain as they exist to be inventoried at some given moment of time. In a SPAN ontology we have perduring entities such as processes and their parts and aggregates. We argue that both kinds of ontology are required, together with the meta-ontology which joins them together. On the level of meta-ontology we are able to impose constraints on ontologies of a sort which can support efficient processing of large amounts of data. (shrink)

This dissertation aims to determine the optimal level of granularity for the variables used in probabilistic causal models. These causal models are useful for generating explanations in a number of scientific contexts. In Chapter 1, I argue that there is rarely a unique level of granularity at which a given phenomenon can be causally explained, thereby rejecting various causal exclusion arguments. In Chapter 2, I consider several recent proposals for measuring the explanatory power of causal explanations, and show that these (...) measures fail to track the comparative depth of explanations given at different levels of granularity. In Chapter 3, I offer a pragmatic account of how to partition the measure space of a causal variable so as to optimally explain its effect. My account uses the decision-theoretic notion of value of information, and indexes the relative depth of an explanation to a particular agent faced with a particular decision problem. Chapter 4 applies this same decisiontheoretic framework to answer the epistemic question of how to discover constitutive relationships in nature. In Chapter 5, I describe the formal details of the relationship between random variables that are meant to be coarse-grained and fine-grained representations of the same type of phenomenon. I use this formal framework to rebut a popular argument for the view that special science probabilities can be objective chances. Chapter 6 discusses challenges related to the causal interpretation of Bayes nets that use imprecise rather than precise probabilities. (shrink)

We provide a methodology for the creation of ontological partitions in biomedicine and we test the methodology via an application to the phenomenon of blood pressure. An ontology of blood pressure must do justice to the complex networks of intersecting pathways in the organism by which blood pressure is regulated. To this end it must deal not only with the anatomical structures and physiological processes involved in such regulation but also with the relations between these at different levels of (...) granularity. For this purpose our ontology offers a variety of distinct partitions � of substances, processes and functions � and integrates these together within a single framework via transitive networks of part-whole and dependence relations among the entities in each of these categories. The paper concludes with a comparison of this methodology with the approaches of GOTM, KEGG, DIP and BIND and provides an outline of how the methodology is currently being applied in the field of biomedical database integration. (shrink)

We propose a new account of vagueness and approximation in terms of the theory of granular partitions. We distinguish different kinds of crisp and non-crisp granular partitions and we describe the relations between them, concentrating especially on spatial examples. We describe the practice whereby subjects use regular grid-like reference partitions as a means for tempering the vagueness of their judgments, and we demonstrate how the theory of reference partitions can yield a natural account of (...) this practice, which is referred to in the literature as ‘approximation’. (shrink)

If Mont Blanc is a vague object, then its vagueness will depend on the context in which reference is made. In a geological context the mountain might include only rock, perhaps together with a certain amount of air in the crevices and tunnels which have been formed beneath its surface. In a context of soil chemistry we might include also a surrounding thin layer of organic matter. In a skiing context we might include some snow. This essay sketches in informal (...) terms the theory of granular partitions, which is designed to do justice to this context-dependence of vagueness by means of what might be described as a contextualized supervaluationism. Granularity and vagueness, it is argued, are two sides of a single coin: what is vague at one level of granularity may appear crisp at another. The resultant theory can be shown to resolve certain problems in our description of perceptual content, for example when John says that he sees the wall, but that he does not see the molecules by which the wall is constituted. (shrink)

The truthmaker theory rests on the thesis that the link between a true judgment and that in the world to which it corresponds is not a one-to-one but rather a one-to-many relation. An analogous thesis in relation to the link between a singular term and that in the world to which it refers is already widely accepted. This is the thesis to the effect that singular reference is marked by vagueness of a sort that is best understood in supervaluationist terms. (...) In what follows we show that the supervaluationist approach to singular reference, when wedded to the truthmaker idea, yields a framework of surprising power, which offers a uniform set of solutions to a range of problems regarding identity, reference and knowledge, problems which have hitherto been dealt with on an ad hoc basis. (shrink)

The truthmaker theory rests on the thesis that the link between a true judgment and that in the world to which it corresponds is not a one-to-one but rather a one-to-many relation. An analogous thesis in relation to the link between a singular term and that in the world to which it refers is already widely accepted. This is the thesis to the effect that singular reference is marked by vagueness of a sort that is best understood in supervaluationist terms. (...) In what follows we show that the supervaluationist approach to singular reference, when wedded to the truthmaker idea, yields a framework of surprising power, which offers a uniform set of solutions to a range of problems regarding identity, reference and knowledge, problems which have hitherto been dealt with on an ad hoc basis. (shrink)

Mereotopology faces problems when its methods are extended to deal with time and change. We offer a new solution to these problems, based on a theory of partitions of reality which allows us to simulate (and also to generalize) aspects of set theory within a mereotopological framework. This theory is extended to a theory of coarse- and fine-grained histories (or finite sequences of partitions evolving over time), drawing on machinery developed within the framework of the so-called ‘consistent histories’ (...) interpretation of quantum mechanics. (shrink)

The Unified Medical Language System and the Gene Ontology are among the most widely used terminology resources in the biomedical domain. However, when we evaluate them in the light of simple principles for wellconstructed ontologies we find a number of characteristic inadequacies. Employing the theory of granular partitions, a new approach to the understanding of ontologies and of the relationships ontologies bear to instances in reality, we provide an application of this theory in relation to an example drawn (...) from the context of the pathophysiology of hypertension. This exercise is designed to demonstrate how, by taking ontological principles into account we can create more realistic biomedical ontologies which will also bring advantages in terms of efficiency and robustness of associated software applications. (shrink)

The wanton dissemination of network pseudohealth information has brought great harm to people’s health, life, and property. It is important to detect and identify network pseudohealth information. Based on this, this paper defines the concepts of pseudohealth information, data block, and data block integration, designs an architecture that combines the latent Dirichlet allocation algorithm and data block update integration, and proposes the combination algorithm model. In addition, crawler technology is used to crawl the pseudohealth information transmitted on the Sina Weibo (...) platform during the “epidemic situation” from February to March 2020 for the simulation test on the experimental case dataset. The research results show that the LDA model can deeply mine the semantic information of network pseudohealth information, obtain the features of document-topic distribution, and classify and train topic features as input variables; the dataset partitioning method can effectively block data according to the text attributes and class labels of network pseudohealth information and can accurately classify and integrate the block data through the data block reintegration method; and considering that the combination model has certain limitations on the detection of network pseudohealth information, the support vector machine model can extract the granularity content of data blocks in pseudohealth information in real time, thus greatly improving the recognition performance of the combination model. (shrink)

The theory of rough sets starts with the notion of an approximation space , which is a pair ( U , R ), U being the domain of discourse, and R an equivalence relation on U . R is taken to represent the knowledge base of an agent, and the induced partition reflects a granularity of U that is the result of a lack of complete information about the objects in U . The focus then is on approximations of concepts (...) on the domain, in the context of the granularity. The present article studies the theory in the situation where information is obtained from different sources. The notion of approximation space is extended to define a multiple-source approximation system with distributed knowledge base , which is a tuple , where N is a set of sources and P ranges over all finite subsets of N . Each R P is an equivalence relation on U satisfying some additional conditions, representing the knowledge base of the group P of sources. Thus each finite group of sources and hence individual source perceives the same domain differently (depending on what information the group/individual source has about the domain), and the same concept may then have approximations that differ with the groups. In order to express the notions and properties related with rough set theory in this multiple-source situation, a quantified modal logic LMSAS D is proposed. In LMSAS D , quantification ranges over modalities, making it different from modal predicate logic and modal logic with propositional quantifiers. Some fragments of LMSAS D are discussed and it is shown that the modal system KTB is embedded in LMSAS D . The epistemic logic is also embedded in LMSAS D , and cannot replace the latter to serve our purpose. The relationship of LMSAS D with first and second-order logics is presented. Issues of expressibility, axiomatization and decidability are addressed. (shrink)

It has been generally assumed that certain categories of numerical expressions, such as ‘more than n’, ‘at least n’, and ‘fewer than n’, systematically fail to give rise to scalar implicatures in unembedded declarative contexts. Various proposals have been developed to explain this perceived absence. In this paper, we consider the relevance of scale granularity to scalar implicature, and make two novel predictions: first, that scalar implicatures are in fact available from these numerical expressions at the appropriate granularity level, and (...) second, that these implicatures are attenuated if the numeral has been previously mentioned or is otherwise salient in the context. We present novel experimental data in support of both of these predictions, and discuss the implications of this for recent accounts of numerical quantifier usage. (shrink)

Possible-worlds accounts of mental or linguistic content are often criticized for being too coarse-grained. To make room for more fine-grained distinctions among contents, several authors have recently proposed extending the space of possible worlds by "impossible worlds". We argue that this strategy comes with serious costs: we would effectively have to abandon most of the features that make the possible-worlds framework attractive. More generally, we argue that while there are intuitive and theoretical considerations against overly coarse-grained notions of content, the (...) same kinds of considerations also prohibit an overly fine-grained individuation of content. An adequate notion of content, it seems, should have intermediate granularity. However, it is hard to construe a notion of content that meets these demands. Any notion of content, we suggest, must be either implausibly coarse-grained or implausibly fine-grained (or both). (shrink)

Rough sets are used in numerous knowledge representation contexts and are then empowered with varied ontologies. These may be intrinsically associated with ideas of rationality under certain conditions. In recent papers, specific granular generalisations of graded and variable precision rough sets are investigated by the present author from the perspective of rationality of approximations (and the associated semantics of rationality in approximate reasoning). The studies are extended to ideal-based approximations (sometimes referred to as subsethood-based approximations). It is additionally shown (...) that co-granular or point-wise approximations defined by σ-ideals/filters (for an arbitrary relation σ) fit easily into the entire scheme. Concepts of the rationality of objects (vague or crisp) and their types are introduced and are shown to be applicable to most general rough sets by the present author. Surprising results on these are proved on these by her in this part of the research paper. The present paper is the first of a three part study on the topic. (shrink)

Grounding is a hyperintensional notion: necessarily equivalent sentences need not be equivalent from a ground-theoretic perspective. How fine-grained, exactly, is grounding? There is a striking lack of consensus on this question. In this chapter, I try to systematize and review the main options that have been put forward in the literature. For reasons that have to do with both naturalness and convenience, I for the most part take the question to be about what is sometimes called, following Kit Fine’s (2012a) (...) terminology, strict full grounding, and I take for granted a conception of grounding as a relation that is many-to-one and non-factive. I discuss the consequences of making alternative assumptions only in the very last section. (shrink)

The ability to represent emotional experiences in a precise fashion with language, termed emotional granularity, is related to a number of beneficial outcomes. However, the emotion granularity construct and operationalization are rooted in the lens of so-called Western, Educated, Industrialized, Rich and Democratic (WEIRD) societies that focus on emotions as a mental-state phenomena. Using evidence from multiple non-WEIRD societies, we illustrate that people's everyday vernacular often emphasizes bodily over, or in addition to, mental states. This suggests that granularity focused only (...) on mental state language may limit our ability to operationalize this construct in a culturally valid manner. To overcome this limitation, we propose capturing “experiential” granularity and a concept model on how it may link to regulation and health. (shrink)

Mathematical proofs generally allow for various levels of detail and conciseness, such that they can be adapted for a particular audience or purpose. Using automated reasoning approaches for teaching proof construction in mathematics presupposes that the step size of proofs in such a system is appropriate within the teaching context. This work proposes a framework that supports the granularity analysis of mathematical proofs, to be used in the automated assessment of students' proof attempts and for the presentation of hints and (...) solutions at a suitable pace. Models for granularity are represented by classifiers, which can be generated by hand or inferred from a corpus of sample judgments via machine-learning techniques. This latter procedure is studied by modeling granularity judgments from four experts. The results provide support for the granularity of assertion-level proofs but also illustrate a degree of subjectivity in assessing step size. (shrink)

Witnessing in merging biological, social and algorithmic realities is crucial to trust, as modelled in the YUTPA framework. Being witness and bearing witness is fundamental to human interaction. System participation in human communities of practice challenges the notion of witnessing and therefore the ability to build trust. Nevertheless, through trial and error, people in a variety of practices have found ways to establish the presence and develop trust in merging realities. This paper presents the results of 20 in-depth interviews with (...) professionals from a variety of disciplines and nations. The conclusion of cumulative analysis is that systems do not witness themselves, but their output deeply affects the mental maps that human beings make of each other, the world around them and their own self. Essential qualities human beings seek when being involved with other beings are defined by granularity and reciprocity in the design of time (duration of engagement, synchronizing performance, integrating rhythms and moments to signify), place (body sense, material interaction, emotional space and situated agency), relation (shared meaning, engagement, reputation and use) and action (tuning, reciprocity, negotiation and quality of deeds). By designing granular interaction in 4 dimensions, reciprocity in witnessing obtains significance and the basis for establishing trust in a variety of presences emerges while human agency acquires potential. (shrink)

We propose an ontological theory that is powerful enough to describe both complex spatio-temporal processes (occurrents) and the enduring entities (continuants) that participate therein. The theory is divided into two major categories of sub-theories: (sub-) theories of type SPAN and (sub-)theories of type SNAP. These theories represent two complementary perspectives on reality and result in distinct though compatible systems of categories. In SNAP we have enduring entities such as substances, qualities, roles, functions; in SPAN we have perduring entities such as (...) processes and their parts and aggregates. We argue that both kinds of ontological theory are required in order to give a non-reductionism account of complex domains of reality. (shrink)

This book presents an original exploration of philosophical questions pertaining to the ways we grasp the Absolute by bringing together the Buddhist notion of interpermeation of all phenomena into contemporary strains of thought in continental philosophy. This text introduces an ontological concept, granularity, deploying it to probe questions concerning the intersection of ontology, ethics, and education. A wide range of issues in metaphysics are covered—including being, nothingness, unity, plurality, truth, change, transformation, subjectivity, contradiction, coherence, potentiality—from the perspective of thinkers such (...) as Hegel, Heidegger, Badiou, Meillassoux, Malabou, Žižek, and Harman. The text deploys granularity in arguing for an ethics of unconditional hospitality within education. This volume is intended for students and researchers working in the areas of philosophy of education, philosophy of religion, and continental philosophy. (shrink)

Lewis sender‐receiver games illustrate how a meaningful term language might evolve from initially meaningless random signals (Lewis 1969; Skyrms 2006). Here we consider how a meaningful language with a primitive grammar might evolve in a somewhat more subtle sort of game. The evolution of such a language involves the co‐evolution of partitions of the physical world into what may seem, at least from the perspective of someone using the language, to correspond to canonical natural kinds. While the evolved language (...) may allow for the sort of precise representation that is required for successful coordinated action and prediction, the apparent natural kinds reflected in its structure may be purely conventional. This has both positive and negative implications for the limits of naturalized metaphysics. (shrink)

This is a re-look at the (Indian) Partition event through the lens of Advaita Vedanta.

Explaining the reasons behind the success of various kinds of fiction is important, but how granular should our explanations be? I suggest that using a less granular, more general hypothesis would allow avoiding some pitfalls, such as using the concept of “imaginary world,” which eludes precise definitions.

In the present version of these lecture notes only a number of typos and a few glaring mistakes have been corrected. Thanks to Paul Dekker for his help in this respect. No attempt has been been made to update the original text or to incorporate new insights and approaches. For a more recent overview, see our ‘Questions’ in the Handbook of Logic and Language (edited by Johan van Benthem and Alice ter Meulen, Elsevier, 1997).

A sand table is an intentional structure that is an early, indeed ancient, interactive platform for visualization and simulation. An intellectual furnishing that is also a tangible instance of speculative infrastructure, the sand table offers a tactile space for the rehearsal of tactics, staccato words whose roots lie in haptics and arrangement. While common in military settings, sand tables have also been used to teach the blind, train wilderness firefighters, conduct therapy for trauma victims, illustrate stories to children, and play (...) imaginative games. Today there is a direct line from this seemingly modest technology—an implementation of what has been called elemental media—to augmented reality and other tangible interfaces. Part media history, part media archaeology, this article argues that sand tables belong to the lineage of platforms for speculative thinking and world-building that culminated in the rise of the digital computer amid a Cold War complex of scenario-driven futurology (whose centerpiece was the so-called situation room). It also suggests that sand, in its literal granularity—the physical affordances of the minute particulars of its particulate matter—offers an alternative to the binary regimen of ones and zeros that is the extractive product of the refined silica out of which semiconductors are still made. (shrink)

We show that Miller partition forcing preserves selective independent families and P-points, which implies the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {u}=\mathfrak {i}<\mathfrak {a}_T=\omega _2$. In addition, we show that Shelah’s poset for destroying the maximality of a given maximal ideal preserves tight mad families and so we establish the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {i}=\omega _1<\mathfrak {u}=\mathfrak {a}_T=\omega _2$.

Two partition-theorems are proved for a particular causal decision theory. One is restricted to a certain kind of partition of circumstances, and analyzes the utility of an option in terms of its utilities in conjunction with circumstances in this partition. The other analyzes an option's utility in terms of its utilities conditional on circumstances and is quite unrestricted. While the first form seems more useful for applications, the second form may be of theoretical importance in foundational exercises. Comparisons are made (...) with theorems of Richard Jeffrey, Brad Armendt, and Peter Fishburn. (shrink)

The last section made it clear that an analysis which at first seems to fail is viable after all. It is viable if we let it depend on a partition function to be provided by the context of conversation. This analysis leaves certain traits of the partition function open. I have tried to show that this should be so. Specifying these traits as Pollock does leads to wrong predictions. And leaving them open endows counterfactuals with just the right amount of (...) variability and vagueness. (shrink)

Nineteenth and twentieth century philosophies of science have consistently failed to identify any rational basis for the compelling character of scientific analogies. This failure is particularly worrisome in light of the fact that the development and diffusion of certain scientific analogies, e.g. Darwin’s analogy between domestic breeds and naturally occurring species, constitute paradigm cases of good science. It is argued that the interactivist model, through the notion of a partition epistemology, provides a way to understand the persuasive character of compelling (...) scientific analogies without consigning them to an irrational or arational context of discovery. (shrink)

Let κ be a cardinal which is measurable after generically adding ${\beth_{\kappa+\omega}}$ many Cohen subsets to κ and let ${\mathcal G= ( \kappa,E )}$ be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value ${r_m^+}$ such that the set [κ] m can be partitioned into classes ${\langle{C_i:i (...) G}$ in ${\mathcal G}$ such that ${[\mathcal{G} ^\ast ] ^m\cap C_i}$ is monochromatic. It follows that ${\mathcal{G}\rightarrow (\mathcal{G} ) ^m_{<\kappa/r_m^+}}$ , that is, for any coloring of ${[ \mathcal {G} ] ^m}$ with fewer than κ colors there is a copy ${\mathcal{G} ^{\prime}}$ of ${\mathcal{G}}$ such that ${[\mathcal{G} ^{\prime} ] ^{m}}$ has at most ${r_m^+}$ colors. On the other hand, we show that there are colorings of ${\mathcal{G}}$ such that if ${\mathcal{G} ^{\prime}}$ is any copy of ${\mathcal{G}}$ then ${C_i\cap [\mathcal{G} ^{\prime} ] ^m\not=\emptyset}$ for all ${i 2 we have ${r_m^+ > r_m}$ where r m is the corresponding number of types for the countable Rado graph. (shrink)

Data analytics, particularly the current rhetoric around “Big Data”, tend to be presented as new and innovative, emerging ahistorically to revolutionize modern life. In this article, we situate one branch of Big Data analytics, spatial Big Data, through a historical predecessor, geodemographic analysis, to help develop a critical approach to current data analytics. Spatial Big Data promises an epistemic break in marketing, a leap from targeting geodemographic areas to targeting individuals. Yet it inherits characteristics and problems from geodemographics, including a (...) justification through the market, and a process of commodification through the black-boxing of technology. As researchers develop sustained critiques of data analytics and its effects on everyday life, we must so with a grounding in the cultural and historical contexts from which data technologies emerged. This article and others develop a historically situated, critical approach to spatial Big Data. This history illustrates connections to the critical issues of surveillance, redlining, and the production of consumer subjects and geographies. The shared histories and structural logics of spatial Big Data and geodemographics create the space for a continued critique of data analyses’ role in society. (shrink)

Several canonical partition theorems are obtained, including a simultaneous generalization of Neumer's lemma and the Erdos-Rado theorem. The canonical partition relation for infinite cardinals is completely determined, answering a question of Erdos and Rado. Counterexamples are given showing that in several ways these results cannot be improved.

Rigorous formal proof is one of the key techniques in the natural sciences, engineering, and of course also in the formal sciences. Progress in automated reasoning increasingly enables computer systems to support, and even teach, users to conduct formal a.

We consider several kinds of partition relations on the set ${\mathbb{R}}$ of real numbers and its powers, as well as their parameterizations with the set ${[\mathbb{N}]^{\mathbb{N}}}$ of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition (...) of ${\mathbb{R}^\mathbb{N}}$ there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of ${[\mathbb{N}]^{\mathbb{N}} \times \mathbb{R}^{\mathbb{N}}}$ there is ${X \in [\mathbb{N}]^{\mathbb{N}}}$ and a sequence ${\{P_{k} : k \in \mathbb{N}\}}$ of perfect sets such that the product ${[X]^{\mathbb{N}} \times \prod_{k}^{\infty}P_{k}}$ lies in one piece of the partition, where ${[X]^{\mathbb{N}}}$ is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. (shrink)

We consider finite partitions of the closure F¯ of an ωα-uniform barrier F. For each partition, we get a homogeneous set having both the same combinatorial and topological structure as F¯, seen as a subspace of the Cantor space 2N.

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on Pκ that satisfy the square brackets partition relation equation image, where μ is a cardinal ≥2.