For any Boolean Algebra A, let cmm be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which cmm exceeds any preassigned number, a rigid interval algebra with that property, and the construction (...) of an interval algebra in which every well-ordered chain has size less than cmm. (shrink)

MV-algebras stand for the many-valued Łukasiewicz logic the same as Boolean algebras for the classical logic. States on MV-algebras were first mentioned [20] in probability theory and later also introduced in effort to capture a notion of `an average truth-value of proposition' [15] in Łukasiewicz many-valued logic. In the presented paper, an integral representation theorem for finitely-additive states on semisimple MV-algebra will be proven. Further, we shall prove extension theorems concerning states defined on sub-MV-algebras and normal partitions of unity (...) generalizing in this way the well-known Horn-Tarski theorem for Boolean algebras. (shrink)

According to the Aristotelian account of substantial change, that is, the corruption of one substance and the generation of another, prime matter must be found at the starting and at the end point of change, as that which persists throughout the change. But knowing that matter remains as the substrate of change tells us little about the nature of this matter, which constitutes both the corrupted substance and the new generated substance. Among the questions we can ask about its nature (...) are whether prime matter is an entity of its own apart from form; whether it is one in all those individual material substances or is somehow divided or partitioned into different parts. Early commentaries on the Physics and Metaphysics were interested in precisely these questions about the unity and divisibility of prime matter and about what kind of unity this may be. This article aims to investigate how authors from the English commentary tradition in the period 1250–1270 answered these questions. (shrink)

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A, B, C such that A (...) ↔, that is, A can be split into two independent parts B and C, and the aforementioned topological notions give rise to a number of splittings involving highly natural A, B, C. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory RCA 0. We obtain similarly “exceptional” results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice. (shrink)

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function f is a (...) McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] n →[0,1] are in Free n . The elements of Free n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of Łukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free n. (shrink)

If κω is a cardinal, a complete Boolean algebra is called κ-dependent if for each sequence bβ: β<κ of elements of there exists a partition of the unity, P, such that each pP extends bβ or bβ′, for κ-many βκ. The connection of this property with cardinal functions, distributivity laws, forcing and collapsing of cardinals is considered.

In 1997, Rusmir Mahmutćehajić, one of Bosnia’s leading public intellectuals, was scheduled to lecture on Bosnia at Stanford University but was unexpectedly denied an entry visa by American authorities. This book, first published in Bosnia in 1998, is an expanded version of that lecture. It is an indictment of the partition of Bosnia, formalized in 1995 by the Dayton Accord. It is also a plea for Bosnia’s communities to reject ethnic segregation and restore mutual trust. For the first time, (...) English-speaking readers can hear this important voice of dissent from within Bosnia-Herzegovina. Mahmutćehajić argues for the history and reality of a Bosnia-Herzegovina based upon a model of “unity in diversity.” He shows that ethnic and religious cultures have coexisted in Bosnia for centuries. Partitioning of Bosnia, therefore, should have been unthinkable except that a multi-ethnic, multi-faith Bosnia stood squarely in the way of Croatian and Serbian leaders determined to enact their own nationalist programs. The decisive moment came when the international community accepted the Serb-Croat argument that ancient ethnic hatreds were endemic to Bosnia. At that point, ethnic segregation became not only acceptable but desirable. With the complicity of Western powers, Serbs and Croats proceeded to carve out ethnically cleansed states. Mahmutćehajić examines the reasons why Western liberal democracies have regarded with sympathy the struggles of Serbia and Croatia for national recognition, while viewing Bosnia’s multicultural society with suspicion. As one of Bosnia's former political leaders in the early peace talks, he describes with authority how the parties were often physically aligned during formal talks, with Bosniak negotiators on one side of the table and everybody else—Serb, Croat, and international representatives—on the other. In the end, justice was subverted and the final solution justified on the basis of an intractable “conflict of civilizations." Mahmutćehajić confronts the religious dimension of the Bosnian dilemma with refreshing honesty. As a Bosniak committed to interreligious dialogue, he calls for more than simple toleration among Serbs, Croats, and Bosnians. He remembers that Judaism, Christianity, and Islam all share the same deity, and it is this common transcendent perspective that should open the door to the acceptance and celebration of religious diversity. Only in this way will Bosnia reclaim its unique civilization. _The Denial of Bosnia_ has dire implications for the future of a Europe searching for a viable post–Cold War order. Will Europe accept ethnic segregation as a solution to the contradictions of ethnic diversity or find a way to protect and build upon this diversity? Bosnia, though currently divided and shaken to its foundations, could become a model for European progress. The greatest danger is for Bosnia to be declared just another ethnoreligious entity, in this case a “Muslim State” ghettoized inside of Europe. If protected and allowed to develop, however, Bosnia too could find a place in the new European order. (shrink)

Ernst Mayr''s contributions to 20th century biology extend far beyond his defense of certain elements in evolutionary theory. At the center of mid-century efforts in American evolutionary studies to build large research communities, Mayr spearheaded campaigns to create a Society for the Study of Evolution and a dedicated journal,Evolution, in 1946. Begun to offset the prominence ofDrosophila biology and evolutionary genetics, these campaigns changed course repeatedly, as impediments appeared, tactics shifted, and compromises built a growing coalition of support. Preserved, however, (...) were designs to balance the community and journal with careful equation of status and explicit partitioning of responsibilities within the working coalition. Choice terms such as cooperation and unity carried a strong political message. Mayr''s editorship ofEvolution provides a superb example of these balancing efforts. The mid-century infrastructural activities described herein also represented aggressive attempts to leverage control across several layers of community. Leaders of these campaigns sought: (1) to promote evolutionary studies as a modernized research discipline and place it at the center of American biology, (2) to promote evolutionary studies within existing disciplines — e.g. systematics, genetics, and paleontology, (3) to foster certain research styles within evolutionary studies, and (4) to emphasize certain solutions to prominent research questions. Throughout, Mayr interjected his priorities, tactics and energy. (shrink)

Chapter 1 First Person Access to Mental States. Mind Science and Subjective Qualities -/- Abstract. The philosophy of mind as we know it today starts with Ryle. What defines and at the same time differentiates it from the previous tradition of study on mind is the persuasion that any rigorous approach to mental phenomena must conform to the criteria of scientificity applied by the natural sciences, i.e. its investigations and results must be intersubjectively and publicly controllable. In Ryle’s view, philosophy (...) of mind needs to adopt an antimentalist stance to achieve this aim. Antimentalism not only definitively rejects the idea that mind is a substance separated from the body, it also denies that mental phenomena radically differ from physical phenomena by virtue of several unique features. Most problematically, mental phenomena have a conscious character (mental states are related to specific qualitative feelings) and are accessible only to the first-person (only the subject knows directly what s/he is experiencing inside his/her mind). Ryle takes a strong stance on antimentalism going so far as to maintain that an approach to mind which aims to meet the criteria of scientificity set by the natural sciences must avoid any reference to internal, unobservable mental states. In his view (which is considered a specifically philosophical version of psychological behaviorism and also addresses questions put forward in psychological research), mental states can be redescribed in terms of behavioral dispositions. In this chapter, we address the historical roots of the antimentalist view and analyze its relation to the later tradition of research on mind. We show that, compared to the antimentalist stance, functionalism and cognitivism take a step back when they maintain that direct reference to mental states is necessary since mental states cause and therefore explain human behavior. This step backwards is often interpreted as a return to mentalism. However, this is only partially true. Indeed, we suggest that these later traditions retain one important element of Ryle’s antimentalism, i.e. the idea that mental states must be uniquely identified using external and publicly observable criteria, while excluding any reference to introspection and those qualitative dimensions of a mental state, which are accessible only to the first-person. According to the perspective we put forward, this epistemological stance has continued to influence contemporary research on mind and current philosophical and psychological theories which both tend to exclude the subjective qualities of human experience from their accounts of how the mind works. The issue we raise here is whether this is legitimate or whether subjective qualities do play a role with respect to the way our mind works. The conclusion of this chapter anticipates the argument the book makes in in favor of this latter position, starting from a particular angle, i.e. the problem of how we categorize concepts related to our internal states. -/- Chapter 2 The Misleading Aspects of the Mind/Computer Analogy. The Grounding Problem and the Thorny Issue of Propriosensitive Information -/- Abstract. After the crisis of behaviorism, cognitivism and functionalism became the predominant models in the field of psychology and of philosophy, respectively. Their success is mainly due to the new key they use for interpreting mental processes: the mind/computer analogy. On the basis of this analogy, mental operations are seen as cognitive processes based on computations, i.e. on manipulations of abstract symbols which are in turn understood as informational unities (representations). This chapter identifies two main problems with this model. The first is how these symbols can relate to and communicate with perception and thus allow us to identify and classify what we perceive through the senses. Here we limit ourselves to presenting this issue in relation to the classical symbol grounding problem originally put forward by Harnad on the basis of Searle’s Chinese room argument. An attempt to address the problem raised here will be made in chap. 3. The second point we discuss in relation to the mind/computer analogy concerns the idea of information it fosters. Indeed, following this analogy, information is something available in the external world which can be captured by the senses and transmitted to the central system without being influenced or modified by the procedures of transmission. This perspective does not take into account that – unlike computers – in living beings information is acquired by means of the body. As Ulric Neisser has already pointed out, the body is itself an informational source that provides us with additional sensory experience that influences (modifies or complements) the information extracted from the external world by the senses. To develop this line of analysis and to determine exactly what information is provided by the body and how this might influence cognition, we examine Sherrington’s and Gibson’s positions. Moving on from their views, we qualify bodily information in terms of ‘proprioception’. We use ‘proprioception’ in a broad sense to describe any kind of experience we have of our internal states (including postural information as well as sensations related to the general state of the body and its parts). Following Damasio’s and Craig’s studies, we further elaborate this position, arguing that living beings are equipped with an internal propriosensitive monitoring system which maps all the changes that constantly occur in our body and that give us perceptual (‘proprioceptive’ or propriosensitive) information about what happens inside us. Moreover, relying on Goldie’s and Ratcliffe’s view, we show that emotional information can also be considered as a form of ‘proprioception’ which contributes to determining everything we perceive. This analysis leads us to the second main thesis of this book: ‘proprioception’ is a form of internal perception and it is an essential component of the sensory information we can access and use for all cognitive purposes. -/- Chapter 3 Semantic Competence from the Inside: Conceptual Architecture and Composition -/- Abstract. Concepts are essential constituents of thought: they are the instruments we use to categorize our experience, i.e. to classify things and group them together in homogeneous sets. Here we define concepts as the internal mental information (representations) that allows us, among other things, to master words in natural language. By analyzing the way in which individuals master word meanings we explore a number of hypotheses regarding the nature of concepts. Following Diego Marconi’s research, we differentiate between two kind of abilities that underpin lexical competence – so-called ‘referential’ and ‘inferential competence’ – and we suggest that, in order to support these abilities, concepts must also include two corresponding kinds of information, i.e. inferential and referential information. We point out that the most widely used and acknowledged theories of concepts do not make this distinction, instead broadly characterizing the information used for categorization in terms of propositionally described feature lists. However, we show that while feature lists can explain inferential competence, they do not account for referential competence. To address the issue of referential competence we examine Ray Jackendoff’s hypothesis that to account for the possibility of linking perceptual and conceptual information we need to assume the existence of a (visual) representation that encodes the geometric and topological properties of objects and bridges the gap between the percept and the concept. Furthermore, we analyze the extension of this work by Jesse Prinz who introduced the notion of a proxytype, a perceptual representation of a class of objects that incorporates structural and parametric information related to their appearance. However, as we point out, proxytypes can only explain the relationship between perception and concepts with respect to instances that can be perceived through the senses and that belong to the same class by virtue of their physical similarity. We suggest that this notion be extended to include larger conceptual classes. To accomplish this, we further develop Mark Johnson, George Lakoff and Jean Mandler’s idea of a schematic image and argue that conceptual representations include a perceptual schema. Perceptual schemata are non-linguistic, structured experiential gestalts (patterns or maps) that make use of information taken from all sensory modalities, including body perception. They accomplish a quasi-conceptual function: they allow us to recognize and to classify different instances. In this work, we hypothesize that perceptual schemata are an essential component of concepts, but not identical to them. Instead, we suggest that concepts include both perceptual and propositional information with perceptual schemata providing the ‘perceptual core concept’ that grounds related propositional information. -/- Chapter 4 In the Beginning There Were Categories. The Bodily Origin of Prelinguistic Categorical Organization: The Example of Folkbiological Taxonomies -/- Abstract. Studies of categorization in psychology and the cognitive sciences have made use of the notions of ‘category’ and ‘concept’ without precisely defining what is meant by either; in fact, often these terms have been used as synonyms, making it difficult to address specific issues related to conceptual development. This chapter begins by discussing the definitions of, as well as the distinctions between, ‘categories’ and ‘concepts’ in the classical philosophical tradition (Aristotle, Kant and Husserl). We introduce our view of categories with reference to Husserl. Categorization is defined as the way in which our experience is originally (pre-linguistically) organized in a passive and fully unconscious manner on the basis of universal structuring principles. Conceptualization is explained as a later process in which the earlier categorical macro-classes are further subdivided into more specific and detailed sets; this later process also relies on linguistic learning and exposure to culture. On the basis of Ray Jackendoff, Jean Mandler, George Lakoff and Mark Johnson’s work, we hypothesize that categorization is a two-step process that begins with the formation of homogeneous sets of entities partitioned into regions that describe the ontological boundaries of the objects that humans perceive (categories) and continues at a later stage with the development of more specific classifications (concepts). In this chapter, we mainly address three related issues: Why should we assume that there is categorical organization which precedes the development of a conceptual system? How do categories and concepts relate to each other? Shall we hypothesize that categories are innate or that they are formed before concepts on the basis of information and organizational structure available at a very early developmental stage? We show that categorical partitions are necessary for categorization and, following Mandler, that the general categories we form at an early age do not match our adult superordinate concepts. As for the third issue, we argue that there might be no need to assume that categories are innately present in the human mind, since their formation can be explained – at least in certain cases – by basic mechanisms that work on body (propriosensitive) information. This hypothesis will not be discussed in general, but in relation to a particularly relevant example of categorical partition, i.e. the folk-biological dichotomy between ANIMATE/INANIMATE. This is compared with other dichotomies that derive from it, but are not directly categorical such as LIVING/NON-LIVING and BIOLOGICAL/NON-BIOLOGICAL. -/- Chapter 5 Internal States: From Headache to Anger. Conceptualization and Semantic Mastery -/- Abstract. Here we ask if we can also apply the distinction between referential and inferential competence we introduced in Chapter 3 to words that do not refer to things that are perceived using the external senses, especially to words/concepts that denote bodily experiences (such as pain, thirst, hunger, etc.) or emotions. We introduce and discuss the hypothesis that – even though such words/concepts do not refer to intersubjectively identifiable entities in the external world – they do have a kind of referent that can be accessed via direct perception, more specifically ‘proprioception’, as we have defined it in terms of all propriosensitive information we can consciously access. In the first part of the chapter, we specifically consider terms denoting bodily experiences such as ‘pain’ or ‘hunger’ and argue that their referents are identified and classified from a first-personal point of view on the basis of four main characteristics: their specific intensity, their localization in the body, their co-occurrence with other signals and above all their specific qualitative sensations. Emotions are addressed in the second part of the chapter. We suggest that there is a continuity between bodily experiences and emotions. In particular, we argue for a perceptual theory of emotions in line with that proposed by James and Lange at the end of the 19th century and developed more recently by authors such as Damasio (see also chap. 2§5, §6). The hypothesis we put forward is that the referential information that supports the categorization of emotions and therefore also our mastery of terms referring to emotions consists in the perception (i.e. the ‘proprioception’) of those bodily states and changes in bodily states which constitute our emotional experience. In the context of this discussion we examine some objections to this line of reasoning that arise from a cognitivist perspective and following authors such as Oatley, Johnson-Laird and Frijda, we distinguish between basic emotions that can be identified and classified solely on the basis on how they feel and complex emotions whose identification and classification additionally depends on cognitive factors. To describe how emotions are identified and classified on the basis of how they feel, we rely on Marcel and Lambie’s distinction between an ‘emotion state’ and an ‘emotion experience’. Both notions indicate kinds of feelings that we consciously experience. However, they describe first-order and second order emotion awareness respectively. The emotion state is the feeling we have of the bodily states and changes that occur when we are experiencing an emotion, while the emotion experience is the fully developed and integrated emotion we both experience and are, with reflection, aware of experiencing. On the basis of this differentiation, we also show that the same characteristics that aid in the identification and classification of bodily experiences (specific qualitative sensations; somatic localization; specific intensity; presence/absence of specific concomitant sensations) can also be used for the identification and classification of emotions – at least basic emotions. In the last two sections of the chapter we present some clinical evidence on the semantic competence of people who suffer from Alexithymia and Autism Spectrum Disorder which supports the conclusions of our preceding analyses. -/- Chapter 6 The ‘Proprioceptive’ Component of Abstract Concepts -/- Abstract. In this chapter, we address the issue of whether the mastery of abstract words requires only inferential knowledge and thus, if the concepts that support the mastery of abstract words include only linguistic information. We start by differentiating the notions of ‘abstract’ and ‘general’ which are often erroneously confused. We then identify a strict definition of abstract, as contrasted with ‘concrete’, that applies to words or concepts whose referent cannot be experienced by the senses. We argue that abstract words/concepts would be better described as theoretical, because they are usually conceived as structured sets of inferential knowledge expressed linguistically; that is, as small theories. Pointing out parallels with Carnap’s analysis of this issue in philosophy of science, we hypothesize that words/concepts denoting non-observable entities are not all ‘equally theoretical’, because their link to sensory experience can be stronger or weaker. We revive the distinction, inspired by Quine, between theoretical and intertheoretical concepts/words. This distinction relies on the fact that the former – unlike the latter – retains a strong, although indirect, connection with perception. In Quine’s discussion, perception is understood uniquely in terms of observability, i.e. of external sensory experience. Here we argue, however, that bodily, ‘proprioceptive’ (i.e. propriosensitive) experience can also serve to referentially ground theoretical (i.e. abstract) concepts/words. We frame this issue using the example of the theoretical concept ‘freedom’ and Lakoff’s hypothesis that this concept is developed on the basis of bodily information. We contrast this with the example of ‘democracy’ which more closely resembles an intertheoretical concept/word. Furthermore, we show that one of the classical views put forward in psycholinguistic research to explain how abstract concepts are mentally represented – i.e. Paivio’s Dual Coding Theory – points in the same direction as our analysis. The same is true of Barsalou’s work suggesting that we use internal information to understand at least some abstract words. To sustain this position, we put forward two lines of evidence: the first comes from psycholinguistic studies while the second examines deficits of semantic competence exhibited by people with Autism Spectrum Disorder. On the basis of our analysis, we put forward a classification that distinguishes between different kinds of concreteness and different degrees of abstraction: concepts/words referring to body experiences and basic emotions are described as analogous to concrete concepts/words because they are grounded in perceptual (i.e. propriosensitive) experience, while abstract concepts/words are considered more or less abstract depending on whether they are intratheoretical (and rely entirely on inferential information) or theoretical (and are partially grounded in perceptual – or more often in propriosensitive perceptual – information). In the last section of the chapter we consider two scales that have been used in psycholinguistic research to measure the degree of concreteness vs. abstractness of words and we show that – used conjointly – they can provide a measure of the internal vs. external grounding of specific words. (shrink)

Louveau, A., S. Shelah and B. Velikovi, Borel partitions of infinite subtrees of a perfect tree, Annals of Pure and Applied Logic 63 271–281. We define a notion of type of a perfect tree and show that, for any given type τ, if the set of all subtrees of a given perfect tree T which have type τ is partitioned into two Borel classes then there is a perfect subtree S of T such that all subtrees of S of type (...) τ belong to the same class. This result simultaneously generalizes the partition theorems of Galvin-Prikry and Galvin-Blass. The key ingredient of the proof is the theorem of Halpern-Laüchli on partitions of products of perfect trees. (shrink)

The harmonious growth and cell‐to‐cell uniformity of steady‐state bacterial populations indicate the existence of a well‐regulated cell cycle, responding to a set of internal signals. In Escherichia coli, the key events of this cycle are the initiation of DNA replication, nucleoid segregation and the initiation of cell division. The replication initiator is the DnaA protein. In nucleoid segregation, the MukB protein, required for proper partitioning, may be a member of the myosin‐kinesin superfamily of mechanoenzymes. In cell division, the FtsZ protein (...) has a tubulin motif, is a GTPase and polymerizes in a ring around midcell during septation; the FtsA protein has an actin‐like structure. The nature of the internal signals triggering these events is not known but candidates include cell mass, the superhelical density of the chromosome and the concentration of two regulatory nucleotides, cyclic AMP and ppGpp. The involvement of cytoskeletal‐like proteins in key cycle events encourages the notion of a fundamental biological unity in cell cycle regulation in all organisms. (shrink)

We prove that all algebras P/IR, where the IR-'s are ideals generated by partitions of W into finite and arbitrary large elements, are isomorphic and homogeneous. Moreover, we show that the smallest size of a tower of such partitions with respect to the eventually-refining preordering is equal to the smallest size of a tower on ω.

It is known that in an infinite very weakly cancellative semigroup with size $\kappa $, any central set can be partitioned into $\kappa $ central sets. Furthermore, if $\kappa $ contains $\lambda $ almost disjoint sets, then any central set contains $\lambda $ almost disjoint central sets. Similar results hold for thick sets, very thick sets and piecewise syndetic sets. In this article, we investigate three other notions of largeness: quasi-central sets, C-sets, and J-sets. We obtain that the statement applies (...) for quasi-central sets. If the semigroup is commutative, then the statement holds for C-sets. Moreover, if $\kappa ^\omega = \kappa $, then the statement holds for J-sets. (shrink)

A desirable property of a diversity index is strict concavity. This implies that the pooled diversity of a given community sample is greater than or equal to but not less than the weighted mean of the diversity values of the constituting plots. For a strict concave diversity index, such as species richness S, Shannon''s entropy H or Simpson''s index 1-D, the pooled diversity of a given community sample can be partitioned into two non-negative, additive components: average within-plot diversity and between-plot (...) diversity. As a result, species diversity can be summarized at various scales measuring all diversity components in the same units. Conversely, violation of strict concavity would imply the non-interpretable result of a negative diversity among community plots. In this paper, I apply this additive partition model generally adopted for traditional diversity measures to Aczél and Daróczy''s generalized entropy of type . In this way, a parametric measure of -diversity is derived as the ratio between the pooled sample diversity and the average within-plot diversity that represents the parametric analogue of Whittaker''s -diversity for data on species relative abundances. (shrink)

This article, taking a social semiotic approach, analyses two pieces of music written, shared and exalted by two pre-1945 European fascist movements – the German NSDAP and the British Union of Fascists. These movements, both political and cultural, employed mythologies of unity, common identity and purpose in order to elide the realities of social distinction and political–economic inequalities between bourgeois and proletarian groups in capitalist societies. Visually and inter-personally, the fascist cultural project communicated a machine-like certainty about a vision (...) for a new society based on discipline, conformity and the might of the nation. In this article, we are interested in the ways that these very same discourses are also communicated through sound and music in two songs: The Horst Wessel Lied and the BUF marching song, two songs that used the same melody. We analyse the discourses communicated by the semiotic choices made in melody, arrangements, sound qualities, rhythms as well as in lyrics. The article first identifies some of the underlying semiotic resources for meaning making in sound and then shows how these are used in order to communicate specific ideas, values and attitudes. (shrink)

This work is a study of Plato's ethical theory and in particular his paradoxical theses that virtue is knowledge and vice is ignorance and that no one voluntarily does evil. In the opening chapters O'Brien discusses Plato's debt to Homer and Hesiod and to the historians and tragedians. The succeeding chapters are devoted to an analysis of the ethical doctrines of over a dozen dialogues ranging from the Apology to the Laws. The principles of interpretation employed in the analysis are: (...) 1) each doctrine must be understood in relation to every other as well as to the whole, 2) we must give a literary analysis of the dialogues in addition to a textual and philosophical one, since Plato often prefers to suggest his view through his literary art rather than give them explicit statement, and 3) statements must be interpreted in context. From an application of these principles, Professor O'Brien concludes that Plato's ethical theory constitutes a unified set of doctrines. Elements which are often thought to be antithetic, i.e., the Socratic paradoxes and the partition of the soul, are seen to be complementary. The one affirms the rational unity of human nature and action and the other, the irrational multiplicity of both. In his words, they affirm "... virtue is knowledge, and it is more than knowledge." The book contains a long bibliography and an extensive index of topics discussed and texts cited. As a result, O'Brien's view on nearly every topic and text is readily accessible.—O. J. L. (shrink)

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering two types of (...) questions for the case when k = 3. First, if domatic 3-partitions exist in a computable graph, how complicated can they be? Second, a decision problem: given a graph, how difficult is it to decide whether it has a domatic 3-partition? We will completely classify this decision problem for highly computable graphs, locally finite computable graphs, and computable graphs in general. Specifically, we show the decision problems for these kinds of graphs to be ${\Pi^{0}_{1}}$ -, ${\Pi^{0}_{2}}$ -, and ${\Sigma^{1}_{1}}$ -complete, respectively. (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)

In the Zermelo–Fraenkel set theory (ZF), |$|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$| for any infinite set |$A$|⁠, where |$\textrm {fin}(A)$| is the set of finite subsets of |$A$|⁠, |$2^{|A|}$| is the cardinality of the power set of |$A$| and |$\textrm {Part}(A)$| is the set of partitions of |$A$|⁠. In this paper, we show in ZF that |$|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$| for any set |$A$| with |$|A|\geq 5$|⁠, where |$\textrm {Part}_{\textrm {fin}}(A)$| is the set of partitions of |$A$| whose members are finite. We (...) also show that, without the Axiom of Choice, any relationship between |$|\textrm {Part}_{\textrm {fin}}(A)|$| and |$2^{|A|}$| for an arbitrary infinite set |$A$| cannot be concluded. (shrink)

These days we hear a lot about the way our eschatological belief can affect our social action. Indeed it can: but do contemporary evangelicals satisfactorily show us how? In this article it is argued that our exact beliefs about the world's future should affect our present activity less than people think. The proposal is made that we distinguish between love and hope as springs of social action, not by rejecting hope but by showing its limitations. One advantage of this suggestion (...) is that it keeps us from over-emphasizing eschatological differences as we pursue a theology of social action. (shrink)

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a maximal (...) family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ.• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products.• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA.• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras. (shrink)

The paper sets out to give in broad outline a comprehensive interpretation of some features of the Derveni text. In some important issues, this text should be seen in the context of the fourth century, rather than the late fifth century. Unfortunately, this 4th c. BCE literature has disappeared almost without a trace, the best characterization of this literature we have is in Plato’s polemical descriptions, most particularly, the one he puts forth as a refutation of different forms of atheism (...) in Laws X.Although on some issues the Derveni author holds theses similar to the ones criticized by Plato, on his physical exegesis of the Orphic poem the world is the result of the teleological action of the divine intellect. Moreover, it is part of this very order that it is providential, and that within this providential order human beings have their place and are provided for. Moreover, the order that Zeus brings about is not only a wise and providential physical order, but also a just order which is maintained by Zeus in such a way that every wrong is set right.Setting out the stages of the teleological activity of the divine intellect, during which the divinity bears different names, the paper closes with some considerations about the extent of the Derveni text, suggesting that it may well be plausible to assume that the author’s text could come to an end on a single roll. (shrink)

The Unity Argument states that the ability to think of one's perceptions as perceptions of objects in the weighty sense is necessary for unity of consciousness. A distinction is drawn between two versions of the Unity Argument. The first turns on the idea that unity of consciousness involves what P. F. Strawson calls ’transcendental’ self‐consciousness. The second turns on the idea that unity of consciousness involves ’personal’ self‐consciousness.

According to mentalism some existing things are endowed with (subjectively) conscious minds. According to physicalism all existing things consist entirely of physical particles in fields of force. Searle holds that mentalism and physicalism are compatible and true.

From 1918 to the late 1940s, a host of influential scientists and intellectuals in Europe and North America were engaged in a number of far-reaching unity of science projects. In this period of deep social and political divisions, scientists collaborated to unify sciences across disciplinary boundaries and to set up the international scientific community as a model for global political co-operation. They strove to align scientific and social objectives through rational planning and to promote unified science as the driving (...) force of human civilization and progress. This volume explores the unity of science movement, providing a synthetic view of its pursuits and placing it in its historical context as a scientific and political force. Through a coherent set of original case studies looking at the significance of various projects and strategies of unification, the book highlights the great variety of manifestations of this endeavour. These range from unifying nuclear physics to the evolutionary synthesis, and from the democratization of scientific planning to the utopianism of H.G. Wells's world state. At the same time, the collection brings out the substantive links between these different pursuits, especially in the form of interconnected networks of unification and the alignment of objectives among them. Notably, it shows that opposition to fascism, using the instrument of unified science, became the most urgent common goal in the 1930s and 1940s. In addressing these issues, the book makes visible important historical developments, showing how scientists participated in, and actively helped to create, an interwar ideology of unification, and bringing to light the cultural and political significance of this enterprise. (shrink)

Generalizing model companions from model theory we define companions of pieces of canonical partitions of Polish G-spaces. This unifies several constructions from logic. The central problem of the paper is the existence of companions which form a G-orbit which is a Gδ-set. We describe companions of some typical G-spaces.

In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.

What makes a visually appealing landscape? How can the design and use of a landscape be harmonised? In this significantly revised and updated third edition of Simon Bell's seminal text, he further explores the answers to these questions by interrogating a range of design principles, applications and ideas. Written for students, instructors and professionals, the book unveils a visual design vocabulary for anyone involved with landscape aesthetics including landscape architects, architects, planners, urban designers, landscape managers, foresters, geographers and ecologists. Structured (...) around key design terms, which are explained and illustrated using an extensive range of examples from around the world, including North America, Europe and Asia, this book enables you to describe, debate and design the visual landscape. It starts with basic elements, before moving onto variable design components, and then the ways these elements are organized into compositions, in order to demonstrate how landscapes are created and how meanings and patterns are perceived within them. This new full colour edition contains over 240 images; an updated introduction; examples from China, Vietnam, and central Asia; a chapter on how to read and understand visual design elements in the landscape; a teaching model for instructors; and expanded appendix materials including a glossary, references and further reading. (shrink)

We show that a version of Ramsey’s theorem for trees for arbitrary exponents is equivalent to the subsystem ${{\sf ACA}^\prime_{0}}$ of reverse mathematics.

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)

In The Principles of Mathematics, Bertrand Russell famously puzzled over something he called the unity of the proposition. Echoing Russell, many philosophers have talked over the years about the question or problem of the unity of the proposition. In fact, I believe that there are a number of quite distinct though related questions all of which can plausibly be taken to be questions regarding the unity of propositions. I state three such questions and show how the theory (...) of propositions defended in my recent book The Nature and Structure of Content answers them. (shrink)

My aim in this paper is to explore some metaphysical and psychological implications of the (contentious) idealist interpretation of the belief in external objects ("bodies") Hume ascribes to us in the Treatise. More specifically, I will argue that the interpretation commits Hume to the claim that space is spatially fragmented, both synchronically and (even more so) diachronically, and renders Hume incapable of allowing for all the spatial thoughts we think we can have. But (perhaps surprisingly) it does not impugn Hume's (...) view of causation. (shrink)