We prove that every convexly ordered valuation ring has a unique completion as a uniform space, which furthermore is a convexly ordered valuation ring. In addition, we give a model theoretic characterisation of complete convexly ordered valuation rings, and give a necessary and sufficient condition for the completion of a convexly ordered valuation ring to be a real closed ring.

We recall the characterisation of positive definite polynomial functions over a real closed ring due to Dickmann, and give a new proof of this result, based upon ideas of Abraham Robinson. In addition we isolate the class of convexly ordered valuation rings for which this characterisation holds.

In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VC-minimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, (...) show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density. (shrink)

In quantum logics, the notions of strong and full order determination and unitality for states on orthomodular posets are well known. These notions are defined for hypergraphs and their state spaces in a consistent manner and the relations between them and to the notions defined for orthomodular posets are discussed. The state space of a hypergraph is a polytope. This polytope is a simplex if and only if every superposition of pure states is a mixture of these same pure states. (...) Isomorphic hypergraphs have convexly isomorphic state spaces. A class of hypergraphs is given whose group of automorphisms is group-isomorphic to the group of convex automorphisms of their state spaces. (shrink)

The Ecclesiastical History of Orderic Vitalis has been called `the greatest of all medieval chronicles'. Written in Normandy between 1114 and 1141, it is a detailed history of the Norman people and their conquests, full of vivid, often penetrating portraits of the lives and characters of kings and queens, lords and bishops, simple knights, and humble villagers. The chronicle gives a unique, authentic picture of feudal society during a period of rapid change in church and state which saw the emergence (...) of the Anglo-Norman realm, the spread of new forms of monasticism, and the launching of the Crusades. Published in 1968, volume II of Marjorie Chibnall's six-volume edition of the Ecclesiastical History is now available for the first time in paperback. `a superbly edited Latin text and a unique English translation of the work of a major historian' American Historical Review `Mrs Chibnall is giving us... her edition of a classical medieval text which will itself become a classic of medieval scholarship.' Times Literary Supplement. (shrink)

Betrifft die Handschrift Cod. 555 der Burgerbibliothek Bern.

Edited with a facing-page English translation from the Latin text by: Chibnall, Marjorie.

It is known that the spaces of orders on orderable computable fields can represent all Π10 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π10 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes.

Tarski [5] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w (X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation |P(X)| (...) = |Y|. (shrink)

We consider the equationally orderable quasivarieties and associate with them deductive systems defined using the order. The method of definition of these deductive systems encompasses the definition of logics preserving degrees of truth we find in the research areas of substructural logics and mathematical fuzzy logic. We prove several general results, for example that the deductive systems so defined are finitary and that the ones associated with equationally orderable varieties are congruential.

After the three families divided up the state of Jin and the Tian family took over Qi, the political situation in the fourth century B.C.E. appeared even more chaotic. Wei conquered Chu's Luyang and Qin's Xihe, Qin defeated Wei at Shimen , and again at Shaoliang , and Wei moved its capital to Daliang. During the mid-Warring States period, Qin became dominant in the west, Qi in the east, Chu in the south, and Wei in the center. Rapid changes occurred (...) one after another; trickery and conflict were rampant. However, the shi , the intelligentsia, actually managed to expand their spheres of existence and activity accordingly in this state of tension between countries, states, and men. A few major states assembled quite a number of literati; and those men of the villages, who had in earlier times merely waited passively for opportunities to fall into their laps, crisscrossed the various feudal states; literati who previously had no cause to use military force became, one by one, the honored guests of the feudal lords. With the breakdown of the previously simple and obvious order, the gradually dismantling of the common body of the ancestral clan of the past, and the increasing complexities of social relationships after the dissolution of blood ties, social order became a matter of concern. Or, to reword this most practical issue: How could this universally chaotic society go from disorder to order? (shrink)

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.

The BRAHMASUTRA of BADARAYANA represents the fundamental text of exegesis of Vedanta. The intent of Badarayana - the sage that for authority and realization of consciousness has been identified with Vyasa, the Rsi who ordered the texts of the Vedas - is that of providing the right perspective in the interpretation of the most profound and meaningful contents of the Upanisads. This had proven necessary in order to rectify some unilateral aspects propounded by several schools of thought, both orthodox and (...) non-orthodox. The Brahmasutra presents, in their simplicity and incisiveness, the assertions of the Sruti and of the Smrti, showing their concordance in the recognition of the Nirguna Brahman as the ultimate Realty. In his notes, RAPHAEL underlines the fact that Badarayana does not oppose the various philosophical schools, but the Rsi allows us to comprehend that their postulates cannot represent the ultimate Truth as expounded in the Vedas and in the Upanisads. Moreover, at times Raphael focuses on certain aspects of the Advaita Doctrine with references to the Western Tradition, and - making the relevant parallels to the philosophy of Parmenides, Plato, Plotinus, and so on - highlights the unity of the sole universal Tradition at the metaphysical level. (shrink)

A theory of higher-order vagueness for partially-defined, context-sensitive predicates like is blue is offered. According to the theory, the predicate is determinately blue means roughly is an object o such that the claim that o is blue is a necessary consequence of the rules of the language plus the underlying non-linguistic facts in the world. Because the question of which rules count as rules of the language is itself vague, the predicate is determinately blue is both vague and partial in (...) a sense analogous (though not completely identical) to the sense in which is blue is vague and partial. However, higher-order vagueness stops there. Although the anti-extension of is determinately blue differs from that of is blue, further iterations of determinately have no semantic effect. The paper closes with a discussion of the need to posit sharp and precise lines dividing the range of potential application of a vague predicate into semantically distinct categories. It is argued that this is less of a problem for the present theory than for many others, because the categories into which the range is divided by the present theory -- like the category of counting as blue because the rules of the language offer speakers no discretion, as opposed to the category of counting as blue because speakers have invoked an absolute minimum of discretion in adjusting the contextually sensitive boundaries of the predicate -- are not those that the linguistic competence of ordinary speakers equips them to make highly accurate and reliable judgments about. (shrink)

You have higher-order uncertainty iff you are uncertain of what opinions you should have. I defend three claims about it. First, the higher-order evidence debate can be helpfully reframed in terms of higher-order uncertainty. The central question becomes how your first- and higher-order opinions should relate—a precise question that can be embedded within a general, tractable framework. Second, this question is nontrivial. Rational higher-order uncertainty is pervasive, and lies at the foundations of the epistemology of disagreement. Third, the answer is (...) not obvious. The Enkratic Intuition---that your first-order opinions must “line up” with your higher-order opinions---is incorrect; epistemic akrasia can be rational. If all this is right, then it leaves us without answers---but with a clear picture of the question, and a fruitful strategy for pursuing it. (shrink)

In this seminal work, acclaimed philosopher Bernhard Waldenfels deals with the problem of the nature of order after the “shattering of the world,” and the loss of the idea of a universal or fundamental order._ _ Order in the Twilight__ unites phenomenological methodology with recent work on the theory of order, normativity, and dialogue, as well as structuralism and Gestalt theory. Philosophically stringent, it expresses a more optimistic attitude than much modern philosophy, especially deconstruction._ Waldenfels passes the question of order (...) through numerous defining aspects, and concludes that there is not one global order, but rather various conflicting domains of order. Whenever the boundary of a vital or experiential domain is crossed, a discourse speaks at the boundary, not about it, and across a threshold without abolishing it. The rest is rationalization, i.e., an attempt to find a place in the respective order for what is to-be-ordered. But why, the author concludes, should a theory be more unambiguous than reality? _ Order in the Twilight__ is an important book at this time, because it may help lift the humanities out of the skeptical, relativistic disarray in which they have been embroiled in recent decades. Waldenfels does not attempt to dictate what reality should be; rather, he is open to any valid evidences. His book offers a solid footing to the human and social sciences as they seek to escape from deconstructive irrationalism. (shrink)

Many academics extol chaos theory and the science of complexity as significant scientific advances with application in such diverse fields as biology, anthropology, economics, and history. In this paper we focus our attention on structure‐within‐chaos and the dynamic self‐organization of complex systems in the context of social philosophy. Although the modern formulation of the science of complexity has developed out of late‐twentieth‐century physics and computational mathematics, its roots may extend much deeper into classical thinking. We argue here that the essential (...) ideas and predictions of the science of complexity are found within the social ordering principle of li (the rites) in Confucius's Analects. (shrink)

This work describes the concept of bond order. It shows that covalent bond energy is correlated to bond order. Simple expressions which included bond order are introduced to calculate bond energies of homo-nuclear and hetero-nuclear bonds. Calculated values of bond energies are compared with literature values and show there is very good agreement between and calculated and experimental values in the vast majority of cases. Bond order reveals the strength of a bond and shows the number of bonds in both (...) homo-nuclear and hetero-nuclear covalent bonds. (shrink)

On at least one of its uses, ‘higher-order evidence’ refers to evidence about what opinions are rationalized by your evidence. This chapter surveys the foundational epistemological questions raised by such evidence, the methods that have proven useful for answering them, and the potential consequences and applications of such answers.

An overview of higher-order representational theories of consciousness. Representational theories of consciousness attempt to reduce consciousness to “mental representations” rather than directly to neural or other physical states. This approach has been fairly popular over the past few decades. Examples include first-order representationalism (FOR) which attempts to explain conscious experience primarily in terms of world-directed (or first-order) intentional states (Tye 2005) as well as several versions of higher-order representationalism (HOR) which holds that what makes a mental state M conscious is (...) that it is the object of some kind of higher-order mental state directed at M. The primary focus of this entry is on HOR and especially higher-order thought (HOT) theory. The key question that should be answered by any theory of consciousness is: What makes a mental state a conscious mental state? Section 1 introduces the overall representationalist approach to consciousness and briefly discuss Tye’s FOR. Section 2 presents three major versions of HOR: higher-order thought theory, dispositional higher-order thought theory, and higher-order perception theory. In section 3, a number of common and important objections and replies are presented. Section 4 briefly outlines a close connection between HOT theory and conceptualism, that is, the claim that the representational content of a perceptual experience is entirely determined by the conceptual capacities the perceiver brings to bear in her experience. Section 5 examines several hybrid higher-order and “self-representational” theories of consciousness which all hold that conscious states are self-directed in some way. Section 6 addresses the potentially damaging claim that HOT theory requires neural activity in the prefrontal cortex (PFC) in order for one to have conscious states. (shrink)

This chapter provides an introduction to higher-order metaphysics as well as to the contributions to this volume. We discuss five topics, corresponding to the five parts of this volume, and summarize the contributions to each part. First, we motivate the usefulness of higher-order quantification in metaphysics using a number of examples, and discuss the question of how such quantifiers should be interpreted. We provide a brief introduction to the most common forms of higher-order logics used in metaphysics, and indicate a (...) number of questions which can be raised in such systems using logical vocabulary alone. Using a further example, we return to applications of higher-order logics in metaphysics. We also mention key developments in the history of higher-order logic as it pertains to metaphysics. Finally, we mention certain arguments which have been raised against the use of higher-order logic, and some ways of responding to them. (shrink)

The gap between the limited human sense of how living entities change over time and the ways in which living systems change is one of the most potent uncertainties in ecological knowledge. This gap remains a significant source of problems and errors for those working with the transformations of living systems. This article foregrounds Mary Catherine Bateson’s cybernetic practice of working with narratives in order to cultivate better understanding and responses to change, at the level of both societies and individuals. (...) To do so, I investigate some significant moments, encounters and projects that connect her practice to the ecological ideas of her father, Gregory Bateson and situate their ideas on abduction, metaphor, recursion and narrative within broader discussions of ecological change. The Batesons did not seek a particular attitudinal change towards the transformation of selves, cities and worlds but rather sought to change the very understanding of what an attitude towards change should be, a distinction that is worth pondering given the challenges of attending to change within the present ecological crisis. (shrink)

A. Poleev. Constitutional order in Russia. Enzymes, Revised print edition 2020.

A critical survey of recent work in epistemology on higher-order evidence. It discusses the nature of higher-order evidence, some puzzles it raises, responses to those puzzles, and problems facing them. It concludes by indicating connections between debates concerning higher-order evidence in epistemology and parallel debates in ethics and aesthetics.

Due to their reliance on constitutive higher-order representing to generate the qualities of which the subject is consciously aware, I argue that the major existing higher-order representational theories of consciousness insulate us from our first-order sensory states. In fact on these views we are never properly conscious of our sensory states at all. In their place I offer a new higher-order theory of consciousness, with a view to making us suitably intimate with our sensory states in experience. This theory relies (...) on the idea of ‘quoting’ sensory qualities, so is dubbed the ‘quotational higher-order thought theory’. I argue that it can capture something of the idea that we are ‘acquainted’ with our conscious states without slipping beyond the pale for naturalists, whilst also providing satisfying treatments of traditional problems for higher-order theories concerning representational mismatch. The theory achieves this by abandoning a representational mechanism for mental intentionality, in favour of one based on ‘embedding’. (shrink)

Ambitious higher-order theories of consciousness aim to account for conscious states when these are understood in terms of what-it-is-like-ness. This paper considers two arguments concerning this aim, and concludes that ambitious theories fail. The misrepresentation argument against HO theories aims to show that the possibility of radical misrepresentation—there being a HO state about a state the subject is not in—leads to a contradiction. In contrast, the awareness argument aims to bolster HO theories by showing that subjects are aware of all (...) their conscious states. Both arguments hinge on how we understand two related notions which are ubiquitous in discussions of consciousness: those of what-it-is-like-ness and there being something it is like for a subject to be in a mental state. This paper examines how HO theorists must understand the two crucial notions if they are to reject the misrepresentation argument but assert the awareness argument. It shows that HO theorists can and do adopt an understanding—the HO reading—which seems to give them what they want. But adopting the HO reading changes the two arguments. On this reading, the awareness argument tells us nothing about those states there is something it is like to be in, and so offers no support to ambitious HO theories. And to respond to the misrepresentation understood according to the HO reading is to simply ignore the argument presented, and so to give no response at all. As things stand, we should deny that HO theories can account for what-it-is-like-ness. (shrink)

Question order effects are commonly observed in self-report measures of judgment and attitude. This article develops a quantum question order model (the QQ model) to account for four types of question order effects observed in literature. First, the postulates of the QQ model are presented. Second, an a priori, parameter-free, and precise prediction, called the QQ equality, is derived from these mathematical principles, and six empirical data sets are used to test the prediction. Third, a new index is derived from (...) the model to measure similarity between questions. Fourth, we show that in contrast to the QQ model, Bayesian and Markov models do not generally satisfy the QQ equality and thus cannot account for the reported empirical data that support this equality. Finally, we describe the conditions under which order effects are predicted to occur, and we review a broader range of findings that are encompassed by these very same quantum principles. We conclude that quantum probability theory, initially invented to explain order effects on measurements in physics, appears to be a powerful natural explanation for order effects of self-report measures in social and behavioral sciences, too. (shrink)

Word Order Universals is a detailed account of word order universals and their role in theories of historical change. The starting point is the Greenberg data set, which is comprised of a sample of 142 languages for certain limited co-occurrences of basic word orders, and a 30-language sample for more detailed information. In the Language Index, the 142 have been expanded to some 350 languages. Using the original Greenberg samples and the Expanded Sample, an alternative set of descriptive word order (...) statements is provided. Comprised of eight chapters, this book begins with an introduction to the theory of word order universals, encompassing topics such as word order variation across languages and theories of universal grammar. The reader is then introduced to the work of Joseph Greenberg and Theo Vennemann on word order universals; implicational universals in Greenberg's data and the Expanded Sample; and the predictions made by implicational and distributional universals for word order change. Reformulated universals for historical reconstruction are also discussed, along with some laws of reconstruction derived from synchronic universals. The final chapter is devoted to the Expanded Sample, with particular reference to its quantities as well as its typological and genetic classification. This monograph will be a useful resource for specialists in grammar and linguistics. (shrink)

Recent authors have drawn attention to a new kind of defeating evidence commonly referred to as higher-order evidence. Such evidence works by inducing doubts that one’s doxastic state is the result of a flawed process – for instance, a process brought about by a reason-distorting drug. I argue that accommodating defeat by higher-order evidence requires a two-tiered theory of justification, and that the phenomenon gives rise to a puzzle. The puzzle is that at least in some situations involving higher-order defeaters (...) the correct epistemic rules issue conflicting recommendations. For instance, a subject ought to believe p, but she ought also to suspend judgment in p. I discuss three responses. The first resists the puzzle by arguing that there is only one correct epistemic rule, an Über-rule. The second accepts that there are genuine epistemic dilemmas. The third appeals to a hierarchy or ordering of correct epistemic rules. I spell out problems for all of these responses. I conclude that the right lesson to draw from the puzzle is that a state can be epistemically rational or justified even if one has what looks to be strong evidence to think that it is not. As such, the considerations put forth constitute a non question-begging argument for a kind of externalism. (shrink)

A logic is called higher order if it allows for quantification over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14].1 While classical type theory has since long been overshadowed by set theory as a foundation of mathematics, recent decades have shown remarkable (...) comebacks in the fields of mechanized reasoning (see, e.g., Benzm¨. (shrink)