The present contribution argues that transitive reasoning, as exemplified in paradigms of linear order construction in mental space, is associated with spatial effects. Starting from robust findings from the early 70s, research so far has widely discussed the symbolic distance effect. This effect shows that after studying pairs of relations, e.g., “A > B,” “B > C,” and “D > E,” participants are more correct, and faster in correct responding, the wider the “distance” between two elements within the chain (...) A > B > C > D > E. The SDE has often been given spatial interpretations, but alternatively, non-spatial models of the effect are also viable on the empirical basis so far, which means the question about spatial contributions to the construction of analog representations of rank orders is still open. We suggest here that laterality effects can add the necessary additional information to support the idea of spatial processes. We introduce anchoring effects in terms of showing response advantages for congruent versus incongruent pairings of presentation location on a screen on the one hand, and the hypothetical spatial arrangement of the order in mental space, on the other hand. We report pertinent findings and discuss anchoring paradigms with respect to their internal validity as well as their being rooted in basic mechanisms of trained reading/writing direction. (shrink)

The central aim of the present investigation is to shed light on Nietzsche’s understanding of happiness and a good life, starting from Nietzsche’s appropriation of Plato’s and Aristotle’s doctrine of a hierarchy of human beings and forms of life.

We show that Mitchell order on downward closed extenders below rank-to-rank type is wellfounded.

In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma /\Delta $ for each definable convex subgroup $\Delta $, and sorts for the quotient groups $\Gamma /(\Delta + \ell \Gamma )$ where $\Delta $ is a definable convex subgroup and $\ell \in \mathbb {N}_{\geq (...) 2}$. We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\ell \Gamma $ in $\Gamma $, where $\ell \in \mathbb {N}_{\geq 2}$. (shrink)

O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable theories. This is the first in an anticipated series of papers whose aim is the development of model theory for ordered structures of rank greater than one. A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here. These (...) include all ordered structures definable in o-minimal structures. The principal result in this paper, Theorem 5.1, asserts roughly that a decomposable structure [Formula: see text] can be partitioned into finitely many definable subsets such that on each set the restriction of < is a "twisted lexicographic" order. As a consequence, for all n and linear orders ≺ definable on a subset X ⊆ Mn in an o-minimal structure [Formula: see text], there is a definable partition of X such that the restriction of ≺ to each set in the partition is "lexicographic". (shrink)

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ2, the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in +“φ2 is well-ordered” and, over , implies +“φ2 is well-ordered”.

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call (...) reflection rank, to any$\Pi ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles. (shrink)

Rankings have dramatically proliferated over the past several decades. An often-overlooked impact of this proliferation is that it has facilitated the institutionalization of an imaginary of the modern world as a stratified order, whose actors are imagined as continuously striving to perform better than others. To better understand this impact, we need to take a closer look at rankings’ premises and the way these resonate with the broader institutional environment of which rankings are a part.

Partial order optimality theory is a conservative generalization of classical optimality theory that makes possible the modeling of free variation and quantitative regularities without any numerical parameters. Solving the ranking problem for PoOT has so far remained an outstanding problem: allowing for free variation, given a finite set of input/output pairs, i.e., a dataset, \ that a speaker S knows to be part of some language L, how can S learn the set of all grammars G under some constraint (...) set C compatible with \?. Here, allowing for free variation, given the set of all PoOT grammars GPoOT over a constraint set C, for an arbitrary \, I provide set-theoretic means for constructing the actual set G compatible with \. Specifically, I determine the set of all STRICT ORDERS of C that are compatible with \. As every strict total order is a strict order, our solution is applicable in both PoOT and COT, showing that the ranking problem in COT is a special instance of a more general one in PoOT. (shrink)

Different evaluators typically disagree how to rank different candidates due to their idiosyncratic concerns for the various qualities of the candidates. Our ranking mechanism asks all evaluators to submit individual bids assigning a monetary amount for each possible rank order. The rules specify for all possible vectors of such individual bids the collectively binding rank order of candidates and the payments, due to the different evaluators. Three requirements uniquely determine procedurally fair ranking rules as a (...) game form. Only when additionally assuming exogenously given true evaluations of collective rankings, which may be commonly or only privately known, and—in case of private information—beliefs concerning the evaluations by others, this game form determines proper games and allows for equilibrium analysis. After an illustration, the approach is adjusted to situations where one wants to rank only acceptable sets of candidates thus rendering the mechanism even more attractive. (shrink)

The Spohnian paradigm of ranking functions is in many respects like an order-of-magnitude reverse of subjective probability theory. Unlike probabilities, however, ranking functions are only indirectly—via a pointwise ranking function on the underlying set of possibilities W —defined on a field of propositions A over W. This research note shows under which conditions ranking functions on a field of propositions A over W and rankings on a language L are induced by pointwise ranking functions on W and the set (...) of models for L, ModL, respectively. (shrink)

We define some variations of the Scott rank for countable models and obtain some inequalities involving the ranks. For mono-unary algebras we prove that the game rank of any subtree does not exceed the game rank of the whole model. However, similar questions about linear orders remain unresolved.

We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any (...) dependence relation for superstable homogeneous models. (shrink)

We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any (...) dependence relation for superstable homogeneous models. (shrink)

Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible positive Deissler rank complexity. The proof shows that this complexity naturally reflects the internal structure (...) of the injective extension in the form of the levels of the Matlis hierarchy. (shrink)

From 1827 to 1831 the German historian Leopold von Ranke travelled through Germany, Austria, and Italy, hunting for documents and archives. During this journey Ranke developed a new model for historical research that transformed the archive into the most important site for the production of historical knowledge. Within the archive, Ranke claimed, the trained historian could forget his personal predispositions and political loyalties, and write objective history. This essay critically examines Ranke's model for historical research through a study of the (...) obstacles, frustrations, and joys that he encountered on his journey. It shows how Ranke's archival experiences inspired him to re-evaluate his own identity as a historian and as a human being, and investigates some of the affiliations between his model for historical research and the political realities of Prince Metternich's European order. Finally, the essay compares Ranke's historical discipline to other nineteenth-century disciplines, such as anthropology and archaeology. (shrink)

In this paper, I investigate the relationship between preference and judgment aggregation, using the notion of ranking judgment introduced in List and Pettit. Ranking judgments were introduced in order to state the logical connections between the impossibility theorem of aggregating sets of judgments and Arrow’s theorem. I present a proof of the theorem concerning ranking judgments as a corollary of Arrow’s theorem, extending the translation between preferences and judgments defined in List and Pettit to the conditions on the aggregation (...) procedure. (shrink)

The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the 0-stable case which share many of the formal properties of forking. This (...) is achieved by considering a rank for this framework which is bounded when the diagram D is 0-stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental first-order theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results as well as negative results . We also give a proof of a Two Cardinal Theorem for this context. (shrink)

In this paper, we introduce the notion of $${\textbf{K}} $$ -rank, where $${\textbf{K}} $$ is a strong amalgamation Fraïssé class. Roughly speaking, the $${\textbf{K}} $$ -rank of a partial type is the number “copies” of $${\textbf{K}} $$ that can be “independently coded” inside of the type. We study $${\textbf{K}} $$ -rank for specific examples of $${\textbf{K}} $$, including linear orders, equivalence relations, and graphs. We discuss the relationship of $${\textbf{K}} $$ -rank to other ranks in model (...) theory, including dp-rank and op-dimension (a notion coined by the first author and C. D. Hill in previous work). (shrink)

While numerous studies have shown diverse effects of rankings, rather little is known about their production. This article contributes to a broader understanding of rankings in society, and does so by focusing on underlying worldviews. I argue that the existence of a ranking and its concrete methodology can be explained by the producer’s paradigmatic assumptions about a world-to-be-ranked. Referring to the sociology of knowledge and studies on commensuration, comparisons, quantification and valuation, I provide a general heuristic to analyze this relation (...) between underlying worldviews and observational regimes through which order is constructed systematically. Presenting empirical results on a ranking for the most famous artists in the world, I show how the review device’s initial problem and its construction of order derive from consistent assumptions about contemporary art, its symbolic structures and its social embeddedness. These findings have implications for both research on rankings and sociology of the arts. (shrink)

We define an appropriate analog of the Morley rank in a totally transcendental homogeneous model with type diagram D. We show that if RM[p] = α then for some 1 ≤ n < ω the type p has n, but not n + 1, distinct D-extensions of rank α. This is surprising, because the proof of the statement in the first-order case depends heavily on compactness. We also show that types over (D,ℵ₀)-homogeneous models have multiplicity (Morley degree) (...) 1. (shrink)

In this paper I revisit nineteenth-century debates over historical objectivity and the political functions of historiography. I focus on two central contributors to these debates: Leopold von Ranke and Johann Gustav Droysen. In their takes on objectivity and subjectivity, impartiality and political engagement I reveal diametrically opposed solutions to shared concerns: how can historians reveal history to be meaningful without taking recourse to speculative philosophy? And how can they produce a knowledge that is relevant to the present when the project (...) of “exemplary” history has been abandoned? I put a special focus on the relativist themes in Ranke’s and Droysen’s answers to these questions. Ranke’s demand for impartiality leads him to think of all historical epochs as equally valid, while Droysen’s emphasis on the historian’s subjectivity relativizes historical truth. In order to explain why Ranke and Droysen nevertheless remained unfazed by the problem of historical relativism, I analyze their normative conceptions of the historian’s disciplinary ethos. I show that Ranke and Droysen think of objective impartiality and subjective partiality not only in methodological terms, but also in terms of justice and ethic duty. This normative element secures the professional study of history an ethical-political relevance for the present. (shrink)

We apply a cognitive modeling approach to the problem of measuring expertise on rank ordering problems. In these problems, people must order a set of items in terms of a given criterion (e.g., ordering American holidays through the calendar year). Using a cognitive model of behavior on this problem that allows for individual differences in knowledge, we are able to infer people's expertise directly from the rankings they provide. We show that our model-based measure of expertise outperforms self-report (...) measures, taken both before and after completing the ordering of items, in terms of correlation with the actual accuracy of the answers. These results apply to six general knowledge tasks, like ordering American holidays, and two prediction tasks, involving sporting and television competitions. Based on these results, we discuss the potential and limitations of using cognitive models in assessing expertise. (shrink)

The theory of ranked partial structures allows a reinterpretation of several of the standard results of model theory and first-order logic and is intended to provide a proof-theoretic method which allows for the intuitions of model theory. A version of the downward Löwenheim-Skolem theorem is central to our development. In this paper we will present the basic theory of ranked partial structures and their logic including an appropriate version of the completeness theorem.

Sustainable development calls for choices among alternative policy options. It is a common view that such choices can be justified by appealing to an evaluative ranking of the options with respect to how their consequences affect a broad range of prudential and moral values. Three philosophically motivated proposals for analysing evaluative rankings are discussed: the measured merits model (e.g. Chang), the ordered values model (e.g. Griffin), and the permissible preference orderings model (Rabinowicz). The analysis focuses on the models’ potential for (...) making transparent how an evaluative ranking can contribute to a justified choice among options, particularly in situations that involve diverse values as typically found in debates on sustainable development. Such transparency plays a crucial role when policy rankings are going to be used as arguments in political decision processes. The measured merits model is found to have questionable consequences for the concept of sustainability, while the ordered values model calls for an axiological framework that cannot plausibly be spelled out for sustainability. The permissible preference orderings model is more promising. Its formal structure and its ability to deal with value-pluralism provide an interesting re-structuring of the problem of justifying choices in sustainability issues. (shrink)

The importance, legitimacy and role of second-order probabilities are discussed. Two descriptive models of the use of second-order probabilities in decisions are presented. The results of two empirical studies of the effects of second-order probabilities upon the rank orderings of bets are summarized briefly. The bets were of three basic types and involved a wide variety of first- and second-order probabilities as subjectively assessed by the subjects. Support was obtained for the assumption that the majority (...) of subjects make use of the one model or the other. It is suggested that greater attention should be paid to second-order probabilities, both from a normative and descriptive standpoint. (shrink)

This paper establishes a conjecture of Steel [7] regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, studied in [5] and [7]. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and (...) a variant of the Mitchell order called the internal relation. (shrink)

While much attention has been given to the use of principles in biomedical ethics and increasing attention is given to alternative theoretical approaches, relatively little attention has been devoted to the critical task of how one resolves conflicts among competing principles. After summarizing the system of principles and some problems in conceptualizing the principles, several strategies for reconciling conflicts among principles are examined including the use of single-principle theories (pure libertarianism, pure utilitarianism, and pure Hippocratism), balancing theories, conflicting appeals theories, (...) and lexical ordering. Then a mixed strategy is proposed in which consequentialist principles are balanced between themselves as are nonconsequentialist principles, after which the result of balancing the nonconsequentialist principles is lexically ranked over the result of balancing the consequentialist ones. Finally, strategies involving specifying and rule generation are discussed concluding that most current specification and rule-generating theories must involve some degree of lexical ordering of principles. (shrink)

A rank function for a directed graph G assigns elements of a well ordering to the vertices of G in a fashion that preserves the order induced by the edges. While topological sortings require a one-to-one matching of vertices and elements of the ordering, rank functions frequently must assign several vertices the same value. Theorems stating basic properties of rank functions vary significantly in logical strength. Using the techniques of reverse mathematics, we present results that require (...) the subsystems ${\ensuremath{\vec{RCA}_0}}$ , ${{\vec{ACA}_0}}$ , ${{\vec{ATR}_0}}$ , and ${{\vec{\Pi^1_1 - CA}_0}}$. (shrink)

Empirical evidence from both utility and psychophysical experiments suggests that people respond quite differently—perhaps discontinuously—to stimulus pairs when one consequence or signal is set to `zero.' Such stimuli are called unitary. The author's earlier theories assumed otherwise. In particular, the key property of segregation relating gambles and joint receipts (or presentations) involves unitary stimuli. Also, the representation of unitary stimuli was assumed to be separable (i.e., multiplicative). The theories developed here do not invoke separability. Four general cases based on two (...) distinctions are explored. The first distinction is between commutative joint receipts, which are relevant to utility, and the non-commutative ones, which are relevant to psychophysics. The second distinction concerns how stimuli of the form (x, C; y) and the operation of joint receipt are linked: by segregation, which mixes stimuli and unitary ones, and by distributivity, which does not involve any unitary stimuli. A class of representations more general than rank-dependent utility (RDU) is found in which monotonic functions of increments U(x)-U(y), where U is an order preseving representation of gambles, and joint receipt play a role. This form and its natural generalization to gambles with n > 2 consequences, which is also axiomatized, appear to encompass models of configural weights and decision affect. When joint receipts are not commutative, somewhat similar representations of stimuli arise, and joint receipts are shown to have a conjoint additive representation and in some cases a constant bias independent of signal intensity is predicted. (shrink)

Traditional accounts of belief change have been criticized for placing undue emphasis on the new belief provided as input. A recent proposal to address such issues is a framework for non-prioritized belief change based on default theories (Ghose and Goebel, 1998). A novel feature of this approach is the introduction of disbeliefs alongside beliefs which allows for a view of belief contraction as independently useful, instead of just being seen as an intermediate step in the process of belief revision. This (...) approach is, however, restrictive in assuming a linear ordering of reliability on the received inputs. In this paper, we replace the linear ordering with a preference ranking on inputs from which a total preorder on inputs can be induced. This extension brings along with it the problem of dealing with inputs of equal rank. We provide a semantic solution to this problem which contains, as a special case, AGM belief change on closed theories. (shrink)

We understand from the oldest sources that have reached us that according to their status, racial characteristics, culture, religion etc. people called their adressees with many different names besides their own names. The Arabic nicknames and titles, which are the main subject of our research result of this necessity. Before the formation of Islamic culture and civilization, different titles were used in all civilizations, especially in the Byzantine and Sassanid empires, for the members of the group, which were considered as (...) the elite class, apart from their names. Of course, the Arab society of jahiliyyah was also greatly influenced by his. In the political and bureaucratic structure created by the rapid spread of Islam and the interaction from different civilizations, a new naming, or rather nicknames, were made by using religious figures. This happened to such extend that the original names of many rulers remained in the shadows. The strict centralist structure of the Umayyad state required an authoritarian state administration that would not need excessive credit or definition. Therefore, the caliph did not need an extra nickname or title to define himself. At the same time, it would be appropriate to say that they never had any thoughts of praising the politicians and bureaucrats affiliated with them. The radical change in the political understanding with the Abbasid revolution also led to a great change in naming strategies. In fact, it is possible to explain the differentiation in this management approach through titles in the shortest way. The famous names of the Abbasid caliphs are nothing but nicknames. These names, which are adorned with the words of Allah, are actually an effort to show religious sensitivity to the subjects. The sensitivity in the language of diplomacy has come to the fore even more with the special divans established. The language of religion and the language of politics have been intertwined, and the caliph, who is the representative of the creator on earth, has been giving nicknames/titles around him. We see the actual situation of this practice as concrete in relation to the vassal states. During the Abbasid period, the level of relations between the states, which first emerged as semi-independent and later independent, and the caliphate differed according to time and conditions. In proportion to the political power of the Abbasid state, these heads of state were given different ranks. In this research, we try to reveal the background of the ranks given to the heads of vassal state established in the Khorasan and Transoxiana region and carrying out very important activities in the political sense. Additionally, we aim reveal to find out the political, religious and social structure of Tahirids, Saffarids, and Samanids, who are also very important in terms of Turkish history and serve as a bridge between the region and the center of the caliphate, Baghdad. We think that the subject discussed is important in order to discover the unknown aspects of the historical process in the political and bureaucratic formation of the Turkish empires established after the collapse of these three vassal states. In fact, it can be said that even today, the historical anatomy of the ranks used in terms of status determination and the arrangement of the political field in different Islamic geographies is being explored. Most importantly, in this study, we try discover the cornerstones of the communication dimension of the mawali with the Arab power. The location of the Transoxiana and Khorasan geographies before the Islamic conquests and the population fluidity accelerated by the Turkish migrations were an important factor in the political formation of the region during the Abbasid period. The historical background was taken into consideration in the titles given to the vassal state rulers. It is remarkable that the caliph gave the old Turkish title “ıhsid” to a Turkish leader of a state established in Egypt. There are many other examles such as this with regard to the vassal states in the East. The conclusion we have obtained from our study is that titles play a decisive role in the political connection between the center of the caliphate and the affiliated states. On the one hand, this regulates the balance between the center and local administrators, on the other hand, it determines the moral as well as the material rank of the sultans in the eyes of the people. In fact, the successes in the conquests led the caliphs to rank the vassal sultans with titles. This situation is a kind of password that determines the balance between the religious authority, the caliphate, and the worldly authority. (shrink)

Within universities, a tension exists between research and teaching and learning, where research is often accorded a higher status creating a Cinderella effect by rendering teaching and learning of lesser importance. World university rankings, also referred to as global rankings, are contentious although they have become a permanent feature of the higher education system internationally. Lindsay argues that institutions have emphasized the importance of publications and prestige, which contribute to national and institutional reputation. Publications increase rankings thereby contributing to the (...) lower prestige given to excellent teachers as compared to excellent researchers. This has consequently led to a decline in the attention given to teaching. Pressure exists in the HE system to play the ranking game without acknowledging that there are expertise and pockets of teaching excellence in universities. Through their performance criteria, global rankings are transforming universities into becoming similar and conforming to a single hierarchy. In order to compete in the global HE market and improve their rankings, some universities have adopted a business model to mitigate the effects of globalisation and have begun to view themselves and higher education as a business. Global ranking systems thus have the agency either to perpetuate teaching and learning as the Cinderella function in HE, through the ranking game or to holistically focus their performance criteria on all the core functions of universities. Ethical policy decision-making in higher is thus essential. (shrink)

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable (...) structure and the latter the archetypal stable structure. In this sense we try here to situate our work ono-minimal structures [PS] in a general topological context. Note, however, that thep-adic numbers, and structures definable therein, will also fit into our analysis.In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions. (shrink)

If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. (...) If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text]. (shrink)

Many philosophical theories make comparisons between objects, events, states of affairs, worlds, or systems, and many such theories deliver plausible verdicts only if some of the elements they compare are ranked as ‘best.’ When the relevant ordering does not have such ‘best’ or ‘tied for best’ elements the theory wrongly falls silent or gives badly counterintuitive results. This paper develops ordering supervaluationism---a very general technique that allows any such theory to handle these problematic cases. Just as ordinary supervaluation helps us (...) save and generalize ‘uniqueness assuming’ theories, ordering supervaluationism helps us save and generalize ‘limit assuming’ theories. With so many otherwise attractive limit assuming theories, this is a sensible, methodologically conservative approach. (shrink)

In this paper the general notion of Bourbaki structures, interpreted in terms of Suppes predicates, will be used to axiomatize a system of meta-rankings in the sense introduced by A. K. Sen. It will be argued that this axiomatization must take place in a Kantian-ruled world in order to provide a link between meta-rankings and individual actions.Dedicated to Prof. Francisco A. Doria on his 50th birthday.

The purpose of this article is to delineate the process of evolution of know­ledge organization systems (KOSs) through identification of principles of unity such as internal and external unity in organizing the structure of KOSs to achieve content storage and retrieval purposes and to explain a novel method used in ranking of KOSs by proposing the principle of rank unity. Different types of KOSs which are addressed in this article include dictionaries, Roget’s thesaurus, thesauri, micro, macro, and meta-thesaurus, ontologies, (...) and lower, middle, and upper-level ontologies. This article relied on dialectic models to clarify the ideas in Kant’s know­ledge theory. This is done by identifying logical relationships between categories (i.e., Thesis, antithesis, and synthesis) in the creation of data, information, and know­ledge in the human mind. The Analysis has adapted a historical methodology, more specifically a documentary method, as its reasoning process to propose a conceptual model for ranking KOSs. The study endeavors to explain the main elements of data, information, and know­ledge along with engineering mechanisms such as data, information, and know­ledge engineering in developing the structure of KOSs and also aims to clarify their influence on content storage and retrieval performance. KOSs have followed related principles of order to achieve an internal order, which could be examined by analyzing the principle of internal unity in know­ledge organizations. The principle of external unity leads us to the necessity of compatibility and interoperability between different types of KOSs to achieve semantic harmonization in increasing the performance of content storage and retrieval. Upon introduction of the principle of rank unity, a ranking method of KOSs utilizing cognition states as criteria could be considered to determine the position of each know­ledge organization with respect to others. The related criteria of the principle of rank unity- cognition states- are derived from Immanuel Kant’s epistemology. The research results showed that KOSs, while having defined positions in cognition states, specific principles of order, related operational mechanisms, and related principles of unity in achieving their specific purposes, have benefited from the developmental experiences of previous KOSs, and further, their developmental processes owe to the experiences and methods of their previous generations. (shrink)

We present a new proof of soundness/completeness of tableaux with respect to dialogical games in Classical First-Order Logic. As far as we know it is the first thorough result for dialogical games where finiteness of plays is guaranteed by means of what we call repetition ranks.

We prove that every type of finite Cantor-Bendixson rank over a model of a first-order theory without the strict order property is definable and has a unique nonforking extension to a global type. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets ⟨x, y⟩ = {{x}, {x, y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a recursive definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater lenght. There are many advantages to the new definition, for it allows for uniform definitions working (...) equally well in a wide range of models for set theories. In ZFC and closely related theories, the of an ordered pair of two infinite sets under the new definition turns out to be equal to the maximum of the ranks of the sets. (shrink)