We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.

Finitary quasivarieties are characterized categorically by the existence of colimits and of an abstractly finite, regularly projective regular generator G. Analogously, infinitary quasivarieties are characterized: one drops the assumption that G be abstractly finite. For (finitary) varieties the characterization is similar: the regular generator is assumed to be exactly projective, i.e., hom(G, –) is an exact functor. These results sharpen the classical characterization theorems of Lawvere, Isbell and other authors.

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ⊨Δ associated with an infinitary variety Δ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ⊨Δ, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ⊢Δ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion (...) of consequence associated with ⊢Δ coincides with ⊨Δ. (shrink)

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...) (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5). (shrink)

Herbrand's Theorem, in the form of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\exists } $$ -inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the classes of provably total functions include the elements of the Polynomial Hierarchy, the Grzegorczyk Hierarchy, and the extended Grzegorczyk Hierarchy $\mathfrak{E}^\alpha $ , α < ε0. A subsidiary aim of the paper is (...) to show that the proof theoretic methods used here are distinguished by technical elegance, conceptual clarity, and wide-ranging applicability. (shrink)

This volume presents the proceedings of the workshop CSL '91 (Computer Science Logic) held at the University of Berne, Switzerland, October 7-11, 1991. This was the fifth in a series of annual workshops on computer sciencelogic (the first four are recorded in LNCS volumes 329, 385, 440, and 533). The volume contains 33 invited and selected papers on a variety of logical topics in computer science, including abstract datatypes, bounded theories, complexity results, cut elimination, denotational semantics, infinitary queries, (...) Kleene algebra with recursion, minimal proofs, normal forms in infinite-valued logic, ordinal processes, persistent Petri nets, plausibility logic, program synthesis systems, quantifier hierarchies, semantics of modularization, stable logic, term rewriting systems, termination of logic programs, transitive closure logic, variants of resolution, and many others. (shrink)

In this dissertation we study two-cardinal phenomena—both of the admitting cardinals variety and of the Chang’s Conjecture variety—under the assumption that all our models have stable theories. All our results involve two, relatively widely accepted generalizations of the traditional definitions in this area. First, we allow the relevant subsets of our models to be picked out by (perhaps infinitary) partial types; second we consider δ-cardinal problems as well as two-cardinal problems.

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover (...) system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions. (shrink)

The aims of this paper are twofold. First, I offer a new insight into Shepherd’s theory of mind by demonstrating that she distinguishes a threefold ‘Variety of Intellect’, that is, three kinds of minds grouped according to their cognitive limitations. Following Shepherd, I call them (i) minds afflicted with idiocy, (ii) inferior understandings, and (iii) sound understandings. Second, I show how Shepherd’s distinction informs her theory of education. While Shepherd claims that her views serve to improve educational practices, she (...) does not explain how her threefold ‘Variety of intellect’ does so. I argue that Shepherd’s distinction contributes to her aim by providing pedagogues with a more comprehensive grasp of the various cognitive limitations of the people in their care. This allows the former to adjust their expectations and to provide need-oriented guidance. (shrink)

I discuss the connections between rationality and intentional action, emphasising that different kinds of action are rational an intentional in different ways.

López-Escobar, E. G. K. Introduction.--Kueker, D. W. Back-and-forth arguments and infinitary logics.--Green, J. Consistency properties for finite quantifier languages.--Cunningham, E. Chain models.--Gregory, J. On a finiteness condition for infinitary languages.

It is widely accepted that necessity comes in different varieties, often called ‘kinds': metaphysical necessity, logical necessity, natural necessity, conceptual necessity, moral necessity, to name but a few – and the same goes for the varieties of possibility. What is usually not fully appreciated, however, is that modal variety is not simply ‘unidimensional': it does not only involve one main variable – kind, whose values are the particular kinds of necessity. Rather, I argue, it is ‘bidimensional', involving two distinct (...) variables – domain and strength. In the first part of the paper, I introduce and develop the proposed bidimensional picture of modal variety, defending it against the common, unidimensional one. In the second part, I consider how the main available accounts of necessities and their relations rely, at least to a significant extent, on the latter picture, pointing out important limitations that they face as a result. I also show how, accordingly, alternative accounts based on a clear and systematic distinction between domain and strength would overcome those limitations. I conclude that, beyond the particular bidimensionalist view defended, our understanding of the modal realm may benefit from more direct debate on whether and how it is multidimensional. (shrink)

In this study, the age-old distinction between decision method and criterion of rightness, commonly employed in normative ethics, was used to attain a detailed understanding of inter- and intrapersonal variety in students' moral reasoning. A total of 24 Swedish students, 12–15 years of age, were interviewed. Inter- and intrapersonal varieties in and between the two dimensions of moral reasoning were found, constituting six novel forms of varieties. We describe several explanations proposed within the field of social-cognitive domain theory, and (...) argue that these do not suffice to explain all the found forms of varieties in students' moral reasoning. Some limitations of the present study are discussed, and some suggestions for future research are given. (shrink)

The aim of my paper is to assess in a critical way the views presented by Graeme Smith in his book A Short History of Secularism as well as in his paper Talking to Ourselves: An Investigation into the Christian Ethics Inherent in Secularism. According to Smith, secular Western societies are underpinned by Christian ethics. An example of a moral norm that – in Smith’s opinion – derives from medieval Christianity and shapes the moral condition of the members of contemporary (...) societies, is the concern about the poor. My criticism of Smith’s thesis is based on the distinction between moral norms and the ways of justifying them. Referring to this distinction, my objective is to show that certain norms which appear to be the same cannot be treated as identical due to the significant differences in their justification. (shrink)

This contribution offers reflections about Chilean Cybersyn, 50 years ago. In recent years, Cybersyn, has received significant attention. It was the brainchild of Stafford Beer, who conceived it to support the transformation of the Chilean economy from its bureaucratic history to hopefully create a vibrant and modern society, driven by cybernetic tools. These aspects have received much attention in recent times; however, in this contribution, I want to discuss how working in Cybersyn influenced my work after the coup of 1973. (...) Perhaps, for me, its major influence was in the management of complexity, through what I refer here as variety engineering and through the Viable System Model VSM as a paradigm to the management of relationships with implications to enterprises, society and the economy. After the 1973 coup major interest was in technological aspects of Cybersyn such as real-time management and its contribution to decision support and executive information systems. In the late 70s I was personally influenced by information management, but by the early 1980s my work moved towards methodological aspects of how to use the VSM. By 1989 I had created the VIPLAN method. Key questions I attempted to answer were, how to model the complexity of enterprises and their interactions with environmental agents. Later on, in the 1990s and 2000s, the main direction of my work was epistemological, ontological and methodological towards second-order cybernetics and relationships. Only in recent decades the political transformations proposed by Cybersyn have captured the imagination of many commentators. The confluence of social and cultural changes with information technology, data models, artificial intelligence, algorithms and several additional technological developments have challenged the excesses of capitalism, particularly after the banking crisis of 2008–2009. The purpose in this paper is discussing this evolution in the light of those early days in Chile. (shrink)

This paper extends the AGM theory of belief revision to accommodate infinitary belief change. We generalize both axiomatization and modeling of the AGM theory. We show that most properties of the AGM belief change operations are preserved by the generalized operations whereas the infinitary belief change operations have their special properties. We prove that the extended axiomatic system for the generalized belief change operators with a Limit Postulate properly specifies infinite belief change. This framework provides a basis for (...) first-order belief revision and the theory of revising a belief state by a belief state. (shrink)

This paper deals with the infinitary modal propositional logic Kω1, featuring countable disjunctions and conjunc- tions. It is known that the natural infinitary extension LK.

This paper describes the process of reception of Catholic Modernism in Poland as well as the Polish contribution to this movement. It shows the Polish antimodernist perspective on modernistic thought. The neglect of Polish modernism was caused by the nationalistic character of the Polish theology and has resulted in absence of historical studies of Polish Catholic Modernism. Based on the results of archival and literature research the paper presents a variety of Polish Catholic Modernists and non-Catholic supporters of the (...) modernist thought. A unique place among Polish modernists belongs to Marian Zdziechowski (1861–1938) who was the only Polish participant of the international intellectual debate on the “modernisation” of Roman Catholicism. The paper analyses the development of Zdziechowski’s thought and shows that his main demand throughout the modernist debates was to create a new, more efficient apologetics, which would be grounded in the religious experience of the individual. (shrink)

This paper describes the process of reception of Catholic Modernism in Poland as well as the Polish contribution to this movement. It shows the Polish antimodernist perspective on modernistic thought. The neglect of Polish modernism was caused by the nationalistic character of the Polish theology and has resulted in absence of historical studies of Polish Catholic Modernism. Based on the results of archival and literature research the paper presents a variety of Polish Catholic Modernists and non-Catholic supporters of the (...) modernist thought. A unique place among Polish modernists belongs to Marian Zdziechowski (1861–1938) who was the only Polish participant of the international intellectual debate on the “modernisation” of Roman Catholicism. The paper analyses the development of Zdziechowski’s thought and shows that his main demand throughout the modernist debates was to create a new, more efficient apologetics, which would be grounded in the religious experience of the individual. (shrink)

Dynamic topological logic is a multi-modal logic that was introduced for reasoning about dynamic topological systems, i.e. structures of the form $\langle{\mathfrak{X}, f}\rangle $, where $\mathfrak{X}$ is a topological space and $f$ is a continuous function on it. The problem of finding a complete and natural axiomatization for this logic in the original tri-modal language has been open for more than one decade. In this paper, we give a natural axiomatization of $\textsf{DTL}$ and prove its strong completeness with respect to (...) the class of all dynamic topological systems. Our proof system is infinitary in the sense that it contains an infinitary derivation rule with countably many premises and one conclusion. It should be remarked that $\textsf{DTL}$ is semantically non-compact, so no finitary proof system for this logic could be strongly complete. Moreover, we provide an infinitary axiomatic system for the logic ${\textsf{DTL}}_{\mathcal{A}}$, i.e. the $\textsf{DTL}$ of Alexandrov spaces, and show that it is strongly complete with respect to the class of all dynamical systems based on Alexandrov spaces. (shrink)

The wide range of interpretations of aoristic and imperfective aspect in Ancient Greek cannot be attributed to unambiguous aspectual operators but suggest an analysis in terms of coercion in the spirit of de Swart (Nat Lang Linguist Theory 16:347–385, 1998). But since such an analysis cannot explain the Ancient Greek data, we combine Klein’s (Time in language, 1994) theory of tense and aspect with Egg’s (Flexible semantics for reinterpretation phenomena, 2005) aspectual coercion approach. Following Klein. (grammatical) aspect relates the runtime (...) of an eventuality and the current time of reference (topic time). We claim that these relations can trigger aspectual selection restrictions (and subsequent aspectual coercions) just like e.g. aspectually relevant temporal adverbials, and are furthermore susceptible to the Duration Principle of Egg (Flexible semantics for reinterpretation phenomena, 2005): Properties of eventualities must be compatible with respect to the duration they specify for an eventuality. The Duration Principle guides the selection between different feasible coercion operators in cases of aspectual coercion but can also trigger coercions of its own. We analyse the interpretations of aorist and imperfective as cases of coercion that avoid impending violations of aspectual selection restrictions or of the Duration Principle, which covers cases that are problematic for de Swart’s (Nat Lang Linguist Theory 16:347–385, 1998) analysis. (shrink)

The purpose of this paper is to show an application of variety engineering in the social realm. It focuses on reducing environmental complexity by catalysing self-organizing processes. This catalysis is based on the use of Sen and Nussbaum’s capabilities approach. By doing this an organization may improve the quality of the relations with their clients by transforming environmental agents into new suppliers. This approach opens a new dimension of social responsibility for organizations. A particular case is presented in which (...) a regional university in Colombia faces the challenge of attending low-income students coming from small municipalities. They designed a strategy to offer low-cost housing and a daily balanced meal without using the university's own resources. Instead, they managed to involve some agents of the community by developing their capabilities. This kind of variety engineering shows the possibility of new forms of social responsibility in universities by incorporating the capabilities approach into their managerial practices. This is something that is currently under study in other universities. (shrink)

We study which [Formula: see text]-distributive forcing notions of size [Formula: see text] can be embedded into tree Prikry forcing notions with [Formula: see text]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [Formula: see text]-distributive forcing notion of size [Formula: see text] be extended to a [Formula: see text]-complete ultrafilter.

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

This study concentrates on an investigation on how the variable of collaboration moderates the relationship between cultural typology of variety and work outcome in the cross-cultural work settings. The author predicts that collaboration will have impact on the relationship between variety of cultural character of gender egalitarian and task satisfactory. The empirical study conducted in the multinational companies located in China supported the assumptions. The result shows that by using the moderator of collaboration, gender variable is no longer (...) the cultural character of separation reflecting conflict in cultural diversity team, instead, under the circumstance of collaboration, the attribute of cultural variety – coexistence of symmetry and asymmetry (e.g. Gender egalitarian) demonstrates more inclusive solution that benefit work outcomes in cultural diversity team. (shrink)

This article aims to find the elements that are required for a common ethical approach that is suitable for the different perspectives adopted in integrative biodiversity conservation research. A general reflection on the integrity of research is a priority worldwide, with a common aim to promote good research practice. Beyond the relationship between researcher and research subject, the integrity of research is considered in a broader perspective which entails scientific integrity towards society. In research involving a variety of disciplines (...) and a diversity of legal and ethical frameworks, there is a need of harmony between different sets of values. The notion of congruence reflects the consistency of ethics in research within the biodiversity conservation’s community of researchers. It also bears on the coherence of values shared between the scientific community and society. We examine the notion of research integrity in a broad sense. This examination is to be conducted in relation to the goal of protecting ecological integrity, which is at the core of biodiversity conservation. The notion of integrity constraints should be investigated further to develop a pragmatic response to the need for integrity and congruence in research for biodiversity conservation. (shrink)

In the current research, we show that low-power state promotes variety-seeking. We argue that this effect occurs because consumers in low-power state desire control and variety-seeking as a way to restore sense of control. The effect of power on variety-seeking is reduced when consumer knowledge in a certain consumption domain is high because knowledge is an alternative way to retain sense of control. Three experiments provide systematic evidence for this effect across different product categories. These findings contribute (...) to the literature on how variety-seeking can be used as a way to compensate and enhance our understanding of power and consumer knowledge. (shrink)

The variety-of-evidence thesis has been criticized by Bovens and Hartmann. This article points to two limitations of their Bayesian model: the conceptualization of unreliable evidential sources as randomizing and the restriction to comparing full independence to full dependence. It is shown that the variety-of-evidence thesis is rehabilitated when unreliable sources are reconceptualized as systematically biased. However, it turns out that allowing for degrees of independence leads to a qualification of the variety-of-evidence thesis: as Bovens and Hartmann claimed, (...) more independence does not always imply stronger confirmation. (shrink)

Let C be a small Barr-exact category, Reg the category of all regular functors from C to the category of small sets. A form of M. Barr's full embedding theorem states that the evaluation functor e : C →[Reg, Set ] is full and faithful. We prove that the essential image of e consists of the functors that preserve all small products and filtered colimits. The concept of κ-Barr-exact category is introduced, for κ any infinite regular cardinal, and the natural (...) generalization to κ-Barr-exact categories of the above result is proved. The treatment combines methods of model theory and category theory. Some applications to module categories are given. (shrink)