We study the Katětov order on Borel ideals. We prove two structural theorems, one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals.

We continue with the study of the Katětov order on MAD families. We prove that Katětov maximal MAD families exist under $\mathfrak {b=c}$ and that there are no Katětov-top MAD families assuming $\mathfrak {s\leq b}.$ This improves previously known results from the literature. We also answer a problem form Arciga, Hrušák, and Martínez regarding Katětov maximal MAD families.

A family $$\mathcal {I}$$ I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal $$\mathcal {I}$$ I on X is below an ideal $$\mathcal {J}$$ J on Y in the Katětov order if there is a function $$f{: }Y\rightarrow X$$ f : Y → X such that $$f^{-1}[A]\in \mathcal {J}$$ f - 1 [ A ] ∈ J for every $$A\in \mathcal (...) {I}$$ A ∈ I. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent. (shrink)

We show that there is a $$ \varvec{\Sigma }_4^0$$ ideal such that it’s neither extendable to any $$ \varvec{\Pi }_3^0$$ ideal nor above the ideal $$ \textrm{Fin}\times \textrm{Fin} $$ in the sense of Katětov order, answering a question from M. Hrušák.

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size.

Minami–Sakai :883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \ ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders. In (...) the course of the proof of the latter result, we also prove that for any analytic ideal \ there is a Borel ideal \ with \. (shrink)

We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of Fréchet-Urysohn property and, in a special case, using sequential properties of the space of finitely-supported probability measures with the weak* topology. We investigate consistency of some inequalities between these invariants and classical ones, and other related combinatorial questions. At last, we discuss maximality properties of almost disjoint families related to certain (...) ordering on ideals. (shrink)

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices (...) of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (shrink)

We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$ -destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$ -positive set which has finite intersection with every $A\in \mathcal {I}\cap V$. Also, we discuss the associated variants $$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y| \omega $ ; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$, and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$ -destroys $\mathcal {I}$ ; (...) and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal. (shrink)

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study the 1—1 or (...) constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

Knížka je určena každému, kdo se zajímá o logickou stavbu a základy matematiky a o začátky moderní matematické logiky. Aniž předpokládala zvláštních věcných znalostí, navazujíc na tradiční středoškolskou logiku, přibližuje čtenáři výraz a methodu logiky moderní. V podstatě jde o řešení dvou otázek: jak se odvozují matematické věty a jaké jsou logické základy matematiky. Probírá spojování výroků, výrokové vzorce, obecné a existenční výroky, logickou dedukci adruhy důkazů, některé matematické pojmy (množina, vztah, zobrazení, funkce), definice a jejich vlastnosti i systémy axiomů (...) a požadavky, jež na ně klademe. (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.

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)

Introduction -- Part 1. The nature of order in world politics: the concept of order in world politics; does order exist in world politics?; how is order maintained in world politics?; order versus justice in world politics -- Part 2. Order in the contemporary international system: the balance of power and international order; international law and international order; diplomacy and international order; war and international order; the great powers and international (...) order -- Part 3. Alternative paths to world order: alternatives to the contemporary states system; the decline of the states system; the obsolescence of the states system; the reform of the states system; conclusion. (shrink)

Like the latter, it unites into one and the same function the possibility of giving things a sign, of representing one thing by another, and the possibility of causing a sign to shift in relation to what it designates. The four functions that define the ...

In his classic work, _Wholeness and the Implicate Order_, David Bohm develops a theory of quantum physics which treats the totality of existence, including matter and consciousness, as an unbroken whole. David Bohm presents a rational and scientific theory which explains cosmology and the nature of reality; written clearly, and without the use of technical jargon, it is essential reading for those interested in physics, philosophy, psychology and the connection between consciousness and matter. David Bohm was one of the foremost (...) scientific thinkers and distinguished theoretical physicists of his generation. His best-known works include: _Quantum Theory_,_Causality and Chance in Modern Physics_ and _The Undivided Universe_. (shrink)

The persistent interpretation problem for quantum mechanics may indicate an unwillingness to consider unpalatable assumptions that could open the way toward progress. With this in mind, I focus on the work of David Bohm, whose earlier work has been more influential than that of his later. As I’ll discuss, I believe two assumptions play a strong role in explaining the disparity: 1) that theories in physics must be grounded in mathematical structure and 2) that consciousness must supervene on material processes. (...) I’ll argue that the first assumption appears to lead us toward Everett’s many worlds interpretation, which suggests a red flag. I’ll also argue that the second assumption is suspect due to the persistent explanatory gap for consciousness. Later, I explore ways that Bohm’s later work holds some promise in providing a better fit with our world, both phenomenologically and empirically. Also, I’ll address the possible problem of realism. (shrink)

Why do we remember the past and not the future? What does it mean for time to "flow"? Do we exist in time or does time exist in us? In lyric, accessible prose, Carlo Rovelli invites us to consider questions about the nature of time that continue to puzzle physicists and philosophers alike. For most readers this is unfamiliar terrain. We all experience time, but the more scientists learn about it, the more mysterious it remains. We think of it as (...) uniform and universal, moving steadily from past to future, measured by clocks. Rovelli tears down these assumptions one by one, revealing a strange universe where at the most fundamental level time disappears. He explains how the theory of quantum gravity attempts to understand and give meaning to the resulting extreme landscape of this timeless world. Weaving together ideas from philosophy, science and literature, he suggests that our perception of the flow of time depends on our perspective, better understood starting from the structure of our brain and emotions than from the physical universe. (shrink)

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that (...) is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

First published in 1984, this is a study of categorization practices: how people categorize each other and their actions; how they describe, infer, and judge. The book presents a sociological analysis and description of practical activities and makes a cogent contribution to the study of how the moral order actually works in practical communicative contexts. Among the issues dealt with are: collectivity categorizations, the organization of lists and descriptions, moral attribution and inferences, and the relationship between standards of morality (...) and standards of rationality. (shrink)

If we speak of degrees of similarity, what kinds of judgment are we assuming to make sense? It will be argued that the necessary and sufficient condition for there to be degrees of similarity is that there should be a four-termed relation of comparative similarity — w resembles x at least as much as y resembles z—obeying certain constraints. Of course, nothing turns on how we use the words 'degree of similarity'. Rather, the point is to distinguish the different levels (...) of ideologicalcommitment which different kinds of judgment of similarity involve. (shrink)

In The Order of Things (OT), Foucault recounts the birth and imminent death of Man as an object of study for science and philosophy. Foucault's point is that this very notion of “Man” is dependent on a particular transformation in the history of Being. The mere formulation of this hypothesis opens up a whole series of questions. First, is it true that Man has only become an object of concern in the late eighteenth century. Secondly, if Man has indeed (...) only recently become an object of theoretical interest, is Foucault's description and explanation of this process correct. Thirdly, why should we overcome the concept of “ Man”. Fourthly, in what sense was structuralism deemed to help us overcoming the limitations of this paradigm. In conclusion, if OT has to be overcome, it seems that it should be on grounds slightly different from those usually advanced. (shrink)

First published in 1984, this is a study of categorization practices: how people categorize each other and their actions; how they describe, infer, and judge. The book presents a sociological analysis and description of practical activities and makes a cogent contribution to the study of how the moral order actually works in practical communicative contexts. Among the issues dealt with are: collectivity categorizations, the organization of lists and descriptions, moral attribution and inferences, and the relationship between standards of morality (...) and standards of rationality. (shrink)

This paper argues that higher-order doubt generates an epistemic dilemma. One has a higher-order doubt with regards to P insofar as one justifiably withholds belief as to what attitude towards P is justified. That is, one justifiably withholds belief as to whether one is justified in believing, disbelieving, or withholding belief in P. Using the resources provided by Richard Feldman’s recent discussion of how to respect one’s evidence, I argue that if one has a higher-order doubt with (...) regards to P, then one is not justified in having any attitude towards P. Otherwise put: No attitude towards the doubted proposition respects one’s higher-order doubt. I argue that the most promising response to this problem is to hold that when one has a higher-order doubt about P, the best one can do to respect such a doubt is to simply have no attitude towards P. Higher-order doubt is thus much more rationally corrosive than non-higher-order doubt, as it undermines the possibility of justifiably having any attitude towards the doubted proposition. (shrink)

Aristotle attaches particular significance to the homomyny of many of the central concepts in philosophy and science: that is, to the diversity of ways of being that are denoted by a single concept. Shields here investigates and evaluates Aristotle's approach to questions about homonymy, characterizing the metaphysical and semantic commitments necessary to establish the homonymy of a given concept. Then, in a series of case studies, he examines in detail some of Aristotle's principal applications of homonymy--to the body, sameness and (...) and oneness, life, goodness, and being. This first full-length study of a central aspect of Aristotle's thought will interest philosophers working in a number of areas. (shrink)

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)

This chapter discusses how phenomenologies of pregnancy challenge traditional philosophical accounts of a subject that is seen as autonomous, rational, genderless, unified, and independent from other subjects. Pregnancy defies simple incorporation into such universal accounts since the pregnant woman and her unborn child are incapable of being subsumed into traditional theories of the subject. Phenomenological descriptions of the experience of pregnancy lead one to question if philosophy needs to reject the subject altogether as central, or rather to revise traditional descriptions (...) of the subject. The chapter examines both options and argues for the later. The exploration of pregnancy in feminist theory upholds the value of working from the subject’s lived experience, but indicates that it is possible without viewing the subject as a disembodied universal agent. Finally, it discusses how phenomenologies of pregnancy are attuned to discussing difference thereby aiding philosophies that take into account the political, historical, and cultural conditioning that shape experience and theory. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency (...) about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

Second-order logic and modal logic are both, separately, major topics of philosophical discussion. Although both have been criticized by Quine and others, increasingly many philosophers find their strictures uncompelling, and regard both branches of logic as valuable resources for the articulation and investigation of significant issues in logical metaphysics and elsewhere. One might therefore expect some combination of the two sorts of logic to constitute a natural and more comprehensive background logic for metaphysics. So it is somewhat surprising to (...) find that philosophical discussion of secondorder modal logic is almost totally absent, despite the pioneering contribution of Barcan. (shrink)

One of the most striking innovations in Aristotle’s philosophical writing is also one of its most characteristic features. That feature is Aristotle’s idea that terms central to philosophy, including ‘cause’ [aition], ‘good’, and even the verb ‘to be’, are, as he likes to put it, “said in many ways.” To be sure, philosophers before Aristotle give some evidence of having recognized the phenomenon of being said in many ways. Plato, in particular, suggests that things in this world that we call (...) “just” are only namesakes of the Just Itself. But, beyond that, homonymy seems to have been, even for Plato, no more than a source of ambiguity for wordplay. Aristotle is apparently the first philosopher to propose a taxonomy of homonymy and to view the identification and analysis of homonymies as a genuinely philosophical project. (shrink)

I ate my broccoli, though my broccoli did not eat me. The eating relation, like many other relations, differentiates between its arguments. The fact that eating holds between a and b does not entail that it holds between b and a. How are we to make sense of this? The standard view is that relations are sensitive to the order of their arguments. As natural as this view is, it has been the target of a powerful objection from Kit (...) Fine. I examine Fine’s objection and defend the standard view. (shrink)