We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H , be the set of order types of ℵ 1 -dense homogeneous subsets of (...) R with the relation of embeddability. We prove that for every finite model L , : Con iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power ℵ 1 , and { U 0 ,\h.;, U n−1 } is a cover of D ▪ X x X -} x,x> ¦ x ϵ X } consisting of symmetric open sets, then X can be partitioned into { X i \brvbar; i ϵ ω } such that for every i ϵ ω there is l such that D ⊇ U l ”. We also prove that MA+OCA [xrArr] 2 ℵ 0 = ℵ 2. (shrink)

We prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for PC classes. Let $M \models Q^nx_1 \cdots x_n \varphi(x_1 \cdots x_n)$ mean that there is an uncountable subset A of |M| such that for every $a_1, \ldots, a_n \in A, M \models \varphi\lbrack a_1, \ldots, a_n\rbrack$ . Theorem 1.1 (Shelah) $(\diamond_{\aleph_1})$ . For every n ∈ ω the class $K_{n + 1} = \{\langle A, R\rangle \mid \langle A, R\rangle \models \neg Q^{n + (...) 1} x_1 \cdots x_{n + 1} R(x_1, \ldots, x_{n + 1})\}$ is not an ℵ 0 -PC-class in the logic L n , obtained by closing first order logic under Q 1 , ..., Q n . I.e. for no countable L n -theory T, is K n + 1 the class of reducts of the models of T. Theorem 1.2 (Rubin) $(\diamond_{\aleph_1}).^3$ . Let $M \models Q^E x y\varphi(x, y)$ mean that there is $A \subseteq |M|$ such that $E_{A, \varphi} = \{\langle a, b \rangle \mid a, b \in A$ and $M \models \varphi\lbrack a, b\rbrack\}$ is an equivalence relation on A with uncountably many equivalence classes, and such that each equivalence class is uncountable. Let $K^E = \{\langle A, R\rangle\mid \langle A, R\rangle\models \neg Q^Exy R(x, y)\}$ . Then K E is not an ℵ 0 -PC-class in the logic gotten by closing first order logic under the set of quantifiers {Q n ∣ n ∈ ω} which were defined in Theorem 1.1. (shrink)

THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ 1 ) If (...) B is an atomic ℵ 1 -saturated infinite BA, ψ ε L ω 1ω and $\langle B, \operatorname{Aut} (B)\rangle \models\psi$ then there is B 1 such that $|\operatorname{Aut}(B_1)| \leq |B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut}(B_1)\rangle\models\psi$ . In particular if B is 1-homogeneous so is B 1 . (b) (a) holds for B = P(ω) even if we assume only CH. (shrink)

Assuming GCH, we prove that for every successor cardinal μ > ω1, there is a superatomic Boolean algebra B such that |B| = 2μ and |Aut B| = μ. Under ◊ω1, the same holds for μ = ω1. This answers Monk's Question 80 in [Mo].

For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal α = α(T) such that We show that for some and for all other T‘s, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize them by invariants similar (...) to the Tarski’s invariants. (shrink)

A Boolean algebra B that has a well-founded sublattice L which generates B is called a well-generated Boolean algebra. If in addition, L is generated by a complete set of representatives for B , then B is said to be canonically well-generated .Every WG Boolean algebra is superatomic. We construct two basic examples of superatomic non well-generated Boolean algebras. Their cardinal sequences are 1,0,1,1 and 0,0,20,1.Assuming MA , we show that every algebra with one of the cardinal sequences , α<1, (...) λ<20, or 0,20,1,1 is CWG.Assuming CH, or alternatively assuming MA , we determine which cardinal sequences admit only WG Boolean algebras.We find a necessary and sufficient condition for the canonical well-generatedness of algebras whose cardinal sequence has the form , α<1. We conclude that if such an algebra is CWG, then all of its quotients are CWG. We show that the above is not true for general Boolean algebras. We also conclude that if the cardinality of such an algebra is less than the cardinal defined below, then it is CWG. The cardinal is the least cardinality of an unbounded subset of {f f : ω → ω}.We investigate questions concerning embeddability, quotients and subalgebras of WG and CWG Boolean algebras, and construct various counter-examples. (shrink)

For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B1 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then $\langle M_T, \leq\rangle$ is well-quasi-ordered. ■ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that $B\upharpoonright (...) a$ is an atomic Boolean algebra and $B\upharpoonright s$ is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that $\langle A, \leq\rangle$ is a partial well-quasi-ordering, if it is a partial quasi-ordering and for every $\{a_i\mid i \in \omega\} \subseteq A$ , there are $i < j < \omega$ such that ai ≤ aj. Theorem 2. $\langle M_{T_\omega}, \leq\rangle$ contains a subset M such that the partial orderings $\langle M, \leq \upharpoonright M \rangle$ and $\langle\mathscr{P}(\omega), \subseteq\rangle$ are isomorphic. ■ Let M'0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M'0, let B1 ≤' B2 mean that B1 is embeddable in B2. Remark. $\langle M'_0, \leq'\rangle$ is well-quasi-ordered. ■ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering. (shrink)

Let B be a superatomic Boolean algebra (BA). The rank of B (rk(B)), is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B - {0}, then the rank of a in B (rk(a)), is defined to be the rank of the Boolean algebra $B b \upharpoonright a \overset{\mathrm{def}}{=} \{b \in B: b \leq a\}$ . The rank of 0 B is defined to be -1. An element a ∈ B - {0} is (...) a generalized atom $(a \in \widehat{At}(B))$ , if the last nonzero cardinal in the cardinal sequence of B $\upharpoonright$ a is 1. Let a,b $\in\widehat{At}$ (B). We denote a ∼ b, if rk(a) = rk(b) = rk(a · b). A subset H $\subseteq \widehat{At}$ (B) is a complete set of representatives (CSR) for B, if for every a $\in \widehat{At}$ (B) there is a unique h ∈ H such that h ∼ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B. THEOREM 1. Let B be a Boolean algebra with cardinal sequence $\langle\aleph_0: i . If B is CWG, then every subalgebra of B is CWG. A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1. Theorem 1 follows from Theorem 2.9, which is the main result of this work. For an ESL BA B we define a set F B of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent. (1) Every subalgebra of B is CWG; and (2) F B is bounded. THEOREM 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated. (shrink)

We prove that the poset algebra of every scattered poset with finite width is embeddable in the poset algebra of a well ordered poset.

During multiple testing, researchers often adjust their alpha level to control the familywise error rate for a statistical inference about a joint union alternative hypothesis (e.g., “H1,1 or H1,2”). However, in some cases, they do not make this inference. Instead, they make separate inferences about each of the individual hypotheses that comprise the joint hypothesis (e.g., H1,1 and H1,2). For example, a researcher might use a Bonferroni correction to adjust their alpha level from the conventional level of 0.050 to 0.025 (...) when testing H1,1 and H1,2, find a significant result for H1,1 (p < 0.025) and not for H1,2 (p > .0.025), and so claim support for H1,1 and not for H1,2. However, these separate individual inferences do not require an alpha adjustment. Only a statistical inference about the union alternative hypothesis “H1,1 or H1,2” requires an alpha adjustment because it is based on “at least one” significant result among the two tests, and so it refers to the familywise error rate. Hence, an inconsistent correction occurs when a researcher corrects their alpha level during multiple testing but does not make an inference about a union alternative hypothesis. In the present article, I discuss this inconsistent correction problem, including its reduction in statistical power for tests of individual hypotheses and its potential causes vis-à-vis error rate confusions and the alpha adjustment ritual. I also provide three illustrations of inconsistent corrections from recent psychology studies. I conclude that inconsistent corrections represent a symptom of statisticism, and I call for a more nuanced inference-based approach to multiple testing corrections. (shrink)

Terence Horgan and Mark Timmons? Moral Twin Earth thought experiment shows that realist ethical naturalism entails a kind of conceptual relativism about moral predicates. This conceptual relativism implies, further, that Earthlings and Twin Earthlings do not express substantive disagreement with one another. Because this latter implication clashes with considered linguistic intuitions, Horgan and Timmons conclude that we should reject realist ethical naturalism. Against this, several critics recommend that realists ?bite the bullet? with respect to Moral Twin Earth: despite our intuitions, (...) we should accept that the speakers do not disagree. These critics argue that the intuitions elicited by the thought experiment are either the result of a failure on the part of readers to appreciate hidden details of the Moral Twin Earth case, or else they are mere artifacts of readers? epistemic limitations. In this paper, I show that the Moral Twin Earth argument prevails against this line of attack. (shrink)

At last, Russia has begun to speak in a truly original voice." So said Anatoly Vaneev, a Soviet dissident who became Karsavin's disciple in the Siberian gulag where the philosopher spent his last two years. The book traces the unusual trajectory of this inspiring voice: Karsavin started his career as Russia's brightest historian of Catholic mysticism; however, his radical methods - which were far ahead of their time - shocked his conservative colleagues. The shock continued when Karsavin turned to philosophy, (...) writing flamboyant and dense essays in a polyphonic style, which both Marxists and religious traditionalists found provocative. There was no let-up after he was expelled by Lenin from Soviet Russia: in exile, he became a leading theorist in the Eurasian political movement, combining Orthodox theology with a left-wing political orientation. (shrink)

莊子為什麼不屬於道家？ 如果他不是道家人物，他的思想如何定位？ 一個新的莊子圖像對於我們今日了解儒家及中國哲學， 能夠提供什麼樣的新視野？ 莊子道家說始於漢代，但這種學派歸屬在歷代常受到修正，楊儒賓最新論著《儒門內的莊子》可視為第三波修正運動中的一環。繼承晚明王夫之、方以智的莊子觀，透過重新解讀莊子的語言、隱喻、技術、身體、氣化諸概念，主 張莊子提供一種新的主體範式，這種新的主體範式奠定了非定命論的氣化世界觀。本書指出，莊子和孔子有特殊的地理風土與文化風土的淵源，孔子是《莊子》內篇的典範人物，兩人在中國文化史上的軸心時代，有另類的人文精 神的傳承關係。.

"Dr. Rubin has brought cognitive psychology into a wholly unprecedented dialogue with studies in oral tradition. The result is a truly new perspective on memory and the processes of oral tradition." --John Miles Foley, University of Missouri.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:An Exposition of The Divine Names, The Book of Blessed Dionysius by Thomas AquinasMichael J. Rubin, Elizabeth C. Shaw, and Staff*AQUINAS, Thomas. An Exposition of The Divine Names, The Book of Blessed Dionysius. Translated and edited with an introduction by Michael A. Augros. Merrimack, N.H.: Thomas More College Press, 2021. xxv + 549 pp. Cloth, $65.00The profound influence that Pseudo-Dionysius had on Aquinas’s thought, especially in his (...) metaphysics of participation and his apophatic theology, has been known for some time now, thanks to the excellent work of such scholars as Fran O’Rourke. One would therefore think that Aquinas’s Exposition of the Divine Names (his only commentary on the works of Pseudo-Dionysius) would have been translated a lot sooner. This delay is assuredly due not just to the difficulty of translating this work but also to a certain embarrassment felt by Thomists over Aquinas’s sincere belief that the “Dionysius” who wrote the Divine Names was the Areopagite whose conversion at the teaching of St. Paul is mentioned in the Acts of the Apostles. That the author calling himself “Dionysius” was in fact a theologian writing in the fifth or sixth centuries has been well established by modern scholarship. Hence, Thomists may have felt tempted to discount or at least pass over the influence of this author on Aquinas since it is discomfiting to think that the Church’s greatest theologian accorded such authority to an imposter.Michael Augros tackles these concerns head-on in the excellent introduction to his new translation of the Divine Names commentary. First, Augros points out that in past centuries a pseudonym was usually not an act of forgery but, rather, a way of concealing the author’s identity, which an early Christian theologian might well decide to do either out of humility or out of a desire “to avoid provoking certain prejudices” in those who disagreed with him in contemporary theological disputes. Moreover, it is likely that the work was never intended for the general public (who could easily mistake the pseudonym as genuine) but, rather, was written for those “already initiated in a particular Neoplatonic school of Christian theology,” who would recognize the author as being “one of their own” but without knowing his precise identity. Finally, in taking the particular name of “Dionysius,” the author of the Divine Names was indicating his intention for the work: Just as the biblical Dionysius was a Greek philosopher who converted to Christianity, so the works of Pseudo-Dionysius [End Page 345] are an attempt to reconcile Greek philosophy with the Christian faith.It therefore seems likely, as Augros observes, that Aquinas would still have accorded a great deal of respect to the Divine Names even if he had learned the truth of the author’s identity. The work would not have had the same deuterocanonical authority for him that comes with being written by a direct disciple of St. Paul himself, but “the inherent cogency, coherence, and beauty” of Pseudo-Dionysius’s theology would have been the same. Hence, Pseudo-Dionysius would surely have earned from Aquinas the “docile attitude” appropriate “for a philosopher and theologian of the first order, such as Boethius.”That Aquinas was right to see the author of the Divine Names as a master philosopher and theologian becomes evident when one dives into the text itself, in which Aquinas sees a truly dazzling order, both externally and internally. According to Aquinas, the Divine Names discusses those truths about God of which there are intelligible likenesses in creatures, such as goodness or wisdom, whereas the truths about God of which there are sensible likenesses (for example, “when God is called a lion, or a rock”) are treated by Pseudo-Dionysius in The Symbolic Theology; meanwhile, in the lost or fictional work The Divine Characteristics, Dionysius claims to have treated those truths about God (such as the Trinity) that cannot be known through any likenesses in creatures.Aquinas finds a still more intricate order within the Divine Names itself. Since this work is dedicated to “the divine names by which the processions of God to creatures are manifested,” the plan of the work... (shrink)

"Dr. Rubin has brought cognitive psychology into a wholly unprecedented dialogue with studies in oral tradition. The result is a truly new perspective on memory and the processes of oral tradition." --John Miles Foley, University of Missouri.

There is increasing interest in determining what impact having women in management positions may have on corporate social responsibility initiatives. Various authors suggest that gender equality practices should be factored into the broader framework of CSR. This paper examines how the presence of women on corporate boards, in top and middle management and as heads of CSR departments, influences gender equality practices in the field of CSR. Using information collected from companies that have signed up to Women's Empowerment Principles in (...) Spain, we show that the presence of women in the aforesaid posts has a positive impact on CSR activities with gender equality objectives. We thus supplement the justice, business and moral arguments with further arguments in support of the incorporation of women into not only corporate boards but all management positions. Finally, we provide a view of how gender equality can be included in the broader framework of CSR. (shrink)

Yoga 365 presents a year's worth of daily readings that invite yoga lovers of every skill level to bring the inspiration they experience on their mats into their everyday lives. Each entry explores a mind-body theme such as balance, strength, and resilience in a short, illuminating paragraph that can be enjoyed in the morning or at bedtime, incorporated into a yoga session, or read on the go. Featuring a serenely beautiful hardcover and a spacious, color-washed interior, the package is as (...) calming in the hand as the readings are to the eye. Yogis will find it to be a motivating tool for personal growth and a lighthearted, gift-worthy way to share the joys of their practice with others. (shrink)

本書為漢語世界第一部探討「自然」概念在中國文化史上流變的著作， 從概念史角度出發，探究「自然」概念的多重淵源與多元脈絡。 「自然」是當代漢語學術詞彙中語義極多源也極多元的重要概念。自《老子》提出「道法自然」以來，「自然」概念的流變始終與中國思想史的發展脈絡並行，而如同「道」、「太極」、「涅槃」一般，是極為重要的理論概念， 在中國文化史各領域亦扮演關鍵性的角色。 本書是漢語世界第一部從概念史的觀點出發，探討「自然」概念在中國文化史上流變的著作。全書由十二個單元組成，撰述的角度是對「自然」一詞的後設反省，內容跨越的年代由先秦直至20世紀，作者分別來自臺灣、香港、 美國的人文學者，議題新穎，觀點多元，一編在手，可大致瀏覽「自然」在中國文化史上的流變過程。.

A structure has a (finite-string) automatic presentation if the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

This paper attempts to show how certain approaches in Values Based Practice duplicate key moves in liberal political theory.

Traduação -/- O positivismo jurídico de Kelsen é frequentemente acusado de submeter o judiciário alemão ao direito nazista. Sobretudo a insistência do autor na separação entre direito e moral foi considerada uma deficiência crucial. Rejeito essa crítica. Meu argumento consiste na afirmação de que a tese de Kelsen, da distinção entre direito e moral em duas esferas normativas próprias, refuta tal acusação, sabendo que os juristas do programa nazista almejavam a ‘unificação do direito e da moral’ com o fim de (...) expandir a autoridade e o poder do regime. Critico, entretanto, o relativismo moral de Kelsen, posição essa que o torna vulnerável às objeções pós-guerra, as quais acusavam o positivismo jurídico de insuficiente para prover a devida salvaguarda contra a perversão do direito pelos nazistas. (shrink)