We give an alternative and more informative proof that every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree is the meet of two incomparable ${\Sigma^{0}_{2}}$ -degrees, which allows us to show the stronger result that for every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree a, there exist enumeration degrees x 1 and x 2 such that a, x 1, x 2 are incomparable, and for all b ≤ a, b = (b ∨ x 1 ) ∧ (b ∨ x 2 ).

We study effectively inseparable prelattices $\wedge, \vee$ are binary computable operations; ${ \le _L}$ is a computably enumerable preordering relation, with $0{ \le _L}x{ \le _L}1$ for every x; the equivalence relation ${ \equiv _L}$ originated by ${ \le _L}$ is a congruence on L such that the corresponding quotient structure is a nontrivial bounded lattice; the ${ \equiv _L}$ -equivalence classes of 0 and 1 form an effectively inseparable pair of sets). Solving a problem in we show, that if (...) L is an e.i. prelattice then ${ \le _L}$ is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relation R there exists a computable function f reducing R to ${ \le _L}$, i.e., $xRy$ if and only if $f\left{ \le _L}f\left$, for all $x,y$. In fact ${ \le _L}$ is locally universal, i.e., for every pair $a{ < _L}b$ and every c.e. preordering relation R one can find a reducing function f from R to ${ \le _L}$ such that the range of f is contained in the interval $\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also ${ \le _L}$ is uniformly dense, i.e., there exists a computable function f such that for every $a,b$ if $a{ < _L}b$ then $a{ < _L}f\left{ < _L}b$, and if $a{ \equiv _L}a\prime$ and $b{ \equiv _L}b\prime$ then $f\left{ \equiv _L}f\left$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. preordering relation on ${{\rm{\Sigma }}_n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s system R or Q is locally universal and uniformly dense; and the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$$\end{document} in question, d > 1, we fix a natural extension of Peano Arithmetic, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T \supseteq \sf{PA}}$$\end{document}, that proves the corresponding second-order sentence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{WPO}\left({\rm (...) S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$$\end{document}. Having this we consider the following parametrized first-order slow well-partial-ordering sentence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| 1}$$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$$\end{document} is not provable in T.Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _{1}^{1}\sf{CA}}$$\end{document}, if d = 2 and d = 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f: \mathbb{N} \rightarrow \mathbb{R}_{+}}$$\end{document} and prove analogous theorems. (In the sequel we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}_{+}}$$\end{document} the set of EFA-provably computable positive reals). In the basic case T = PA we strengthen the basic phase transition result by adding the following static threshold clause\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$$\end{document} is still provable in T = PA (actually in EFA). Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1α given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}$$\end{document} being the familiar fast growing Hardy function and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}$$\end{document} the corresponding slowly growing inversion. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha < \varepsilon _{0}}$$\end{document}, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$$\end{document} is provable in T = PA. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$$\end{document} is not provable in T = PA. We conjecture that this pattern is characteristic for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T\supseteq \sf{PA}}$$\end{document} under consideration and their proof-theoretical ordinals o (T ), instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon _{0}}$$\end{document}. (shrink)

Let A be an infinite Δ₂⁰ set and let K be creative: we show that K≤Q A if and only if K≤Q₁ A. (Here ≤Q denotes Q-reducibility, and ≤Q₁ is the subreducibility of ≤Q obtained by requesting that Q-reducibility be provided by a computable function f such that Wf(x)∩ Wf(y)=∅, if x \not= y.) Using this result we prove that A is hyperhyperimmune if and only if no Δ⁰₂ subset B of A is s-complete, i.e., there is no Δ⁰₂ subset (...) B of A such that \overline{K}≤s B, where ≤s denotes s-reducibility, and \overline{K} denotes the complement of K. (shrink)

The influence of physician judgment on the disclosure, competency, understanding, voluntariness, and decision aspects of informed consent for bone marrow transplantation are described. Ethical conflicts which arise from the amount and complexity of the information to be disclosed and from the barriers of limited time, patient anxiety and lack of prior relationship between patient and physician are discussed. The role of the referring physician in the decision-making is considered. Special ethical issues which arise with use of healthy related bone marrow (...) donors are discussed, as is the physician's discretion in raising questions of competency. It is concluded that in this setting, regardless of the theoretical goals of the physician, patients appear to utilize informed consent discussions to assess their capacity to trust the physician rather than as a time to weigh the large amount of relevant data. The conscientious physician best serves the patient with recommendation of the best medical alternative rather than with attempts to remain neutral. (shrink)

Urban landscapes in the Roman world were covered in written text, from monumental building inscriptions to smaller, more personal texts of individual accomplishment and commemoration. In the East, Greek dominated these written landscapes, but Latin also appeared with some frequency, especially in places where a larger Roman audience was expected, such as major cities and Roman colonies. When Latin and Greek appear alongside each other, whether in the same inscription or across a single monumental space, we might ask what benefits (...) the sponsor of the monument hoped to gain from such a bilingual presentation, and whether each language was serving the same function. This paper considers the monumental entrance to the Pamphylian city of Perge as a case study for exploring this relationship between bilingual inscriptions and civic space. By surveying the display of both Greek and Latin on this entrance, examining how the entrance interacted with the broader linguistic landscape of Perge, and considering the effects that each language would have had on the viewer, I show that the use of language, and the variation between the languages, served not only to communicate membership in both Greek and Roman societies but also to delineate civic space from imperial space, both physically and symbolically. (shrink)

A complex world like ours demands for the teachers and professors to command intercultural competences in order to avoid the instrumentalization of the alterities. It is precisely the professionals of education who, given their social function, have the responsibility of forming the citizens of the future in attitudes and behaviours adjusted to plural communities. This article presents the first part of a research project carried out by researchers from Barcelona, Marseille, Rabat and Beirut on the complex world of the respect (...) necessary to face prejudices and stereotypes. (shrink)

M. Bonnefond-Coudry has performed a great service by compiling a list of senators who are known to have spoken in the senate in the first century b.c. Yet her list for the year 50 invites a thoroughgoing revision. Beside the rubric ‘supplicatio à Cicéron’ she gives the following list: Cato, Hirrus, Balbus, Lentulus , Domitius , Scipio, Favonius. She also notes that Pompey spoke at a session late in the year , and maintains that Scipio spoke on 1 December.

The praetorship of L. Rupilius is of great importance only to the biography of L. Rupilius. His consular candidacy has a wider significance, since his repulsa represents a reverse for his most prominent supporter, Scipio Aemilianus. As the praetorship is not explicitly mentioned in the sources, its terminus non post quem is fixed by the consular candidacy. Scholarly treatment of the question is hard to come by. The terminus post quem for the candidacy of Lucius is his brother's candidacy ; (...) the terminus ante quem, Scipio's death : Cicero tells us that Scipio brought about the election of P. Rupilius, but failed to make his brother Lucius consul. Broughton classified L. Rupilius as ‘Pr. by 133 ’, and in this seems to have followed Münzer, who indeed considered L. Rupilius ‘Praetor gegen 620 = 134’, but added that he was a candidate ‘urn das Consulat für 623 =131 oder noch eher für 624 = 130’. Of course, if L. Rupilius was a candidate in 131 for a consulship of 130, then he was praetor by 133. A candidacy in 1294 can be safely ruled out: Scipio died in the first half of the year, in spring or early summer. But we cannot rule out candidacy in 130 for 129: Scipio might have preferred L. Rupilius to either consul of that year. The latest possible date for the praetorship of L. Rupilius is therefore 132. (shrink)

M. Bonnefond-Coudry has performed a great service by compiling a list of senators who are known to have spoken in the senate in the first century b.c. Yet her list for the year 50 invites a thoroughgoing revision. Beside the rubric ‘supplicatio à Cicéron’ she gives the following list: Cato, Hirrus, Balbus, Lentulus, Domitius, Scipio, Favonius. She also notes that Pompey spoke at a session late in the year, and maintains that Scipio spoke on 1 December.

The praetorship of L. Rupilius is of great importance only to the biography of L. Rupilius. His consular candidacy has a wider significance, since his repulsa represents a reverse for his most prominent supporter, Scipio Aemilianus. As the praetorship is not explicitly mentioned in the sources, its terminus non post quem is fixed by the consular candidacy. Scholarly treatment of the question is hard to come by. The terminus post quem for the candidacy of Lucius is his brother's candidacy ; (...) the terminus ante quem, Scipio's death : Cicero tells us that Scipio brought about the election of P. Rupilius, but failed to make his brother Lucius consul. Broughton classified L. Rupilius as ‘Pr. by 133 ’, and in this seems to have followed Münzer, who indeed considered L. Rupilius ‘Praetor gegen 620 = 134’, but added that he was a candidate ‘urn das Consulat für 623 =131 oder noch eher für 624 = 130’. Of course, if L. Rupilius was a candidate in 131 for a consulship of 130, then he was praetor by 133. A candidacy in 1294 can be safely ruled out: Scipio died in the first half of the year, in spring or early summer. But we cannot rule out candidacy in 130 for 129: Scipio might have preferred L. Rupilius to either consul of that year. The latest possible date for the praetorship of L. Rupilius is therefore 132. (shrink)

Background: Previous studies have investigated the effects of dance interventions on Parkinson’s motor and non-motor symptoms in an effort to develop an integrated view of dance as a therapeutic intervention. This within-subject study questions whether dance can be simply considered a form of exercise by comparing a Dance for Parkinson’s class with a matched-intensity exercise session lacking dance elements like music, metaphorical language, and social reality of art-partaking.Methods: In this repeated-measure design, 7 adults with Parkinson’s were tested four times; before (...) and after a Dance for Parkinson’s class, as well as before and after a matched-intensity exercise session. Physiological measures included heart rate and electrodermal activity. Self-reported affect and body self-efficacy were collected. Gait symmetry and dual task cost were analyzed using the 6 min walking test and Timed-Up-and-Go test, respectively.Results: Average heart rate was the same for both conditions, while electrodermal activity was higher during Dance for Parkinson’s. Significant differences were found in body self-efficacy, beauty subscale, symmetry of gait, and dual task performance.Conclusion: Dance, compared to an exercise intervention of matched intensity, yields different outcomes through the means of intrinsic artistic elements, which may influence affective responses, the experience of beauty, self-efficacy, and gait performance. (shrink)

Our study focuses on the correlation of concept map (CMap) structures and learning success tested with short answer tests, taking into particular account the complexity of the subject matter. Novice sixth grade students created CMaps about two subject matters of varying difficulty. The correlation of the complexity of CMaps with the post-test was small but highly significant in both subject matters. The complexity of the CMaps correlated with the long-term knowledge in the difficult subject matter but not in the context (...) of the easy one. Furthermore, the high number of technical errors makes it close to impossible to estimate students? knowledge. In summary, CMaps do not provide an adequate alternative to conventional short answer knowledge tests, but together with them they may offer a better comprehension of a student?s knowledge structure and aid in the preparation of further instruction tailored to individual needs. (shrink)

My article critically evaluates five key claims in Kerr’s interpretation of Aquinas’s De Ente et Essentia, ch. 4, proof for God. The claims are: the absolutely considered essence is a second intention, or cognitional being; à la John Wippel, the real distinction between essence and existence is known before the proof; contra David Twetten, Aristotelian form is not self-actuating and so requires actus essendi; the De Ente proof for God uses the Principle of Sufficient Reason; an infinite regress must be (...) eliminated before concluding to God. This author wonders if these questionable claims are traceable to the mindset of analytic philosophy which values precision and discreteness and so can fail to appreciate crucial paradoxes in Aquinas’s metaphysics. (shrink)

By providing a phenomenological presentation of Aquinas’s duplex operatio intellectus, the author argues that a reader is better equipped to understand where and when Aquinas arrives at the real distinction between essence and existence in the much disputed De Ente et Essentia, chapter four. “Phenomenological presentation” means an honest description of one’s own mental life as it conducts the duplex operatio. From phenomenological observations in the Thomistic texts, the author argues that a penetrative and rebounding movement of attention upon some (...) initially presented multiplicity characterizes the duplex operatio. When this dynamic is conducted upon the multiplicity of a real thing juxtaposed to itself cognitionally existing in sensation, the rebound of the secunda operatio presents the real existence in a sui generis relation of priority to the individual thing understood as existence neutral. Unfortunately, the rebound of attention is too quick to discern accurately the nature of the borderline between the attribute of existence and its subject, the individual thing. The attribute of existence may actuate the thing by shading into it or by remaining distinct from it. Fortunately, the phenomenological situation is sufficient to leave phenomenology and to initiate the third intellectual operation of reasoning. Reasoning concludes to a first cause of phenomenologically observable attributive existence. In this first cause existence is the thing itself. In order for this first cause to have its proper effect corresponding to what it is, the author argues that the second of the above two alternatives for phenomenologically observable attributive existence must be the true one. After linking these reflections to stages of De Ente, chapter 4, the author critically relates them to other interpretations of the text. These interpretations include those of Cahalan, McDonald, Wippel, Dewan, Patt, Kenny, and Owens. (shrink)

Inspired by a discussion about whether John Paul II grounded human dignity in a Kantian way, viz., emphasizing the person as an end unto itself, the author considers: (1) the relations between Kant and Aquinas on the topic of the philosophical basis of human dignity, and (2) John Paul II’s remarks on Kant’s ethics. He concludes that: (1) both Kant and Aquinas ground human dignity upon human freedom, but both understand the human freedom differently; (2) for Kant, human freedom is (...) self-legislating and so exercised without rational direction; (3) the Thomistic notion of freedom is compatible with rational direction which consists, e.g., in the human understood as an intellector of being or as a willer of the good, though neither seem to be exploited by Wojtyla. (shrink)