I first seriously contemplated writing a book on degree theory in 1976 while I was visiting the University of Illinois at Chicago Circle. There was, at that time, some interest in ann-series book about degree theory, and through the encouragement of Bob Soare, I decided to make a proposal to write such a book. Degree theory had, at that time, matured to the point where the local structure results which had been the mainstay of the earlier papers in the area (...) were finding a steadily increasing number of applications to global degree theory. Michael Yates was the first to realize that the time had come for a systematic study of the interaction between local and global degree theory, and his papers had a considerable influence on the content of this book. During the time that the book was being written and rewritten, there was an explosion in the number of global theorems about the degrees which were proved as applications of local theorems. The global results, in turn, pointed the way to new local theorems which were needed in order to make further progress. I have tried to update the book continuously, in order to be able to present some of the more recent results. It is my hope to introduce the reader to some of the fascinat ing combinatorial methods of Recursion Theory while simultaneously showing how to use these methods to prove some beautiful global theorems about the degrees. (shrink)

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.

This book presents a unifying framework for using priority arguments to prove theorems in computability. Priority arguments provide the most powerful theorem-proving technique in the field, but most of the applications of this technique are ad hoc, masking the unifying principles used in the proofs. The proposed framework presented isolates many of these unifying combinatorial principles and uses them to give shorter and easier-to-follow proofs of computability-theoretic theorems. Standard theorems of priority levels 1, 2, and 3 are chosen to demonstrate (...) the framework's use, with all proofs following the same pattern. The last section features a new example requiring priority at all finite levels. The book will serve as a resource and reference for researchers in logic and computability, helping them to prove theorems in a shorter and more transparent manner"--Provided by publisher. (shrink)

A choice set for a computable linear ordering is a set which contains one element from each maximal block of the ordering. We obtain a partial characterization of the computable linear order-types for which each computable model has a computable choice set, and a full characterization in the relativized case; Every model of the linear order-type α of degree ≤ d has a choice set of degree ≤ d iff α can written as a finite sum of order-types, each of (...) which either has finitely many blocks, or has order-type n · η for some integer n. (shrink)

Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number of maximal sets whose many-one degrees are pairwise (...) incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′. The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x ∈, where M is a maximal set, and M is its complement. We showed that / 1 ≡ / 2 if, and only if, 1 = m 2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree. (shrink)

We exhibit a finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees. Our method promises to lead to a full characterization of the finite lattices embeddable into the enumerable Turing degrees.

We investigate homomorphisms of degree structures with various relations, functions and constants. Our main emphasis is on pseudolattices, i.e., partially ordered sets with a join operation and relations simulating the meet operation. We show that there are no finite quotients of the pseudolattice of degrees or of the pseudolattice of degrees 0′, but that many finite distributive lattices are pseudolattice quotients of the pseudolattice of computably enumerable degrees.

We show that the lattice L 20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (J). We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π 2 necessary and sufficient condition for embedding a finite pseudolattice into J.

The degrees of unsolvability were introduced in the ground-breaking papers of Post [20] and Kleene and Post [7] as an attempt to measure theinformation contentof sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally in mathematics; in particular, they wished to know when a solution to one decision problem contained the information necessary to solve a second decision problem. As decision problems can be coded by sets of natural numbers, this question (...) is equivalent to: Given a computer with access to an oracle which will answer membership questions about a setA, can a program be written which will correctly compute the answers to all membership questions about a setB? If the answer is yes, then we say thatBisTuring reducibletoAand writeB≤TA. We say thatB≡TAifB≤TAandA≤TB. ≡Tis an equivalence relation, and ≤Tinduces a partial ordering on the corresponding equivalence classes; the poset obtained in this way is called thedegrees of unsolvability, and elements of this poset are calleddegrees.Post was particularly interested in computability from sets which are partially generated by a computer, namely, those for which the elements of the set can be enumerated by a computer. (shrink)

We show the decidability of the existential theory of the recursively enumerable degrees in the language of Turing reducibility, Turing reducibility of the Turing jumps, and least and greatest element.

We describe the motivation for the construction of a general framework for priority arguments, the ideas incorporated into the construction of the framework, and the use of the framework to prove theorems in computability theory which require priority arguments.

Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite $\Pi _1^0 $ chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable (...) antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite $\Pi _1^0 $ chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite $\Pi _1^0 $ antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA₀ to WSCAC, the corresponding principle for weakly stable posets. (shrink)

We give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are (...) disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theory answers a question raised by Shore and Slaman in [21]. (shrink)

We divide the class of infinite computable trees into three types. For the first and second types, 0' computes a nontrivial self-embedding while for the third type 0'' computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite Π01 antichain. This result is optimal and has connections (...) to the program of reverse mathematics. (shrink)

Robert I. Soare, Automorphisms of the Lattice of Recursively Enumerable Sets. Part I: Maximal Sets.Manuel Lerman, Robert I. Soare, $d$-Simple Sets, Small Sets, and Degree Classes.Peter Cholak, Automorphisms of the Lattice of Recursively Enumerable Sets.Leo Harrington, Robert I. Soare, The $\Delta^0_3$-Automorphism Method and Noninvariant Classes of Degrees.

William Molyneux was born in Dublin, studied in Trinity College Dublin, and was a founding member of the Dublin Philosophical Society (DPS), Ireland’s counterpart to the Royal Society in London. He was a central figure in the Irish intellectual milieu during the Early Modern period and – along with George Berkeley and Edmund Burke – is one of the best-known thinkers to have come out of that context and out of Irish thought more generally. In 1688, when Molyneux wrote the (...) letter to Locke in which he posed the now famous question about a man born blind made to see, he was an active member of the DPS and was on familiar terms with several other key figures in Irish philosophy at the time. For the most part, Irish thinkers in the seventeenth and eighteenth centuries answered “no” to Molyneux’s question of whether a blind man made to see could distinguish, by sight alone, between a cube and a sphere. (shrink)

The artificial intelligence models with machine learning that exhibit the best predictive accuracy, and therefore, the most powerful ones, are, paradoxically, those with the most opaque black-box architectures. At the same time, the unstoppable computerization of advanced industrial societies demands the use of these machines in a growing number of domains. The conjunction of both phenomena gives rise to a control problem on AI that in this paper we analyze by dividing the issue into two. First, we carry out an (...) epistemological examination of the AI’s opacity in light of the latest techniques to remedy it. And second, we evaluate the rationality of delegating tasks in opaque agents. (shrink)

ver since the traditional criteria for medical decision-making capacity (understanding, appreciation, reasoning, evidencing a choice) were formulated, they have been criticized for not taking sufficient account of emotions or values that seem, according to the critics and in line with clinical experiences, essential to decision-making capacity. The aim of this paper is to provide a nuanced and structured overview of the arguments provided in the literature emphasizing the importance of these factors and arguing for their inclusion in competence evaluations. Moreover, (...) a broader reflection on the findings of the literature is provided. Specific difficulties of formulating and measuring emotional and valuational factors are discussed inviting reflection on the possibility of handling relevant factors in a more flexible, case-specific, and context-specific way rather than adhering to a rigid set of operationalized criteria. (shrink)

Until quite recently, almost no philosophers trained in the continental tradition saw anything of value in realism. The situation in analytic philosophy was always different, but in continental philosophy realism was usually treated as a pseudo-problem. That is no longer the case. In this provocative new book, two leading philosophers examine the remarkable rise of realism in the continental tradition. While exploring the similarities and differences in their own positions, they also consider the work of others and assess rival trends (...) in contemporary philosophy. They begin by discussing the relation between realism and materialism, which DeLanda links closely but which Harman tries to separate. Part Two covers the many different meanings of realism, with the two authors working together to develop an expanded definition of the term. Part Three features a spirited exchange on the respective virtues and drawbacks of DeLanda's realism of attractors and singularities and Harman's object-oriented theory. Part Four shifts to the question of the knowability of the real, as the authors discuss whether scientific knowledge does full justice to reality. In Part Five, they shift the focus to space, time, and science more generally, and here Harman offers a defence of actor-network theory despite its obvious anti-realist elements. Lively, accessible and engaging, this book is the best attempt so far to clarify the different paths for realism in continental philosophy. It will be of great value to students and scholars of continental philosophy and to anyone interested in the cutting-edge debates in philosophy and critical theory today. (shrink)

In this paper I argue for a novel interpretation of Shepherd’s notion of selfhood. In distinction to Deborah Boyle’s interpretation, I contend that Shepherd differentiates between the mind and the self. The latter, for Shepherd, is an effect arising from causal interactions between mind and body – specifically those interactions that give rise to our present stream of consciousness, our memories, and that can unite these two. Thus, the body plays a constitutive role in the formation of the self. The (...) upshot of this interpretation is that it can dissolve the problem of individuating mind that Boyle identifies. Briefly, the problem consists in Shepherd seemingly individuating minds in terms of their associated bodies and bodies in terms of the minds they are united with. My interpretation, however, allows to see that Shepherd neither wants nor needs to individuate the mind in isolation of the body and to read the passages, in which the problem seems to arise, as being about what makes living beings individual – with mind and body both playing a crucial role in this context. (shrink)

This paper defends a version of Davidson’s demonstrative theory of quotation and against against the Fregean identity theory (IT henceforth) as articulated and defended by Corey Washington (1992). On the Fregean view, when an expression is referred to by means of quotation the quoted material itself is a linguistic referring expression. Quotation-marks are not needed; when they are used, they serve to make clearer the shift in syntactic and semantic properties effected on the quoted material by its occupying that linguistic (...) context: whatever its usual syntactic function, the quoted material functions as a singular term; whatever its usual semantic function, in that linguistic environment the quoted material refers to itself. On DT, in contrast, quotation-marks themselves are the linguistic bearers of reference, functioning like a demonstrative; the quoted material merely plays the role of a demonstratum. In the version I argue for the referent is obtained through some contextually suggested relation; in the default case the relation will be: … instantiates the linguistic type __, but there are other possibilities. In this way, the view can deal with the fact that we do not merely refer with quotations to expression-types, but also to other entities related in some way to the relevant token we use: features exhibited by the token distinct from those constituting its linguistic type, features exhibited by other tokens of the same type but not by the one actually used (as when, by using a graphic token, we refer to its phonetic type), or even other related tokens. (shrink)

The main aim of this article is to reflect on the impact of biases related to artificial intelligence (AI) systems developed to tackle issues arising from the COVID-19 pandemic, with special focus on those developed for triage and risk prediction. A secondary aim is to review assessment tools that have been developed to prevent biases in AI systems. In addition, we provide a conceptual clarification for some terms related to biases in this particular context. We focus mainly on nonracial biases (...) that may be less considered when addressing biases in AI systems in the existing literature. In the manuscript, we found that the existence of bias in AI systems used for COVID-19 can result in algorithmic justice and that the legal frameworks and strategies developed to prevent the apparition of bias have failed to adequately consider social determinants of health. Finally, we make some recommendations on how to include more diverse professional profiles in order to develop AI systems that increase the epistemic diversity needed to tackle AI biases during the COVID-19 pandemic and beyond. (shrink)

Alternative Learning System (ALS) has been adopted in Philippine basic education, yet there is no academic institution in the region prepares ALS teachers in teaching life skills. ALS teachers graduated from different programs of teacher education for formal education. In response, an extension project was conceptualized and implemented to enhance the teaching capacity and effectiveness of ALS teachers. Case study was conducted to evaluate the effectiveness of the project. It explored the transfer of life skills among ALS teachers. Data were (...) collected from multiple sources. Interview and written responses, and feedbacks were analyzed using thematic narrative analysis. Other data were analyzed using document and descriptive analyses. The project had achieved its objectives and rated best by participants. These indicate that the project was implemented effectively and successfully. ALS teachers have acquired life skills and transferred them to others. Thus, the project has made a difference in the lives of these teachers. (shrink)

Tracing graduates has become an imperative for higher education institutions much more during the pandemic. This tracer determined the employment and employability status of the 2019 BSE graduates and identified the competencies they adequately acquired and deemed vital for work. It used descriptive design, and data were collected from the 103 graduates through a google form with open and closed-ended questions administered between November and December 2020. Results revealed that most of the graduates had been employed in teaching and teaching-related (...) jobs but mostly in contractual arrangements within the first and second six months after graduation. Many had their first jobs with meager salaries from the private sector. Communication, pedagogy, information communication technology, time management, and flexibility were the top competencies they adequately acquired and were beneficial in work. The study concluded that these graduates had acquired 21st-century skills in their respective degree programs. These results have corresponding implications for future research in confirming the most employable skills in secondary teaching. As recommended, classroom instruction might emphasize the development of these skills. Eventually, these become the competitive advantage and employability capitals of future graduates. Administering the licensure examination and the release of its results can be done within the first three months after graduation to lessen the cost of waiting. (shrink)

Culturally Responsive Teaching (CRT) weaves together rigor and relevance while it improves student achievement and engagement. The Philippine Department of Education implemented Indigenous People’s education to respond to the demands for culturally responsive teaching. Teacher education graduates are expected to articulate the rootedness of education in sociocultural contexts in creating a learning environment that recognizes respect, connectedness, choice, personal relevance, challenges, engagement, authenticity, and effectiveness. Practice teachers need relevant exposure and immersion to fully develop their competence in CRT. This scenario (...) necessitates attention to assessing the competence of 191 practice teachers in CRT, correlating their competence across dimensions, and verifying the role of gender and degree programs in competence development. This investigation provides evidence on the role of gender in competence development toward culturally responsive teaching. Such competence is vital in practice teachers, especially because they are exposed to learners with diverse backgrounds during their teaching internship. This evidence informs supervising instructors in molding practice teachers. (shrink)

‘British Empiricism’ is a name traditionally used to pick out a group of eighteenth-century thinkers who prioritised knowledge via the senses over reason or the intellect and who denied the existence of innate ideas. The name includes most notably John Locke, George Berkeley, and David Hume. The counterpart to British Empiricism is traditionally considered to be Continental Rationalism that was advocated by Descartes, Spinoza, and Leibniz, all of whom lived in Continental Europe beyond the British Isles and all embraced innate (...) ideas. This article characterizes empiricists more broadly as those thinkers who accept Locke’s Axiom that there is no idea in the mind that cannot be traced back to some particular experience. It includes British-Irish Philosophy from the seventeenth, eighteenth, and nineteenth century. As well as exploring the traditional connections among empiricism and metaphysics and epistemology, it examines how British empiricists dealt with issues in moral philosophy and the existence and nature of God. The article identifies some challenges to the standard understanding of British Empiricism by including early modern thinkers from typically marginalised groups, especially women. Finally, in showing that there is nothing uniquely British about being an empiricist, it examines a particular case study of the eighteenth-century philosopher Anton Wilhelm Amo, the first African to receive a doctorate in Europe. (shrink)

This is a short scholarly note about my retrieval an original copy of the Daily Post-Boy issue no. 7024 from September 9th,1732 from a private seller. In this issue we find an anonymous letter addressed to Berkeley which gave rise to him writing the Theory of Vision Vindicated. While Berkeley Berkeley appended a copy of the anonymous critic’s letter to TVV, until now an original copy of The Daily Post-Boy issue had yet to be discovered. -/- I have donated the (...) original copy to the Marsh's Library in Dublin, but you will find a scan of the original document attached to the note. -/- . (shrink)

The demand in the workplace is rapidly changing brought about by the educational reforms and the emergence of disruptive technology. The changes increase the importance of employability skills and literacy that would ensure career success and degree program relevance. On this premise, a study was carried out using a scoping review to examine the existing literature that published information related to employability skills of Teacher Education graduates in the Philippines. The review covered fifteen published articles that qualified in inclusion and (...) exclusion criteria. Results revealed the top employable skills which include communication; information and communication technology; problem-solving and critical thinking; collaboration, cooperation, and teamwork; research; and leadership. This study offers these skills as a framework in the curriculum enhancement of teacher education programs. It was recommended to Teacher Education Institutions to revisit their curriculum and find out if these competencies were given considerable emphasis in instruction. The partially employable skills of graduates from teacher education programs imply the furtherance of the inquiry to confirm if these skills are employable or not. Future tracer studies may take the lens of the framework in evaluating the responsiveness of the curriculum. (shrink)

Recently, research capability has received an overwhelming and remarkable interest among academics and practitioners. This is timely since the Department of Education had institutionalized research and encouraged teachers to engage in it to support evidence-based practice, decision-making, policy, and program development. On these premises, a study was carried out to assess the research capability of public teachers in Malaybalay City, determine its correlates and determinants. It utilized descriptive, correlational, and explanatory designs. It administered survey questionnaires to 92 participants. Data were (...) analyzed using descriptive and inferential statistics. The results revealed that teachers were slightly capable of conducting research and having neutral attitudes toward it, motivated to write research, had a high level of difficulties in research processes, and moderately capable of action planning. They also show evidence of potential in mentoring. Further, mentoring and action planning skills, motivation to write research, attitudes toward research, and the number of studies completed by teachers were the correlates of their research capability at different magnitudes of the relationship. Notably, the research capability of teachers had a low, negative but significant relationship with their age and accumulated years of service. Thus, this capability deteriorates as they age and accumulate years of service. The motivation to write research, the number of studies completed, and age were the determinants of research capability. It was concluded that research capability can be determined and predicted by how motivated teachers are to write research, how productive they are in research, and how young they are when they engage in this rigorous endeavor. The study identified and recommended topics for continuing professional development. (shrink)

Higher education institutions are expected to produce quality and competitive graduates for the job market and nation-building. In realizing this role, Bukidnon State University needs to ensure that graduates may land a job-relevant and align with their education and training. With this, a tracer study was conducted to verify whether the three batches of graduates are employed and are employable. It ascertained their employability based on their work experience from graduation to the present job. It employed a cross-sectional method and (...) data mining for the information of 326 graduates. The results revealed that the majority of graduates had jobs relevant to their education and training; yet, there was a significant difference in the employability of graduates across batches, except in terms of gender. Biological Science, Social Studies, and Math graduates were employable within the first six and twelve months compared to graduates from other curricula. It was claimed that the BukSU had prepared BSE graduates for employment. However, these results have implications for strategic options in improving the programs. The study made some recommendations for future tracer initiatives. (shrink)

Recently, research capability has received an overwhelming and remarkable interest among academics and practitioners. This is timely since the Department of Education had institutionalized research and encouraged teachers to engage in it to support evidence-based practice, decision-making, policy, and program development. On these premises, a study was carried out to assess the research capability of public teachers in Malaybalay City, determine its correlates and determinants. It utilized descriptive, correlational, and explanatory designs. It administered survey questionnaires to 92 participants. Data were (...) analyzed using descriptive and inferential statistics. The results revealed that teachers were slightly capable of conducting research and having neutral attitudes toward it, motivated to write research, had a high level of difficulties in research processes, and moderately capable of action planning. They also show evidence of potential in mentoring. Further, mentoring and action planning skills, motivation to write research, attitudes toward research, and the number of studies completed by teachers were the correlates of their research capability at different magnitudes of the relationship. Notably, the research capability of teachers had a low, negative but significant relationship with their age and accumulated years of service. Thus, this capability deteriorates as they age and accumulate years of service. The motivation to write research, the number of studies completed, and age were the determinants of research capability. It was concluded that research capability can be determined and predicted by how motivated teachers are to write research, how productive they are in research, and how young they are when they engage in this rigorous endeavor. The study identified and recommended topics for continuing professional development. (shrink)