Postmodernism is often characterized, among other things, as the belief in the unattainability of objective truth and as a rejection of teleological and reductionist, or essentialist, forms of thought. For instance, in his provocative book The Rhetoric of Economics, Donald McCloskey sketches the implications for economic methodology of Richard Rorty's rejection of the modernist quest for Truth, as represented by various rationalist and empiricist epistemologies. McCloskey describes modernist methodology as displaying a desire to predict and control, a search for objective–;which (...) often means measurable–;knowledge, and an attempt to develop a value-free inquiry, among other characteristics. This “methodological correctness,” McCloske suggests, is discredited by the postmodern dissatisfaction with traditional epistemology. Thus, in place of the modernist belief in a rule-guided path to truth, he advocates a “free market” approach to knowledge, in which participants in the variety of theoretical conversations agree to be earnest and listen politely to one another. (shrink)

We show that every nontrivial interval in the recursively enumerable degrees contains an incomparable pair which have an infimum in the recursively enumerable degrees.

. We prove that every countable relation on the enumeration degrees, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}, is uniformly definable from parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}. Consequently, the first order theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document} is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first (...) order theory of the enumeration degrees of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Sigma^0_2$\end{document} sets is not decidable. (shrink)

Posner [6] has shown, by a nonuniform proof, that every ▵ 0 2 degree has a complement below 0'. We show that a 1-generic complement for each ▵ 0 2 set of degree between 0 and 0' can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above $\varnothing'$ . In the second half of the paper, we show that the complementation (...) of the degrees below 0' does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems. (shrink)

We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair $M\subset N$ of models of set theory implying that every perfect set in N has an element in N which is not in M.

A spate of recent articles on Thomas Reid’s aesthetic theory constitutes a valuable commentary on both Reid’s own theory and on eighteenth-century aesthetics. However, while these articles provide a generally sympatheic introduction to Reid’s position, they are primarily expository in nature and uncritical in tone. I shall therefore address the plausibility of both Reid’s general aesthetic theory and the arguments advanced for the theory. I contend that his theory, however much an improvement over those offered by his contemporaries, is fatally (...) flawed by internal confusion and inconsistency; he cannot justify his claim that beauty and sublimity are objective qualities of objects. My analysis focuses on Reid’s mature statement on aesthetics, the essay “Of Taste” found at the end of the Essays on the Intellectual Powers of Man. (shrink)

A spate of recent articles on Thomas Reid’s aesthetic theory constitutes a valuable commentary on both Reid’s own theory and on eighteenth-century aesthetics. However, while these articles provide a generally sympatheic introduction to Reid’s position, they are primarily expository in nature and uncritical in tone. I shall therefore address the plausibility of both Reid’s general aesthetic theory and the arguments advanced for the theory. I contend that his theory, however much an improvement over those offered by his contemporaries, is fatally (...) flawed by internal confusion and inconsistency; he cannot justify his claim that beauty and sublimity are objective qualities of objects. My analysis focuses on Reid’s mature statement on aesthetics, the essay “Of Taste” found at the end of the Essays on the Intellectual Powers of Man. (shrink)

Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X', its Turning jump, is recursive in $\varnothing'$ and high if X' computes $\varnothing''$ . Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of $A \bigoplus W$ is recursive in the jump of A. We prove that there are no (...) deep degrees other than the recursive one. Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turning functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that $(W \bigoplus A)'$ is forced to disagree with Φ(-; A'). The conversion has some ambiguity; in particular, A cannot be found uniformly from W. We also show that there is a "moderately" deep degree: There is a low nonzero degree whose join with any other low degree is not high. (shrink)

Let A and B be subsets of the reals. Say that A κ ≥ B, if there is a real a such that the relation "x ∈ B" is uniformly Δ 1 (a, A) in L[ ω x,a,A 1 , x,a,A]. This reducibility induces an equivalence relation $\equiv_\kappa$ on the sets of reals; the $\equiv_\kappa$ -equivalence class of a set is called its Kleene degree. Let K be the structure that consists of the Kleene degrees and the induced partial order (...) K ≥. A substructure of K that is of interest is P, the Kleene degrees of the Π 1 1 sets of reals. If sharps exist, then there is not much to P, as Steel [9] has shown that the existence of sharps implies that P has only two elements: the degree of the empty set and the degree of the complete Π 1 1 set. Legrand [4] used the hypothesis that there is a real whose sharp does not exist to show that there are incomparable elements in P; in the context of V = L, Hrbacek has shown that P is dense and has no minimal pairs. The Hrbacek results led Simpson [6] to make the following conjecture: if V = L, then p forms a universal homogeneous upper semilattice with 0 and 1. Simpson's conjecture is shown to be false by showing that if V = L, then Godel's maximal thin Π 1 1 set is the infimum of two strictly larger elements of P. The second main result deals with the notion of jump in K. Let A' be the complete Kleene enumerable set relative to A. Say that A is low-n if A (n) has the same degree as $\varnothing^{(n)}$ , and A is high-n if A (n) has the same degree as $\varnothing^{(n + 1)}$ . Simpson and Weitkamp [7] have shown that there is a high (high-1) incomplete Π 1 1 set in L. They have also shown that various other Π 1 1 sets are neither high nor low in L. Legrand [5] extended their results by showing that, if there is a real x such that x # does not exist, then there is an element of P that, for all n, is neither low-n nor high-n. In § 2, ZFC is used to show that, for all n, if A is Π 1 1 and low-n then A is Borel. The proof uses a strengthened version of Jensen's theorem on sequences of admissible ordinals that appears in [7, Simpson-Weitkamp]. (shrink)

J. Łoś raised the following question: Under what conditions can a countable partially ordered set be extended to a dense linear order merely by adding instances of comparability ? We show that having such an extension is a Σ 1 l -complete property and so there is no Borel answer to Łoś's question. Additionally, we show that there is a natural Π 1 l -norm on the partial orders which cannot be so extended and calculate some natural ranks in that (...) norm. (shrink)

Harvey Siegel argues that minimum competency testing (MCT) is incompatible with strong sense critical thinking. His arguments are reviewed and contrasted with positions held by John E. McPeck and Michael Scriven. Siegel's arguments seem directed against the prevailing form of MCT. However, alternative formats which allow for the aggregate and context-sensitive nature of critical thinking are not doomed to the arbitrariness Siegel finds. MCT may be a legitimate and useful means for furthering critical thinking as one of our educational ideals.

We exhibit a structural difference between the truth-table degrees of the sets which are truth-table above 0′ and the PTIME-Turing degrees of all sets. Though the structures do not have the same isomorphism type, demonstrating this fact relies on developing their common theory.