2,952 research outputs found
Conservation theorems for the Cohesiveness Principle
We prove that the Cohesiveness Principle (COH) is conservative over
and over for all by
recursion-theoretic means. We first characterize COH over
as a `jumped' version of Weak K\"{o}nig's Lemma (WKL) and develop suitable
machinery including a version of the Friedberg jump-inversion theorem. The main
theorem is obtained when we combine these with known results about WKL. In an
appendix we give a proof of the conservativity of WKL over by
way of the Superlow Basis Theorem and a new proof of a recent jump-inversion
theorem of Towsner
Caveats for using statistical significance tests in research assessments
This paper raises concerns about the advantages of using statistical
significance tests in research assessments as has recently been suggested in
the debate about proper normalization procedures for citation indicators.
Statistical significance tests are highly controversial and numerous criticisms
have been leveled against their use. Based on examples from articles by
proponents of the use statistical significance tests in research assessments,
we address some of the numerous problems with such tests. The issues
specifically discussed are the ritual practice of such tests, their dichotomous
application in decision making, the difference between statistical and
substantive significance, the implausibility of most null hypotheses, the
crucial assumption of randomness, as well as the utility of standard errors and
confidence intervals for inferential purposes. We argue that applying
statistical significance tests and mechanically adhering to their results is
highly problematic and detrimental to critical thinking. We claim that the use
of such tests do not provide any advantages in relation to citation indicators,
interpretations of them, or the decision making processes based upon them. On
the contrary their use may be harmful. Like many other critics, we generally
believe that statistical significance tests are over- and misused in the social
sciences including scientometrics and we encourage a reform on these matters.Comment: Accepted version for Journal of Informetric
Borderline vs. unknown: comparing three-valued representations of imperfect information
International audienceIn this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and González-RodrĂguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility–necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions
The Consistency of Arithmetic
This paper offers an elementary proof that formal arithmetic is consistent. The system that will be proved consistent is a first-order theory R♯, based as usual on the Peano postulates and the recursion equations for + and ×. However, the reasoning will apply to any axiomatizable extension of R♯ got by adding classical arithmetical truths. Moreover, it will continue to apply through a large range of variation of the un- derlying logic of R♯, while on a simple and straightforward translation, the classical first-order theory P♯ of Peano arithmetic turns out to be an exact subsystem of R♯. Since the reasoning is elementary, it is formalizable within R♯ itself; i.e., we can actually demonstrate within R♯ (or within P♯, if we care) a statement that, in a natural fashion, asserts the consistency of R♯ itself.
The reader is unlikely to have missed the significance of the remarks just made. In plain English, this paper repeals Goedel’s famous second theorem. (That’s the one that asserts that sufficiently strong systems are inadequate to demonstrate their own consistency.) That theorem (or at least the significance usually claimed for it) was a mis- take—a subtle and understandable mistake, perhaps, but a mistake nonetheless. Accordingly, this paper reinstates the formal program which is often taken to have been blasted away by Goedel’s theorems— namely, the Hilbert program of demonstrating, by methods that everybody can recognize as effective and finitary, that intuitive mathematics is reliable. Indeed, the present consistency proof for arithmetic will be recognized as correct by anyone who can count to 3. (So much, indeed, for the claim that the reliability of arithmetic rests on transfinite induction up to ε0, and for the incredible mythology that underlies it.
A Computable Economist’s Perspective on Computational Complexity
A computable economist.s view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called "Post's Program of Research for Higher Recursion Theory". Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix.
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