The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the (...) mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field. (shrink)

• We define a notion of order of indiscernibility type of a structure by analogy with Mitchell order on measures; we use this to define a hierarchy of strong axioms of infinity defined through normal filters, the α-weakly Erdős hierarchy. The filters in this hierarchy can be seen to be generated by sets of ordinals where these indiscernibility orders on structures dominate the canonical functions.• The limit axiom of this is that of greatly Erdős and we use it to calibrate (...) some strengthenings of the Chang property, one of which, CC+, is equiconsistent with a Ramsey cardinal, and implies that where K is the core model built with non-overlapping extenders — if it is rigid, and others which are a little weaker. As one corollary we have:TheoremIf then there is an inner model with a strong cardinal. • We define an α-Jónsson hierarchy to parallel the α-Ramsey hierarchy, and show that κ being α-Jónsson implies that it is α-Ramsey in the core model. (shrink)

In philosophical logic necessity is usually conceived as a sentential operator rather than as a predicate. An intensional sentential operator does not allow one to express quantified statements such as 'There are necessary a posteriori propositions' or 'All laws of physics are necessary' in first-order logic in a straightforward way, while they are readily formalized if necessity is formalized by a predicate. Replacing the operator conception of necessity by the predicate conception, however, causes various problems and forces one to reject (...) many philosophical accounts involving necessity that are based on the use of operator modal logic. We argue that the expressive power of the predicate account can be restored if a truth predicate is added to the language of first-order modal logic, because the predicate 'is necessary' can then be replaced by 'is necessarily true'. We prove a result showing that this substitution is technically feasible. To this end we provide partial possible-worlds semantics for the language with a predicate of necessity and perform the reduction of necessities to necessary truths. The technique applies also to many other intensional notions that have been analysed by means of modal operators. (shrink)

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...) sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

Axiomatic set theory is the concern of this book. More particularly, the authors prove results about the coding of models M, of Zermelo-Fraenkel set theory together with the Generalized Continuum Hypothesis by using a class 'forcing' construction. By this method they extend M to another model L[a] with the same properties. L[a] is Gödels universe of 'constructible' sets L, together with a set of integers a which code all the cardinality and cofinality structure of M. Some applications are also considered. (...) Graduate students and research workers in set theory and logic will be especially interested by this account. (shrink)

We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...) can be -complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus. (shrink)

Modern scientific theories of emotional behavior, almost without exception, trace their origin to Charles Darwin, and his publications On the Origin of Species (1859) and The Expression of the Emotions in Man and Animals (1872). The most famous evolutionary dilemma Darwin acknowledged as a challenge to his theory of natural selection was the incomplete sub Cambrian fossil record. However, Darwin struggled with two other rarely referenced theoretical and scientific dilemmas that confounded his theories about emotional behavior. These included (1) the (...) origin of social instincts (e.g., altruism, empathy, reciprocity and cooperation) and the reasons for their conservation in evolution and (2) the peripheral control of heart rate vis-a-vis emotional behavior outside of consciousness. Darwin acknowledged that social instincts are critical to the survival of some species, but had difficulty aligning them with his theory of natural selection in humans. Darwin eventually proposed that heart rate and emotions are controlled via one’s intellect and cortical mechanisms, and that instinctive behavior is genetically-programmed and inherited. Despite ongoing efforts, these two theoretical dilemmas are debated to this day. Simple testable hypotheses have yet to emerge for the biological mechanisms underlying instinctive behavior or the way heart rate is controlled in infants. In this paper, we review attempts to resolve ¬these issues over the past 160 years. We posit that research and theories that supported Darwin’s individualistic brain-centric and genetic model have become an “orthodox” Western view of emotional behavior, one that produced the prevailing behavioral construct of attachment as developed by John Bowlby. We trace research and theories that challenged this orthodoxy at various times, and show how these challenges were repeatedly overlooked, rejected, or misinterpreted. We review two new testable theories, emotional connection theory and calming cycle theory, which we argue resolve the two dilemmas We show emerging scientific evidence from physiology and a wide variety of other fields, as well from clinical trials among prematurely born infants, that supports the two theories. Clinical implications of the new theories and possible new ways to assess risk and intervene in emotional, behavioral and developmental disorders are discussed. (shrink)

The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a "Σ₂-machine" in that it uses a Σ₂ Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σ n rule. Such machines either halt or enter an infinite loop by stage ζ(n) = df μζ(n)[∃Σ(n) > ζ(n) L ζ(n) ≺ Σn L Σ(n) ], again generalising precisely the ITTM case. The collection of (...) such machines taken together computes precisely those reals of the least model of analysis. (shrink)

We give the proof of a theorem of Jensen and Zeman on the existence of a global □ sequence in the Core Model below a measurable cardinal κ of Mitchell order ) equal to κ++, and use it to prove the following theorem on mutual stationarity at n.Let ω1 denote the first uncountable cardinal of V and set to be the class of ordinals of cofinality ω1.TheoremIf every sequence n m. In particular, there is such a model in which for (...) all sufficiently large m<ω, the class of measurables λ with oM≥ωm is, in V, stationary below m+2. (shrink)

In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding (...) sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel. (shrink)

This article is concerned with reflection principles in the context of Cantor’s conception of the set-theoretic universe. We argue that within such a conception reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity.

In this article, we consider music and noise in terms of vibrational and transferable energy as well as from the evolutionary significance of the hearing system of Homo sapiens. Music and sound impinge upon our body and our mind and we can react to both either positively or negatively. Much depends, in this regard, on the frequency spectrum and the level of the sound stimuli, which may sometimes make it possible to set music apart from noise. There are, however, two (...) levels of description: the physical-acoustic description of the sound and the subjective-psychological reactions by the listeners. Starting from a vibrational approach to sound and music, we first investigate how sound may activate the sense of touch and the vestibular system of the inner ear besides the sense of hearing. We then touch upon distinct issues such as the relation between low-frequency sounds and annoyance, the harmful effect of loud sound and noise, the direct effects of overstimulation with sound, the indirect effects of unwanted sounds as related to auditory neurology, and the widespread phenomenon of liking loud sound and music, both from the point of view of behavioral and psychological aspects. (shrink)

We have previously proposed that mothers and infants co-regulate one another’s autonomic state through an autonomic conditioning mechanism, which starts during gestation and results in the formation of autonomic socioemotional reflexes following birth. Theoretically, autonomic physiology associated with the ASR should correlate concomitantly with behaviors of mother and infant, although the neuronal pathway by which this phenomenon occurs has not been elucidated. In this paper, we consider the neuronal pathway by which sensory stimuli between a mother and her baby/child affect (...) the physiology and emotional behavior of each. We divide our paper into two parts. In the first part, to gain perspective on current theories on the subject, we conduct a 500-year narrative history of scientific investigations into the human nervous system and theories that describe the neuronal pathway between sensory stimulus and emotional behavior. We then review inconsistencies between several currently accepted theories and recent data. In the second part, we lay out a new theory of emotions that describes how sensory stimuli between mother and baby unconsciously control the behavior and physiology of both. We present a theory of mother/infant emotion based on a set of assumptions fundamentally different from current theories. Briefly, we propose that mother/infant sensory stimuli trigger conditional autonomic socioemotional reflexes, which drive cardiac function and behavior without the benefit of the thalamus, amygdala or cortex. We hold that the ASR is shaped by an evolutionarily conserved autonomic learning mechanism that forms between mother and fetus during gestation and continues following birth. We highlight our own and others research findings over the past 15 years that support our contention that mother/infant socioemotional behavior is driven by mutual autonomic state plasticity, as opposed to cortical trait plasticity. We review a novel assessment tool designed to measure the behaviors associated with the ASR phenomenon. Finally, we discuss the significance of our theory for the treatment of mothers and infants with socioemotional disorders. (shrink)

We show how in the hierarchies${F_\alpha }$of Fieldian truth sets, and Herzberger’s${H_\alpha }$revision sequence starting from any hypothesis for${F_0}$ that essentially each${H_\alpha }$ carries within it a history of the whole prior revision process.As applications we provide a precise representation for, and a calculation of the length of, possiblepath independent determinateness hierarchiesof Field’s construction with a binary conditional operator. We demonstrate the existence of generalized liar sentences, that can be considered as diagonalizing past the determinateness hierarchies definable in Field’s recent (...) models. The ‘defectiveness’ of such diagonal sentences necessarily cannot be classified by any of the determinateness predicates of the model. They are ‘ineffable liars’. We may consider them a response to the claim of Field that ‘the conditional can be used to show that the theory is not subject to “revenge problems”.’. (shrink)

After an introduction to the major issues confronting intellectuals, this book explores the various aspects of the intellectual's role including: * philosophers and academics who have tried to define the function of the intellectual * how intellectuals have assumed the status of the conscience of the nation and the voice of the oppressed * the interaction of intellectuals with Marxism * the place of the intellectual in American society Covering regions as diverse as Israel, Algeria, Britain, Ireland, central Europe and (...) America, this collection considers the question of whether the intellectual can still lay claim to the language of truth. In answering, this study tells us much about the modern world in which we live. _Coverage includes the following thinkers:_ Gramsci, Weber, Yeats, Auden, Levy, Mailer, Walzer, Marx and many more. (shrink)

We develop the concepts of recursively nowhere dense sets and sets that are recursively of first category and study closed sets of points in light of Baire's Category Theorem. Our theorems are primarily concerned with exdomains of recursive quantum functions and hence with avoidable points . An avoidance function is a recursive function which can be used to expel avoidable points from domains of recursive quantum functions. We define an avoidable set of points to be an arbitrary subset of the (...) avoidable points of a single avoidance function, and we study an effective union of such sets which we call a piecemeal avoidable set. We note that each of ‘the set of recursive points’ and ‘the set of avoidable points’ is of first category but not recursively of first category. We show an exdomain exists which is recursively nowhere dense, as well as one that is nowhere dense but not recursively nowhere dense. After establishing that every exdomain is recursively of first category, we prove that given any fixed exdomain, there is another exdomain, which while dense in the underlying space, is disjoint from the fixed exdomain. Finally, we show how to build a recursive sequence of recursive quantum functions that have mutually disjoint, dense exdomains. (shrink)

The paper explores the handling of singular analogy in quantitative inductive logics. It concentrates on two analogical patterns coextensive with the traditional argument from analogy: perfect and imperfect analogy. Each is examined within Carnap’s λ-continuum, Carnap’s and Stegmüller’s λ-η continuum, Carnap’s Basic System, Hintikka’s α-λ continuum, and Hintikka’s and Niiniluoto’s K-dimensional system. Itis argued that these logics handle perfect analogies with ease, and that imperfect analogies, while unmanageable in some logics, are quite manageable in others. The paper concludes with a (...) modification of the K-dimensional system that synthesizes independent proposals by Kuipers and Niiniluoto. (shrink)

PurposeThe purpose of this paper is to explore a particular philosophical underpinning for Information Systems (IS) research – critical systemic thinking (CST). Drawing upon previous work, the authors highlight the principal features of CST within the tradition of critical research and attempt to relate it to trends in the Italian school of IS research in recent years, as exemplified by the work of Claudio Ciborra but also evident in work by, e.g. Resca, Jacucci and D'Atri.Design/methodology/approachThis is a conceptual paper which (...) explores CST, characterised by a focus on individual uniqueness, and socially‐constructed, individual worldviews as generators of human knowing.FindingsThe paper draws on work by Heinz Klein in which he elaborated three constitutive stages in critical research: interpretive, genealogical and constructive. The authors introduce a fourth, reflective stage and discuss five categories of critical research, reflecting different perspectives on emancipation, culminating in emergent expressionism, associated with Ciborra and the Italian school more generally.Research limitations/implicationsThis paper discusses approaches to CST and how they might have practical implications in IS development. The distinction between approaches founded in logical empiricism and those founded in hermeneutic dialectics are considered and the development of critical and systems strands are discussed.Practical implicationsThe paper addresses CST as an approach to development of information systems. Such approaches enable users to explore their individually unique understandings and create a constructive dialogue with one another, which emancipates and empowers users to own and control their own development processes and hence build more productive and usable systems.Social implicationsA focus on research which is oriented towards emancipation in the tradition of critical social theory.Originality/valueThe paper draws on extensive theoretical research carried out by the authors over a period of more than ten years in CST and synthesises the practical implications. (shrink)

We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. (...) The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second-order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query ? n H ? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime, which is thus a universal constant of that spacetime. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any spacetime there will be a countable ordinal upper bound, , on the complexity of questions in the Borel hierarchy computable in it. Introduction 1.1 History and preliminaries Hyperarithmetic Computations in MH Spacetimes 2.1 Generalising SADn regions 2.2 The complexity of questions decidable in Kerr spacetimes An Upper Bound on Computational Complexity for Each Spacetime CiteULike Connotea Del.icio.us What's this? (shrink)

We show that Leitgeb’s dependence operator of Leitgeb is a \-operator and that this is best possible.

A correction is needed to our paper: to the definition contained within the statement of Lemma 1.5 and thus arguments around it in §3.

In this paper we first give a variant of a theorem of Jockusch–Lewis– Remmel on existence of a computable, degree-preserving homeomorphism between a bounded strong ${\Pi^0_2}$ class and a bounded ${\Pi^0_1}$ class in 2 ω . Namely, we show that for mathematically common and interesting topological spaces, such as computably presented ${\mathbb{R}^n}$ , we can obtain a similar result where the homeomorphism is in fact the identity mapping. Second, we apply this finding to give a new, priority-free proof of existence (...) of a tree of shadow points computable in 0′. (shrink)

This paper continues our study of computable point-free topological spaces and the metamathematical points in them. For us, a point is the intersection of a sequence of basic open sets with compact and nested closures. We call such a sequence a sharp filter. A function fF from points to points is generated by a function F from basic open sets to basic open sets such that sharp filters map to sharp filters. We restrict our study to functions that have at (...) least all computable points in their domains.We follow Turing's approach in stating that a point is computable if it is the limit of a computable sharp filter; we then define the Turing degree Deg of a general point x in an analogous way. Because of the vagaries of the definition, a result of J. Miller applies and we note that not all points in all our spaces have Turing degrees; but we also show a certain class of points do. We further show that in ℝn all points have Turing degrees and that these degrees are the same as the classical Turing degrees of points defined by other researchers.We also prove the following: For a point x that has a Turing degree and lies either on a computable tree T or in the domain of a computable function fF, there is a sharp filter on T or in dom converging to x and with the same Turing degree as x. Furthermore, all possible Turing degrees occur among the degrees of such points for a given computable function fF or a complete, computable, binary tree T. For each x ∈ dom for which x and fF have Turing degrees, Deg) ≤ Deg. Finally, the Turing degrees of the sharp filters convergent to a given x are closed upward in the partial order of all Turing degrees. (shrink)

Most investigations of musical performance anxiety have employed quantitative methodologies. Whereas such methodologies can provide useful insights into the measurable aspects of the experience in a larger group of participants, the complexity, subtlety and individuality of the emotional experience and the importance of the individual’s interpretation of it are often overlooked. This study employed a phenomenological approach to investigate the lived, subjective experience of performance anxiety, as described in professional musicians’ narratives. Semistructured interviews with four professional musicians (two males, two (...) females) specializing in Western classical and jazz music genres were conducted and analyzed using Interpretative Phenomenological Analysis (IPA). The analysis revealed the presence of four overarching themes: (1) Intensity of performance anxiety experience, (2) perceived effects, (3) development of coping strategies and (4) achieving release from anxiety. Findings suggest that the lived experience of performance anxiety is multifaceted, characterized by a physical and a psychological dimension. Interpretative Phenomenological Analysis is a useful research tool that can facilitate our understanding of the subjective experience of performance anxiety (how it is felt and understood at an individual level) and can thus be useful in the development of tailor-made intervention programs for musicians. (shrink)