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)

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)

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)

This paper is a culmination of our new foundations for recursive analysis through recursive topology as reported in Kalantari and Welch 125; 98 87). While in those papers we developed groundwork for an approach to point free analysis and applied recursion theory, in this paper we blend techniques of recursion theory with those of topology to establish new findings. We present several new techniques different from existing ones which yield interesting results. Incidental to our work is a unifying explanation (...) of various schools of study for recursive analysis. (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)

In this paper we continue our work of Kalantari and Welch . There we introduced machinery to produce a point-free approach to points and functions on topological spaces and found conditions for both which lend themselves to effectivization. While we studied recursive points in that paper, here, we present two useful classes of recursive functions on topological spaces, apply them to the reals, and find precise accounting for the nature of the properties of some examples that exist in the (...) literature. We end with a construction of a recursive function on a small subset of the unit interval which is strongly nonextendible. (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)

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)

We describe the solution of the Limit Rule Problem of Revision Theory and discuss the philosophical consequences of the fact that the truth set of Revision Theory is a complete 1/2 set.

We show that the predicate “x is the power set of y” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has (...) a cardinal strong up to one of its ℵ‐fixed points, and, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ. (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)

We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, (...) if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (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)

Background: There are no universally agreed rules of healthcare ethics. Ethical decisions and standards tend to be linked to professional codes of practice when dealing with complex issues. Objectives: This paper aims to explore the ethical complexities on who should decide to give infants born on the borderline of viability lifesaving treatment, parents or the healthcare professionals. Method: The paper is a discussion using the principles of ethics, professional codes of practice from the UK, Nursing Midwifery Council and UK legal (...) case law and statute. Healthcare professionals' experiences that influence parental decision are also considered. Findings & Discussion: There are considerable barriers to an effective discussion taking place in an environment where clinical decisions have to be made quickly once the baby is born. This is compounded by the need and respect for parental autonomy and the difficulties they face when making a best interest's decision knowing that this could cause more harm than good for their infant child and balancing any decision they make with quality of life. Conclusion: On deciding whether to give lifesaving treatment born at the borderline of viability, it should be a joint decision between the parents and the neonatal team. (shrink)

Do factive mental states come in degrees? If so, what is their underlying structure, and what is their theoretical significance? Many have observed that ‘knows that’ is not a gradable verb and have taken this to be strong evidence that propositional knowledge does not come in degrees. This paper demonstrates that the adjective ‘aware that’ passes all the standard tests of gradability, and thus strongly motivates the idea that it refers to a factive mental state that comes in degrees. We (...) will explore a range of structural questions that have been overlooked in regard to our awareness of facts, showing that the degreed structure of awareness has significant implications for our understanding of knowledge. (shrink)

The study of metaphor is now firmly established as a central topic within cognitive science and the humanities. We marvel at the creative dexterity of gifted speakers and writers for their special talents in both thinking about certain ideas in new ways, and communicating these thoughts in vivid, poetic forms. Yet metaphors may not only be special communicative devices, but a fundamental part of everyday cognition in the form of 'conceptual metaphors'. An enormous body of empirical evidence from cognitive linguistics (...) and related disciplines has emerged detailing how conceptual metaphors underlie significant aspects of language, thought, cultural and expressive action. Despite its influence and popularity, there have been major criticisms of conceptual metaphor. This book offers an evaluation of the arguments and empirical evidence for and against conceptual metaphors, much of which scholars on both sides of the wars fail to properly acknowledge. (shrink)

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)

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)

Interpreting Figurative Meaning critically evaluates the recent empirical work from psycholinguistics and neuroscience examining the successes and difficulties associated with interpreting figurative language. There is now a huge, often contradictory literature on how people understand figures of speech. Gibbs and Colston argue that there may not be a single theory or model that adequately explains both the processes and products of figurative meaning experience. Experimental research may ultimately be unable to simply adjudicate between current models in psychology, linguistics and philosophy (...) of how figurative meaning is interpreted. Alternatively, the authors advance a broad theoretical framework, motivated by ideas from 'dynamical systems theory', that describes the multiple, interacting influences which shape people's experiences of figurative meaning in discourse. This book details past research and theory, offers a critical assessment of this work and sets the stage for a new vision of figurative experience in human life. (shrink)

Since James defined emotion as consciousness of bodily reactions and Cannon and others detailed the nature of these reactions, there has been an increasing tendency among behaviorists to equate emotions with visceral reactions and to neglect some of the genetic and adaptive aspects of emotion which had been discussed by Darwin.

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$ , provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$ . There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.

When a belief is self-fulfilling, having it guarantees its truth. When a belief is self-defeating, having it guarantees its falsity. These are the cases of “self-impacting” beliefs to be examined below. Scenarios of self-defeating beliefs can yield apparently dilemmatic situations in which we seem to lack sufficient reason to have any belief whatsoever. Scenarios of self-fulfilling beliefs can yield apparently dilemmatic situations in which we seem to lack reason to have any one belief over another. Both scenarios have been used (...) independently to challenge Evidentialism, on which what we may rationally believe is all and only what fits our current evidence. Here we tie the two scenarios together and explore what a knowledge-sensitive evidentialist approach to one implies for the other. (shrink)

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$. Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

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 prove that a form of the $Erd\H{o}s$ property (consistent with $V = L\lbrack H_{\omega_2}\rbrack$ and strictly weaker than the Weak Chang's Conjecture at ω1), together with Bounded Martin's Maximum implies that Woodin's principle $\psi_{AC}$ holds, and therefore 2ℵ0 = ℵ2. We also prove that $\psi_{AC}$ implies that every function $f: \omega_1 \rightarrow \omega_1$ is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum (...) fails. (shrink)

Measured inequality has increased tremendously between the 1960s and 1990s, not only in the United States but throughout the majority of industrial nations. Wages among people of the same race and gender have become less equal. The hours worked by men have fallen, and the drop has been more pronounced among those who earn lower wages—as a result, inequality in labor income, which is the product of the wage rate and hours worked, has increased relative to inequality in wage rates. (...) Moreover, among married couples, employment of the wives of high-income men has increased until these wives are approximately as likely to be employed outside the home as are the wives of low-income men, who have always worked for wages. In addition, due to assortative mating, wage rates of husbands and wives are positively correlated, and it is clear that the growth in inequality of labor incomes among families has outstripped the growth in inequality in individual labor income. Finally, only the highest-income families have savings in excess of home equity and company-sponsored pensions, which implies that inequality in wealth among families has been exacerbated by the growth in stock prices. (shrink)