In a study involving 62 Danish children with autism spectrum disorder, we obtained results showing that the mastery of linguistic recursion is a significant predictor of success in second-order false belief tasks. The same study also showed that the mastery of linguistic recursion was not significantly correlated with success in a task involving three heavily used Danish discourse particles. This calls for further explanation, as the reasoning involved in both types of tasks seems similar. In this paper, we discuss (...) second-order false belief reasoning, the reasoning underlying the use of the three Danish discourse particles, say what we know about them experimentally, and discuss what they do have in common. (shrink)

We describe a strongly minimal theory S in an effective language such that, in the chain of countable models of S, only the second model has a computable presentation. Thus there is a spectrum of an -categorical theory which is neither upward nor downward closed. We also give an upper bound on the complexity of spectra.

We employ an infinite-signature Hrushovski amalgamation construction to yield two results in Recursive Model Theory. The first result, that there exists a strongly minimal theory whose only recursively presentable models are the prime and saturated models, adds a new spectrum to the list of known possible spectra. The second result, that there exists a strongly minimal theory in a finite language whose only recursively presentable model is saturated, gives the second non-trivial example of a spectrum produced in (...) a finite language. (shrink)

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, ranging (...) from small time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers and mathematicians.

Alex has written an excellent thesis in the area of computable model theory. The latter is a subject that nicely combines model-theoretic ideas with delicate recursiontheoretic constructions. The results demand good knowledge of both fields. In his thesis, Alex begins by reviewing the essential model-theoretic facts, especially the Baldwin-Lachlan result about uncountably categorical theories. This he follows with a brief discussion of recursion theory, including mention of the priority method. The deepest part of the thesis concerns the study of the (...) recursive spectrum of an uncountably categorical theory, i.e. the set of n ≤ ! such that the n-th model of the theory (in the Baldwin-Lachlan sense) has a computable presentation. This is a deep and very active area of conteporary research in computable model theory which Alex discusses in considerable detail. The exposition is very good and therefore the thesis makes a nice introduction to the subject for a wide community of logicians. (Sy-David Friedman, Professor of Mathematical Logic, Director of the Kurt Godel Research Center for Mathematical Logic, University of Vienna). (shrink)

Review by: Alexander G. Melnikov The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 400-401, September 2013.

Let T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2 λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least $\min(2^\lambda,\beth_2)$ resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, (...) 2 ω } there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ. (shrink)

We build a new spectrum of recursive models (SRM(T)) of a strongly minimal theory. This theory is non-disintegrated, flat, model complete, and in a language with a finite structure.

In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years (...) pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open. (shrink)

We construct a recursive model , a recursive subset R of its domain, and a Turing degree x 0 satisfying the following condition. The nonrecursive images of R under all isomorphisms from to other recursive models are of Turing degree x and cannot be recursively enumerable.

With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they (...) prove that R is intrinsically r.e. if and only if a natural recursive-syntactic condition is satisfied. We show that, while a recursive non-intrinsically r.e. relation may have a two element degree spectrum, a non-intrinsically r.e. relation which satisfies the Ash–Nerode decidability condition has an infinite degree spectrum. We also study several related decidability conditions and their effects on the degree spectra, including some conditions which are sufficient to obtain every r.e. degree in a spectrum. (shrink)

By compiling an experimental method combining both dialectic and rhetoric, 'Dialectic, Rhetoric and Contrast: The Infinite Middle of Meaning' demonstrates how singular meanings can be rendered in a spectrum of 12 repeating concepts that are in a continuum, gradated and symmetrical. The ability to arrange meaning into this pattern opens enquiry into its ontology, and presents meaning as closer to the sensation of colours or musical notes than the bivalent oppositions depicted in classical logic. However, the experiment does not (...) assert that this pattern suggests some sort of constant or absolute principle; instead, it theorises on the ways in which meaning can be considered to be recursive. To explain this, the book explores the concept of contrast itself. No exactitude on the precise existence of contrast can ever be struck because the answer varies infinitely depending upon the scale of measurement used to gauge the meeting point. This characteristic of contrast helps to define a whole new dimension from which sensation, meaning, cognition and consciousness can be analogised to the infinite forms between forms. At a time when the widest consensus in philosophy is the exhaustion of its central themes, the significance of such a hypothesis provides fresh impetus to revise some of the key meanings and concepts underpinning contemporary thought. To do this, the method explores the opposing themes of idealism and realism that run throughout western philosophy from Plato to the Speculative turn. This book will be of interest to professional academic audiences in the humanities and social sciences, from graduates to senior scholars. It will also be an interesting read to anyone wishing to keep abreast of developments in continental philosophy, epistemology, metaphysics, and the sociology of knowledge. (shrink)

A central feature of complexity is that it is based on non-linear, recursive relations. However, in most current accounts of complexity such relations, while non-linear, are based on the reductive relations of a Newtonian onto-epistemological framework. This means that the systems that are emergent from the workings of such relations are a narrowly reduced spectrum of complex systems. It is argued that John Dewey’s trans-actional relations, relations that are characterized by an irreducible internal distinction, can function as an (...) exemplar of the less reductive relations seen in living complex systems. Conceptualizing complex relations and complex systems in a less reductive way allows indeterminate concepts such as epistemological limits and the generative production of meaning, which are inextricably linked, to be considered. Methodologically, such lessened reduction highlights functional systems, revealing aspects of the biopsychosocial human ‘complex system of complex systems’ that are hidden from view in a Newtonian framework. (shrink)

This paper is a full version of the author’s talk at “Procedural Logic and Philosophy of Consciousness”, a discussion dedicated to A.V. Smirnov’s book “The Logic of Mean­ing as a Philosophy of Consciousness: An Invitation to Reflection” (2021). The proposed content can be viewed as a co-reflection on the key concepts of the logical-semantic con­cept of consciousness: coherence, integrity, constriction, extension as the foundations and methods of conceptualization in their existential perspective. It is stressed that the logi­cal-semantic approach of A.V. (...) Smirnov reveals new facets in the understanding of philo­sophical rationality, personal and cultural subjectivity, experience, consciousness and their formation. To demonstrate the general philosophical, methodological status of the system of concepts introduced by the author, analogies are drawn with the understanding of exis­tential experience as a constriction of meaning-forming experiences that are unfolded by the subject in communication, creativity, self-understanding. The concepts that have re­ceived a conceptual justification in the works of A.V. Smirnov bring the origins of experi­ence to another level of comprehension, in accordance with which the spectrum of under­standing of human reflexive capabilities also changes. If we “start with integrity”, as the author advises us, then the idea of recursiveness of experience as a pulsating process of comprehension and self-reflection in the process of human interaction with the world or his “flickering” being-in-the-world is strengthened and acquires new angles. Of partic­ular importance here is the very style of author’s reflection, which is associated with the revision of established philosophical categories and their interpretations, the departure from the usual interpretations to discover one’s own philosophical language: one’s own – both in terms of the individual author’s approach and in aspects of the formation of the modern domestic philosophical tradition. (shrink)

We prove two antibasis theorems for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} classes. The first is a jump inversion theorem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} classes with respect to the global structure of the Turing degrees. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P\subseteq 2^\omega}$$\end{document}, define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists \documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \in P}$$\end{document} of degree a. For any degree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf a \geq 0'}}$$\end{document}, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$$\end{document}. We prove that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf a \geq 0'}}$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} class P, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$$\end{document} then P contains a member of every degree. For any degree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf a \geq 0'}}$$\end{document} such that a is recursively enumerable (r.e.) in 0', let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$$\end{document}. The second theorem concerns the degrees below 0'. We prove that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf a\geq 0'}}$$\end{document} which is recursively enumerable in 0' and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} class P, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$$\end{document} then P contains a member of every degree. (shrink)

This work is a self-contained elementary exposition of the theory of recursive functionals, that also includes a number of advanced results.

On a widely held characterization, triadic joint attention is the capacity to perceptually attend to an object or event together with another subject. In the last four decades, research in developmental psychology has provided increasing evidence of the crucial role that this capacity plays in socio-cognitive development, early language acquisition, and the development of perspective-taking. Yet, there is a striking discrepancy between the general agreement that joint attention is critical in various domains, and the lack of theoretical consensus on how (...) to account for it. This paper pursues three interrelated aims: (i) it examines the contrast between reductive and non-reductive views of (triadic) joint attention, by bringing into focus the notion of recursive mindreading; (ii) it assembles, advances, and discusses a number of arguments against reductive views; (iii) finally, in dialogue with some prominent non-reductive views, it outlines the case for a non-reductive view that gives pride of place to the idea that co-attenders relate to one another as a ‘you’. (shrink)

Larry Temkin famously argues that what he calls spectrum arguments yield strong reason to reject Transitivity, according to which the ‘all-things-considered better than’ relation is transitive. Spectrum arguments do reveal that the conjunctions of independently plausible claims are inconsistent with Transitivity. But I argue that there is very strong independent reason to reject such conjunctions of claims, and thus that the fact that they are inconsistent with Transitivity does not yield strong reason to reject Transitivity.

The Spectrum of Consciousness (1977)--one of the founding texts of transpersonal psychology--introduces the full-spectrum model, showing how the psychological systems of the West can be integrated with the contemplative traditions of the East. No Boundary: Eastern and Western Approaches to Personal Growth (1979) is a simple yet comprehensive guide to psychologies and therapies available from both Western and Eastern sources. Several important early articles: "The Psychologia Perennis," "Are the Chakras Real?" and "Where It Was, I Shall Become.".

The current literature standardly conceives of voting power in terms of decisiveness: the ability to change the voting outcome by unilaterally changing one’s vote. I argue that this classic conception of voting power, which fails to account for partial decisiveness or efficacy, produces erroneous results because it saddles the concept of voting power with implausible microfoundations. This failure in the measure of voting power in turn reflects a philosophical mistake about the concept of social power in general: a failure to (...) recognize that an agent can exercise individual social power with others’ assistance, in virtue of the group’s collective power, sometimes even when she could not unilaterally scuttle the group’s collective power. I therefore develop a conception of efficacy that admits of degrees and defend a Recursive Measure of voting power that takes partial efficacy into account. (shrink)

ABSTRACT I consider paradoxical spectrum arguments involving transitive relations like ‘better than’. I argue that, despite being formally different from sorites arguments, at least some spectrum arguments arise from vagueness, and that vagueness might often be the most natural diagnosis.

Scientific realism is typically associated with metaphysics. One current incarnation of such an association concerns the requirement of a metaphysical characterization of the entities one is being a realist about. This is sometimes called “Chakravartty’s Challenge”, and codifies the claim that without a metaphysical characterization, one does not have a clear picture of the realistic commitments one is engaged with. The required connection between metaphysics and science naturally raises the question of whether such a demand is appropriately fulfilled, and how (...) metaphysics engages with science in order to produce what is called “scientific metaphysics”. Here, we map some of the options available in the literature, generating a conceptual spectrum according to how each view approximates science and metaphysics. This is done with the purpose of enlightening the current debate on the possibility of epistemic warrant that science could grant to such a metaphysics, and how different positions differently address the thorny issue concerning such a warrant. (shrink)

This volume contains papers from the recursion theory session of the Kazan Workshop on Recursion and Complexity Theory. Recursion theory, the study of computability, is an area of mathematical logic that has traditionally been particularly strong in the United States and the former Soviet Union. This was the first workshop ever to bring together about 50 international experts in recursion theory from the United States, the former Soviet Union and Western Europe.

This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.

Conceptual engineering is the method for assessing and improving our representational devices. On its ‘broad-spectrum’ version, it is expected to be appropriately applicable to any of our representation-involving cognitive activities, with major consequences for our whole cognitive life. This paper is about the theoretical foundations of conceptual engineering thus characterised. With a view to ensuring the actionability of conceptual engineering as a broad-spectrum method, it addresses the issue of how best to construe the subject matter of conceptual engineering (...) and successively defends the theses that conceptual engineering should be: (i) About concepts, (ii) psychologically theorised, (iii) as multiply realised functional kinds. Thereby, I claim to theoretically secure and justify the maximum scope, flexibility, and impact for the method of conceptual engineering on our representational devices in our whole cognitive life—in other words, a broad-spectrum version of conceptual engineering. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure - Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if computable is replaced by primitive recursive (p. r., for short), these definitions lead to a number of (...) different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in (...) the limit if, after some finite point in the sequence, all the guesses are correct and the same. Functions, sets, and functionals which are decidable by such infinite algorithms will be calledlimiting recursive. These, together with classes of objects which can beidentified in the limit, are the subjects of this report.Without qualification,setwill mean set of numbers;functionwill mean number-theoretic function of 1 variable, possibly partial;functionalswill take numerical values and have any number of numerical and/or function variables, the latter ranging solely over total functions of 1 variable. Thus a function is a special case of a functional,xwill invariably stand for a numerical variable;φfor a function variable;gfor a guess ;nfor the numerical variable which indexes the guesses, referred to as thetime. (shrink)

It has been claimed that recursion is one of the properties that distinguishes human language from any other form of animal communication. Contrary to this claim, a recent study purports to demonstrate center‐embedded recursion in starlings. I show that the performance of the birds in this study can be explained by a counting strategy, without any appreciation of center‐embedding. To demonstrate that birds understand center‐embedding of sequences of the form AnBn (such as A1A2B2B1, or A3A4A5B5B4B3) would require not only that (...) they discriminate such patterns from other patterns, but that they appreciate that elements must be bound from the outside in (thus, in the above examples, A1B1, A2B2, A3B3, A4B4, A5B5 are bound pairs). This has not been shown in nonhuman species, and sentences with this structure are difficult even for humans to parse. There appears to be no evidence to date that nonhuman species understand recursion. (shrink)

Martine Nida-Rümelin (1996) argues that color science indicates behaviorally undetectable spectrum inversion is possible and raises this possibility as an objection to functionalist accounts of visual states of color. I show that her argument does not rest solely on color science, but also on a philosophically controversial assumption, namely, that visual states of color supervene on physiological states. However, this assumption, on the part of philosophers or vision scientists, has the effect of simply ruling out certain versions of functionalism. (...) While Nida-Rümelin is quite right to search for empirical tests for claims about the nature of visual states, philosophical issues remain pivotal in determining the correctness of these claims. (shrink)

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega}$. $\mathsf{PROVI}$ supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and Shoenfield’s (...) unramified forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and the extension of a provident $M$ by a set-generic $\mathcal{G}$ is the provident closure of $M\cup\{\mathcal{G}\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$. (shrink)

Conceptual engineering is the method for assessing and improving our representational devices. On its ‘broad‐spectrum’ version, it is expected to be appropriately applicable to any of our representation‐involving cognitive activities, with major consequences for our whole cognitive life. This paper is about the theoretical foundations of conceptual engineering thus characterised. With a view to ensuring the actionability of conceptual engineering as a broad‐spectrum method, it addresses the issue of how best to construe the subject matter of conceptual engineering (...) and successively defends the theses that conceptual engineering should be: (i) About concepts, (ii) psychologically theorised, (iii) as multiply realised functional kinds. Thereby, I claim to theoretically secure and justify the maximum scope, flexibility, and impact for the method of conceptual engineering on our representational devices in our whole cognitive life—in other words, a broad‐spectrum version of conceptual engineering. (shrink)

This almost self-contained introduction to higher recursion theory is essential reading for all researchers in the field.

I explore the view that the imago Dei is essential to us as humans but accidental to us as persons. To image God is to resemble God, and resemblance comes in degrees. This has the straightforward—and perhaps disturbing—implication that we can be more or less human, and possibly cease to be human entirely. Hence, I call it the spectrum view. I argue that the spectrum view is complementary to the Biblical data, helps explain the empirical reality of horrendous (...) evil, and offers an elegant rapprochement between the traditional view of hell and its rivals. (shrink)

It is well known that any finitary operation is recursive in a suitable total numeration. A. Orlicki showed that there is an ω-operation not recursive in any total numeration. We will show that any ω-operation is recursive in a partial numeration.

Wilber's groundbreaking synthesis of religion, philosophy, physics, and psychology started a revolution in transpersonal psychology.

This fascinating popular science journey explores key concepts in information theory in terms of Conway's "Game of Life" program. The author explains the application of natural law to a random system and demonstrates the necessity of limits. Other topics include the limits of knowledge, paradox of complexity, Maxwell's demon, Big Bang theory, and much more. 1985 edition.

This chapter examines the spectrum inversion hypothesis as an argument against certain kinds of account of what it’s like to be conscious of color. The hypothesis aims to provide a counterexample to accounts of what it’s like to be conscious of color in non-qualitative terms, as well as to accounts of what it’s like to be conscious of color in terms of the representational content of conscious visual states (which, according to some philosophers, is in turn given an account (...) in non-qualitative terms). I explain the hypothesis’s reasoning in sections 1 and 2. I then take up background issues: since the counterexample provided by the hypothesis is typically given as a possibility rather than an actuality, I briefly discuss the kind of possibility involved in section 3, and the methodologies used to evaluate the plausibility of possibilities in section 4. In section 5, I describe some general considerations that are commonly used against the spectrum inversion hypothesis. I take up attempts to support the hypothesis with findings from color science in section 6. I end with comments skeptical of both the motivation for the spectrum inversion hypothesis and the methodology used to support it. (shrink)

The series is devoted to the publication of high-level monographs on all areas of mathematical logic and its applications. It is addressed to advanced students and research mathematicians, and may also serve as a guide for lectures and for seminars at the graduate level.

This article shows that introducing the positive comparative relation parity only helps one block so‐called “Spectrum Arguments” in order to avoid their unsavoury implications if one specifies parity in a specific way with respect to its persistence. The article illustrates how parity must both admit of persistency and be weakly non‐persistent for parity to block Spectrum Arguments, and identifies some consequences of that discovery for the general debate on Spectrum Arguments, value theory and comparability problems.

Even if spectrum inversion of various sorts is possible, spectrum inversion without a difference in representation is not. So spectrum inversion does not pose a challenge for the intentionalist thesis that, necessarily, within a given sense modality, if two experiences are alike with respect to content, they are also alike with respect to their phenomenal character. On the contrary, reflection on variants of standard cases of spectrum inversion provides a strong argument for intentionalism. Depending on one’s (...) views about the possibility of spectrum inversion, the impossibility of spectrum inversion without a difference in representation can also be used as an argument against a variety of reductive theories of mental representation. (shrink)

Formerly a spectral apparition that haunted behaviorism and provided a puzzle about our knowledge of other minds, the inverted spectrum possibility has emerged as an important challenge to functionalist accounts of qualia. The inverted spectrum hypothesis raises the possibility that two individuals might think and behave in the same way yet have different qualia. The traditional supposition is of an individual who has a subjective color spectrum that is inverted with regard to that had by other individuals. (...) When he looks at red objects, this individual has the qualia normally produced in others by blue objects. And when presented with a blue object, this individual experiences qualia that most persons experience only when presented with red objects. And so forth - the Invert's color spectrum is the inverse of normal; there are systematic inter-subjective differences in qualia. (shrink)

Language sciences have long maintained a close and supposedly necessary coupling between the infinite productivity of the human language faculty and recursive grammars. Because of the formal equivalence between recursion and non-recursive iteration; recursion, in the technical sense, is never a necessary component of a generative grammar. Contrary to some assertions this equivalence extends to both center-embedded relative clauses and hierarchical parse trees. Inspection of language usage suggests that recursive rule components in fact contribute very little, and (...) likely nothing significant, to linguistic creativity. Further than this, if the productivity of human language is considered as not rigidly bound, but not infinite, then the need for any sort of iteration in generative grammars vanishes and can be replaced with a Non-iterative Explicit Alternatives Rule grammar. The knock-on effects of dispensing with recursive grammar components are: that language diversity can simply be based on rule and lexicon combinatorics with no potentially infinite dimensions derived from recursive or iterative components of rules; the oddity of a vast, multiply-infinite competence set of ‘grammatical but unacceptable’ productions is gone; and the development of a language faculty based on rules that eschew iterative rule components avoids any need for explaining ‘special’ mechanisms. On the broader front of searching for the mechanisms of mind, our analysis can be similarly applied to the proposals for a recursive basis for mind as an explanation for humanity’s great leap forward. (shrink)