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 the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (...) (second-order number theory with a -comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a operator can be introduced in other settings. (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)

We locate winning strategies for various ${\mathrm{\Sigma }}_{3}^{0}$ -games in the L-hierarchy in order to prove the following: Theorem 1. KP+Σ₂-Comprehension $\vdash \exists \alpha L_{\alpha}\ models"\Sigma _{2}-{\bf KP}+\Sigma _{3}^{0}-\text{Determinacy}."$ Alternatively: ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ "there is a β-model of ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17 em}}{\mathrm{\Sigma }}_{3}^{0}$ -Determinacy." The implication is not reversible. (The antecedent here may be replaced with ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\mathrm{\Pi }}_{3}^{1}\right)-{\mathrm{C}\mathrm{A}}_{0}:\text{\hspace{0.17em}}{\mathrm{\Pi }}_{3}^{1}$ instances of Comprehension with only ${\mathrm{\Pi }}_{3}^{1}$ -lightface definable parameters—or even weaker theories.) Theorem 2. KP +Δ₂-Comprehension +Σ₂-Replacement + ${\mathrm{\Sigma }}_{3}^{0}\phantom{\rule{0ex}{0ex}}$ -Determinacy. (Here AQI (...) is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: $\Delta _{3}^{1}{\rm CA}_{0}+{\rm AQI}\nvdash \Sigma _{3}^{0}$ -Determinacy. Hence the theories: ${\mathrm{\Pi }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0},\text{\hspace{0.17em}}{\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\text{\hspace{0.17em}}{\mathrm{\Sigma }}_{3}^{0}-\mathrm{D}\mathrm{e}\mathrm{t}\phantom{\rule{0ex}{0ex}}$ -Det, ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\mathrm{A}\mathrm{Q}\mathrm{I}$ , and ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ are in strictly descending order of strength. (shrink)

We represent truth sets for a variety of the well known semantic theories of truth as those sets consisting of all sentences for which a player has a winning strategy in an infinite two person game. The classifications of the games considered here are simple, those over the natural model of arithmetic being all within the arithmetical class of $\Sum_{3}^{0}$.

We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set. We give a simplified account of varied revision sequences-as a generalised algorithmic theory of truth. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.

We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise the revision theoretically definable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta’s theory of truth over arithmetic using \emph{fully varied revision} sequences yields a complete \Pi13 set of integers; (iii) the set of \emph{stably categorical} sentences using their revision operator ψ is similarly \Pi13 and which (...) is complete in Gödel’s universe of constructible sets L; (iv) give an alternative account of a theory of truth—realistic variance that simplifies full variance, whilst at the same time arriving at Kripkean fixed points. (shrink)

The titles of these lectures were "In Search of the Miraculous" and "The Problems of Death." In these lectures, which were to serve as an introduction to a book on my travels it was my intention to write, I said that in India the "miraculous" was ...

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated (...) with the jump operator satisfy a Spector criterion, and correspond to the L ζ -stables. It also implies that the machines devised are "Σ 2 Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. (shrink)

The fact that psychic life is not merely given externally and as mutual externality, but is given in its nexus, given by self-knowledge, by internal experience, constitutes the basic difference between psychological knowledge and knowledge of nature.

Experimental evidence strongly supports the view that the subdivision of organ anlagen into smaller structural units is an autonomous process. Dalcq &Pasteels' hypothesis which says that the boundaries between the different areas into which a morphogenetic field differentiates are determined by “Threshold values” in the “potential” of the field in question, is inconsistent with our present knowledge of biochemical reaction systems. Threshold values may only be used indescribing the spatial differentiation of a morphogenetic field. It is suggested that the latter (...) concept be replaced by the concept of “optimal values” in the intensity of a morphogenetic field-specific for the various chemo-differentiations characterizing the different regions into which a morphogenetic field will split up-and that each optimal value isdeterminative for the appearance of one particular differentiation within the morphogenetic field, each differentiation being formed within a certain range of intensity of the field. On the basis of the hypothesis formulated above a “virginal” morphogenetic field will necessarily differentiate into a number of smaller fields of second order, which in their turn will split up into fields of third order, etc. ” Pattern formation” may be defined as the differentiation of a morphogenetic field into an increasing number of smaller fields of higher order of qualitative different chemical and structural composition. Since differentiation of a morphogenetic field proceeds autonomously, the capacity for pattern formation is an intrinsic property of a morphogenetic field. In the light of these considerations it is suggested that a morphogenetic field action is responsible for the first differentiation of the developing embryo into regions with predominantly ectodermal, mesodermal or endodermal differentiation tendencies. The present knowledge about the characteristic pattern formation taking place in the axial mesoderm as well as in the overlying neural plate seems to be in good agreement with the implications of the present morphogenetic field concept. It is emphasized, however, that if the present morphogenetic field concept cannot be characterized further biochemically and biophysically, it will have to be replaced by another concept which can be defined more sharply and is more easily accessible to experimental analysis. Les preuves expérimentales confirment que la subdivision des ébauches des organes en unités structurelles plus petites est un processus autonome. L'hypothèse deDalcq &Pasteels disant que les bornes entre les différents domaines formées par la différenciation d'un champ morphogénétique sont déterminées par des „valeurs seuil” dans le „potentiel” du champ en question, n'est pas en harmonie avec notre connaissance actuelle des systèmes biochimiques. Des “valeurs seuil” ne peuvent être employés qu'en décrivant la différenciation spatiale d'un champ morphogénétique. On prend en considération de remplacer le concept des “valeurs seuil” par le concept des “valeurs optimales” dans l'intensité d'un champ morphogénétique. Ce seraient les valeurs spécifiques aux différenciations chimiques diverses qui caractérisent les différents domaines dans lesquels un champ morphogénétique se différencie. En outre, chaque valeur optimale serait déterminative à la formation d'une différenciation spéciale dans le champ morphogénétique, chaque différentiation se formant dans un certain trajet de l'intensité du champ. Par suite de l'hypothèse formulée ci-dessus un champ morphogénétique “virginal” va nécessairement se différencier en un nombre de champs plus petits de deuxième ordre, qui à leur tour se divisent en champs d'ordre troisième, etc. ... La régionalisation peut être définie comme la différenciation d'un champ morphogénétique dans un nombre de plus en plus élevé de champs plus petits d'ordre supérieur de composition chimique et structurelle qualitativement différente. Parce que la différenciation d'un champ morphogénétique se produit de façon autonome, le pouvoir de régionalisation est une propriété intrinsique d'un champ morphogénétique. Ces considérations nous mènent à proposer que l'action d'un champ morphogénétique est responsable de la première différenciation de l'embryon en régions qui possèdent des tendences de différenciation principalement ectodermiques, mésodermiques, ou bien endodermiques. La connaissance actuelle concernant la régionalisation caractéristique qui se produit dans le mésoderme axial ainsi que dans la plaque neurale superposée paraît se conformer aux conséquences du concept de champ morphogénétique. Cependant on souligne le fait que dans le cas òu le concept actuel du champ morphogénétique ne pourra pas être plus amplement caractérisé de façon biochimique et biophysique, il faudra le remplacer par un autre concept qui puit être défini plus nettement et qui est mieux accessible à l'analyse expérimentale. Die Auffassung, dass die Aufteilung von Organanlagen in kleineren Struktureinheiten ein selbständiger Prozess ist, findet Bestätigung in den experimentellen Ergebnissen. Die Hypothese vonDalcq &Pasteels , dass die Grenzen zwischen den unterschiedenen Gebieten, in denen ein morphogenetisches Feld sich differenziert, durch “Schwellenwerte” im “Potential” des genannten Feldes bestimmt werden, ist unvereinbar mit unserem heutigen Wissen über biochemischen Reaktionssystemen. Der Gebrauch des Begriffs “Schwellenwert” ist nur gestattet bei der Beschreibung der räumlichen Differenzierung eines morphogenetischen Feldes. Statt dieses Begriffs wird der Begriff “Optimaler Wert” vorgeschlagen; Werte also in der Intensität eines morphogenetischen Feldes, optimal und spezifisch für die verschiedene Chemodifferenzierungen, welche die Gebiete kennzeichnen, in denen ein morphogenetisches Feld sich differenzieren wird. Jeder optimaler Wert soll dann das Erscheinen einer bestimmten Differenzierung innerhalb des morphogenetischen Feldes veranlassen, welche Differenzierung sich bildet innerhalb eines bestimmten Bereichs der Feldintensität. Auf Grund der obigen Hypothese wird ein “virginelles” morphogenetisches Feld zwangsläufig sich differenzieren in eine Anzahl kleinerer Feldern zweiten Grades; diese in Felder dritten Grades, usw. Musterbildung könnte definiert werden als die Differenzierung eines morphogenetischen Feldes in eine zunehmende Anzahl kleinerer Felder höheren Grades von qualitativ verschiedener chemischen und strukturellen Zusammensetzung, Weil diese Differenzierung autonom vor sich geht, ist das Vermögen zur Musterbildung eine innewohnende Eigenschaft eines morphogenetischen Feldes. Im Lichte dieser Betrachtungen ist es naheliegend anzunehmen dass eine morphogenetische Feldwirkung verantwortlich ist für die erste Differenzierung des sich entwickelnden Embryos in Gebiete überwiegend ektodermalen, mesodermalen oder entodermalen Differenzierungstendenzen. Die heutigen Kenntnisse über die charakteristische Musterbildung im axialen Teil des Mesoderms und in der überliegenden Neuralplatte scheinen in guter Übereinstimmung zu sein mit den des morphogenetischen Feldbegriffs. Dennoch wird benachdruckt, dass, falls das gegenwärtige morphogenetische Feldbegriff nicht weiter biochemisch und biophysisch gekennzeichnet werden kann, anstatt dessen ein anderer Begriff entwickelt werden muß, der schärfer definiert werden kann und besser zugänglich ist für experimentelle Analyse. (shrink)

The Argument from Inductive Risk (AIR) is taken to show that values are inevitably involved in making judgements or forming beliefs. After reviewing this conclusion, I pose cases which are prima facie counterexamples: the unreflective application of conventions, use of black-boxed instruments, reliance on opaque algorithms, and unskilled observation reports. These cases are counterexamples to the AIR posed in ethical terms as a matter of personal values. Nevertheless, it need not be understood in those terms. The values which load a (...) theory choice may be those of institutions or past actors. This means that the challenge of responsibly handling inductive risk is not merely an ethical issue, but is also social, political, and historical. (shrink)

We follow [8] in asking when a set of ordinals $X \subseteq \alpha$ is a countable union of sets in K, the core model. We show that, analogously to L, and X closed under the canonical Σ 1 Skolem function for K α can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.

Some philosophers understand natural kinds to be the categories which are constraints on enquiry. In order to elaborate the metaphysics appropriate to such an account, I consider the complicated history of scurvy, citrus, and vitamin C. It may be tempting to understand these categories in a shallow way (as mere property clusters) or in a deep way (as fundamental properties). Neither approach is adequate, and the case instead calls for middle-range ontology: starting from categories which we identify in the world (...) and elaborating their structure, but not pretending to jump ahead to a complete story about fundamental being. (shrink)

We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent Approachable Free Subsets Property (AFSP) in his work on the $${\text {pcf}}$$ pcf -conjecture. A recent related property is the Approachable Bounded Subset Property (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish:TheoremIf ABSP holds for an ascending sequence $$ \langle \aleph (...) _{n_{m}} \rangle _{m}$$ ⟨ ℵ n m ⟩ m $$( n_{m} \in \omega )$$ ( n m ∈ ω ) then there is an inner model with measurables $$\kappa < \aleph _{\omega }$$ κ < ℵ ω of arbitrarily large Mitchell order below $$\aleph _{\omega }$$ ℵ ω, that is: $$\sup \left\{ \alpha \mid {\exists }\kappa < \aleph _{\omega } o ( \kappa ) \ge \alpha \right\} = \aleph _{\omega }$$ sup α ∣ ∃ κ < ℵ ω o ( κ ) ≥ α = ℵ ω. A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales. (shrink)

There is a history of thinking that photographs provide a special kind of access to the objects depicted in them, beyond the access that would be provided by a painting or drawing. What is included in the photograph does not depend on the photographer’s beliefs about what is in front of the camera. This feature leads Kendall Walton to argue that photographs literally allow us to see the objects which appear in them. Current generative algorithms produce images in response to (...) users’ text prompts. Depending on the parameters, the output can resemble specific people or things which are named in the prompt. This resemblance does not depend on the user’s beliefs, so generated images are in this sense like photographs. Given this parallel, how should we think about AI-generated images? (shrink)

Threshold- and signal-detection-based models have dominated theorizing about recognition memory. Building upon these theoretical frameworks, we have argued for a dual-process model in which conscious recollection and familiarity contribute to memory performance. In the current paper we assessed several memory models by examining the effects of levels of processing and the number of presentations on recognition memory receiver operating characteristics . In general, when the ROCs were plotted in probability space they exhibited an inverted U shape; however, when they were (...) plotted inzspace they exhibited a U shape. An examination of the ROCs showed that the dual-process model could account for the observed ROCs, but that models based solely on either threshold or signal-detection processes failed to provide a sufficient account of the data. Furthermore, an examination of subjects' introspective reports using the remember/know procedure showed that subjects were aware of recollection and familiarity and were able to consistently report on their occurrence. The remember/know data were used to accurately predict the shapes of the ROCs, and estimates of recollection and familiarity derived from the ROC data mirrored the subjective reports of these processes. (shrink)

Wikipedia is a free encyclopedia that is written and edited entirely by visitors to its website. I argue that we are misled when we think of it in the same epistemic category with traditional general encyclopedias. An empirical assessment of its reliability reveals that it varies widely from topic to topic. So any particular claim found in it cannot be relied on based on its source. I survey some methods that we use in assessing specific claims and argue that the (...) structure of the Wikipedia frustrates them. (shrink)

We sketch a broadening of the Gupta-Belnap notion of a circular or revision theoretic definition into that of a more generalized form incorporating ideas of Kleene’s generalized or higher type recursion. This thereby connects the philosophically motivated, and derived, notion of a circular definition with an older form of definition by recursion using functionals, that is functions of functions, as oracles. We note that Gupta and Belnap’s notion of ‘categorical in L’ can be formulated in at least one of these (...) schemes. (shrink)

This little volume shows a remarkable familiarity with both continental and Anglo-American literature on the subjects of probability and induction. Discussion is both expository and critical. The author argues in favor of a psychological-pragmatic interpretation of the principle of induction and the theory of confirmation.--A. P. D. M.

According to the standard narrative, natural kind is a technical notion that was introduced by John Stuart Mill in the 1840s and the recent craze for natural kinds, launched by Putnam and Kripke, is a continuation of that tradition. I argue that the standard narrative is mistaken. The Millian tradition of kinds was not particularly influential in the 20th-century, and the Putnam-Kripke revolution did not clearly engage with even the remnants that were left of it. The presently active tradition of (...) natural kinds is less than half a century old. Recognizing this might help us better appreciate both Mill and natural kinds. (shrink)

Many enhancement technologies are distributed by healthcare professionals—by physicians—who are held to the Hippocratic Oath and the goals of medicine. While the ethics of enhancement has been widely discussed with regard to the social justice, humanism, morals and normative values of these interventions, their place in medicine has not attracted a great deal of attention. This paper investigates the potential for enhancement technologies to fulfil the goals of medicine, arguing that they play a role in promoting the health of individuals, (...) and thus, an unavoidable place in medicine. It also warns of potential dangers, suggesting a set of guidelines to initiate conversations regarding the role and responsibilities of physicians practising in an era of enhancement. (shrink)

A small large cardinal upper bound in V for proving when certain subsets of ω 1 (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that $\forall B \subseteq \omega_1 \lbrack B$ is universally Baire $\Leftrightarrow B \in L\lbrack r \rbrack$ for some real r] is preserved under set-sized forcing extensions if and only if there (...) are arbitrarily large "admissibly measurable" cardinals. (shrink)

Medical ethics is that branch of applied philosophy which considers issues of values raised by medical practice, and should not be equated with 'principlism'. Clarification of facts/values distinctions is an important part of this work. The notion that medical philosophy can flourish in the hands of medical 'generalists' without specialist philosophers, is misguided. Both must work together to promote right reason and right action in medical education and practice.

More than half of the book is devoted to what is almost a catalogue exposition of Western moral philosophy from Kant to Nowell-Smith. Although the author claims in the preface to have included "suitable references to Indian points of view, including the Buddhist," what references there are are usually too general, and never fully explained to be of any use to Western readers. Exposition tends to be disjointed, and the English leaves much to be desired. On the whole there is (...) much eclecticism and little originality.--A. P. D. M. (shrink)

The problem of the logic of understanding history is discussed. We propose a model of understanding history as a conversation, which the person thinking about history has with interlocutors (sources) of previous epochs. The epoch is interpreted as a special way of problematizing human's understanding of him (her) self and the world around. At the same time, three gifts — attention, recognition and name — underlie the connection of the historian in conversation with interlocutors (sources) from other eras. They are (...) not given in advance but are results of a struggle for the recognition. In choosing his (her) interlocutor (source), the historian chooses him (her) self. By giving this interlocutor (the source) the opportunity to be heard, the historian, at the same time, receives a gift from the source — the opportunity to express what would otherwise remain unspoken in the historian's own mind. The logic of conversation, designated as transduction, is not the connection through the identification of the beginning and the end of the discourse, which is appropriate and necessary within the speech of each of the conversationalists, but the connection of the beginnings of these discordant discourses among themselves. (shrink)

It is now commonly held that values play a role in scientific judgment, but many arguments for that conclusion are limited. First, many arguments do not show that values are, strictly speaking, indispensable. The role of values could in principle be filled by a random or arbitrary decision. Second, many arguments concern scientific theories and concepts which have obvious practical consequences, thus suggesting or at least leaving open the possibility that abstruse sciences without such a connection could be value-free. Third, (...) many arguments concern the role values play in inferring from evidence, thus taking evidence as given. This paper argues that these limitations do not hold in general. There are values involved in every scientific judgment. They cannot even conceivably be replaced by a coin toss, they arise as much for exotic as for practical sciences, and they are at issue as much for observation as for explicit inference. (shrink)

It seems obvious that a community of one thousand scientists working together to make discoveries and solve puzzles should arrange itself differently than would one thousand scientist-hermits working independently. Because of limited time, resources, and attention, an independent scientist can explore only some of the possible approaches to a problem. Working alone, each hermit would explore the most promising approaches. They would needlessly duplicate the work of others and would be unlikely to develop approaches which look unpromising but really have (...) tremendous potential. Contrariwise, a large community can more rigorously explore the space of possible approaches. Most scientists should work on the most promising approaches, but a smaller number can be committed to approaches that initially look less promising. Exploratory work can reveal if one of those initially unpromising approaches has unrealized potential, and more scientists can adopt it once its potential becomes more apparent. (shrink)