We show that many principles of first-order arithmetic, previously only known to lie strictly between [Formula: see text]-induction and [Formula: see text]-induction, are equivalent to the well-foundedness of [Formula: see text]. Among these principles are the iteration of partial functions of Hájek and Paris, the bounded monotone enumerations principle by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Péter function. With this we show that the well-foundedness of [Formula: see text] is a (...) far more widespread than usually suspected. Further, we investigate the [Formula: see text]-iterated version of the bounded monotone iterations principle, and show that it is equivalent to the well-foundedness of the -height [Formula: see text]-tower [Formula: see text]. (shrink)

Let the chain antichain principle (CAC) be the statement that each partial order on $\mathbb{N}$ possesses an infinite chain or an infinite antichain. Chong, Slaman, and Yang recently proved using forcing over nonstandard models of arithmetic that CAC is $\Pi^1_1$-conservative over $\text{RCA}_0+\Pi^0_1\text{-CP}$ and so in particular that CAC does not imply $\Sigma^0_2$-induction. We provide here a different purely syntactical and constructive proof of the statement that CAC (even together with WKL) does not imply $\Sigma^0_2$-induction. In detail we show using a (...) refinement of Howard's ordinal analysis of bar recursion that $\text{WKL}_0^\omega+\text{CAC}$ is $\Pi^0_2$-conservative over PRA and that one can extract primitive recursive realizers for such statements. Moreover, our proof is finitary in the sense of Hilbert's program. CAC implies that every sequence of $\mathbb{R}$ has a monotone subsequence. This Bolzano-Weierstraß}-like principle is commonly used in proofs. Our result makes it possible to extract primitive recursive terms from such proofs. We also discuss the Erdős-Moser principle, which—taken together with CAC—is equivalent to $\text{RT}^2_2$. (shrink)

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher-order arithmetic. Let [Formula: see text] be the statement that a non-principal ultrafilter on ℕ exists and let [Formula: see text] be the higher-order extension of ACA0. We show that [Formula: see text] is [Formula: see text]-conservative over [Formula: see text] and thus that [Formula: see text] is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly (...) [Formula: see text] statement ∀ f ∃ g A qf in [Formula: see text] a realizing term in Gödel's system T can be extracted. This means that one can extract a term t ∈ T, such that ∀ f A qf). (shrink)

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle and a weak variant of it.We show that BW is instance-wise equivalent to the weak König’s lemma for Σ01-trees . This means that from every bounded sequence of reals one can compute an infinite Σ01-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d ≫ 0′ are exactly those containing (...) an accumulation point for all bounded computable sequences.Let BWweak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence . We show that BWweak is instance-wise equivalent to the cohesive principle and—using this—obtain a classification of the computational and logical strength of BWweak. Especially we show that BWweak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BWweak. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

In this paper we study with proof-theoretic methods the function(al) s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functional provably recursive from $RCA_0 + COH + \Pi _1^0 - CP$ are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that $WKL_0 + \Pi _1^0 - (...) CP + COH$ is $\Pi _1^0 $ -conservative over PRA. Recent work of the first author showed that $\Pi _1^0 + CP + COH$ is equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey's theorem for pairs and two colors $\left( {RT_2^2 } \right)$ we obtain the upper bounded that the type 2 functionals provable recursive relative to $RCA_0 + \sum\nolimits_2^0 {IA + RT_2^2 } $ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT2..... Moreover, this yields a new proof of the fact that $WKL_0 + \sum\nolimits_2^0 {IA} + RT_2^2 $ is $\Pi _1^0 $ -conservative over $RCA_0 + \sum\nolimits_2^0 { - IA} $ . The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using $\Pi _1^0 $ -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for $RT_2^2 $. (shrink)

Let ${2-\textsf{RAN}}$ be the statement that for each real X a real 2-random relative to X exists. We apply program extraction techniques we developed in Kreuzer and Kohlenbach (J. Symb. Log. 77(3):853–895, 2012. doi:10.2178/jsl/1344862165), Kreuzer (Notre Dame J. Formal Log. 53(2):245–265, 2012. doi:10.1215/00294527-1715716) to this principle. Let ${{\textsf{WKL}_0^\omega}}$ be the finite type extension of ${\textsf{WKL}_0}$ . We obtain that one can extract primitive recursive realizers from proofs in ${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN}}$ , i.e., if ${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} (...) + 2-\textsf{RAN} \, {\vdash} \, \forall{f}\, {\exists}{x} A_{qf}(f,x)}$ then one can extract from the proof a primitive recursive term t(f) such that ${A_{qf}(f,t(f))}$ . As a consequence, we obtain that ${{\textsf{WKL}_0}+ \Pi^0_1 - {\textsf{CP}} + 2-\textsf{RAN}}$ is ${\Pi^0_3}$ -conservative over ${\textsf{RCA}_0}$. (shrink)

We show how one can obtain solutions to the Arzelà‐Ascoli theorem using suitable applications of the Bolzano‐Weierstraß principle. With this, we can apply the results from and obtain a classification of the strength of instances of the Arzelà‐Ascoli theorem and a variant of it. Let be the statement that each equicontinuous sequence of functions contains a subsequence that converges uniformly with the rate and let be the statement that each such sequence contains a subsequence which converges uniformly but possibly without (...) any rate. We show that is instance‐wise equivalent, over, to the Bolzano‐Weierstraß principle and that is instance‐wise equivalent, over, to, and thus to the strong cohesive principle (). Moreover, we show that over the principles, and are equivalent. (shrink)

We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete metric space cannot be decomposed into countably many nowhere dense pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of (...) the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets. (shrink)

The familiar Vendler-Kenny scheme of verb-types, viz., performances (further differentiated by Vedler into accomplishments and achievements), activities, and states, is too narrow in two important respects. First, it is narrow linguistically. It fails to take into account the phenomenon of verb aspect. The trichotomy is not one of verbs as lexical types but of predications. Second, the trichotomy is narrow ontologically. It is a specification in the context of human agency of the more fundamental, topic-neutral trichotomy, event-process-state.The central component in (...) this ontological trichotomy, event, can be sharply differentiated from its two flanking components by adapting a suggestion by Geoffrey N. Leech and others that the contrast between perfective and imperfective aspect in verbs corresponds to the count/mass distinction in the domain of nouns. With the help of two distinctions, of cardinal count adverbials versus frequency adverbials, and of occurrence versus associated occasion, two interrelated criteria for event predication are developed. Accordingly, Mary capsized the boat is an event predication because (a) it is equivalent to There was at least one capsizing of the boat by Mary, or (b) because it admits cardinal count adverbials, e.g., at least once, twice, three times. Ontologically speaking, events are defined as those occurrences that are inherently countable. (shrink)

Achieving space situational awareness requires, at a minimum, the identification, characterization, and tracking of space objects. Leveraging the resultant space object data for purposes such as hostile threat assessment, object identification, and conjunction assessment presents major challenges. This is in part because in characterizing space objects we reference a variety of identifiers, components, subsystems, capabilities, vulnerabilities, origins, missions, orbital elements, patterns of life, operational processes, operational statuses, and so forth, which tend to be defined in highly heterogeneous and sometimes inconsistent (...) ways. The Space Domain Ontologies are designed to provide a consensus-based realist framework for formulating such characterizations in a way that is both consistent and computable. Space object data are aligned with classes and relations in a suite of ontologies built around the existing Space Object Ontology. They are stored in a dynamically updated Resource Description Framework triple store, which can be queried to support space situational awareness and the needs of spacecraft operators and analysts. This paper provides an overview of the Space Domain Ontologies and their development and use. It presents the motivation for and advantages of the Space Domain Ontologies, including the benefits they provide for enhancing and maintaining long-term space situational awareness. (shrink)

Achieving space domain awareness requires the identification, characterization, and tracking of space objects. Storing and leveraging associated space object data for purposes such as hostile threat assessment, object identification, and collision prediction and avoidance present further challenges. Space objects are characterized according to a variety of parameters including their identifiers, design specifications, components, subsystems, capabilities, vulnerabilities, origins, missions, orbital elements, patterns of life, processes, operational statuses, and associated persons, organizations, or nations. The Space Object Ontology provides a consensus-based realist framework (...) for formulating such characterizations in a computable fashion. Space object data are aligned with classes and relations in the Space Object Ontology and stored in a dynamically updated Resource Description Framework triple store, which can be queried to support space domain awareness and the needs of spacecraft operators. This paper presents the core of the Space Object Ontology, discusses its advantages over other approaches to space object classification, and demonstrates its ability to combine diverse sets of data from multiple sources within an expandable framework. Finally, we show how the ontology provides benefits for enhancing and maintaining longterm space domain awareness. (shrink)

Analyzes the poem "On Nature" by Parmenides, arguing that is actually a philosophical argument disguised as Homer-like mythological journey. Original.

We have begun work on two separate but related ontologies for the study of neurological diseases. The first, the Neurological Disease Ontology (ND), is intended to provide a set of controlled, logically connected classes to describe the range of neurological diseases and their associated signs and symptoms, assessments, diagnoses, and interventions that are encountered in the course of clinical practice. ND is built as an extension of the Ontology for General Medical Sciences — a high-level candidate OBO Foundry ontology that (...) provides a set of general classes that can be used to describe general aspects of medical science. ND is being built with classes utilizing both textual and axiomatized definitions that describe and formalize the relations between instances of other classes within the ontology itself as well as to external ontologies such as the Gene Ontology, Cell Ontology, Protein Ontology, and Chemical Entities of Biological Interest. In addition, references to similar or associated terms in external ontologies, vocabularies and terminologies are included when possible. Initial work on ND is focused on the areas of Alzheimer’s and other diseases associated with dementia, multiple sclerosis, and stroke and cerebrovascular disease. Extensions to additional groups of neurological diseases are planned. The second ontology, the Neuro-Psychological Testing Ontology (NPT), is intended to provide a set of classes for the annotation of neuropsychological testing data. The intention of this ontology is to allow for the integration of results from a variety of neuropsychological tests that assay similar measures of cognitive functioning. Neuro-psychological testing is an important component in developing the clinical picture used in the diagnosis of patients with a range of neurological diseases, such as Alzheimer’s disease and multiple sclerosis, and following stroke or traumatic brain injury. NPT is being developed as an extension to the Ontology for Biomedical Investigations. (shrink)

A lavishly decorated handbook of medicine was conceived for the lay public on topics such as human health, healing, medicine, and household management.

I am grateful to the editors of the Canadian Journal of Philosophy for inviting me to write a comment on Kathleen Gill’s ‘On the Metaphysical Distinction Between Processes and Events’. I readily concede that she is right in the central criticism she makes of my 1978 paper: that a properly metaphysical or ontological distinction between processes and events, if it is to be made at all, cannot be sustained on the basis of the informal linguistic criteria I offered in ‘Events, (...) Processes, and States.’ My main concern in that paper was to show that discussions of the Kenny-Vendler typology of activities, performances, and states had focused too narrowly on verb types, on adverbial phrases, on the domain of human action, and on English-language intuitions. Mine was one of several voices urging in the late 1970s that issues of tense logic and of the logic of verb types had to be placed in the broader context of verb aspect — a salient and pervasive structure in natural languages generally, and one that had been studied by philologists and linguists since the nineteenth century. Captivated, as I and others had become at the time, by evidence that the distinction between count nouns and mass nouns appears to be mirrored in verb predications, I went on to make some rather venturesome ‘ontological’ claims. It is satisfying, of course, that Gill accepts not only that there is ‘substantial evidence in favor of recognizing a grammatical distinction between the language of processes and [the language of] events’ but also that ‘similarities with count and mass expressions will allow us to apply techniques for han dling nominal expressions to verbal expressions’. It is more than satisfying that the phenomenon of verb aspect is now noticed and investigated by philosophers as well as by linguists. (shrink)

This article discusses Xenophanes' “cloud astro-physics”. It analyses and explains all heavenly and meteorological phenomena in terms of clouds. It provides a view of this newer Xenophanes, who is now being recognized as an important philosopher-scientist in his own right and a crucial figure in the development of critical thought about human knowledge and its objects in the next generation of Presocratic thinkers. Xenophanes' account has been preserved in Aëtius, the doxographic compendium reconstructed by Hermann Diels late in the nineteenth (...) century mainly from two sources that show extensive parallelism: pseudo-Plutarch Placita Philosophorum or Epitome of Physical Opinions ; and Ioannes Stobaeus' Eclogae Physicae or Physical Extracts. In the Stobaeus version, which is also the one printed in the standard edition of the Pre-socratics. (shrink)

In the work of interpreting Parmenides we have witnessed in the ’sixties and ’seventies, in English language scholarship, that rarest of phenomena in the study of ancient philosophy, the emergence of a consensus. Four interpretive theses now seem quite widely shared: Parmenides deliberately suppresses the subject of esti, “is,” or einai, “to be,” in his statement of the two “routes” in B2, his intention being to allow the subject to become gradually specified as the argument unfolds. The negative route, ouk (...) esti, “is not,” or mē einai, “not to be,” is banned because sentences that adhere to it fail to refer to actual entities—the latter to be understood broadly, as will shortly be stated in thesis. The argument does not depend on a confusion between the “is” of predication and the “is” of existence. In the relevant contexts, esti and einai involve a “fused” or “veridical” use of the verb “to be”; in other words, esti or einai have the force of “is actual” or “obtains,” or “is the case,” envisaging a variable subject x that ranges over states-of-affairs. (shrink)

This collection introduces readers to some of the most respected Pre-Socratic scholarship of the twentieth century. It includes translations of important works from European scholars that were previously unavailable in English and incorporates the major topics and approaches of contemporary scholarship. Here is an essential book for students and scholars alike. "Students of the Pre-Socratics must be grateful to Mourelatos and his publishers for making these essays available to a wider public."--T. H. Irwin, American Journal of Philology "Mourelatos is a (...) superb editor, and teaching Pre-Socratics in the future with this collection on the reading list will not only be easier but also better."--Jorgen Mejer, The Classical World "The editor has done his work judiciously. It would be difficult to devise a better balance between different parts of the subject."--Edward Hussey, Archives internationales d'histoire des sciences "[This book] will undoubtedly become an indispensable aid for beginning and advanced students of the Pre-Socratics."--David E. Hahm, Isis. (shrink)

To understand how individuals adapt to and anticipate each other in joint tasks, we employ a bidirectional delay–coupled dynamical system that allows for mutual adaptation and anticipation. In delay–coupled systems, anticipation is achieved when one system compares its own time‐delayed behavior, which implicitly includes past information about the other system’s behavior, with the other system’s instantaneous behavior. Applied to joint music performance, the model allows each system to adapt its behavior to the dynamics of the other. Model predictions of asynchrony (...) between two simultaneously produced musical voices were compared with duet pianists’ behavior; each partner performed one voice while auditory feedback perturbations occurred at unpredictable times during live performance. As the model predicted, when auditory feedback from one musical voice was removed, the asynchrony changed: The pianist’s voice that was removed anticipated (preceded) the actions of their partner. When the auditory feedback returned and both musicians could hear each other, they rapidly returned to baseline levels of asynchrony. To understand how the pianists anticipated each other, their performances were fitted by the model to examine change in model parameters (coupling strength, time‐delay). When auditory feedback for one or both voices was removed, the fits showed the expected decrease in coupling strength and time‐delay between the systems. When feedback about the voice(s) returned, the coupling strength and time‐delay returned to baseline. These findings support the idea that when people perform actions together, they do so as a coupled bidirectional anticipatory system. (shrink)

The Covid-19 pandemic reveals a new phenomenon, unaddressed by the existing literature on “price gouging” in times of emergency. While merchants – getting large remuneration for providing desperately needed goods – evoke public moral outrage for assumed “price gouging”, employees – getting large remuneration for providing desperately needed services – do not cause such outrage but rather experience moral appraisal for their valuable commitment. To address this inherent inconsistency of moral judgment, we propose to embrace insights from research on folk (...) economics. By understanding the folk perception underlying public outrage at “price gougers,” business ethics might better enlighten the moral legitimacy of anti-“price gouging” measures. (shrink)

Louis Loeb has identified a “nasty problem” in connection with Hume’s theory of meaning. The problem is that Hume seemingly claims we lack ideas corresponding to key metaphysical terms, such as terms like “substance” and “necessary connection,” but he then proceeds to explain why philosophers believe in the existence of entities denoted by such terms. In short, Hume seems motivated to explain belief in the existence of certain entities, despite his claiming we have no ideas corresponding to them. In this (...) paper, I strive to solve the problem by noting the important role of clear and distinct perception in his thought. In particular, I argue Hume only wishes to deny that we have clear and distinct ideas of substance and necessary connection, and not that we altogether lack any idea of substance and necessary connection, traditionally conceived. (shrink)

Mourelatos’ study of the fragments of Parmenides’ poem combines traditional philological reconstruction with the approaches of literary criticism and philosophical analysis in order to reveal the thought structure and expressive unity of the best preserved and most important, influential, and coherent text of Greek philosophy before Plato. Through philosophical, philological, and literary analysis, Mourelatos examines the morphology of images and metaphors in Parmenides’ text with the aim of articulating and interpreting the poem’s key concepts and component arguments. Relevant antecedents and (...) parallels from the tradition of epic poetry, especially from Homer’s Odyssey, are explored in depth. (shrink)

This book presents some of the most recent trends and developments in Presocratic scholarship. A wide range of topics are covered - from the metaphysical to the moral to the methodological - as well as a broad a range of authors: from recognized figures such as Heraclitus and Parmenides to Sophistic thinkers whose place has traditionally been marginalized, such as Gorgias and the author of the Dissoi Logoi. Several of the pieces are concerned with the later reception and influence of (...) the Presocratics on ancient philosophy, an area of study important both for the light it sheds on our evidence for Presocratic thought and for understanding the philosophical power of their ideas. Drawing together contributions from distinguished authorities and internationally acclaimed scholars of ancient philosophy, this book offers new challenges to traditional interpretations in some areas of Presocratic philosophy and finds new support for traditional interpretations in other areas. (shrink)

A response to James S. Spiegel's article in THINK 43 in which he argues that same-sex marriage is unjust.

The main argument in Parmenides’ didactic poem begins with these remarks by the unnamed goddess who delivers the revelation (B2 in Diels-KranzDie Fragmente der Vorsokratiker):Come now and I shall tell you, and you listen to the account and carry it forth, which routes of inquiry (ơδοί…διζησιος, B2.2) alone are for knowing: the one (μέν, B2.3), that (…) is and that it is not possible (for …) not to be ὅπως ἔστιν τε ϰαὶ ὼς οὐϰ ἔστι μὴ είναι, B2.3) is the (...) course of Persuasion, for it attends truth; the other ἠδ; B2.5 that (…) is not and that it is right (for …) not to be (ὡς οὐϰ ἔστιυ τε ϰαὶ ὡς χρεών ἐστι μὴ εἶναι, B2.5) that one I mark for you as being a byway from which no tidings ever come (παναπευθέα ἔμμεν άταρπόυ, B2.6) For you could neither come to know (Υυοίμς, aorist, B2.7) the thing itself which is not (τό Υε μὴ έόυ), for it cannot be consummated (οὐ Υὰρ ἀνυστόυ), nor could you point it out (φράσαις, aorist, B2.8). (shrink)

A natural reading of Hume’s distinction between impressions and ideas is that impressions are forceful perceptions whereas ideas are faint. A problem emerges, however, when Hume countenances the possibility of faint impressions and forceful ideas. In this paper, I attempt a resolution to the problem. I argue that Hume characterizes impressions and ideas intensionally and extensionally, and sometimes uses the term in only one of the two senses. I argue that Hume intensionally defines impressions and ideas as forceful perceptions and (...) weak perceptions, respectively, but takes these to be extensionally equivalent to original and copied perceptions, respectively. Hume recognizes that his two characterizations—the intensional and extensional—don’t perfectly match up, and that there are exceptions to the purported equivalences (the exceptions being disease, sleep, madness, and enthusiasm). Nonetheless, I argue that Hume’s willing to proceed with his definitions. (shrink)