Results for 'Conceptual necessity'

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  1. Conceptual necessity, causality and self-ascriptions of sensation.Frederik Kaufman - 1990 - International Studies in Philosophy 22 (3):3-11.
  2.  27
    The conceptual necessity of quantum probabilities in cognitive psychology.Reinhard Blutner & Peter Beim Graben - 2013 - Behavioral and Brain Sciences 36 (3):280-281.
  3. Are there a posteriori conceptual necessities?Daniel Dohrn - 2011 - Philosophical Studies 155 (2):181-197.
    I critically assess Stephen Yablo’s claim that cassinis are ovals is an a posteriori conceptual necessity. One does not know it simply by mastering the relevant concepts but by substantial empirical scrutiny. Yablo represents narrow content by would have turned out -conditionals. An epistemic reading of such conditionals does not bear Yablo’s claim. Two metaphysically laden readings are considered. In one reading, Yablo’s conditionals test under what circumstances concepts remain the same while their extensions diverge. As an alternative, (...)
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  4. Myth of the Conceptual Necessity of Truth-Directed Communication.Kensuke Ito - 2024 - In Adam C. Podlaskowski & Drew Johnson (eds.), Truth 20/20: How a Global Pandemic Shaped Truth Research. Synthese Library. pp. 173-185.
    In 4.062 of the Tractatus, Wittgenstein repudiates the possibility of “mak[ing] ourselves understood with false propositions just as we have done up until now with true ones”. This chapter challenges this claim by distinguishing two possible cases in which people communicate truth-apt information through false propositions. The first case presents a semantic option, where false propositions become true “construed in the new way” while people aim at conveying truths, whereas the second case presents a pragmatic option, where false propositions remain (...)
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  5.  27
    Æternus Est: Divinity as a Conceptual Necessity in the Principle of Causation.Larry Hunt - 2018 - Philosophia 46 (4):895-910.
    The modern belief that mindless forces can be ultimate efficient causes of natural events is a conceptual impossibility. The logically ultimate cause of any change, the something that is ultimately making it occur in the present moment, is either a mind or not. More specifically, the cause either chooses to act or it does not. By choice here, I mean an act of free will in the libertarian sense. Where there is choosing in this sense there must be a (...)
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  6.  83
    Logical Necessity, Conceptual Necessity, and the Ontological Argument.C. Anthony Anderson - 2015 - In Mirosław Szatkowski (ed.), God, Truth, and Other Enigmas. Berlin: De Gruyter. pp. 3-14.
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  7. Constructed and Wild Conceptual Necessities in Contemporary Jurisprudence.Stefan Sciaraffa - 2015 - Jurisprudence 6 (2):391-406.
  8. Divine and Conceptual Necessity.Adel Daher - 1982 - Philosophical Studies (Dublin) 29:34-47.
  9. Is language of thought a conceptual necessity?Olga Markic - 2001 - Acta Analytica 16 (26):53-60.
  10. Part I. introduction: Traveling without moving: The conceptual necessity of survive-minimalism.Michael T. Putnam & Thomas S. Stroik - 2009 - In Towards a Derivational Syntax: Survive-Minimalism. John Benjamins Pub. Company.
     
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  11.  49
    Conceptual Truth, Necessity, and Negation.Jean-Philippe Narboux - 2020 - The Monist 103 (4):468-480.
    Throughout his philosophical career, Hilary Putnam was preoccupied with the question of what survives of the traditional notion of a priori truth in light of the recurring historical phenomenon, made prominent by the scientific revolutions of the early decades of the twentieth century, through which “something that was literally inconceivable has turned out to be true”. Impugning the analytic-synthetic dichotomy, Putnam’s redefinition of “conceptual truth” in terms of “quasi-necessity relative to a conceptual scheme” is meant to accommodate (...)
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  12. Conceptual and natural necessity.Rom Harre & E. H. Madden - 2008 - In Ruth Groff (ed.), Revitalizing causality: realism about causality in philosophy and social science. New York: Routledge.
  13.  7
    Necessity for finite rational minds– Kant on empirical nomic necessity and the conceptual purposiveness of nature.Ido Geiger - 2025 - Synthese 205 (1):1-28.
    Kant claims that empirical laws of nature “carry with them an expression of necessity” (Kant, 1998 ; A159/B198). What precisely is this “expression of necessity” and what grounds it? The metaphysical necessitation approach asks “What are laws of nature for Kant?” It answers that, for Kant, empirical laws possess necessity and are grounded in the properties and causal powers essential to natural kinds. The epistemological systematization approach claims that empirical natures, kinds and causal laws are rightly viewed (...)
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  14.  17
    Conceptual Truth and Metaphysical Necessity.Robert Stalnaker - 2007 - In Ways a World Might Be. Oxford University Press Uk.
    This paper discusses the basic two-dimensional framework and identifies the requirements to apply it to the phenomena of speech and thought. It then contrasts two kinds of interpretation — two ways of applying the abstract two-dimensional framework — that suggest different pictures of the phenomena Kripke brought to light. It shows what must be assumed about how the problem of intentionality must be solved to sustain the kind of general reduction of necessary a posteriori truth that Jackson and Chalmers propose.
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  15. Metaphysical Necessity and Conceptual Truth.Eli Hirsch - 1986 - Midwest Studies in Philosophy 11 (1):243-256.
  16.  90
    Conceptual Truths, Strong Possibilities and Our Knowledge of Metaphysical Necessities.Christian Nimtz - 2012 - Philosophia Scientiae 16 (2):39-58.
    Dans mon article, je soutiens qu'il existe une voie epistemique fiable qui mène de la connaissance des verités conceptuelles a celle des nécessites métaphysiques. Dans un premier temps, je montre que nous pouvons prétendre connaitre des vérites conceptuelles dans la mesure ou nous savons à quelles conditions nos termes (ou du moins un grand nombre d'entre eux) s'appliquent. Je défends notamment cette idée face a un argument récent que Williamson adresse a la conception épistémique de l'analyticite. Dans un second temps, (...)
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  17.  18
    The Necessity of Private Language: A Conceptual Confusion Latent in Diverse Episodes of the History of Psychology / Potrzeba języka prywatnego: zamieszanie pojęciowe zawarte w różnorodnych wątkach historii psychologii.José María Ariso - 2014 - Annales Umcs. Sectio I 39 (1):7-19.
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  18. Knowledge of Possibility and of Necessity.Bob Hale - 2003 - Proceedings of the Aristotelian Society 103 (1):1-20.
    I investigate two asymmetrical approaches to knowledge of absolute possibility and of necessity--one which treats knowledge of possibility as more fundamental, the other according epistemological priority to necessity. Two necessary conditions for the success of an asymmetrical approach are proposed. I argue that a possibility-based approach seems unable to meet my second condition, but that on certain assumptions--including, pivotally, the assumption that logical and conceptual necessities, while absolute, do not exhaust the class of absolute necessities--a necessity-based (...)
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  19. Learning and the Necessity of Non-Conceptual Content in Sellars's Empiricism and the Philosophy of Mind.David Forman - 2006 - In Michael P. Wolf & Mark Norris Lance (eds.), The Self-Correcting Enterprise: Essays on Wilfrid Sellars. Rodopi. pp. 115-145.
    For Sellars, the possibility of empirical knowledge presupposes the existence of "sense impressions" in the perceiver, i.e., non-conceptual states of perceptual consciousness. But this role for sense impressions does not implicate Sellars' account in the Myth of the Given: sense impressions do not stand in a justificatory relation to instances of perceptual knowledge; their existence is rather a condition for the possibility of the acquisition of empirical concepts. Sellars suggests that learning empirical concepts presupposes that we can remember certain (...)
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  20.  11
    Necessity and Apriority.Eric Loomis - 2017 - In Hans-Johann Glock & John Hyman (eds.), A Companion to Wittgenstein. Chichester, West Sussex, UK: Wiley-Blackwell. pp. 346–358.
    The nature of necessary truth was a central concern of Ludwig Wittgenstein. It was present in his early reflections on logic, a core motif of the Tractatus, and a topic he returned to over and again in his reflections on language, logic, and mathematics. This chapter explores the aspects of Wittgenstein's account of necessity and apriority, beginning with the Tractatus, where many of his core insights received their first expression. It discusses two more contemporary accounts of these topics: (...) role semantic theories of the a priori, and the arguments for divorcing apriority from necessity. Necessity in the Tractatus appears at two locations. One is at the level of individual elementary propositions. The other is at the level of the inferential connections between propositions. The chapter looks at the Tractatus theory of the proposition. Wittgenstein saw an important difference between the formulations of the rules of grammar and descriptions of states of affairs. (shrink)
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  21.  46
    The paradox of medical necessity.Samantha Godwin & Brian D. Earp - 2023 - Clinical Ethics 18 (3):281-284.
    The concept of medical necessity is often used to explain or justify certain decisions—for example, which treatments should be allowed under certain conditions—as though it had an obvious, agreed-upon meaning as well as an inherent normative force. In introducing this special issue of Clinical Ethics on medical necessity, we argue that the term, as used in various discourses, generally lacks a definition that is clear, non-circular, conceptually plausible, and fit for purpose. We propose that future work on this (...)
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  22.  68
    When Language Gives Out: Conceptualization, and Aspect‐Seeing as a Form of Judgment.Reshef Agam-Segal - 2014 - Metaphilosophy 45 (1):41-68.
    This article characterizes aspect-perception as a distinct form of judgment in Kant's sense: a distinct way in which the mind contacts world and applies concepts. First, aspect-perception involves a mode of thinking about things apart from any established routine of conceptualizing them. It is thus a form of concept application that is essentially reflection about language. Second, this mode of reflection has an experiential, sometimes perceptual, element: in aspect-perception, that is, we experience meanings—bodies of norms. Third, aspect-perception can be “preparatory”: (...)
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  23. Necessity and triviality.Ross P. Cameron - 2010 - Australasian Journal of Philosophy 88 (3):401-415.
    In this paper I argue that there are some sentences whose truth makes no demands on the world, being trivially true in that their truth-conditions are trivially met. I argue that this does not amount to their truth-conditions being met necessarily: we need a non-modal understanding of the notion of the demands the truth of a sentence makes, lest we be blinded to certain conceptual possibilities. I defend the claim that the truths of pure mathematics and set theory are (...)
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  24.  38
    Local Explanations via Necessity and Sufficiency: Unifying Theory and Practice.David S. Watson, Limor Gultchin, Ankur Taly & Luciano Floridi - 2022 - Minds and Machines 32 (1):185-218.
    Necessity and sufficiency are the building blocks of all successful explanations. Yet despite their importance, these notions have been conceptually underdeveloped and inconsistently applied in explainable artificial intelligence, a fast-growing research area that is so far lacking in firm theoretical foundations. In this article, an expanded version of a paper originally presented at the 37th Conference on Uncertainty in Artificial Intelligence, we attempt to fill this gap. Building on work in logic, probability, and causality, we establish the central role (...)
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  25. The status of charity I: Conceptual truth or a posteriori necessity?Kathrin Glüer - 2006 - International Journal of Philosophical Studies 14 (3):337 – 359.
    According to Donald Davidson, linguistic meaning is determined by the principle of charity. Because of Davidson's semantic behaviourism, charity's significance is both epistemic and metaphysical: charity not only provides the radical interpreter with a method for constructing a semantic theory on the basis of his data, but it does so because it is the principle metaphysically determining meaning. In this paper, I assume that charity does determine meaning. On this assumption, I investigate both its epistemic and metaphysical status: is charity (...)
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  26. On the Reduction of Necessity to Essence.Fabrice Correia - 2012 - Philosophy and Phenomenological Research 84 (3):639-653.
    In his influential paper ‘‘Essence and Modality’’, Kit Fine argues that no account of essence framed in terms of metaphysical necessity is possible, and that it is rather metaphysical necessity which is to be understood in terms of essence. On his account, the concept of essence is primitive, and for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all things. Fine also proposes a reduction of conceptual and (...)
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  27.  71
    The Necessity of Gibbsian Statistical Mechanics.David Wallace - unknown
    In discussions of the foundations of statistical mechanics, it is widely held that the Gibbsian and Boltzmannian approaches are incompatible but empirically equivalent; the Gibbsian approach may be calculationally preferable but only the Boltzmannian approach is conceptually satisfactory. I argue against both assumptions. Gibbsian statistical mechanics is applicable to a wide variety of problems and systems, such as the calculation of transport coefficients and the statistical mechanics and thermodynamics of mesoscopic systems, in which the Boltzmannian approach is inapplicable. And the (...)
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  28.  87
    Τεχνητή νοημοσύνη και εκφραστικότητα: η αναγκαιότητα περάσματος της εννοιολογικής γνώσης από το μονοπάτι της αισθητικής εποπτείας (Artificial Intelligence and expressiveness: The necessity for conceptual knowledge to go through the path of aesthetic perception). [REVIEW]Dimitrios Dacrotsis - 2024 - Days of Art in Greece 16:43-55.
    Δ.Δακρότσης, «Τεχνητή νοημοσύνη και εκφραστικότητα: η αναγκαιότητα περάσματος της εννοιολογικής γνώσης από το μονοπάτι της αισθητικής εποπτείας» (Artificial Intelligence and expressiveness: The necessity for conceptual knowledge to go through the path of aesthetic perception), Days of Art in Greece, Τεύχος 18, Φθινόπωρο 2024, σς 26-39.
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  29. The status of charity I: Conceptual truth or aposteriori necessity?Kathrin Glüer-Pagin - manuscript
    in International Journal of Philosophical Studies 14, 2006: 337-359 (special issue on Donald Davidson ed. M. Baghramian/J. Malpas).
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  30.  20
    Is the Conceptual Eurocentrism So Much Frightening?Nataliya A. Kanaeva - 2019 - Russian Journal of Philosophical Sciences 62 (6):70-87.
    The article is a response to the criticism of “conceptual Eurocentrism” expressed in the paper by A.A. Krushinsky at the Round Table on the Geography of Rationality on April 25, 2019. It deals with the main thesis of A.A. Krushinsky that in cross-cultural philosophical studies the Western conceptual matrix currently defines a single conceptual space for all participants, the language of Western philosophy acts as a trans-civilizational language in the world philosophy. The author of the article agrees (...)
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  31.  22
    The necessity and possibilities of playfulness in narrative care with older adults.Bodil H. Blix, Charlotte Berendonk, D. Jean Clandinin & Vera Caine - 2021 - Nursing Inquiry 28 (1):e12373.
    For us, narrative care is grounded in pragmatist philosophy and focused on experience. Narrative care is not merely about acknowledging or listening to people's experiences, but draws attention to practical consequences. We conceptualize care itself as an intrinsically narrative endeavour. In this article, we build on Lugones' understanding of playfulness, particularly to her call to remain attentive to a sense of uncertainty, and an openness to surprise. Playfulness cultivates a generative sense of curiosity that relies on a close attentiveness not (...)
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  32.  87
    Alexy on Necessity in Law and Morals.Dennis Patterson - 2012 - Ratio Juris 25 (1):47-58.
    Robert Alexy has built his original theory of law upon pervasive claims for “necessary” features of law. In this article, I show that Alexy's claims suffer from two difficulties. First, Alexy is never clear about what he means by “necessity.” Second, Alexy writes as if there have been no challenges to claims of conceptual necessity. There have been such challenges and Alexy needs to answer them if his project is to succeed.
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  33. Necessity in the Cosmology of Tommaso Campanella.Emanuele Costa - manuscript
    In this short paper, I offer a reconstruction of Tommaso Campanella's conception of necessity. I particularly focus on those aspects of his theory of necessity which relate to his peculiar cosmology.
     
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  34.  22
    In Defence of Necessity.Suzanne Uniacke - 2023 - Philosophia 51 (5):2317-2325.
    This paper disputes Uwe Steinhoff’s view that a _jus ad bellum_ requirement of necessity can be merged with a condition of proportionality. It argues that the proposed merger detracts from a conceptual and moral understanding of the structure and rationale of both the necessity and the proportionality considerations applicable in a range of moral contexts, including those of war and so-called lesser evils cases, where these conditions are intended as action-guiding.
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  35. Existence and Necessity.John-Michael Kuczynski - 2016 - Amazon Digital Services LLC.
    It is clearly explained: -/- *What it is for a statement to be necessarily true, *Why necessity, possibility, existence, and non-existence are properties of propositions (truths and falsehoods), not of objects or states of affairs, *What conditions a class of expressions must meet if the expressions belonging to it jointly constitute a single language, *The significance for meta-linguistic research of the concepts of systematicity and productivity, as Chomsky defines these terms, and the relevance of these concepts to researches into (...)
     
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  36. Conceptual diversity in epistemology.Richard Foley - 2002 - In Paul K. Moser (ed.), The Oxford Handbook of Epistemology. New York: Oup Usa. pp. 177--203.
    In “Conceptual Diversity in Epistemology,” Richard Foley reflects on such central topics in epistemology as knowledge, warrant, rationality, and justification, with the purpose of distinguishing such concepts in a general theory. Foley uses “warrant” to refer to that which constitutes knowledge when added to true belief and suggests that rationality and justification are not linked to knowledge by necessity. He proceeds to offer a general schema for rationality. This schema enables a distinction between “rationality” and “rationality all things (...)
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  37. Conceptual cartography.Robert Smithson - 2021 - Inquiry: An Interdisciplinary Journal of Philosophy 64 (1):97-122.
    ABSTRACT Certain features of our conceptual scheme seem necessary for subjects with our basic nature: we cannot imagine humans accomplishing their basic projects without having a conceptual scheme with these features. Other aspects of our conceptual scheme seem more contingent: we can imagine communities effectively using a somewhat different conceptual scheme. Conceptual cartography is the project of investigating the necessity and contingency of the various features of conceptual schemes. The project of conceptual (...)
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  38. Local explanations via necessity and sufficiency: unifying theory and practice.David Watson, Limor Gultchin, Taly Ankur & Luciano Floridi - 2022 - Minds and Machines 32:185-218.
    Necessity and sufficiency are the building blocks of all successful explanations. Yet despite their importance, these notions have been conceptually underdeveloped and inconsistently applied in explainable artificial intelligence (XAI), a fast-growing research area that is so far lacking in firm theoretical foundations. Building on work in logic, probability, and causality, we establish the central role of necessity and sufficiency in XAI, unifying seemingly disparate methods in a single formal framework. We provide a sound and complete algorithm for computing (...)
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  39. Key formulations. Critical realism and substance / Roy wood sellars; causality and substance / Roy wood sellars; essence and accident / Irving copi; conceptual and natural necessity / Rom Harre and E.h. Madden; powers and dispositions. [REVIEW]Brian Ellis - 2008 - In Ruth Groff (ed.), Revitalizing causality: realism about causality in philosophy and social science. New York: Routledge.
     
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  40.  66
    Actualist versus Naturalist and Conceptual Realist Interpretations of Hegel's Metaphysics.Paul Redding - 2021 - Hegel Bulletin 42 (1):19-38.
    The understanding of Hegel's metaphysics that is here argued for—that it is a metaphysics of the actual world—may sound trivial or empty. To counter this, in part one the actualist reading of Hegel's idealism is opposed to two other currently popular interpretations, those of the naturalist and the conceptual realist respectively. While actualism shares motivations with each of these positions, it is argued that it is better equipped to capture what both aim to bring out in Hegel's metaphysics, but (...)
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  41.  49
    Regularities, Degrees of Necessity, and Dispositionalism.Xavi Lanao - 2020 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 51 (4):513-524.
    Traditionally, philosophers have cashed out the distinction between law-like and accidental regularities sharply: a regularity is either law-like, and thereby necessary, or accidental. However, Mitchell and Lange have drawn attention to the fact that some law-like regularities come in different degrees of necessity. For instance, the regularity expressed by “all electrons are negatively charged” has a greater degree of necessity than the one expressed by “all mammals are warm-blooded”, even if both of them are true. Moreover, Mitchell argues (...)
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  42. Noncivil Disobedience and the Right of Necessity. A Point of Convergence.Alejandra Mancilla - 2012 - Krisis 3:3-15.
    Given the conceptual gap in the global justice debate today (where most of the talk is about the duties of the rich, but little is said about what the poor may do for themselves), in this article I reintroduce the idea of a right of necessity. I first delineate a normative framework for such a right, inspired by these historical accounts. I then offer a contemporary case where the exercise of the right of necessity would be morally (...)
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  43.  65
    Causal Powers. A Theory of Natural Necessity[REVIEW]B. W. A. - 1976 - Review of Metaphysics 29 (4):735-736.
    This provocative but persuasive book is essentially a radical attack upon the Humean conception of causality and the presentation and defense of a counter-theory, closer to everyday experience and pre-Humean traditional views. As formulated by empiricist philosophers, the Humean approach depends on two basic postulates. The philosophical analysis of any non-empirical concept must be a formal explication; any residue elements have to be accounted for in terms of their psychological origins. The world as experienced can be conceived adequately as a (...)
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  44.  39
    Harming, Rescuing and the Necessity Constraint on Defensive Force.Cécile Fabre - 2022 - Criminal Law and Philosophy 16 (3):525-538.
    In _The Morality of Defensive Force_, Quong defends a powerful account of the grounds and conditions under which an agent may justifiably inflict serious harm on another person. In this paper, I examine Quong's account of the necessity constraint on permissible harming—the RESCUE account. I argue that RESCUE does not succeed. Section 2 describes RESCUE. Section 3 raises some worries about Quong's conceptual construal of the right to be rescued and its attendant duties. Section 4 argues that RESCUE (...)
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  45.  2
    Building (Conceptual) Bridges: Mills’s Non-Ideal Theory and Disciplinary Whitopias.Emmalon Davis - 2025 - In Mark William Westmoreland (ed.), The Philosophy of Charles W. Mills: Race and the Relations of Power. New York: Routledge. pp. 134-152.
    This chapter revisits the metaphilosophical critique offered in The Racial Contract (Mills 1997). My analysis explicates Mills’s characterization of the “Racial Contract”—and non-ideal theory more broadly—as a conceptual bridge. I consider three questions: (a) what is the nature of the domains it connects, (b) what is the function and orientation of the bridge, (c) what is the relationship between once isolated domains after a bridge has been constructed? In answering these questions, I outline several features of the bridge’s construction, (...)
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  46.  57
    Theorizing about patience formation – the necessity of conceptual distinctions.Ole-Jørgen Skog - 2001 - Economics and Philosophy 17 (2):207-219.
    The concept of patience describes a person's ability to make prolonged efforts towards future goals, and his or her ability to consider long-term future consequences. Clearly, patience is a capacity that comes by degrees. On the following pages, a person will be said to be patient to the extent that his actions are motivated by future consequences. Hence, a person is not patient if he has the ability to see long-term consequences, while being unable to take these consequences into consideration (...)
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  47. Physical Systems: Conceptual Pathways Between Spacetime and Matter.Ori Belkind - 2004 - Dissertation, University of Washington
    This dissertation elucidates the notion of physical system which opens new conceptual pathways that connect the three realms of physical theory; spacetime, material bodies and their properties, and the laws of nature which govern their evolution. The notion of physical system includes two presuppositions regarding their structure. The first presupposition is a description of isolated systems and their evolution in time, which amounts to a Paradigm of Uniform Motion. The second presupposition describes how parts of a physical system are (...)
     
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  48.  41
    Naming without Necessity.Yves Gingras - 2010 - Revue de Synthèse 131 (3):439-454.
    The recent discussions on the label “historical epistemology” provide us with an interesting example of branding of concepts, ideas and methods. Given this recent interest in the meaning of the expression “historical epistemology”, a detailed analysis of its genealogy and context of emergence may provide some conceptual clarification in a discussion that is often confused and curiously silent on the long tradition of sociology of knowledge. This essay also sheds light on the difficulty with the international and interdisciplinary circulation (...)
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  49. A Conceptual Model of Forgiveness and Mental Health: A Philosophical Appraisal.R. L. Tripathi - 2024 - Public Health Open Access 8 (2):6.
    This paper explores the nature of hate, forgiveness, and interconnectedness in human relationships. Hatred often arises from conflicts with personal expectations but can be transformed into forgiveness by adopting an impersonal, holistic perspective. Drawing on evolutionary theory, psychological insights, and Buddhist philosophy, the paper argues that forgiveness is essential for individual mental well-being and societal harmony. The Buddhist concept of “two arrows” illustrates that while pain is unavoidable, suffering stems from emotional reactions and can be mitigated. Embracing the interconnected nature (...)
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  50.  31
    Conceptual Foundations of Operational Set Theory.Kaj Børge Hansen - 2010 - Danish Yearbook of Philosophy 45 (1):29-50.
    I formulate the Zermelo-Russell paradox for naive set theory. A sketch is given of Zermelo’s solution to the paradox: the cumulative type structure. A careful analysis of the set formation process shows a missing component in this solution: the necessity of an assumed imaginary jump out of an infinite universe. Thus a set is formed by a suitable combination of concrete and imaginary operations all of which can be made or assumed by a Turing machine. Some consequences are drawn (...)
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