Results for 'Todd Parker'

216 found
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  1. Online Deliberation: Design, Research, and Practice.Todd Davies & Seeta Peña Gangadharan (eds.) - 2009 - CSLI Publications/University of Chicago Press.
    Can new technology enhance purpose-driven, democratic dialogue in groups, governments, and societies? Online Deliberation: Design, Research, and Practice is the first book that attempts to sample the full range of work on online deliberation, forging new connections between academic research, technology designers, and practitioners. Since some of the most exciting innovations have occurred outside of traditional institutions, and those involved have often worked in relative isolation from each other, work in this growing field has often failed to reflect the full (...)
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  2. Default Positions in Clinical Ethics.Parker Crutchfield, Tyler Gibb & Michael Redinger - 2023 - Journal of Clinical Ethics 34 (3):258-269.
    Default positions, predetermined starting points that aid in complex decision-making, are common in clinical medicine. In this article, we identify and critically examine common default positions in clinical ethics practice. Whether default positions ought to be held is an important normative question, but here we are primarily interested in the descriptive, rather than normative, properties of default positions. We argue that default positions in clinical ethics function to protect and promote important values in medicine—respect for persons, utility, and justice. Further, (...)
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  3. The Open Future: Why Future Contingents Are All False.Patrick Todd - 2021 - Oxford: Oxford University Press.
    This book launches a sustained defense of a radical interpretation of the doctrine of the open future. Patrick Todd argues that all claims about undetermined aspects of the future are simply false.
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  4. The problem of future contingents: scoping out a solution.Patrick Todd - 2020 - Synthese 197 (11):5051-5072.
    Various philosophers have long since been attracted to the doctrine that future contingent propositions systematically fail to be true—what is sometimes called the doctrine of the open future. However, open futurists have always struggled to articulate how their view interacts with standard principles of classical logic—most notably, with the Law of Excluded Middle. For consider the following two claims: Trump will be impeached tomorrow; Trump will not be impeached tomorrow. According to the kind of open futurist at issue, both of (...)
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  5. The paradox of self-blame.Patrick Todd & Brian Rabern - 2022 - American Philosophical Quarterly 59 (2):111–125.
    It is widely accepted that there is what has been called a non-hypocrisy norm on the appropriateness of moral blame; roughly, one has standing to blame only if one is not guilty of the very offence one seeks to criticize. Our acceptance of this norm is embodied in the common retort to criticism, “Who are you to blame me?”. But there is a paradox lurking behind this commonplace norm. If it is always inappropriate for x to blame y for a (...)
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  6. The Conditions for Ethical Chemical Restraints.Parker Crutchfield & Michael Redinger - 2024 - American Journal of Bioethics Neuroscience 15 (1):3-16.
    The practice of medicine frequently involves the unconsented restriction of liberty. The reasons for unilateral liberty restrictions are typically that being confined, strapped down, or sedated are necessary to prevent the person from harming themselves or others. In this paper, we target the ethics of chemical restraints, which are medications that are used to intentionally restrict the mental states associated with the unwanted behaviors, and are typically not specifically indicated for the condition for which the patient is being treated. Specifically, (...)
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  7. Imagination, Aesthetic Feelings, and Scientific Reasoning.Cain Todd - 2020 - In Milena Ivanova & Stephen French (eds.), Aesthetics and Science. Routledge.
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  8. A Unified Account of the Moral Standing to Blame.Patrick Todd - 2019 - Noûs 53:347-374.
    Recently, philosophers have turned their attention to the question, not when a given agent is blameworthy for what she does, but when a further agent has the moral standing to blame her for what she does. Philosophers have proposed at least four conditions on having “moral standing”: -/- 1. One’s blame would not be “hypocritical”. 2. One is not oneself “involved in” the target agent’s wrongdoing. 3. One must be warranted in believing that the target is indeed blameworthy for the (...)
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  9. Moral Enhancement Can Kill.Parker Crutchfield - 2018 - Journal of Medicine and Philosophy 43 (5):568-584.
    There is recent empirical evidence that personal identity is constituted by one’s moral traits. If true, this poses a problem for those who advocate for moral enhancement, or the manipulation of a person’s moral traits through pharmaceutical or other biological means. Specifically, if moral enhancement manipulates a person’s moral traits, and those moral traits constitute personal identity, then it is possible that moral enhancement could alter a person’s identity. I go a step further and argue that under the right conditions, (...)
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  10.  50
    Formalizing Mechanical Analysis Using Sweeping Net Methods.Parker Emmerson - 2024 - Journal of Liberated Mathematics 1:12.
    We present a formal mechanical analysis using sweeping net methods to approximate surfacing singularities of saddle maps. By constructing densified sweeping subnets for individual vertices and integrating them, we create a comprehensive approximation of singularities. This approach utilizes geometric concepts, analytical methods, and theorems that demonstrate the robustness and stability of the nets under perturbations. Through detailed proofs and visualizations, we provide a new perspective on singularities and their approximations in analytic geometry.
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  11. Newton on active and passive quantities of matter.Adwait A. Parker - 2020 - Studies in History and Philosophy of Science Part A 84:1-11.
    Newton published his deduction of universal gravity in Principia (first ed., 1687). To establish the universality (the particle-to-particle nature) of gravity, Newton must establish the additivity of mass. I call ‘additivity’ the property a body's quantity of matter has just in case, if gravitational force is proportional to that quantity, the force can be taken to be the sum of forces proportional to each particle's quantity of matter. Newton's argument for additivity is obscure. I analyze and assess manuscript versions of (...)
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  12. Set Size and the Part–Whole Principle.Matthew W. Parker - 2013 - Review of Symbolic Logic (4):1-24.
    Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...)
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  13. Revisiting Reid on Religion.Todd Buras - 2021 - Journal of Scottish Philosophy 19 (3):261-274.
    This paper answers two interpretive questions surrounding belief in God in Thomas Reid’s philosophy, the status question and the detachability question. The former has to do with the type of justification Reid assigns to belief in God – immediate or mediate. The later question is whether anything philosophically significant depends on his belief in God. I argue that, for Reid, belief in God is immediately justified and integral to some parts of his system. Reid’s response to skepticism about God is (...)
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  14. Representation and ephemerality in olfaction.Cain Todd - 2018 - In Thomas Crowther & Clare Mac Cumhaill (eds.), Perceptual Ephemera. New York, NY: Oxford University Press.
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  15. The Conditions For Ethical Application of Restraints.Parker Crutchfield, Tyler Gibb, Michael Redinger, Dan Ferman & John Livingstone - 2018 - Chest 155 (3):617-625.
    Despite the lack of evidence for their effectiveness, the use of physical restraints for patients is widespread. The best ethical justification for restraining patients is that it prevents them from harming themselves. We argue that even if the empirical evidence supported their effectiveness in achieving this aim, their use would nevertheless be unethical, so long as well known exceptions to informed consent fail to apply. Specifically, we argue that ethically justifiable restraint use demands certain necessary and sufficient conditions. These conditions (...)
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  16. What is it like to write philosophy?Matthew W. Parker - 2016 - Lse Philosophy Blog:1-1.
    With essay deadlines looming for many of our students, Matt Parker relives some of the angst involved in writing philosophy. You’re not alone.
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  17. Non-Compensable Harms.Todd Karhu - 2019 - Analysis 79 (2):222–230.
    It is more or less uncontroversial that when we harm someone through wrongful conduct we incur an obligation to compensate her. But sometimes compensation is impossible: when the victim is killed, for example. Other times, only partial compensation is possible. In this article, I take some initial steps towards exploring this largely ignored issue. I argue that the perpetrator of a wrongful harm incurs a duty to promote the impartial good in proportion to the amount of harm that cannot be (...)
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  18. Three concepts of decidability for general subsets of uncountable spaces.Matthew W. Parker - 2003 - Theoretical Computer Science 351 (1):2-13.
    There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: (...)
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  19. (1 other version)Geachianism.Patrick Todd - 2011 - Oxford Studies in Philosophy of Religion 3:222-251.
    The plane was going to crash, but it didn't. Johnny was going to bleed to death, but he didn't. Geach sees here a changing future. In this paper, I develop Geach's primary argument for the (almost universally rejected) thesis that the future is mutable (an argument from the nature of prevention), respond to the most serious objections such a view faces, and consider how Geach's view bears on traditional debates concerning divine foreknowledge and human freedom. As I hope to show, (...)
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  20. Let's See You Do Better.Patrick Todd - 2023 - Ergo: An Open Access Journal of Philosophy 10.
    In response to criticism, we often say – in these or similar words – “Let’s see you do better!” Prima facie, it looks like this response is a challenge of a certain kind – a challenge to prove that one has what has recently been called standing. More generally, the data here seems to point a certain kind of norm of criticism: be better. Slightly more carefully: One must: criticize x with respect to standard s only if one is better (...)
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  21. Empirically Skeptical Theism.Todd DeRose - 2020 - Faith and Philosophy 37 (3):323-335.
    Inspired by Peter van Inwagen’s “simulacra model” of the resurrection, I investigate whether it could be reasonable to adopt an analogous approach to the problem of evil. Empirically Skeptical Theism, as I call it, is the hypothesis that God shields our lives from irredeemable evils surreptitiously (just as van Inwagen proposes that God shields our bodies from destruction surreptitiously). I argue that EST compares favorably with traditional skeptical theism and with eschatological theodicies, and that EST does not have the negative (...)
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  22. Future Contingents and the Logic of Temporal Omniscience.Patrick Todd & Brian Rabern - 2019 - Noûs 55 (1):102-127.
    At least since Aristotle’s famous 'sea-battle' passages in On Interpretation 9, some substantial minority of philosophers has been attracted to the doctrine of the open future--the doctrine that future contingent statements are not true. But, prima facie, such views seem inconsistent with the following intuition: if something has happened, then (looking back) it was the case that it would happen. How can it be that, looking forwards, it isn’t true that there will be a sea battle, while also being true (...)
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  23. The Epistemology of Moral Bioenhancement.Parker Crutchfield - 2016 - Bioethics 30 (5):389-396.
    Moral bioenhancement is the potential practice of manipulating individuals’ moral behaviors by biological means in order to help resolve pressing moral issues such as climate change and terrorism. This practice has obvious ethical implications, and these implications have been and continue to be discussed in the bioethics literature. What have not been discussed are the epistemological implications of moral bioenhancement. This article details some of these implications of engaging in moral bioenhancement. The argument begins by making the distinction between moral (...)
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  24. Strawson, Moral Responsibility, and the "Order of Explanation": An Intervention.Patrick Todd - 2016 - Ethics 127 (1):208-240.
    P.F. Strawson’s (1962) “Freedom and Resentment” has provoked a wide range of responses, both positive and negative, and an equally wide range of interpretations. In particular, beginning with Gary Watson, some have seen Strawson as suggesting a point about the “order of explanation” concerning moral responsibility: it is not that it is appropriate to hold agents responsible because they are morally responsible, rather, it is ... well, something else. Such claims are often developed in different ways, but one thing remains (...)
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  25. Critical Notice: The Modal Future: A Theory of Future-Directed Thought and Talk.Patrick Todd - 2024 - Philosophical Quarterly 74 (3):1026-1035.
    At least since Aristotle's famous discussion of the sea-battle tomorrow in On Interpretation 9, philosophers have been fascinated by a rich set of interconnecte.
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  26. Manipulation Arguments and the Freedom to do Otherwise.Patrick Todd - 2017 - Philosophy and Phenomenological Research 95 (2):395-407.
    I provide a manipulation-style argument against classical compatibilism—the claim that freedom to do otherwise is consistent with determinism. My question is simple: if Diana really gave Ernie free will, why isn't she worried that he won't use it precisely as she would like? Diana's non-nervousness, I argue, indicates Ernie's non-freedom. Arguably, the intuition that Ernie lacks freedom to do otherwise is stronger than the direct intuition that he is simply not responsible; this result highlights the importance of the denial of (...)
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  27. Future Contingents are all False! On Behalf of a Russellian Open Future.Patrick Todd - 2016 - Mind 125 (499):775-798.
    There is a familiar debate between Russell and Strawson concerning bivalence and ‘the present King of France’. According to the Strawsonian view, ‘The present King of France is bald’ is neither true nor false, whereas, on the Russellian view, that proposition is simply false. In this paper, I develop what I take to be a crucial connection between this debate and a different domain where bivalence has been at stake: future contingents. On the familiar ‘Aristotelian’ view, future contingent propositions are (...)
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  28. A role for representations in inflexible behavior.Todd Ganson - 2020 - Biology and Philosophy 35 (4):1-18.
    Representationalists have routinely expressed skepticism about the idea that inflexible responses to stimuli are to be explained in representational terms. Representations are supposed to be more than just causal mediators in the chain of events stretching from stimulus to response, and it is difficult to see how the sensory states driving reflexes are doing more than playing the role of causal intermediaries. One popular strategy for distinguishing representations from mere causal mediators is to require that representations are decoupled from specific (...)
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  29. Ignorance and moral judgment: Testing the logical priority of the epistemic.Parker Crutchfield, Scott Scheall, Mark Justin Rzeszutek, Hayley Dawn Brown & Cristal Cardoso Sao Mateus - 2023 - Consciousness and Cognition 108 (C):103472.
    It has recently been argued that a person’s moral judgments (about both their own and others’ actions) are constrained by the nature and extent of their relevant ignorance and, thus, that such judgments are determined in the first instance by the person’s epistemic circumstances. It has been argued, in other words, that the epistemic is logically prior to other normative (e.g., ethical, prudential, pecuniary) considerations in human decision-making, that these other normative considerations figure in decision-making only after (logically and temporally) (...)
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  30. Foreknowledge requires determinism.Patrick Todd - 2022 - Philosophy and Phenomenological Research 107 (1):125-146.
    There is a longstanding argument that purports to show that divine foreknowledge is inconsistent with human freedom to do otherwise. Proponents of this argument, however, have for some time been met with the following reply: the argument posits what would have to be a mysterious non-causal constraint on freedom. In this paper, I argue that this objection is misguided – not because after all there can indeed be non-causal constraints on freedom (as in Pike, Fischer, and Hunt), but because the (...)
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  31. Burge’s Defense of Perceptual Content.Todd Ganson, Ben Bronner & Alex Kerr - 2012 - Philosophy and Phenomenological Research 88 (3):556-573.
    A central question, if not the central question, of philosophy of perception is whether sensory states have a nature similar to thoughts about the world, whether they are essentially representational. According to the content view, at least some of our sensory states are, at their core, representations with contents that are either accurate or inaccurate. Tyler Burge’s Origins of Objectivity is the most sustained and sophisticated defense of the content view to date. His defense of the view is problematic in (...)
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  32. Future Contingents, Openness, and the Possibility of Omniscience: Defending an Argument Against Relativism and Supervaluationism.Patrick Todd - forthcoming - Theoria.
    Todd and Rabern (2021) mount an argument that – contra both Thomason’s (1970) supervaluationism and MacFarlane’s (2014) relativism – an “open future” view is incompatible with the principle they call “Retro-closure”, according to which today’s rain implies that yesterday it was true that it would rain a day later. In a recent piece, MacFarlane replies. This paper has two aims. First, I argue that MacFarlane’s response to Todd and Rabern is unsuccessful on its own terms. Second, I attempt (...)
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  33. Does God Have the Moral Standing to Blame?Patrick Todd - 2018 - Faith and Philosophy 35 (1):33-55.
    In this paper, I introduce a problem to the philosophy of religion – the problem of divine moral standing – and explain how this problem is distinct from (albeit related to) the more familiar problem of evil (with which it is often conflated). In short, the problem is this: in virtue of how God would be (or, on some given conception, is) “involved in” our actions, how is it that God has the moral standing to blame us for performing those (...)
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  34. The Priority of the Epistemic.Parker Crutchfield & Scott Scheall - 2021 - Episteme 18 (4):726-737.
    Epistemic burdens – the nature and extent of our ignorance (that and how) with respect to various courses of action – serve to determine our incentive structures. Courses of action that seem to bear impossibly heavy epistemic burdens are typically not counted as options in an actor’s menu, while courses of action that seem to bear comparatively heavy epistemic burdens are systematically discounted in an actor’s menu relative to options that appear less epistemically burdensome. That ignorance serves to determine what (...)
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  35. (1 other version)The Rational/Non-Rational Distinction in Plato's Republic.Todd Ganson - 2009 - Oxford Studies in Ancient Philosophy 36:179-197.
    An attempt to show that Plato has a unified approach to the rationality of belief and the rationality of desire, and that his defense of that approach is a powerful one.
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  36. Compulsory moral bioenhancement should be covert.Parker Crutchfield - 2018 - Bioethics 33 (1):112-121.
    Some theorists argue that moral bioenhancement ought to be compulsory. I take this argument one step further, arguing that if moral bioenhancement ought to be compulsory, then its administration ought to be covert rather than overt. This is to say that it is morally preferable for compulsory moral bioenhancement to be administered without the recipients knowing that they are receiving the enhancement. My argument for this is that if moral bioenhancement ought to be compulsory, then its administration is a matter (...)
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  37. Representation in Cognitive Science, by Nicholas Shea. Oxford: Oxford University Press, 2018. Pp. 292.Todd Ganson - 2021 - Mind 130 (517).
    A central component of the cognitive revolution is a commitment to explaining behaviour by reference to internal representations of the world. This core aspect.
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  38. Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system.Matthew W. Parker - 2003 - Philosophy of Science 70 (2):359-382.
    Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in (or d- ) for any measure , which is particularly appropriate for physics and in some ways more intuitive than Ko's (1991) recursive approximability (r.a.). For Lebesgue measure , d- implies r.a. Sets with positive -measure that are sufficiently "riddled" with holes are never d- but are often r.a. This explicates Sommerer (...)
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  39. Ethical Allocation of Remdesivir.Parker Crutchfield, Tyler S. Gibb, Michael J. Redinger & William Fales - 2020 - American Journal of Bioethics 20 (7):84-86.
    As the federal government distributed remdesivir to some of the states COVID-19 hit hardest, policymakers scrambled to develop criteria to allocate the drug to their hospitals. Our state, Michigan, was among those states to receive an initial quantity of the drug from the U.S. government. The disparities in burden of disease in Michigan are striking. Detroit has a death rate more than three times the state average. Our recommendation to the state was that it should prioritize the communities that bear (...)
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  40. On behalf of a mutable future.Patrick Todd - 2016 - Synthese 193 (7):2077-2095.
    Everyone agrees that we can’t change the past. But what about the future? Though the thought that we can change the future is familiar from popular discourse, it enjoys virtually no support from philosophers, contemporary or otherwise. In this paper, I argue that the thesis that the future is mutable has far more going for it than anyone has yet realized. The view, I hope to show, gains support from the nature of prevention, can provide a new way of responding (...)
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  41. On The Open Future: Replies to Rhoda and Rubio.Patrick Todd - forthcoming - International Journal for Philosophy of Religion.
    These are my materials (a short precis, and replies to Alan Rhoda and Daniel Rubio) for an invited symposium on my book _The Open Future: Why Future Contingents are All False_ (OUP, 2021) in IJPR. [The commentaries from Rhoda/Rubio are available on their respective PhilPapers profiles.].
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  42. Visual Prominence and Representationalism.Todd Ganson & Ben Bronner - 2013 - Philosophical Studies 164 (2):405-418.
    A common objection to representationalism is that a representationalist view of phenomenal character cannot accommodate the effects that shifts in covert attention have on visual phenomenology: covert attention can make items more visually prominent than they would otherwise be without altering the content of visual experience. Recent empirical work on attention casts doubt on previous attempts to advance this type of objection to representationalism and it also points the way to an alternative development of the objection.
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  43. (1 other version)Symmetry arguments against regular probability: A reply to recent objections.Matthew W. Parker - 2018 - European Journal for Philosophy of Science 9 (1):8.
    A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking (...)
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  44. Cross-modal Influence on Oral Size Perception.Parker Crutchfield, Connor Mahoney, Cesar Rivera & Vanessa Pazdernik - 2016 - Archives of Oral Biology 61:89-97.
    Objective: Evidence suggests people experience an oral size illusion and commonly perceive oral size inaccurately; however, the nature of the illusion remains unclear. The objectives of the present study were to confirm the presence of an oral size illusion, determine the magnitude (amount) and direction (underestimation or overestimation) of the illusion, and determine whether immediately prior crossmodal perceptual experiences affected the magnitude and direction. Design: Participants (N = 27) orally assessed 9 sizes of stainless steel spheres (1/16 in to 1/2 (...)
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  45. Fitting Feelings and Elegant Proofs: On the Psychology of Aesthetic Evaluation in Mathematics.Cain Todd - 2017 - Philosophia Mathematica:nkx007.
    ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and (...)
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  46.  60
    Optimized Energy Numbers Continued.Parker Emmerson - 2024 - Journal of Liberated Mathematics 1:12.
    In this paper, we explore the properties and optimization techniques related to polyhedral cones and energy numbers with a focus on the cone of positive semidefinite matrices and efficient computation strategies for kernels. In Part (a), we examine the polyhedral nature of the cone of positive semidefinite matrices, , establishing that it does not form a polyhedral cone for due to its infinite dimensional characteristics. In Part (b), we present an algorithm for efficiently computing the kernel function on-the-fly, leveraging a (...)
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  47. Mental Privacy, Cognitive Liberty, and Hog-tying.Parker Crutchfield - forthcoming - Journal of Bioethical Inquiry:1-16.
    As the science and technology of the brain and mind develop, so do the ways in which brains and minds may be surveilled and manipulated. Some cognitive libertarians worry that these developments undermine cognitive liberty, or “freedom of thought.” I argue that protecting an individual’s cognitive liberty undermines others’ ability to use their own cognitive liberty. Given that the threatening devices and processes are not relevantly different from ordinary and frequent intrusions upon one’s brain and mind, strong protections of cognitive (...)
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  48. Illusions of Commutativity: The Case for Conditional Excluded Middle Revisited.Patrick Todd, Brian Rabern & Wolfgang Schwarz - manuscript
    The principle of Conditional Excluded Middle has been a matter of longstanding controversy in both semantics and metaphysics. The principle suggests (among other things) that for any coin that isn't flipped, there is a fact of the matter about how it would have landed if it had been flipped: either it would have landed heads, or it would have landed tails. This view has gained support from linguistic evidence indicating that ‘would’ commutes with negation (e.g., ‘not: if A, would C’ (...)
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  49. How Anselm Separates Morality from Happiness.Parker Haratine - 2024 - American Catholic Philosophical Quarterly 98 (2):195-213.
    Contemporary scholarship is divided over whether Anselm maintains a version of Eudaemonism. The debate centers on the question of whether the will for justice only moderates the will for happiness or, instead, provides a distinct end for which to act. Because of two key passages, various scholars hold that Anselm maintained elements of medieval Eudaemonism. In this article, I argue that Anselm separates morality from happiness, and I provide a sketch of his alternative view. First, I argue against some recent (...)
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  50.  53
    Defining π via Infinite Densification of the Sweeping Net and Reverse Integration.Parker Emmerson - 2024 - Journal of Liberated Mathematics 1 (1):7.
    We present a novel approach to defining the mathematical constant π through the infinite den- sification of a sweeping net, which approximates a circle as the net becomes infinitely dense. By developing and enhancing notation related to sweeping nets and saddle maps, we establish a rigor- ous framework for expressing π in terms of the densification process using reverse integration. This method, inspired by the concept that numbers ”come from infinity,” leverages a reverse integral approach to model the transition from (...)
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