Schmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf {AD}_{\mathbb R}$, which is a much stronger axiom than that asserting all integer games are determined, $\mathsf {AD}$. One of (...) our main results is a general theorem which under the hypothesis $\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $(\alpha,\beta,\rho )$ game on $\mathbb R$ is determined from $\mathsf {AD}$ alone, but on $\mathbb R^n$ for $n \geq 3$ we show that $\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $(\alpha, \beta, \rho )$ game on $\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $\mathsf {AD}$. (shrink)

Regarded as a revisionist historian in the West, Patricia Crone, who wrote many books and articles on various subjects of early Islamic history, was a prolific researcher. The place and role of mawālī, in the Umayyad and Abbāsid society, has been one of the special topics she has been interested in since the very beginning of her academic life. Her works with a revisionist perspective have been faced many criticisms by colleagues, especially the western ones, because of her biased (...) view of the sources written by Muslim authors, her methodology, and her results. However, it is obvious that Crone’s studies on mawālī have positively or negatively affected the Eastern and Western academics in Islamic studies. This article’s main purpose is to reveal Crone’s views on mawālī in Isla-mic society and draw an outline without comment as much as possible. The paper consists of three parts. In the first part, Crone’s revisionist perspective and her works on mawālī have been explained. In the second part, her distinctive views on the mawālī and its origin have been discussed. Finally, in the third part, her findings of the position of mawālī in social, political, military, and economic life during the Rāshidun Caliphs, the Umayyads, and the Abbāsids have been introduced. This study, which tackles Crone’s views on mawālī, aims to encourage the further studies on this subject. (shrink)

Epistemic decision theorists aim to justify Bayesian norms by arguing that these norms further the goal of epistemic accuracy—having beliefs that are as close as possible to the truth. The standard defense of Probabilism appeals to accuracy dominance: for every belief state that violates the probability calculus, there is some probabilistic belief state that is more accurate, come what may. The standard defense of Conditionalization, on the other hand, appeals to expected accuracy: before the evidence is in, one should expect (...) to do better by conditionalizing than by following any other rule. We present a new argument for Conditionalization that appeals to accuracy‐dominance, rather than expected accuracy. Our argument suggests that Conditionalization is a rule of coherence: plans that conflict with Conditionalization don't just prescribe bad responses to the evidence; they also give rise to inconsistent attitudes. (shrink)

In this article, we consider two independently appealing theories—the Growing-Block view and Humean Supervenience—and argue that at least one is false. The Growing-Block view is a theory about the nature of time. It says that past and present things exist, while future things do not, and the passage of time consists in new things coming into existence. Humean Supervenience is a theory about the nature of entities like laws, nomological possibility, counterfactuals, dispositions, causation, and chance. It says that none of (...) these entities are fundamental, since if they were, this would entail the existence of irreducible necessary connections between matters of fact. Instead, these entities supervene on a fundamental, non-nomological ‘Humean mosaic’ of property instances at spacetime points. We will further explain and motivate the Growing-Block view and Humean Supervenience in Sects. 2 and 3, but first, we turn to our master argument. (shrink)

I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A.

The author's motivation for constructing the calculi of this paper\nis so that time and tense can be "discussed together in the same\nlanguage" (p. 282). Two types of enriched propositional caluli for\ntense logic are considered, both containing ordinary propositional\nvariables for which any proposition may be substituted. One type\nalso contains "clock-propositional" variables, a,b,c, etc., for\nwhich only clock-propositional variables may be substituted and that\ncorrespond to instants or moments in the semantics. The other type\nalso contains "history-propositional" variables, u,v,w, etc., for\nwhich only history-propositional variables may (...) be substituted and\nthat correspond to maximally linearly ordered subsets of instants.\nThis second type is for tense logics in which time is not linear\nand where at any moment there may exist alternative future courses\nof events. Quantifiers are included in both types but only for clock-\nand history-propositional variables. However, these quantifiers are\ngiven an essentially substitutional interpretation. In addition,\nbesides each clock-propositional variable being true at exactly one\ninstant, the semantics requires that for each instant there be at\nleast one clock-propositional variable true at that instant, which\nin effect amounts to having each instant of the model "denoted" in\nthe formalism by some clock-propositional variable. Thus it is not\nsurprising that quantification over clock-propositional variables\nturns out to have a "well behaved first-order semantics" (p. 284).\n{One axiom relating history- with clock-propositions seems to need\nrevision: CKTauTbuAUabUba. Since the clock-propositional variables\nmight be true at the same instant the consequent should include an\nalternative, namely, LCab.} Besides the standard truth-functional\nand tense operators G,H,F,P, the author includes L for "it is\nalways the case that". The model-theoretic earlier-than relation\nordering instants and the semantic truth-at (an instant) relation\nare formalized by Uab and Taα (for a wff α) which\nare defined respectively as LCbPa (whenever the clock-proposition\nb is the case the clock-proposition a was the case) and LCaα\n(α is the case whenever the clock-proposition a is the case).\nBecause the future tense operators are definable in terms of U\nand T and the latter are definable in terms of L and P, the\ndual of H, the author notes that his completeness proofs can be\nrestricted to L and H as primitive and with the future tense\noperators used for other interpretations, especially Prior, A.'s\nindeterminist "it will be the case that", where time is not linear.\nThe author concludes with some rather technical observations showing\nthat "in ordinary tense logics the instants are nonstandard elements\nof the models. This provides", according to the author, "a semantic\npressure to build the instants into tense logic, so that they will\nappear as standard elements in the models" (p. 283). (shrink)

We show that five important elements of the ‘nomological package’— laws, counterfactuals, chances, dispositions, and counterfactuals—needn’t be a problem for the Growing-Block view. We begin with the framework given in Briggs and Forbes (in The real truth about the unreal future. Oxford studies in metaphysics. Oxford University Press, Oxford, 2012 ), and, taking laws as primitive, we show that the Growing-Block view has the resources to provide an account of possibility, and a natural semantics for non-backtracking causal counterfactuals. We show (...) how objective chances might ground a more fine-grained concept of feasibility, and furnished a places in the structure where causation and dispositions might fit. The Growing-Block view, thus understood, provides the resources to explain the close link between modality and tense, so that it predicts modal change as time passes. This account lets us capture not only what the future might hold for us, and also what might have been. (shrink)

Aim To study the views of people in a largely Muslim country, Kuwait, of the acceptability of a life-ending action such as physician-assisted suicide (PAS). Method 330 Kuwaiti university students judged the acceptability of PAS in 36 scenarios composed of all combinations of four factors: the patient's age (35, 60 or 85 years); the level of incurability of the illness (completely incurable vs extremely difficult to cure); the type of suffering (extreme physical pain or complete dependence) and the extent to (...) which the patient requests a life-ending procedure, euthanasia or PAS (no request, some form of request, repeated requests). In all scenarios, the patients were women who were receiving the best possible care. The ratings were subjected to cluster analysis and analyses of variance. Results Five clusters were found. For 44%, PAS was always very unacceptable, no matter what the circumstances. For 23%, it was unacceptable, but less so if the patient was older or requested it repeatedly. For 16%, it was unacceptable if the patient was young but was acceptable if the patient was elderly. For 5%, it was unacceptable if the patient had extreme pain but was acceptable if completely dependent. For 11%, it was unacceptable if the patient did not request it but acceptable if she did. Conclusion The majority of the Kuwaiti university students opposed PAS either categorically or with a slight variation according to circumstances. Nonetheless, a minority approved of PAS in some cases, particularly when the patient was elderly. (shrink)

Part of the Studies in Crime and Public Policy series, this book, written by one of the top philosophers of punishment, examines the main trends in penal theorizing over the past three decades. Duff asks what can justify criminal punishment, and then explores the legitimacy of actual practices by examining what would count as adequate justification for them. Duff argues that a "communicative conception of punishment," which he presents as a third way between consequentialist and retributive theories, offers the most (...) fruitful way of understanding punishment's meaning and justification. Duff addresses such questions as how much sentences should be constrained by proportionality requirements; what modalities of punishment best communicate their intended meaning; and what decisionmaking procedures he envisions. This book will appeal to criminologists, philosophers, and others interested in theories of punishment. (shrink)

Gender discrimination and sexual harassment of female police officers by their male counterparts remain areas of liability where police departments appeared to have failed to effectively confront the nagging issues. However, the appreciable level of research conducted on these issues in the global North has not been matched by the South, where issues bordering on sexual violence have cultural underpinnings. Drawing from the case of the Nigeria Police Force, feminist analysis was used to explore the lived reality of 43 female (...) officers in a qualitative study. Participants expressed distress in abiding by structural discriminatory regulations against female officers which include non-admittance of married women into the force, forbidding unmarried officers from getting pregnant, and a stipulated minimum of three-year work period before a female officer can marry. Findings also indicated that participants suffer a variety of sexual harassment, intimidation, and, violence from their male colleagues, while they feel constrained in reporting their victimization. Important policy and practical implications requiring the Nigerian police to address hegemonic masculinity and promote gender equality were provided. (shrink)

Marcia Baron has offered an illuminating and fruitful discussion of extra-legal excuses. What is particularly useful, and particularly important, is her focus on our excusatory practices—on the ways and contexts in which we make, offer, accept, bestow and reject excuses: if we are to reach an adequate understanding of excuses, their implications and their grounds, we must attend to the roles that they can play in our human activities and relationships—and to the complexities and particularities of those roles. However, I (...) want to focus my comments less on the details of Baron’s discussions of excuses in extra-legal contexts than on the implications of her discussion for our understanding of excuses in the criminal law. What light (if any, a sceptic might add) can such analyses of our extra-legal concepts and practices throw on legal concepts and doctrines? (shrink)

In [1, §4], my ‘proof’ that GH1 has the finite model property is incorrect; there are considerable obscurities towards the end of §1, particularly on p. 33; and I should have exhibited the finite models for GH1. In §1 of this paper I expand the analysis of the sub-directly irreducible models for GH1 which I give in §1 of [1]. In §2 I give a correct proof that GH1 has the finite model property. In §3 I exhibit these finite models (...) as models on certain ordered sets. (shrink)

The discovery of the microworld presented a serious trial for many systems of views held by mankind, including its logic. This world was found to lack the familiar solid bodies, the unchanging particles and interrelations between them, the reflection of which, in one way or another, is the logic of the macroscopic world. What elementary particle physics encountered in the microscopic world seemed illogical: the rest-mass of a particle equals zero; a part that is not smaller than the whole; a (...) particle that in diffraction experiments moved like a wave; and much else that was unusual and strange in the properties of elementary particles. One of these properties was actually so named by physicists — "strangeness.". (shrink)

In response to Lawrence Solum's advocacy of a ‘virtue–centred theory of judging’, I argue that there is indeed important work to be done in identifying and characterising those qualities of character that constitute judicial virtues – those qualities that a person needs if she is to judge well (though I criticise Solum's account of one of the five pairs of judicial vices and virtues that he identifies – avarice and temperance). However, Solum's more ambitious claims – that a judge's vice (...) necessarily corrupts her decisions, and that in at least some contexts we must define a legally correct decision as one that would be reached by a virtuous judge – should be rejected: we can undermine the former by attending to the requirements of due process, and the latter by attending to the ways in which a judge would try to justify her decision. (shrink)