We present a modal logic for the class of subset spaces based on discretely descending chains of sets. Apart from the usual modalities for knowledge and effort the standard temporal connectives are included in the underlying language. Our main objective is to prove completeness of a corresponding axiomatization. Furthermore, we show that the system satisfies a certain finite model property and is decidable thus.

This paper is concerned with tracing the implications of two ideas as they affect quantum theory. One, which descends from Leibniz and Mach, is that there is no space-time continuum, but that which are involved are spacial and temporal relations involving the distant matter of the universe. The other is that our universe is finite. The picture of the world to which we are led is that of an enormous space-time Feynman diagram whose vertices are events. A consequence of finiteness (...) is that between each pair of events, along a world line, there can be only finitely many intermediate events. A further change is that we are no longer required to believe that particles need be anywhere between events. The paper takes up nonrelativistic quantum theory in a way that is consistent with these ideas. By considering analogies between the Wiener and the Feynman integrals, and between the Wiener process and related discrete processes, there is obtained a straightforward theory for the Feynman integral. Propagators are worked out for many of the cases relevant to the nonrelativistic theory.The paper shows that, even when there are, along each world-line, no more than one event per Compton wavelength, agreement is good with the usual Schrödinger theory. (shrink)

Life is a self-maintaining process based on metabolism. Something is said to be alive when it exhibits organization and is actively involved in its own continued existence through carrying out metabolic processes. A life is a spatio-temporally restricted event, which continues while the life processes are occurring in a particular chunk of matter (or, arguably, when they are temporally suspended, but can be restarted at any moment), even though there is continuous replacement of parts. Life is organized in discrete packages, (...) particular cells and multicellular organisms with differing degrees of individuality. Biological species, too, have been shown to be individuals, and not classes, as these collections of organisms are spatio-temporally localized, restricted, continuous, and somewhat cohesive entities, with a definite beginning and end. Assuming that all life on Earth has a common origin, all living organisms, cells, and tissues descending from this origin exhibit continuity of the life processes at the cellular level, as well as many of the features that define the individual character of species: spatio-temporal localization and restriction, continuity, historicity, and cohesiveness. Therefore, life on Earth is an ontological individual. Independent origins of life will have produced other such individuals. These provisionally called ‘life-individuals’ constitute a category of organization of life which has seldom been recognized. The discovery of at least one independent life-individual would go a long way toward the project of the universality of biology. (shrink)

Polyhedral functions provide a model for an important class of problems that includes both linear programming and applications in data analysis. General methods for minimizing such functions using the polyhedral geometry explicitly are developed. Such methods approach a minimum by moving from extreme point to extreme point along descending edges and are described generically as simplicial. The best-known member of this class is the simplex method of linear programming, but simplicial methods have found important applications in discrete approximation and (...) statistics. The general approach considered in this text, first published in 2001, has permitted the development of finite algorithms for the rank regression problem. The key ideas are those of developing a general format for specifying the polyhedral function and the application of this to derive multiplier conditions to characterize optimality. Also considered is the application of the general approach to the development of active set algorithms for polyhedral function constrained problems and associated Lagrangian forms. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Diagnostic WannabesJennifer Radden, PhD (bio)Saunders explores challenges for the clinician faced with self-styled sufferers from attention deficit hyperactivity disorder, post-traumatic stress disorder, bipolar disorder, autism spectrum disorder (ASD), and fibromyalgia. The diagnostic system was not meant to be used as “a scaffold for identity,” she points out. Yet wannabe patients now step into the clinic wielding self-proclaimed diagnoses as social identities. Saunders explains the context where such phenomena arise, (...) and offers guidelines for clinicians addressing this new reality. To do so, she enlists Rashed’s innovative normative approach to the so-called boundary problem of assigning, and providing justification for, the contested line between normal and disordered. The boundary problem itself seems to rise to prominence during our current era of increasingly dimensional thinking, it is worth noting, with psychiatric diagnoses taken to refer to points on a continuum, rather than to discrete categorical disease entities.On the likely sources of these ‘diagnostic’ social identities, as well as reasons why certain diagnoses attach to them more commonly than others, Saunders points to social trends (including what she asserts to be a “post-stigma” cultural environment), the failure of efforts to identify neurobiological markers for psychiatric disorder, the ubiquity of social media, the algorithms themselves, and the hyperconnected existence of contemporary times. Such narratives also offer the comforts of a sense of belonging and explanations for perceived inadequacies, she rightly observes.Despite today’s virtual and other inducements, the problem introduced here is not an entirely new one. The first part of the seventeenth century in Europe saw an apparent “epidemic” of what was known as “melancholy,” a condition associated with the spleen, whose symptoms loosely resembled those of today’s depression and anxiety. Much of this suffering was undeniably real, but much was a fashionable, identity-conferring pose. “Every distemper of the body now is complicated with spleen,” the poet John Donne ironically observes in a letter dated 1622, “and when we were young men we scarce ever heard of the spleen. In our declinations now, every accident is accompanied with heavy clouds of melancholy”1 (Gosse, 2019). By then, the Melancholic Man (or homo melancholicus), with his surfeit of spleen, was an unmistakable character type—or social identity, as we would now say. Recognized and saluted in centuries-long literary and illustrative traditions, the type was an anchoring element of the humoral medicine which, despite the gradual emergence of more empirical science, had endured since Galenic times.Melancholy’s closest descendants, affective depressive and anxiety disorders, are today less evident among the troubling identities Saunders discusses—somewhat preempted, she points out, by more cognitively-based diagnoses such as ASD. [End Page 279] This is itself an ironic reminder of the vagaries of cultural attitudes, cautioning us to remember the broader context where these particular social identities are selected for emulation and adoption.The lure of the melancholy man’s2 social identity is well understood: already normalized, within the humoral system in which excess black bile afflicted roughly one in four people, it had been valorized for centuries—arguably by Aristotelian writing (in the dubious ‘Problems’), during the Renaissance by Ficino, and Shakespeare—even by Freud in his 1916 ‘Mourning and Melancholia.’ Through most of Western history, the positive attributes and benefits of the dark moods of melancholy have needed no further explaining or justifying. They glow with glamorous associations.Applying Rashed’s conditions for appropriately acknowledging and recognizing the fit of a person’s chosen social identity, Saunders‘ guidance for clinicians clarifies boundary matters. And it illustrates how to understand, regulate and communicate the appropriate norms in ways that can dissuade those whose personal growth and mental health are not served by claiming such identities.With at least some of the popular social identities Saunders describes, I think we can helpfully continue the comparison with the homo melancholicus. For the ‘neurodiversity’ rhetoric of our own times contains forces promoting both the normalization and the valorization of syndromes such as ASD. Pointing to analogies with differences of gender, ethnicity and culture, neurodiversity theorizing argues for a respect for, and appreciation of, difference. Mental or cognitive variations are equally natural and valuable, it is... (shrink)

This work of ours is about the servant's display of behavior such as ruminating while fulfilling his responsibility to Allah. Since the verse related to our subject is in the Surah Hucurat, an evaluation will be made within the framework of this verse. During this evaluation, verses with similar characteristics that we think may be relevant will also be included. In addition, it will not only touch on the historical dimension of the issue, but also draw attention to what kind (...) of attitudes are exhibited today. Indeed, such behavior It continues in the same way in later periods as it did in the period of the Prophet. The person sent to the world for a test is responsible for fulfilling some obligations in this context. This is not a choice but a necessity. This is because man's being rewarded in the hereafter and living a more peaceful life in this world depend on this. Therefore, it is in his favor to fulfill his duty of servitude. For this reason, it is not morally correct to try to make others feel grateful for the behavior he fulfilled in accordance with the divine command. Because Allah is never in need of a servant's worship. There will always be people who will serve Him. It is the human being that is in need. Since everything he owns is at Allah's discretion, he needs Allah's help and support throughout his life, and even more so in the hereafter. The main goal of our study is to reveal opinions about how people should behave while fulfilling their servitude responsibility. In this context, the position of the creator and the issue of where the servant should stand against it will be explained. While the creator is at the highest position, man is only one of the things that he created. Of course, being given responsibility, unlike other beings, has increased the value of man in the sight of Allah. But this value is valid as long as he fulfills his servitude responsibility. Otherwise, he will descend to a lower level than all creatures. No matter how valuable a person is, he has to be aware of how helpless he is in front of his creator and that he is always in need of him. If he is not aware of this, the worship and good behavior he will perform will have no value in the sight of Allah. Because he could not behave in the way He wanted against the being he was responsible for. If he does this on purpose, of course, the price will be higher. The belief part of the issue is of greater importance. As a matter of fact, a person who appreciates Allah properly and knows His power well will need to determine his attitudes accordingly. In this context, knowing what the attributes of Allah mean and what the dimensions of these attributes are reflected in the servants will make the subject more understandable. Because the way to know Allah is to know His attributes. Another relevant issue is to show the necessary respect to the messengers sent by Allah, both in their lives and in the context of the knowledge they left behind. In this context, it will be necessary in our opinion to act with the knowledge that the hadiths that have reached us authentically are interpretations of the Qur'an. Because they are not people acting on their own, but chosen people who fulfill their duties according to Allah's orders. Therefore, disrespect to them will be considered as disrespect to God. It is seen that the Bedouins acted unconsciously on this issue that we are focusing on. These people who are new Muslims or claim to be Muslims. They had an extreme attitude towards the Prophet. This event did not occur once, but several times in similar ways. As we have just mentioned, their behavior will actually be considered disrespectful to Allah. It is known that they act with purely worldly motives. Because, instead of receiving good news, prayer and advice for the hereafter, they have revealed that they are after worldly interests in material terms. Verses on the subject have been revealed to set an example for people who will follow this attitude and show the same behavior in the future. In response to their behavior. It is admirable that the Prophet kept his calm and behaved tolerantly instead of responding to them in kind. Muslims living in later periods will also need to be more careful not to be exposed to the warning in the verse. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Politica of Justus Lipsius and the Commonplace-BookAnn MossThroughout Western Europe in the sixteenth century, schoolboys and grown men educated in the Latin schools of the humanists would recognize the commonplace-book as an indispensable tool for making sense of the books they read, for assimilating the written culture transmitted to them, and for possessing the means of production in their turn. This handy organizer of information and rather effective (...) retrieval mechanism has been treated with a fair amount of contempt after being sidelined towards the end of the seventeenth century and then falling into public disrepute, its rich stock of precious commonplaces devalued into the small change of mere banalities. Nor indeed does it suit the agenda of post-Romantic scholars and critics looking in a simplistic way for signs of originality and innovation. However, the last few years have seen commonplaces in their more sophisticated guise back on the critical agenda, with several particular studies and, latterly, two major overviews published in a single year. 1 The thesis underlying this paper is that the commonplace-book is central to an understanding of how knowledge was organized in the early modern period. More than that, the commonplace-book keys us into procedures of investigation and debate and into the dialectical and rhetorical modes of articulating thought which were agreed to have persuasive force; and it exhibits the commonality of expectations, the common places, which ensured a common route into a shared area of communication when the cultural consensus of Western Europe was strained to breaking point.One test of this thesis will be whether it can be usefully applied to texts which fit badly into interpretative frames familiar to the modern reader. The particular purpose of this paper is to apply the principles of commonplace-book [End Page 421] reading to just such an awkward case, in order at least to suggest the possibility of recovering an historically probable reading strategy which may more fully recuperate the text for the modern reader. The Politica of the eminent Stoic philosopher and scholar of antiquity, Justus Lipsius (1547–1606), is a network of quotations from ancient authorities linked together by Lipsius’s own words. He himself regarded it as one of his most important achievements. His contemporaries and their seventeenth-century descendants knew perfectly well how to read it, and there is plenty of evidence that they regarded it highly. 2 Modern readers find it an embarrassment, not knowing quite what to do with a book from which the author eclipses himself, yet keeps a dead hand on it: “All of it is mine and nothing is.” 3 Modern critics tend to assume it is all Lipsius, riding roughshod over the quotation indicators 4 ; or they prefer to think that nothing is and treat the text solely as a compilation of extracts 5 ; or they get tied in knots trying to rescue Lipsius’s voice drowned out in his quotations. 6 But Lipsius’s original readers in 1589 knew how to read it, because he told them: “For what is it but a well-arranged register of accounts or COMMONPLACES?” 7If we are going to get anywhere near the original point of reception, we must remind ourselves about commonplaces and commonplace-books. It was Erasmus in De copia who gave the first systematic guidelines for making commonplace-books, but by the time of Lipsius schoolboys in every Latin school in Northern Europe were busy excerpting from their [End Page 422] texts as they read, either under the schoolmaster’s instructions or at their own discretion. The commonplace-book was a probe and an instrument for redistributing text so as to ensure maximal retrievability and optimum application. Here is one of the many enthusiastic promoters of the practice, writing in 1564:So that the more important passages in the authors set for reading and the more brilliant sententiae, exempla, similitudes, words, phrases, and outstanding figurative expressions may be the more readily imprinted on the memory and available and ready for use as occasion requires, it is extremely useful to have the common-places of the main intellectual disciplines arranged in a definite order, under which students may note... (shrink)

In his recent book, Professor J. S. Morrison has brought to a happy conclusion a quarter of a century and more of inspired research into the problem of how the oars of a classical trireme were arranged. The essence of his solution of this perennial problem is that the fifth-century Athenian trireme had her oars and benches alike disposed at three different levels, each rower having his own oar, and each oar its separate thole set at a distance of feet, (...) not inches, from its neighbours. The evidence is marshalled with such mastery that it may be thought unlikely that there will ever be any general recrudescence of the di scaloccio and a zenzile theories that were as fashionable once as they are seen now to have been unhistorical. In his inquiry, however, Professor Morrison has wisely confined himself to the ancient sources, and no more than touched upon the analogy of the Byzantine dromon, the direct descendant of the classical trireme and to some extent the parent of the a zenzile galley. Other protagonists, and notably Tarn, have been far from sharing his discretion, and there is still room perhaps for a brief note calling attention to the possibility that the dromon of the Middle Ages may shed indirect light upon the trireme of fifteen hundred years earlier. (shrink)

In his recent book, Professor J. S. Morrison has brought to a happy conclusion a quarter of a century and more of inspired research into the problem of how the oars of a classical trireme were arranged. The essence of his solution of this perennial problem is that the fifth-century Athenian trireme had her oars and benches alike disposed at three different levels, each rower having his own oar, and each oar its separate thole set at a distance of feet, (...) not inches, from its neighbours. The evidence is marshalled with such mastery that it may be thought unlikely that there will ever be any general recrudescence of the di scaloccio and a zenzile theories that were as fashionable once as they are seen now to have been unhistorical. In his inquiry, however, Professor Morrison has wisely confined himself to the ancient sources, and no more than touched upon the analogy of the Byzantine dromon, the direct descendant of the classical trireme and to some extent the parent of the a zenzile galley. Other protagonists, and notably Tarn, have been far from sharing his discretion, and there is still room perhaps for a brief note calling attention to the possibility that the dromon of the Middle Ages may shed indirect light upon the trireme of fifteen hundred years earlier. (shrink)

Perceptual judgments result from a dynamic process, but little is known about the dynamics of number‐line estimation. A recent study proposed a computational model that combined a model of trial‐to‐trial changes with a model for the internal scaling of discrete numbers. Here, we tested a surprising prediction of the model—a situation in which children's estimates of numerosity would be better than those of adults. Consistent with the model simulations, task contexts led to a clear developmental reversal: children made more adult‐like, (...) linear estimates when to‐be‐estimated numbers were descending over trials (i.e., backward condition), whereas adults became more like children with logarithmic estimates when numbers were ascending (i.e., forward condition). In addition, adults’ estimates were subject to inter‐trial differences regardless of stimulus order. In contrast, children were not able to use the trial‐to‐trial dynamics unless stimuli varied systematically, indicating the limited cognitive capacity for dynamic updates. Together, the model adequately predicts both developmental and trial‐to‐trial changes in number‐line tasks. (shrink)

The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article (...) explains the distinction and why it has proved to be one of the great organizing themes of mathematics. (shrink)

Family relations between Greek eponymous heroes almost certainly reflect political or commonly agreed ethnic relationships between the communities concerned. Pelops and his proverbially numerous descendants are investigated here from this perspective and it is argued that the creation of eponymous Pelopids was primarily due to political motivations. Pelops as one of the most remote ancestors of Sparta was used, already by the 5th century BC, to establish connections on a mythological level with those cities which became the allies of the (...) Lakedaimonians and are commonly known as the members of the Peloponnesian League. It is also suggested that after the battle of Plataea, these poleis erected a victory monument at Olympia on their own, which is commonly referred to as the “Achaean monument”. (shrink)

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set (...) of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism. (shrink)

Advocates of dynamic systems have suggested that higher mental processes are based on continuous representations. In order to evaluate this claim, we first define the concept of representation, and rigorously distinguish between discrete representations and continuous representations. We also explore two important bases of representational content. Then, we present seven arguments that discrete representations are necessary for any system that must discriminate between two or more states. It follows that higher mental processes require discrete representations. We also argue that discrete (...) representations are more influenced by conceptual role than continuous representations. We end by arguing that the presence of discrete representations in cognitive systems entails that computationalism (i.e., the view that the mind is a computational device) is true, and that cognitive science should embrace representational pluralism. (shrink)

We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest type-definable subgroup G00 of bounded index and G/G00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.

We present two discrete dualities for double Stone algebras. Each of these dualities involves a different class of frames and a different definition of a complex algebra. We discuss relationships between these classes of frames and show that one of them is a weakening of the other. We propose a logic based on double Stone algebras.

In this paper, I present a discrete solution for the paradox of Achilles and the tortoise. I argue that Achilles overtakes the tortoise after a finite number of steps of Zeno’s argument if time is represented as discrete. I then answer two objections that could be made against this solution. First, I argue that the discrete solution is not an ad hoc solution. It is embedded in a discrete formulation of classical mechanics. Second, I show that the discrete solution cannot (...) be falsified experimentally. (shrink)

The present study provides evidence that valence focus and arousal focus are important processes in determining whether a dimensional or a discrete emotion model best captures how people label their affective states. Individuals high in valence focus and low in arousal focus fit a dimensional model better in that they reported more co-occurrences among like-valenced affective states, whereas those lower in valence focus and higher in arousal focus fit a discrete model better in that they reported fewer co-occurrences between like-valenced (...) affective states. Taken together, these findings suggest that one static, nomothetic theory may not accurately describe the subjective affective experience of all individuals. (shrink)

Causal connectionists need to provide an account of causal linkage and of causal direction. This chapter distinguishes between two kinds of causal connection, namely, discrete and continuous. Causal connectionists have a number of options for explaining the linkage between causes and effects in the case of discrete causation. The chapter provides some popular options. If some causation is discrete, and the exercise of causal powers provides a direction to discrete causation, then the causal direction of processes can be derived from (...) the causal direction of discrete interaction. David Hume famously made the claim about the knowledge of causation in his Enquiry Concerning Human Understanding. Hume offers, in defense of his principle, a set of thought‐experiments that seem to show that we can always imagine any event's having a different result from the one we in fact observe. Causal processes like locomotion may provide counterexamples to Hume's Epistemic Principle. (shrink)

The discretionary powers of welfare state professionals are in tension with the requirements of the democratic Rechtsstaat. Extensive use of discretion can threaten the principles of the rule of law and relinquish democratic control over the implementation of laws and policies. These two tensions are in principle ineradicable. But does this also mean that they are impossible to come to grips with? Are there measures that may ease these tensions? We introduce an understanding of discretion that adds an epistemic dimension (...) (discretion as a mode of reasoning) to the common structural understanding of discretion (an area of judgment and decision). Accordingly we distinguish between structural and epistemic measures of accountability. The aim of the former is to constrain discretionary spaces or the behaviour within them while the aim of the latter is to improve the quality of discretionary reasoning. The focus in this article is on epistemic measures that are internally related to the main characteristic of accountability, namely justified use of discretionary power. (shrink)

Automated decision-making takes up an increasingly significant place in the administrative state. This article presents a conception of discretion that is helpful for evaluating the proper place of algorithms in public decision-making. I argue that the algorithm itself is not a site of discretion. The threat is that automated decision-making alters the relationships between traditional actors in a way that can cut down discretion and human commitment. Algorithmic decision-makers can serve to fetter the discretion that the legislature and the populace (...) expect to be exercised. We must strive to maintain discretion, moral agency, deliberative ideals, and human commitment through the system that surrounds the use of an algorithm and to develop a new expertise that can retain and exercise the expected discretion. Backing this argument are traditional legal constraints, public expectations, and administrative law principles, tied together through the organizing principle of discretion. (shrink)

Heterocyst spacing in blue -green bacteria is widely assumed to be due to a diffusible inhibitor. The inhibitor, a nitrogen-rich compound, probably glutamine, is produced via the N2-fixing enzymes of the heterocyst and in turn serves to suppress the induction of these enzymes and of the differentiation of vegetative cells to heterocysts. This simple morphogenetic mechanism operating in growing cellular filaments ofAnabaena species is investigated on the basis of a continuous and a discrete cellular model, as well as by cell-by-cell (...) simulation of the inhibitor transport. The resulting distances between heterocysts and kinetics of their production are compared with observations, and the values of physical parameters are estimated from the models. (shrink)

Carroll Izard completed his dissertation in 1952, beginning a career spanning more than six decades that coincided with clinical psychology maturing as a profession, and the birth of clinical science and cognitive neuroscience. Izard’s focus on discrete emotions as evolved systems that organize information, prepare responses, and shape the development of personality and relationships persisted through his career, despite “emotions” often being overshadowed by psychodynamic, behavioral, or cognitive perspectives. His theoretical work anticipated and now integrates contemporary neuroscience and relational perspectives. (...) Exploration of discrete emotions has kept lines of inquiry open that enrich our understanding of psychopathology. Izard also embraced clinical science, combining basic research with effectiveness studies addressing the unmet need for mental health services via emotion-focused interventions. (shrink)

H.L.A. Hart’s lost and found essay ‘Discretion’ has provided new insight into the issue of how legal systems can cope with indeterminacy in the law. The so-called ‘open texture’ of law calls for the exercise of judicial discretion, which, I argue, renders judges susceptible to the problem of dirty hands. To show this, I frame the problem as being open to an array of appropriate emotional responses, namely, various senses of guilt. With these responses in mind, I revise an example (...) from Michael Walzer’s original analysis in a way that highlights purely personal sacrifices in solutions to dirty hands situations. I then turn to an account of moral emotions in legal decision-making and show how judges—in failing to advance all interests—might be left with a unique sense of guilt. With an application of this account to Hart’s legal positivism, it can be seen that a judge’s hands are often dirtied in resolving borderline cases. If discretion leaves judges in situations where they must do wrong in order to do right, Hart’s endorsement of a closure view of wrongdoing will lead to difficulties in how he can explain the presence of moral remainders in jurisprudence. (shrink)

In this paper I argue against grounding being necessarily well-founded, and provide some reasons to think it's actually not well-founded.

Prosecutors in the US legal system have great power to interfere at their discretion in the lives of citizens, and face relatively few checks on the exercise of this discretion. The vast scope of the criminal law provides a pretext for prosecuting nearly anyone. Meanwhile, other features of the legal system, such as the way plea bargains are structured and the doctrine of prosecutorial immunity, further increase prosecutorial power. And existing institutional restraints on prosecutorial abuses, such as democratic accountability, the (...) grand jury system, and the possibility of a selective prosecution defense, are mostly ineffectual. I draw on republican political theory, including insights from Philip Pettit and Elizabeth Anderson, to argue that this state of affairs gives prosecutors dominating, and therefore unjust, power over vast swathes of the public. I then survey some potential institutional changes which might help ameliorate the problem. (shrink)

Elected prosecutors have pledged not to enforce abortion laws, in response to state-level abortion bans. For their pledges to be meaningful, prosecutors must exercise their discretion in cases of individuals who face legal risk, including people who help others self-manage their abortions. With a harm-reduction approach to improving abortion access, prosecutors should aim to reduce abortion helpers’ involvement with the criminal justice system.

Newtonian gravitation is studied from a discrete point of view, in that the dynamical equation is an energy-conserving difference equation. Application is made to planetary-type, nondegenerate three-body problems and several computer examples of perturbed orbits are given.

Recent discussions in the philosophy of science have devoted considerable attention to the analysis of conceptual issues relating to the methodology of explanation and prediction in the sciences. Part of this literature has been devoted to clarifying the very ideas of explanation and prediction. But the discussion has also ranged over various related topics, including the status of laws to be used for explanatory and predictive purposes, the logical interrelationships between explanatory and predictive reasonings, the differences in the strategy of (...) explanatory argumentation in different branches of science, the nature and possibility of teleological explanation, etc. The aim of the present article is to examine the issues involved in such questions from the specialized perspective afforded by one particular kind of physical systems--namely, systems, here to be characterized as discrete state systems, whose behavior has been studied extensively in the scientific literature under the general heading of Markov chains. These systems have been chosen as our focus because their behavior over time can be analyzed at once with great ease and with extraordinary precision. (shrink)

In this field questions arise which are certainly difficult; but as I listened last time to members of the group, I felt that the main difficulty perhaps lay in determining precisely what questions we are trying to answer. I have the conviction that if we could only say clearly what the questions are, the answers to them might not appear so elusive. So I have begun with a simple list of questions about discretion which in one form or another were, (...) as it seemed to me, expressed by the group last time. I may indeed have omitted something and inserted something useless: if so, no doubt I shall be informed of this later. The central questions then seem to me to be the following: 1. What is discretion, or what is the exercise of discretion? 2. Under what conditions and why do we in fact accept or tolerate discretion in a legal system? 3. Must we accept discretion or tolerate discretion, and if so, why? 4. What values does the use of discretion menace, and what values does it maintain or promote? 5. What can be done to maximize the beneficial operation of the use of discretion and to minimize any harm that it does? (shrink)

Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and we prove a discrete duality for them.

This paper argues that liberal tenats that justify intervention to promote the welfare of an incompetent do not suffice as a basis for analyzing parent-child relationships, and that this inadequacy is the basis for many of the problems that arise when thinking about the state's role in resolving family conflicts, particularly when monitoring parental discretion in medical decision-making on behalf of a child. The state may be limited by the best interest criterion when dealing with children, but parents are not. (...) The state's relation with the child is formal while the parental relation is intimate, having its own goals and purposes. While the liberal canons insist on the incompetent one's best interest, parents are permitted to compromise the child's interests for ends related to these familial goals and purposes. Parents decisions should be supervened, in general, only if it can be shown that no responsible mode of thinking warrants such treatment of a child. Keywords: proxy medical consent, children's rights, state's protection of children, parental authority and the state's intervention, paternalism, liberalism, parental values CiteULike Connotea Del.icio.us What's this? (shrink)

Using only uncontentious principles from the logic of ground I construct an infinitely descending chain of ground without a lower bound. I then compare the construction to the constructions due to Dixon and Rabin and Rabern.

In this essay, I argue against a way of approaching the issue of Judicial discretion that finds its clearest exposition and highest development in recent works by Ronald Dworkin. This approach is too narrow. It ignores a kind of Judicial discretion whose existence has been maintained by jurists with discretionist sympathies, and which is philosophically significant. The kind of discretion it ignores raises the issue of the justification of adjudication as clearly as does the kind of discretion that it recognizes. (...) Moreover, discussion of the kind of discretion ignored is in some respects the natural starting place for a discussion of judicial discretion in general. (shrink)

The purpose of the study is to reveal the relationship between the discreteness of historical consciousness and the processes of spiritual and intellectual development of the individual; to offer his vision of what the discreteness of historical consciousness is and to determine the degree of its influence on the process of spiritual and intellectual formation of the individual. The article focuses on the fact that the discreteness of historical consciousness has a destructive effect on public consciousness and its forms. The (...) degree of discreteness of historical consciousness and the level of spiritual and intellectual development of the individual are interdependent and have a mutual influence on each other. As a result of the conducted research, it was revealed that the discrete historical consciousness is a fragmented "torn" historical consciousness, which poses a threat to the spiritual and intellectual formation of the individual. (shrink)

A book on the notion of fundamental length, covering issues in the philosophy of math, metaphysics, and the history and the philosophy of modern physics, from classical electrodynamics to current theories of quantum gravity. Published (2014) in Cambridge University Press.

In this work we investigate the Weihrauch degree of the problem Decreasing Sequence of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem Bad Sequence of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf {DS}$, despite being hard to solve, is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We (...) then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$, $\boldsymbol {\Sigma }^1_1$, $\boldsymbol {\Pi }^1_1$. We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the Baire hierarchy and show that they do not collapse at any finite level. (shrink)

Three classes of decidable discrete linear orders with varying degrees of effectiveness are investigated. We consider how a classical order type may lie in relation to these three classes, and we characterize by their order types elements of these classes that have effective nontrivial self-embeddings.

As the result of the complexity inherent in nature, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often deal with models for structured populations in which individuals are classified regarding their age, size, activity or location, and this structuring of the population leads to high dimensional systems. In many instances, the dynamics of the system is controlled by processes whose time scales are very different from (...) each other. Aggregation techniques take advantage of this situation to build a low dimensional reduced system from which behavior we can approximate the dynamics of the complex original system.In this work we extend aggregation techniques to the case of time dependent discrete population models with two time scales where both the fast and the slow processes are allowed to change at their own characteristic time scale, generalizing the results of previous studies. We propose a non-autonomous model with two time scales, construct an aggregated model and give relationship between the variables governing the original and the reduced systems. We also explore how the properties of strong and weak ergodicity, regarding the capacity of the system to forget initial conditions, of the original system can be studied in terms of the reduced system. (shrink)

Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a continuous (...) variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order. (shrink)

Van Bendegem has recently offered an argument to the effect that, if time is discrete, then there should exist a correspondence between the motions of massive bodies and a discrete geometry. On this basis, he concludes that, even if space is continuous, it should nonetheless appear discrete. This paper examines the two possible ways of making sense of that correspondence, and shows that in neither case van Bendegem’s conclusion logically follows.

Using a neutron interferometer and the phase difference calculated from spatial discrete quantum mechanics, a test for discrete quantum theory may implemented by measuring the X spin polarization and its variation with position.

Leibniz argues against Descartes’s conception of material substance based on considerations of unity. I examine a key premise of Leibniz’s argument, what I call the Plurality Thesis—the claim that matter (i.e. extension alone) is a plurality of parts. More specifically, I engage an objection to the Plurality Thesis stemming from what I call Material Monism—the claim that the physical world is a single material substance. I argue that Leibniz can productively engage this objection based on his view that matter is (...) discrete. The discreteness of matter provides two aspects of support for the Plurality Thesis. First, it indicates that the parts of matter do not share boundaries and are, therefore, independent in an important sense. Second, it indicates that the parts of matter are determinate and are, therefore, ontologically prior to the wholes they compose. (shrink)

Editors of Cicero'sPro Archiahave assumed that Petrarch's lost transcription of the equally lost Liège manuscript that he discovered in 1333 survives in an almost unaltered version in a single Florentine manuscript, while the remaining 265Italireflect another stage of the text, when conjectural corrections by its learned discoverer were introduced into the text. This article proposes a reassessment of that dichotomy, based on a first comprehensive study of the whole transmission.

Democratic institutions and practice depend on trust, in two ways. Citizens must trust each other to abide by shared rules and norms that together govern a political community; it is a feature of democratic states that they direct their resources not to enforcement of rule abidingness, but rather towards providing collective and public goods. Instead, states rely on the semi-voluntary compliance of citizens with these shared norms and laws. Citizens must also trust their political representatives, who via their election are (...) granted discretion to make decisions that will have significant impact on the lives of those they represent. Yet, according to many scholars across many disciplines, this trust is at risk of being undermined, if it hasn’t yet been so. The possible reasons for weakened and broken trust are many, ranging from escalating wealth inequalities, to increasing cultural and ethnic diversity, to political incompetence of both voters and political representatives, to corruption among political representatives. In this article, I examine the trust that underpins the discretion accorded to elected representatives, to identify when the improper use of this discretion may reasonably be thought to be a source of distrust. In particular, I argue that where discretion is used arbitrarily, it may threaten to undermine the trust relations between citizens and their representatives, and therefore the efficient and effective running of democratic states. (shrink)

A general metatheorem is proved which reduces a wide class of statements about continuous time stochastic processes to statements about discrete time processes. We introduce a strong language for stochastic processes, and a concept of forcing for sequences of discrete time processes. The main theorem states that a sentence in the language is true if and only if it is forced. Although the stochastic process case is emphasized in order to motivate the results, they apply to a wider class of (...) random variables. At the end of the paper we illustrate how the theorem can be used with three applications involving submartingales. (shrink)