There is an ongoing investigation by scholars of ethics and economics into whether human decision making and the resultant acts should be guided by rules and procedures or by judgment and discretion. Although each of these modes offers advantages and disadvantages to decision makers, they are by no means neutral in their effect on professional development. The paper presents an in-depth view of discretionary decisions using an Aristotelian-Thomistic framework. This is the first of the series of papers which focus (...) on the application of realistic philosophical principles to ethical professional behaviour and decision making in daily practice. Results indicate that classical philosophical theories which use virtues and truth as indispensable components still may give rise to desirable and moral conduct among individuals who apply them. (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)

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.

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)

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)

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)

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)

: How might we locate originality as emerging from within the "discrete" work of commentary? Because many women have engaged with philosophy in forms (including commentary) that preclude their work from being seen as properly "original," this question is a feminist issue. Via the work of selected contemporary French women philosophers, the author shows how commentary can reconfigure the philosophical tradition in innovative ways, as well as in ways that change what counts as philosophical innovation.

Examining the role of dispositions (potentials and propensities) in both physics and psychology reveals that they are commonly derivative dispositions, so called because they derive from other dispositions. Furthermore, when they act, they produce further propensities. Together, therefore, they appear to form discrete degrees within a structure of multiple generative levels. It is then constructively hypothesized that minds and physical nature are themselves discrete degrees within some more universal structure. This gives rise to an effective dualism of mind and nature, (...) but one according to which they are still constantly related by causal connections. I suggest a few of the unified principles of operation of this more complicated but universal structure. (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)

In various guises (usually referred to as the “basic emotion” or “discrete emotion” approach), scientists and philosophers have long argued that certain categories of emotion are natural kinds. In a recent paper, Colombetti (2009) proposed yet another natural kind account, and in so doing, characterized and critiqued psychological constructionist approaches to emotion, including our own Conceptual Act Model. In this commentary, we briefly address three topics raised by Columbetti. First, we correct several common misperceptions about the discrete emotion approach to (...) emotion. Second, we discuss misconceptions of our Conceptual Act Model. Finally, we briefly comment on Columbetti's Dynamical Discrete Emotion model. (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.

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.

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.

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 (...) class='Hi'>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)

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)

In aggressive interactions, animals often use a discrete set of signals, while the properties being signalled are likely to be continuous, for example fighting ability or value of victory. Here we investigate a particular model of fighting which allows for conventional signalling of subjective resource value to occur. The result shows that neither perfect nor no signalling are evolutionarily stable strategies in the model. Instead, we find ESSs in which partial information is communicated, with discrete displays signalling a range of (...) values rather than a precise one. The result also indicates that communication should be more precise in conflicts over small resources. Signalling strategies can exist in fighting because of the common interest in avoiding injuries, but communication is likely to be limited because of the fundamental conflict over the resource. Our results reflect a compromise between these two factors. Data allowing for a thorough test of the model are lacking; however, existing data seem consistent with the obtained theoretical results. (shrink)

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)

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.

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)

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)

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)

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)

Energy nonconservation is a serious problem of dynamical collapse theories. In this paper, we propose a discrete model of energy-conserved wavefunction collapse. It is shown that the model is consistent with existing experiments and our macroscopic experience.

We define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities ∑ i a i · x i ≥ b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CP- inequalities, we prove a monotone interpolation theorem (...) obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Theorem 5.1]. LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantification is allowed only over the former sort. We shall give a construction of a theory LK(M) for any discretely ordered module M (e.g., LK(Z) extends LK(CP)). The interpolation theorem generalizes to these theories obtained from discretely ordered Z-modules. We shall also discuss a connection to quantifier elimination for such theories. We formulate a communication complexity problem whose (suitable) solution would allow to improve the monotone interpolation theorem and the lower bound for R(CP). (shrink)

A journey of a thousand miles begins, obviously, with a single step. But isn’t it equally obvious that a step of a single metre must begin with a single millimetre? And before you can begin the last micron of that millimetre, don’t you have to get through 999 other microns first? And so ad infinitum? That “ad infinitum” bit is what worried the philosopher Zeno of Elea. Can our every action really consist of sub actions each consisting of sub sub (...) actions ... so that before we can move at all, we have to perform a literally infinite number of distinct, consecutive actions? (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)

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)

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 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)

The well known Master Argument of ancient Stoic logician Diodorus Cronus is an argument in favour of the philosophical doctrine of fatalism. Perhaps in antiquity this argument was a subject of the most celebrated controversy about temporal truth and modality. This argument is a subject of logical analysis, especially in connection with temporal logic, also today. 1 The most elegant tense-logical formulation of the Master Argument has been given by A. N. Prior. Discreteness and irreflexivity of time are semantical assumptions (...) of Prior’s formulation of the argument. The assumption of discreness is troublesome. 2 The problem of this assumption is a subject of M. J. White’s paper “The Necessity of the Past and ModalTense Logic Incompleteness”. 3. (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)

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)

This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding mathematical (...) understanding prevail over mere preemptive reductionism to whole numbers. (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)

Mathematical modeling is a convenient way for characterization of complex ecosystems. This approach was applied to study the dynamics of zooplankton in Lake Sevan (Armenia) at different stages of anthropogenic eutrophication with the use of a novel method called discrete modeling of dynamical systems with feedback (DMDS). Simulation demonstrated that the application of this method helps in characterization of inter- and intra-component relationships in a natural ecosystem. This method describes all possible pairwise inter-component relationships like “plus–plus,” “minus–minus,” “plus–minus,” “plus–zero,” “minus–zero,” (...) and “zero–zero” that occur in most ecosystems. Based on the results, a working hypothesis was formulated. It was found that the sensitivity to weak external influence in zooplanktons was the greatest during the mid period of eutrophication in Lake Sevan, whereas in the final stages of eutrophication, an outbreak in the biomass production of cyanobacteria was evident. To support this approach, a weak external disturbance in the form of magnetic storm was used to see its effect on species Daphnia longispina sevanica. A statistically significant correlation between the frequency of magnetic storms and the number of this species was revealed and an increase in the number of toxic cyanobacteria species as a consequence of eutrophication. This paper, for the first time, suggests a DMDS method, to diagnose impact of anthropogenic eutrophication on environment. (shrink)

Experiential and behavioural aspects of emotions can be measured readily but developing a contactless measure of emotions’ physiological aspects has been a major challenge. We hypothesised that different emotion-evoking films can produce distinctive facial blood flow patterns that can serve as physiological signatures of discrete emotions. To test this hypothesis, we created a new Transdermal Optical Imaging system that uses a conventional video camera to capture facial blood flows in a contactless manner. Using this and deep machine learning, we analysed (...) videos of the faces of people as they viewed film clips that elicited joy, sadness, disgust, fear or a neutral state. We found that each of these elicited a distinct blood flow pattern in the facial epidermis, and that Transdermal Optical Imaging is an effective contactless and inexpensive tool to the reveal physiological correlates of discrete emotions. (shrink)

Discrete Mathematics is designed to serve as a textbook for undergraduate engineering students of computer science and postgraduate students of computer applications. The book would also prove useful to post graduate students of mathematics. It seeks to provide a thorough understanding of the subject and present its practical applications to computer science.

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.

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)

We report that upon excitation by a single pulse, a classical harmonic oscillator immersed in the classical electromagnetic zero-point radiation exhibits a discrete harmonic spectrum in agreement with that of its quantum counterpart. This result is interesting in view of the fact that the vacuum field is needed in the classical calculation to obtain the agreement.

In this paper we give a representation of the free MV-algebra over one generator as a structure of functions having finite domain.