Classical Arguments Research Articles

Pascal’s Wager and Decision-making with Imprecise Probabilities

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations where the decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation.

Steering the longitudinal photoelectron momentum in the above-threshold ionization with two not-quite-collinear laser beams

Strong-field atomic processes, driven by long-wavelength laser beams, are known to be affected by magnetic forces. In such beams, the Lorentz force pushes the photoelectrons along the beam direction and prevents their rescattering or recombination with the parent ions. In high-order harmonic generation (HHG), therefore, the yield of energetic photons is markedly suppressed, rendering x-ray radiation sources from high harmonics so far impractical. To compensate these magnetic forces and to reenable HHG at long wavelengths, a setup of two not quite collinear beams has been suggested recently but not much analyzed beyond classical arguments and with respect to accessible laser parameters. Using the nondipole strong-field approximation, we here investigate when the longitudinal momentum of the photoelectrons vanishes and how this noncollinear setup explicitly depends on the wavelength and intensity of the driving beams. We also demonstrate that an optimal crossing angle ${\ensuremath{\delta}}_{0}$ between these beams always exists for which the fraction of the returning electrons is maximized. This rather simple steering of the longitudinal momentum will allow an efficient HHG with driving beams deep in the midinfrared.

From Function to Structure: A Reverse Solution to the Mind-Brain Problem

Starting from the tension and integration between cognitive neuroscience and cognitive psychology, this paper analyzes the waxing and waning of physicalism versus mentalism over the last century and defends an anti-reductionist ontology of the mind. It first argues that “function logically precedes structure” and then refutes two classic arguments about the mind-brain problem. The paper shows that if structure is treated as the logical premise of function, discussions of the mind-brain problem will inevitably lead to dualism or reductionism. Structure cannot explain its own occurrence; instead, if function, which logically precedes structure, is taken as the starting point of the explanation, then the principle of the “ladder of explanation” will be satisfied. A higher-level theory can explain lower-level observed facts, and lower-level observed facts are the cognitive starting point leading to a theory through induction. We conclude that function logically precedes structure; therefore, logically, “the mind shapes the brain.” The logical sequence from function to structure represents a reverse strategy for solving the mind-brain problem.

Impact of chamber back pressure on the ignition dynamics of hydrogen enriched premixed flames

The impact of chamber back pressure and H2-enrichment on flashback of premixed CH4−H2-Air flames triggered by ignition dynamics is investigated. Different back pressures are imposed by varying the exhaust flow blockage with perforated plates of different porosities. Experiments measuring the ignition transient, from kernel formation to stable flames and flashback, with increasing levels of H2-enrichment covering a wide range of operating conditions are conducted. Time-series of pressure, velocity and heat release rate are examined to characterise the ignition process. It is shown that the final state of the flame is highly modified by the over pressure caused by the volumetric expansion of the hot gases. Three different scenarios are identified: Soft ignition with direct stabilisation, transient flashback and permanent flashback. The magnitude of the over pressure generated at ignition is significantly amplified by small increases in mean back pressure which, in turn, modifies the flowrate through the injector thereby promoting flashback. In addition to its role as a pressure amplifier, increasing the combustor back pressure also affects the time lag between the over pressure and the peak in heat release rate just after ignition. The impact of H2-enrichment on flashback is investigated by keeping both the bulk injection velocity Ub and laminar burning velocity Sl0 constant for flames with 0% to 100%H2-content. It is shown that the magnitude of the over pressure approximately scales with Ub/Sl0 and is independent of the total thermal power and H2 enrichment. Although the mean flow conditions and over pressures are kept constant, the stabilisation mode directly depends on H2-content. This shows that flashback triggered by the ignition dynamics cannot be fully described by classical kinematic arguments and a new mechanism is presented.

Bures–Wasserstein geometry for positive-definite Hermitian matrices and their trace-one subset

In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

ANALYSIS OF VISCOPLASTIC CONTACT PROBLEM

In this paper, we analyze a quasistatic problem modeling frictional contact between a viscoplastic body and an obstacle, the so called foundation. The material constitutive relation is assumed to be non-linear. The boundary conditions of contact and friction are modeled respectively by the \textit{Signorini} conditions and the generalized Coulomb's non-local law. We derive a variational formulation for the problem and prove the existence of its unique weak solution. The proof use, essentially, classical arguments of compactness, variational inequalities and Banach’s fixed point theorem.

The Pedagogy of Bafflement: Bernard Bailyn's History 2910, Fall 1996

The Pedagogy of Bafflement: Bernard Bailyn's History 2910, Fall 1996

Iconological Dualism Re-Thought: A New Variation on Two Old Theories

Abstract This article aims at defending the old theory of iconological dualism that opposes ‘handmade’ pictures to photographic pictures. I defend a new version of that theory, according to which photographs always enable viewers to have singular thoughts on the things photographed, while handmade pictures by themselves never enable viewers to have singular thoughts but only enable them to have what I call ‘thoughts by depiction’. To this end, I defend the old theory according to which singular thoughts require a special relation—called ‘acquaintance’—between the thinker and the object thought about. I put this traditional view forward thanks to a new version of the classic spy argument for the acquaintance constraint on singular thoughts.

On dense intermingling of exact overlaps and the open set condition

AbstractWe prove that certain families of homogenous affine iterated function systems in $\mathbb {R}^{d}$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan–Pollicott–Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb {R}^{d}$ with an adaptation of a classic argument of Erdős on the singularity of certain Bernoulli convolutions. This result encompasses a one-dimensional example due to Kenyon which arises as a special case.

A combinatorial proof of Aldous–Broder theorem for general Markov chains

AbstractAldous–Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph , but it is more general: given an irreducible and reversible Markov chain on started at , the tree rooted at formed by the first entrance steps in each node (different from the root) has a probability proportional to , where the edges are directed toward . In this article we give proofs of Aldous–Broder theorem in the general case, where the kernel is irreducible but not assumed to be reversible (this generalized version appeared recently in Hu, Lyons, and Tang [5]). We provide two new proofs: an adaptation of the classical argument, which is purely probabilistic, and a new proof based on combinatorial arguments. On the way we introduce a new combinatorial object that we call the golf sequences.

Detectability, duality, and surplus extraction

We study surplus extraction in the general environment of McAfee and Reny (1992), and provide two alternative proofs of their main theorem. The first is an analogue of the classic argument of Crémer and McLean (1985, 1988), using geometric features of the set of agents' beliefs to construct a menu of contracts extracting the desired surplus. This argument, which requires a finite state space, also leads to a counterexample showing that full extraction is not possible without further significant conditions on agents' beliefs or surplus, even if the designer offers an infinite menu of contracts. The second argument uses duality and applies with an infinite state space, thus yielding the general result of McAfee and Reny (1992). Both arguments suggest methods for studying surplus extraction in settings beyond the standard model, in which the designer or agents might have objectives other than risk neutral expected value maximization.

Fourier analysis of a time-simultaneous two-grid algorithm using a damped Jacobi waveform relaxation smoother for the one-dimensional heat equation

Abstract In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite difference approximation in space while the semi-discrete problem is discretized in time using the ϑ-scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown. It is shown that the all-at-once system based on an equidistant discretization in space and time stays well conditioned even if the number of blocked time-steps grows arbitrarily. Furthermore, mesh-independent convergence rates of the considered two-grid algorithm are proved by adopting classical Fourier arguments in space without assuming periodic boundary conditions. The rate of convergence with respect to the Euclidean norm does not deteriorate arbitrarily if the number of blocked time steps increases and, hence, underlines the potential of the solution algorithm under investigation. Numerical studies demonstrate why minimizing the spectral norm of the iteration matrix may be practically more relevant than improving the asymptotic rate of convergence.

Dissipation Effects Induced by Internal Relative Motion Between Two Nonlinear σ-Models

We studied the dissipation effect induced by the internal relative motion between two nonlinear [Formula: see text]-models (NLSMs). The internal degrees of freedom (DOF) of the two NLSMs coupled to each other via an interface potential. We performed the path integral of the vacuum persistence amplitude (VPA) and got the quantum action. Then we introduced the relative motion via a boost transformation with velocity parameter [Formula: see text]. Furthermore, we calculated the imaginary part of the boosted quantum action which was proportional to the vacuum decay probability. At last, we calculated the frictional force. The numerical results gave the velocity [Formula: see text] dependency of the vacuum decay probability, and the velocity [Formula: see text] dependency of the frictional force. There was a threshold of relative velocity [Formula: see text] for the vacuum decay probability and the frictional force. We gave a classical argument of the existence of the threshold, and the argument matched the mathematical results.

Bet-hedging strategies determine daily choices in effort allocation for Nile perch fishers of Lake Victoria

Small scale fishers experience high variability in daily catches within a resource space limited in extent by operational constraints, and biophysical factors affect daily choices in spatial effort allocation. We focus on the management consequences of the extent of the individual fisher’s resource space and his options to handle risk arising from daily catch variability through a portfolio of sizes and species from within that resource space. Gillnet and longline Nile perch fishers in south-eastern Lake Victoria, Tanzania were provided with a GPS and recorded their position together with their daily catch and operations. Three different gillnet and longline fishing strategies could be discerned. All had bet-hedging characteristics, differing in the size of their resource space of on average 120 – 141 km2, distance from the shore fished and emphasis on the mix of sizes of Nile perch and other species caught. Daily choice of fishing locations did not relate to previous days’ catch success. Still, fishers used the general inshore to offshore distributional patterns of small and large Nile perch in their daily choices of mesh and hook sizes. The mix of sizes and species reduced day-to-day catch variability due to a portfolio effect. Current mesh and landing size regulations based on classical arguments around growth-overfishing interfere with these strategies and force individual fishers to specialise in size, species, and area. They must accept higher uncertainties by either choosing their fishing locations further offshore or accepting a more variable lower catch, leading to higher personal and occupational risk or ongoing management conflicts when disregarding regulations. Portfolio management of fished resources by compromising on mesh size regulations would make sense by allowing fishers to utilise a certain proportion of smaller mesh or hook sizes that are now illegal as part of their fishing strategies.

Philip Roth

Philip Roth

On a vertex-capturing game

In this paper, we study the recently introduced scoring game played on graphs called the Edge-Balanced Index Game. This game is played on a graph by two players, Alice and Bob, who take turns colouring an uncoloured edge of the graph. Alice plays first and colours edges red, while Bob colours edges blue. The game ends once all the edges have been coloured. A player captures a vertex if more than half of its incident edges are coloured by that player, and the player that captures the most vertices wins.Using classical arguments from the field, we first prove general properties of this game. Namely, we prove that there is no graph in which Bob can win (if Alice plays optimally), while Alice can never capture more than 2 more vertices than Bob (if Bob plays optimally). Through dedicated arguments, we then investigate more specific properties of the game, and focus on its outcome when played in particular graph classes. Specifically, we determine the outcome of the game in paths, cycles, complete bipartite graphs, and Cartesian grids, and give partial results for trees and complete graphs.

Second order local minimal-time mean field games

The paper considers a forward–backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $$\Omega $$ in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon T is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit $$T\rightarrow \infty $$ , and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker–Planck equation on the density of agents and the Hamilton–Jacobi–Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $$\partial \Omega $$ . The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.

The structuralist approach to underdetermination

This paper provides an exposition of the structuralist approach to underdetermination, which aims to resolve the underdetermination of theories by identifying their common theoretical structure. Applications of the structuralist approach can be found in many areas of philosophy. I present a schema of the structuralist approach, which conceptually unifies such applications in different subject matters. It is argued that two classic arguments in the literature, Paul Benacerraf’s argument on natural numbers and W. V. O. Quine’s argument for the indeterminacy of translation, can be analyzed as instances of the structuralist schema. These two applications illustrate different kinds of conclusions that can be drawn through the structuralist approach; Benacerraf’s argument shows that we can derive an ontological conclusion about the given subject matter, while Quine’s structuralist approach leads to a semantic conclusion about how to determine linguistic meanings given radical translation. Then, as a case study, I review a recent debate in metaphysics between Shamik Dasgupta, Jason Turner, and Catharine Diehl to consider the extent to which different instances of the structuralist schema are conceptually unified. Both sides of the debate can be interpreted as utilizing the structuralist approach; one side uses the structuralist approach for an ontological conclusion, while the other side relies on a semantic conclusion. I argue that this has a strong dialectical consequence, which sheds light on the conceptual unity of the structuralist approach.

The actual challenge for the ontological argument

Abstract Many versions of the ontological argument have two shortcomings: First, despite the arguments put forward, for example, by Hugh Chandler and Nathan Salmon, they assume that S5 is the correct modal logic for metaphysical modality. Second, despite the classical arguments put forward, for example, by David Hume and Immanuel Kant or the more recent arguments put forward, for example, by John Findlay and Richard Swinburne, they assume that necessary existence is possible. The aim of the paper is to develop an alternative version of the ontological argument that, by an appeal to the logic of ‘actually’, avoids these two shortcomings. The version helps to leave aside unnecessary debates about the modal logic S5 or about the possibility of necessary existence and to focus on the actual challenge for the debate about the ontological argument: the challenge to evaluate the claim that it is possible that God exists.

Epistemic Modals in Hypothetical Reasoning

Data involving epistemic modals suggest that some classically valid argument forms, such as reductio, are invalid in natural language reasoning as they lead to modal collapses. We adduce further data showing that the classical argument forms governing the existential quantifier are similarly defective, as they lead to a de re–de dicto collapse. We observe a similar problem for disjunction. But if the classical argument forms for negation, disjunction and existential quantification are invalid, what are the correct forms that govern the use of these items? Our diagnosis is that epistemic modals interfere with hypothetical reasoning. We present a modal first-order logic and model theory that characterizes hypothetical reasoning with epistemic modals in a principled manner. One upshot is a sound and complete natural deduction system for reasoning with epistemic modals in first-order logic.