Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained.     

Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 62–68, January, 1998.     

Human Cognitive Closure and Mysterianism: Reply to Kriegel

In this paper, I respond to Kriegel’s criticism of McGinn’s mysterianism (the thesis that humans are cognitively closed with respect to the solution of the mind-body problem). Kriegel objects to a particular argument for the possibility of human cognitive closure and also gives a direct argument against mysterianism. I intend to show that neither the objection nor the argument is convincing.     

A probabilistic approach to modelling uncertain logical arguments

Argumentation can be modelled at an abstract level using a directed graph where each node denotes an argument and each arc denotes an attack by one argument on another. Since arguments are often uncertain, it can be useful to quantify the uncertainty associated with each argument. Recently, there have been proposals to extend abstract argumentation to take this uncertainty into account. This assigns a probability value for each argument that represents the degree to which the argument is believed to hold, and this is then used to generate a probability distribution over the full subgraphs of the argument graph, which in turn can be used to determine the probability that a set of arguments is admissible or an extension. In order to more fully understand uncertainty in argumentation, in this paper, we extend this idea by considering logic-based argumentation with uncertain arguments. This is based on a probability distribution over models of the language, which can then be used to give a probability distribution over arguments that are constructed using classical logic. We show how this formalization of uncertainty of logical arguments relates to uncertainty of abstract arguments, and we consider a number of interesting classes of probability assignments.     

Natural Boundaries for Solutions to a Certain Class of Functional Differential Equations

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation. The pantograph equation contains a linear functional argument. In this paper we generalize this functional argument to include nonlinear polynomials. In contrast to the entire solutions generated by the pantograph equation for the retarded case, we show that in the nonlinear case natural boundaries occur. These boundaries are linked to the Julia set of the polynomial functional argument.     

Parametrization of the Cauchy problem for systems of ordinary differential equations with limiting singular points

The method of solution continuation with respect to a parameter is used to solve an initial value problem for a system of ordinary differential equations with several limiting singular points. The solution is continued using an argument (called the best) measured along the integral curve of the problem. Additionally, a modified argument is introduced that is locally equivalent to the best one in the considered domain. Theoretical results are obtained concerning the conditioning of the Cauchy problem parametrized by the modified argument in a neighborhood of each point of its integral curve.     

An Old Argument Against Co-location

I defend an old argument against co-location—the view that human animals are distinct from, but co-located with human persons. The argument is drawn from St. Thomas Aquinas. In order to respond to the argument, co-locationists have to endorse at least one of a trio of claims, none of which is obviously correct. Further, two of the options do not seem to be the sort of positions that should be flowing out of the acceptance of a general metaphysical position. I conclude that co-locationism is more costly than generally thought.     

On the drawbacks of large committees

This paper aims at identifying possible drawbacks of large committees. I present a model of a committee of experts with career concerns where each expert may observe an argument about the state of the world but be unsure about the argument’s soundness. Experts may remain silent or decide to be the first to announce an argument. In contrast with the existing literature, I focus on the possibility that experts may discover duplicate information. I show that experts become more reluctant to speak in larger committees as a result. This effect can be sufficiently strong to make small groups of experts optimal.     

On the Coherence of Wittgensteinian Constructivism

Michael Dummett presents a modus tollens argument against a Wittgensteinian conception of meaning. In a series of papers, Dummett claims that Wittgensteinian considerations entail strict finitism. However, by a “sorites argument”, Dummett argues that strict finitism is incoherent and therefore questions these Wittgensteinian considerations.In this paper, I will argue that Dummett’s sorites argument fails to undermine strict finitism. I will claim that the argument is based on two questionable assumptions regarding some strict finitist sets of natural numbers. It will be shown that strict finitism entails none of these assumptions. Hence, the argument does not prove that the view is internally incoherent, and consequently, Dummett fails to undermine the Wittgensteinian conception of meaning.     

Triangles, Schemes and Worlds: Reply to Nulty

Nulty proposes a “Davidsonian” argument for metaphysical pluralism, the thesis that there are (or could be) many actual worlds, which appeals to the possibility of alien forms of triangulation. I dispute Nulty’s reading of Davidson on two important points: Davidson’s attack on the notion of a conceptual scheme is not, as Nulty thinks, directed at pluralism, and his understanding of the notions of objective truth and reality is at odds with the conception needed for Nulty’s argument. I also show that the pluralist argument fails on its own terms as it requires an assimilation of worlds to worldviews. But there is much of value in Nulty’s paper despite these important flaws. When the confusions are cleared up, we are left with an intriguing and novel line of argument for conceptual relativism.     

Probabilistic qualification of attack in abstract argumentation

An argument graph is a graph where each node denotes an argument, and each arc denotes an attack by one argument on another. It offers a valuable starting point for theoretical analysis of argumentation following the proposals by Dung. However, the definition of an argument graph does not take into account the belief in the attacks. In particular, when constructing an argument graph from informal arguments, where each argument is described in free text, it is often evident that there is uncertainty about whether some of the attacks hold. This might be because there is some expressed doubt that an attack holds or because there is some imprecision in the language used in the arguments. In this paper, we use the set of spanning subgraphs of an argument graph as a sample space. A spanning subgraph contains all the arguments, and a subset of the attacks, of the argument graph. We assign a probability value to each spanning subgraph such that the sum of the assignments is 1. This means we can reflect the uncertainty over which is the actual subgraph using this probability distribution. Using the probability distribution over subgraphs, we can then determine the probability that a set of arguments is admissible or an extension. We can also obtain the probability of an attack relationship in the original argument graph as a marginal distribution (i.e. it is the sum of the probability assigned to each subgraph containing that attack relationship). We investigate some of the features of this proposal, and we consider the utility of our framework for capturing some practical argumentation scenarios.     

The Argument from Silence

The argument from silence is a pattern of reasoning in which the failure of a known source to mention a particular fact or event is used as the ground of an inference, usually to the conclusion that the supposed fact is untrue or the supposed event did not actually happen. Such arguments are widely used in historical work, but they are also widely contested. This paper surveys some inadequate attempts to model this sort of argument, offers a new analysis using a Bayesian probabilistic framework that isolates the most problematic step in such arguments, illustrates a key problem besetting many uses of the argument, diagnoses the attraction of the argument in terms of a known human cognitive bias affecting the critical step, and suggests a standard that must be met in order for any argument from silence to have more than a very weak influence on historical reasoning.     

Percolation on dual lattices with k‐fold symmetry

Zhang found a simple, elegant argument deducing the nonexistence of an infinite open cluster in certain lattice percolation models (for example, p = 1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only two‐fold (or three‐fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's argument, our extension applies in many contexts; in particular, it enables us to answer a question of Grimmett concerning the anisotropic random cluster model on the triangular lattice. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.     

Oscillatory properties of first order linear functional differential equations

Oscillatory properties of retarded and advanced functional differential equations are investigated.In the first part, the study concerns equations with piecewise constant arguments, which found applications in certain biomedical problems. Then, results of some authors are generalized for general equations with many argument deviations. Finally, applications are given to equations with linear transformations of the argument.     

A Humean Argument for Personal Identity

Considering various arguments in Hume’s Treatise, I reconstruct a Humean argument against personal identity or unity. According to this argument, each distinct perception is separable from the bundle of perceptions to which it belongs and is thus transferable either to the external, material reality or to another psychical reality, another bundle of perceptions. Nevertheless, such transference (Hume’s word!) is entirely illegitimate, otherwise Hume’s argument against causal inference would have failed; furthermore, it violates private, psychical accessibility. I suggest a Humean thought experiment clearly demonstrating that, to the extent that anything within a psychical reality is concerned, no distinction leads to separation or transference and that private, psychical accessibility has to be allowed in the Humean argument for personal identity or unity. Private accessibility and psychical untransferability secure personal identity and unity. Referring to the phenomenon of multiple personality along the lines of the Humean argument for personal identity or unity, I illustrate both private accessibility and a possible notion of one and the same person distinct from his/her alters or psychical parts. Finally, I show why Parfit’s Humean argument against personal identity must fail.     

Reply to Nathan: How to reconstruct the causal argument

Nicholas Nathan tries to resist the current version of the causal argument for sense-data in two ways. First he suggests that, on what he considers to be the correct re-construction of the argument, it equivocates on the sense of proximate cause. Second he defends a form of disjunctivism, by claiming that there might be an extra mechanism involved in producing veridical hallucination, that is not present in perception. I argue that Nathan’s reconstruction of the argument is not the appropriate one, and that, properly interpreted, the argument does not equivocate on proximate cause. Furthermore, I claim that his postulation of a modified mechanism for hallucinations is implausibly ad hoc.     

About a numerical method of successive interpolations for two-point boundary value problems with deviating argument

A new numerical method for two-point boundary value problems with deviating argument is obtained. The method uses the fixed point technique, the trapezoidal quadrature rule, and a Birkhoff interpolation procedure. The convergence of the method is proved without smoothness conditions, the kernel function being only Lipschitzian in each argument. The interpolation procedure is used only on the points where the argument is modified. A stopping criterion of the algorithm is obtained and the accuracy of the method is illustrated on four numerical examples of pantograph type.     

Zur existenz periodischer losungen von pseudodifferential gleichungen mit nacheilendem argument

In this paper an existence theorem for periodic solutions of pseudo-differential equations with delayed argument is proved by using an algebra of pseudo-differential operators. The existence of periodic solutions of the heat equation with delayed argument is included.     

Mathematics Majors' Perceptions of Conviction,Validity, and Proof

In this paper, 28 mathematics majors who completed a transition-to-proof course were given 10 mathematical arguments. For each argument, they were asked to judge how convincing they found the argument and whether they thought the argument constituted a mathematical proof. The key findings from this data were (a) most participants did not find the empirical argument in the study to be convincing or to meet the standards of proof, (b) the majority of participants found a diagrammatic argument to be both convincing and a proof, (c) participants evaluated deductive arguments not by their form but by their content, but (d) participants often judged invalid deductive arguments to be convincing proofs because they did not recognize their logical flaws. These findings suggest improving undergraduates' comprehension of mathematical arguments does not depend on making undergraduates aware of the limitations of empirical arguments but instead on improving the ways in which they process the arguments that they read.