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.     

A natural boundary for solutions to the second order pantograph equation

Pantograph equations are characterized by the presence of a linear functional argument. These equations arise in several applications and often the argument has a repelling fixed point at the origin. Recently, Marshall et al. [J. Math. Anal. Appl. 268 (2002) 157-170] studied a related class of functional differential equations with nonlinear functional arguments and showed that, generically, solutions to such equations have a natural boundary. Their approach uses some well-known properties of the Julia set and relies heavily on the nonlinearity of the functional argument. The method is not directly applicable to pantograph type equations though some of the techniques can be exploited. In this paper we show that solutions to pantograph equations generally have natural boundaries. We focus on a special set of solutions that have the imaginary axis as a natural boundary.     

Non-equilibrium pattern selection in particle sedimentation

In this paper, we explain some well known experimental observations in fluid solid interaction from a thermodynamic perspective. In particular we use the extremum of the rate of entropy production to establish the stability of specific patterns observed in single and multiparticle sedimentation in an infinite fluid and the sedimentation of spheres in the presence of walls. While these phenomena have been explained numerically, there is no known rigorous theoretical argument to establish the stability of the observed configurations. We provide a very convincing theoretical basis using entropy based arguments that are considered by several scientists as the underlying theme of nature, life and evolution. In the absence of many rigorous examples for the entropy production principle, our paper advances this argument and lends it much credibility. In addition to looking at the rate of entropy production, we also put forth a very plausible heuristic argument based on the thermal gradients in the systems being studied, which could be the underlying causal principle for many known patterns in nature.     

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.     

Numerical integration of a class of singular perturbation problems

In this paper, we discuss an approximate method for the numerical integration of a class of linear, singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This method requires a minimum of problem preparation and can be implemented easily on a computer. We replace the original singular perturbation problem by an approximate first-order differential equation with a small deviating argument. Then, we use the trapezoidal formula to obtain the three-term recurrence relationship. Discrete invariant imbedding algorithm is used to solve a tridiagonal algebraic system. The stability of this algorithm is investigated. The proposed method is iterative on the deviating argument. Several numerical experiments have been included to demonstrate the efficiency of the method.The authors wish to express their sincere thanks to Dr. S. M. Roberts for his comments and valuable suggestions.     

A Note about Congruences on Subsemigroups of Groups

A well known elementary argument shows that a totally disconnected normal subgroup of a connected topological group is central, in [2] K. H. Hofmann has shown that by a skillful application of this argument (or variations of it) a much wider class of interesting centrality results can be obtained. In the present paper we offer a modification of the argument also applying to congruences on subsemigroups of topological groups.     

A note about congruences on subsemigroups of groups

A well known elementary argument shows that a totally disconnected normal subgroup of a connected topological group is central, in [2] K. H. Hofmann has shown that by a skillful application of this argument (or variations of it) a much wider class of interesting centrality results can be obtained. In the present paper we offer a modification of the argument also applying to congruences on subsemigroups of topological groups.     

Fate, freedom and contingency

Argument for fatalism attempts to prove that free choice is a logical or conceptual impossibility. The paper argues that the first two premises of the argument are sound: propositions are either true or false and they have their truth-value eternally. But the claim that from the fatalistic premises with the introduction of some innocent further premise dire consequences follow as regards to the possibility of free choice is false. The introduced premise, which establishes the connection between the first two premises (which are about the nature of propositions) and the concept of free choice is not innocent. It creates the impression that the truth of certain propositions can somehow determine the occurrence of certain events. But no proposition can have such an effect since the counterfactuals “If proposition P were true, event E would happen” does not say anything about determination. The argument for fatalism is, however, not a boring sophism. It does reveal something about the nature of propositional representation. It shows that each proposition represents necessarily the fact what it represents, i.e. it shows that propositions have their truth conditions non-contingently. But from this nothing follows as regards to the contingent nature of the facts represented. On the bases of the first two premises of the argument for fatalism we cannot infer to the impossibility of free choice. The argument for fatalism should not be interpreted as an attempt to prove on purely logical or conceptual grounds that we do not have the ability to influence future events by our choices. But it could be used to show something about the nature of propositional representation.     

Continued fractions and the polygamma functions

In a previous study we have shown that the polygamma functions (derivatives of the logarithm of the gamma function) relate to Stieltjes transforms in the square of the argument. These transforms in turn may be converted to Stieltjes continued fractions; in the background is a determined Stieltjes moment problem.In the present study we use the Hamburger form of the Stieltjes integral to produce a set of real monotonic increasing and monotonic decreasing approximants to each of the real and imaginary parts of a polygamma function when the argument is complex. The approximants involve rational fractions which appear to be new.Special attention is given to ln Γ(z) and the psi function.     

Holomorphic solutions to functional differential equations

The pantograph equation is perhaps one of the most heavily studied class of functional differential equations owing to its numerous applications in mathematical physics, biology, and problems arising in industry. This equation is characterized by a linear functional argument. Heard (1973) [10] considered a generalization of this equation that included a nonlinear functional argument. His work focussed on the asymptotic behaviour of solutions for a real variable x as x→∞. In this paper, we revisit Heard's equation, but study it in the complex plane. Using results from complex dynamics we show that any nonconstant solution that is holomorphic at the origin must have the unit circle as a natural boundary. We consider solutions that are holomorphic on the Julia set of the nonlinear argument. We show that the solutions are either constant or have a singularity at the origin. There is a special case of Heard's equation that includes only the derivative and the functional term. For this case we construct solutions to the equation and illustrate the general results using classical complex analysis.     

Pritchard,Revisionism and Warranted Assertability

Against contextualism, Duncan Pritchard has argued that conversational pragmatics give rise to an argument in favour of invariantist neo-Mooreanism. More specifically, he argues that when we conjoin a Moorean view with a warranted assertability manoeuvre, we can satisfy our pre-theoretical intuitions (which are decidedly invariantist), whereas contextualists cannot. In the following paper, I challenge Pritchard’s argument and contend that he is too quick to declare victory for invariantism, for not only does the WAM he employs appear to be ad hoc vis-à-vis DeRose’s plausible criteria, but it also seems to have very implausible pragmatic implications when subjected to close scrutiny.     

Concept nativism and the rule following considerations

In this paper I argue that the most prominent and familiar features of Wittgenstein’s rule following considerations generate a powerful argument for the thesis that most of our concepts are innate, an argument that echoes a Chomskyan poverty of the stimulus argument. This argument has a significance over and above what it tells us about Wittgenstein’s implicit commitments. For, it puts considerable pressure on widely held contemporary views of concept learning, such as the view that we learn concepts by constructing prototypes. This should lead us to abandon our general default hostility to concept nativism and be much more sceptical of claims made on behalf of learning theories.     

Zermelo's discovery of the “Russell Paradox”

In his 1908 paper on the Well-Ordering Theorem, Zermelo claimed to have found “Russell's Paradox” independently of Russell. Here we present a short note, written by E. Husserl in 1902, which contains a detailed exposition of Zermelo's original version of the paradox. We add some comments concerning the date of Zermelo's discovery, the circumstances which caused Husserl to write down Zermelo's argument, and the argument itself.     

How persuaded are you? A typology of responses

Several recent studies have suggested that there are two different ways in which a person can proceed when assessing the persuasiveness of a mathematical argument: by evaluating whether it is personally convincing, or by evaluating whether it is publicly acceptable. In this paper, using Toulmin's (1958) argumentation scheme, we produce a more detailed theoretical classification of the ways in which participants can interpret a request to assess the persuasiveness of an argument. We suggest that there are (at least) five ways in which such a question can be interpreted. The classification is illustrated with data from a study that asked undergraduate students and research-active mathematicians to rate how persuasive they found a given argument. We conclude by arguing that researchers interested in mathematical conviction and proof validation need to be aware of the different ways in which participants can interpret questions about the persuasiveness of arguments, and that they must carefully control for these variations during their studies.     

On the stieltjes procedure for closing gaps in time scales

In the present paper, we show that the theory of dynamic equations on time scales (with an argument with “holes,” i.e., a discontinuous argument) can significantly be simplified and generalized by using Stieltjes integration, which is inverse to the differentiation with respect to Riesz measures.     

Modelling Inference in Argumentation through Labelled Deduction: Formalization and Logical Properties

Artificial Intelligence (AI) has long dealt with the issue of finding a suitable formalization for commonsense reasoning. Defeasible argumentation has proven to be a successful approach in many respects, proving to be a confluence point for many alternative logical frameworks. Different formalisms have been developed, most of them sharing the common notions of argument and warrant. In defeasible argumentation, an argument is a tentative (defeasible) proof for reaching a conclusion. An argument is warranted when it ultimately prevails over other conflicting arguments. In this context, defeasible consequence relationships for modelling argument and warrant as well as their logical properties have gained particular attention. This article analyzes two non-monotonic inference operators C_arg and C_war intended for modelling argument construction and dialectical analysis (warrant), respectively. As a basis for such analysis we will use the LDS_ar framework, a unifying approach to computational models of argument using Labelled Deductive Systems (LDS). In the context of this logical framework, we show how labels can be used to represent arguments as well as argument trees, facilitating the definition and study of non-monotonic inference operators, whose associated logical properties are studied and contrasted. We contend that this analysis provides useful comparison criteria that can be extended and applied to other argumentation frameworks. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03B42.     

Optimization of dialectical outcomes in dialogical argumentation

When informal arguments are presented, there may be imprecision in the language used, and so the audience may be uncertain as to the structure of the argument graph as intended by the presenter of the arguments. For a presenter of arguments, it is useful to know the audience's argument graph, but the presenter may be uncertain as to the structure of it. To model the uncertainty as to the structure of the argument graph in situations such as these, we can use probabilistic argument graphs. The set of subgraphs of an argument graph is a sample space. A probability value is assigned to each subgraph such that the sum is 1, thereby reflecting the uncertainty over which is the actual subgraph. We can then determine the probability that a particular set of arguments is included or excluded from an extension according to a particular Dung semantics. We represent and reason with extensions from a graph and from its subgraphs, using a logic of dialectical outcomes that we present. We harness this to define the notion of an argumentation lottery, which can be used by the audience to determine the expected utility of a debate, and can be used by the presenter to decide which arguments to present by choosing those that maximize expected utility. We investigate some of the options for using argumentation lotteries, and provide a computational evaluation.     

A note on the ‘underadjustment phenomenon’

In recent work in the area of employment discrimination statistics we have noticed that a basic formula quantifying the degree of underadjustment bias in regression coefficients seems not to have been clearly stated and proved. We believe the underadjustment phenomenon to be of great importance in assessing the validity of multiple regression studies of wage disparities in equal employment litigation, and of observational studies involving two populations in general. Since the argument is based entirely on some simple mathematical features of the linear model, the result should be of interest in other fields of application as well.     

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.     

An elementary analysis of a procedure for sampling points in a convex body

In this paper we describe a new method for proving the polynomial-time convergence of an algorithm for sampling (almost) uniformly at random from a convex body in high dimension. Previous approaches have been based on estimating conductance via isoperimetric inequalities. We show that a more elementary coupling argument can be used to give a similar result. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 213–235, 1998