Primitive notions

Definitions – avoiding circularity

Some elementary observations have profound corollaries. Here is an example. Suppose finitely many points are distributed in some space and each point is joined to a number of other points by arrows, forming a complex directed network. We choose a point at random and trace a path, following the direction of the arrows, spoilt for choice at each turn. No matter how skillfully we traverse the network, and no matter how large the network is, we are forced at some stage to return to a point we have already visited. Every road eventually becomes part of a loop, in fact many loops.

A dictionary is a familiar example of such a network. Represent each word by a point and connect it via outwardly pointing arrows to each of the words used in its definition. We see that a dictionary is a dense minefield of circular definitions. In practice it is desirable to make these loops as large as possible, but this is a tactic knowingly founded on denial. The union of all such loops forms the core of the artificial language world of the dictionary, every word there in definable in terms of the loop members.

Cartesian products and relations

Functions and relations

The clarification of the concept of function, even when confined to the subject of analysis, had become a matter of desperate importance in the eighteenth and nineteenth centuries. Fourier's representation of piecewise continuous functions [a, b] → ℝ by trigonometric series and, later, Weierstrass and Riemann's examples of continuous nowhere differentiable functions and functions which are nowhere continuous yet integrable forced a generation of mathematicians (who were accustomed to treating functions of real variables as visualizable geometric entities) to reluctantly revise their understanding. A precise and general definition of function which is not tied to analysis is expressible within the set-theoretic framework.

Functions and relations saturate every part of mathematics. Even in our humble introduction we had to introduce these ideas at an early stage in order to construct models of the basic number systems. Now that the logical prerequisites have been brought into focus it is worth reviewing the formal definitions.

Russell's paradox and some of its relatives

The consequences of the Axiom of Abstraction

In general, a logical paradox is a contradiction, usually expressed in its simplest form Φ ↔ ¬Φ, which reveals a theory to be inconsistent, even though the axioms of the theory seem to be plausible and the rules of inference appear to be valid. The emphasis is on the surprise of the contradiction, the paradox arising unexpectedly in what is intuitively a perfectly sound system. Zermelo, quite early on, suggested the more precise term ‘antinomy’ to describe the phenomenon of contradictions which arise within logical systems, demoting the description ‘paradox’ to those results which are merely counterintuitive but not contradictory.

Contradictions which arise in apparently sensible systems are harsh reminders of our poor intuition regarding large abstract structures. Humans, unfortunately, are very good at cherry-picking attractive sounding ideas without questioning their total lack of empirical foundation and, more to the point, without examining their logical consequences. Even extremely convincing systems can have a sting in the tail, Frege's set theory being an important example. Contradictory systems can be embraced by an individual in the full course of a lifetime without any signs of nausea simply because the system-bearer has not lived long enough, or has not analyzed the system closely enough, to reveal the contradiction.

In the later years of the nineteenth century Georg Cantor discovered that there are different sizes of infinity. What had begun as the study of a concrete problem concerning the convergence of trigonometric series had turned into something far more profound. Further investigation led Cantor to a new theory of the infinite and the mathematical community stood with a mixture of bewilderment and disbelief before an unfamiliar universe. This theory was refined and developed, and continues to this day as axiomatic set theory. The theory of sets is a body of work that I feel should be known to a much wider audience than the mathematicians and mathematical philosophers who hold it in such high esteem.

This book is about the concepts which lie at the logical foundations of mathematics, including the rigorous notions of infinity born in the ground-breaking work of Cantor. It has been my intention to make the heroes of the book the ideas themselves rather than the multitude of mathematicians who developed them, so the treatment is deliberately sparse on biographical information, and on lengthy technical arguments. The aim is to convey the big ideas, to advertise the theory, leaving the enthused to seek further details in what is now a vast literature.

In the previous chapters we have discussed how to represent the operation of critical questions in a formal and computational model that can incorporate argumentation schemes as well as their accompanying critical questions. In order to illustrate how this works the example of the scheme for argument from expert opinion has been used. The problem is to classify the critical questions as assumptions or exceptions in order to properly reflect the distribution of the burden of proof between the party who put forward the argument and the other party, the respondent who is raising critical questions about the argument. Is this problem merely a technical problem of how to model argumentation by the use of defeasible argumentation schemes? Or is it a problem that could arise in a real case of argumentation? In Chapter 4, a legal case concerning how to logically represent critical questions appropriate for argument from witness testimony is studied that illustrates the problem of how to arrive at a decision to properly assign a burden of proof to the one side or the other.

In this case, the Oregon Supreme Court overturned the previous procedures for determining the admissibility of eyewitness identification evidence. The decision to change the law was based on recent research in the social sciences concerning the reliability of eyewitness identification, and by considerations put to the court by the Innocence Network, an organization dedicated to the study of unjust convictions. In some cases it can be quite difficult for the courts to make a decision on burden of proof, and in some of these cases a ruling is made that can act as a precedent when the same kind of decision about burden of proof arises in a comparable case. In Chapter 4, a more challenging kind of case is studied in which a change was made in the normal way of dealing with burden of proof in criminal trials. This change was prompted by a gradually growing body of scientific evidence suggesting that witness testimony evidence is much more fallible in certain respects than was previously thought.

In law, there is a fundamental distinction between two main types of burden of proof (Prakken and Sartor, 2009). One is the setting of the global burden of proof before the trial begins, which is called the burden of persuasion. It does not change during the argumentation stage, and it is the device used to determine which side has won at the closing stage. The other is the local setting of burden of proof at the argumentation stage, often called the burden of production (or the evidential burden, or the burden of going forward with evidence) in law. This burden can shift back and forth as the argumentation proceeds. For example, if one side puts forward a strong argument, the other side must meet the local burden to respond to that argument by criticizing or presenting a counterargument, or otherwise the strong argument will hold, and it will fulfill the burden of persuasion of its proponent unless the respondent puts forward an equally strong objection or counterargument. Otherwise the respondent will lose the trial at that point, and the judge can declare that the trial is over.

According to Williams (2003, 166), considerable confusion has arisen from a failure to distinguish between two distinct kinds of burdens of proof, especially by appeal courts who discuss questions of burden of proof without making it clear whether they are talking about burden of persuasion or evidential burden. Recent ground-breaking work in AI shows great promise for helping law to work toward a more systematic conceptual grasp of the notion of burden of proof by seeing how to model it in a precise way.

In his book on fallacies, Hamblin (1970) built a simple system for argumentation in dialogue he called the Why-Because System with Questions. In his discussion of this system, he replaced the concept of burden of proof with a simpler concept of initiative, which could be described as something like getting the upper hand as the argumentation moves back and forth in the dialogue between the one party and the other. No doubt he realized that the concept of burden of proof was too complex a matter to be dealt with in the limited scope of his chapter on formal dialogue systems. In this chapter, it is shown how an extended version of Hamblin’s dialogue system provides a nice way of modeling the phenomenon of shifting of burden of proof in a dialogue, yielding with a precise way of distinguishing between different kinds of burden of proof, and dealing with fallacies like the argumentum ad ignorantiam (argument from negative evidence).

Over forty years has passed since the publication of Hamblin’s book Fallacies (1970), and there has been much written on the subject of argumentation since that time. One might think that such a book would have long ago ceased to have much value in contributing to the latest research. Such is not the case, however, especially with regard to Hamblin’s remarkably innovative Chapter 8 on formal dialogue systems, a chapter that provided the basis for much subsequent work. To give an example of a formal dialogue system of the kind he recommended in Chapter 8, he built a Why-Because System with Questions. A leading feature of this system is that it has a speech act representing a move in a dialogue in which one party asks the other party to prove, or give an argument to support a claim made by the first party. The Hamblin system has several rules for managing dialogues in which such support request questions are asked, and need to be responded to. It is shown in Chapter 5 how these rules are fundamentally important in attempting to build any formal dialogue system designed to be a framework modeling the operation of burden of proof in rational argumentation.

The notions of burden of proof and presumption are central to law, but as we noted in Chapter 1, they are also said to be the slipperiest of any of the family of legal terms employed in legal reasoning. However, as shown in Chapter 2, recent studies of burden of proof and presumption (Prakken, Reed and Walton, 2005; Prakken and Sartor, 2006; Gordon, Prakken and Walton, 2007; Prakken and Sartor, 2007) offer formal models that can render them into precise tools useful for legal reasoning. In this chapter, the various theories and formal models are comparatively evaluated with the aim of working out a more comprehensive theory that can integrate the components of the argumentation structure on which they are based. It is shown that the notion of presumption has both a logical component and a dialectical component, and the new theory of presumption developed in the chapter, called the dialogical theory, combines these two components. Thus, the aim of Chapter 3 is to build on the clarification of the notion of burden of proof achieved in Chapter 2, and to move forward to show how presumption is related to burden of proof. By this means, the goal is to achieve a better theory of presumption.

According to Ashford and Risinger (1969) there is no agreement among legal writers on the questions of exactly what a presumption is and how presumptions operate. However, they think that there is some general agreement on at least a minimal account of what a presumption is: “Most are agreed that a presumption is a legal mechanism which, unless sufficient evidence is introduced to render the presumption inoperative, deems one fact to be true when the truth of another fact has been established” (165). According to legal terminology, the fact to be proved is called “the fact presumed,” and the fact to be established before this other fact is to be deemed true is called “the fact proved” (Ashford and Risinger, 1969). The analysis of presumption put forward in this chapter takes this minimal account as its basic structure.

Erik Krabbe’s pioneering article on metadialogues (dialogues about dialogues) opened up an important new avenue of research in the field, largely unexplored up to that point. His modest conclusion was that it was too early for conclusions (Krabbe, 2003, 89). Even so, by posing a number of problems along with tentative solutions, his article was a very important advance in the field. Hamblin (1970) was the first to suggest the usefulness of metadialogues in the study of fallacies. He proposed (1970, 283–284) that disputes that can arise about allegations that the fallacy of equivocation has been committed could be resolved by redirecting the dispute to a procedural level. This procedural level would correspond to what Krabbe calls a metadialogue (Krabbe, 2003). Other writers on argumentation (Mackenzie, 1979, 1981; Finocchiaro, 1980, 2005; van Eemeren and Grootendorst, 1992), as noted by Krabbe (2003, 86–87) have tacitly recognized the need to move to a metalevel dialogue framework, but none provided a metadialogue system. The study of metadialogues is turning out to be very important in argumentation theory and in computer science (Wooldridge, McBurney and Parsons, 2005).

In Chapter 6, it is shown how analyzing disputes about burden of proof is an important research topic for investigation in the field of metadialogue theory. It has recently been shown (Prakken, Reed and Walton, 2005) that legal disputes about burden of proof can be formally modeled by using the device of a formal dialogue protocol for embedding a metadialogue about the issue of burden of proof into an ongoing dialogue about some prior issue. In Chapter 8, a general solution to the problem of how to analyze burden of proof is yielded by building on this framework, using three key examples from Chapter 1 to show how disputes about burden of proof can arise. These three examples were presented in Chapter 1 as classic cases of burden of proof disputes, and now in Chapter 6 it is shown how current tools from argumentation theory and artificial intelligence based on metadialogues can be applied to the problems they pose.