Logic of Arguments, and Arguments of Cases R. P. Loui* Box 1045 Washington University St. Louis, Missouri USA loui@ai.wustl.edu http://www.cs.wustl.edu/~loui/ 314-935-6102, 314-935-7302(FAX) Summary: This paper has two aims. The first aim is to describe the new mathematics of argument to a philosophy of law audience. The second is to show how "the rule of the case" might be understood using the new models of argument. *R. P. Loui, Harvard, B.A., 1982, Rochester M.S. in Computer Science, 1985 and Ph.D. in Philosophy and Computer Science, 1987. Currently Associate Professor, Dept. of Computer Science and Affiliate of the Program on Legal Studies. Some of this text has been taken from a paper in joint preparation with Jeff Norman. I Law is fundamentally concerned with argument, adjudication, process, and the application of rules and cases. Formal work in the philosophy of law has inherited logical and philosophical traditions that were created to serve mathematics. Legal reasoning is different from mathematical reasoning. Formalists have thus had limited success in their illuminations of jurisprudence. The formalization of argument begins with different philosophical foundations. Argument has no obligation to be consistent with mathematical logic. The formalization of argument begins anew, by characterizing the important phenomena directly. It does not start by fettering itself to another field's framework. The new ideas can be enumerated: 1. Rules are defeasible. 2. Arguments are not demonstrative. 3. Argument admits counterargument. 4. Some arguments are better than others. 5. Arguments are produced in a process of deliberation. 6. Adjudication of conflicting arguments justifies a claim. In contrast, logical tradition takes all conditionals to be indefeasible, all proofs to be demonstrative, all inference to be monotonic, all derivations to be valid, all truths to be independent of their proof, and all process to be superfluous. The history of logic contains the proper themes for argument; it just does not include them in the work of this century. Logic and law are legitimate and historical allies. Only a narrow understanding of what logic is would lead someone to the opposite conclusion. All six of the main ideas for argument are contained in H.L.A. Hart's Aristotelian Society paper of 1948. Philosophical logic gave flesh those ideas thirty years later with Nicholas Rescher's book, Dialectics. The mathematical investigation of argument has intensified in this decade because of research in artificial intelligence, under the titles "nonmonotonic reasoning," "default logic," "inheritance," and "defeasible reasoning." Some of the main contributors to the current view include, in the seventies, John Pollock, Jon Doyle, and Ray Reiter; in the eighties, Donald Nute and this author; and in the nineties, Guillermo Simari, Gerard Vreeswijk, Henry Prakken, Tom Gordon, Jaap Hage, Giovanni Sartor, Robert Kowalski, and Bart Verheij. II There are three main structures in a logic of argument. First, arguments have structure. Arguments consist of claims and reasons. Picture an argument as a triangle. At the apex of an argument is its main claim. The rest of the argument supports this main claim. Throughout the argument are more claims used for support, organized into a hierarchy. At the argument's base are the claims that are advanced for which no supporting reasons have been given. If some claims sit under some other claim, then the former provide reason for the latter. This structure is like the resolution-refuation tree of an automated theorem-prover. It is unlike Stephen Toulmin's diagram in several respects. First, the labels of the diagram are symbols. Second, the argument tree does not represent what Toulmin called the backing or warrant. Third, the Toulmin diagram conflates this structure for arguments with the dialectical structure described next. Second, dialectic has structure. There is a main claim under dispute. There are counterarguments, rebuttals, ripostes. These form a disputation tree. Dialectic consists of a collection of disputation trees, any one of which can justify the main claim. Each tree contains arguments, so it is a tree of triangles. Atop each tree is an argument for the main claim. Wherever the tree branches, there is an argument at the point of branching and there are counterarguments at the branches. Depending on what kinds of leaves this tree has, it can be determined whether the argument succeeds in justifying the main claim. Third, dialogue has structure. Dialogue consists of an alternation between pro and con. A Kuno Lorenz diagram (from intuitionist logic) has two columns, with lines beginning at the top and lines constantly being added at the bottom. Each line consists of pro's statement in the first column and con's response in the second column. Suppose this diagram is augmented to permit whole disputation trees on each line, instead of just statements. Then the diagram expresses the dialogue of argumentation. The progression from top to bottom represents the process of argumentation; arguments are introduced one at a time. On each line, pro exhibits a disputation tree that justifies the main claim. On that same line, con extends pro's disputation tree so that it is no longer justifying. If either pro or con cannot adhere to this requirement, then the dialogue ends and the last player to move is the winner. The third structure, dialogue, is a process in time. It is a process either because the search for arguments takes time, or because the presentation of arguments is restricted by protocol. If all of the arguments were available to both players at once, there would be no reason to suppose this third structure. The first and second structures would suffice. Most of the work on the logic of argument is directed at mathematically acceptable definitions of these structures. Elaborate and precise protocols have been defined and implemented as computer programs. Clearly a theory of argument that imposes so much structure cannot be said to model legal disputation directly. However, the model has considerable explanatory power, which I hope to demonstrate in what follows. III Joseph Raz ventured in The Authority of Law to describe the case with a set of letters. We can do better by describing the case as a dialectic, a collection of disputation trees. Raz proposed that A reported decision, P, records that in that case where the facts were a, b, c, d, e, g, the decision was based on the rule that whenever A, B, C then X should be decided. The ruling in P, can be summarized as: [P] a, b, c, d, e, g / A, B, C --> X. The novel case, N, is a case of a1, b1, c1, not-e1, f1, and is thus governed by P ... Raz identified two important patterns of legal reasoning which he described symbolically. The first was distinguishing: The rule laid down in P was when A, B, C then X. Since N is a case of a1, b1, c1 the rule applies to it. But the court has a power to distinguish. It can change the rule into A, B, C, E, then X. The second was analogizing: If the facts in the new case N1 include a1, b1, and c1, there is no room for argument by analogy with P. N1 falls directly under the rule in P ... . Suppose however, that the new case is N2 which is a case of a2, b2, not-c2, d2, not-e2. The ruling in P does not apply to the facts of N2. Yet the two cases are similar ... : both cases of A, B, D. Both of these patterns are well appreciated by formalizers of legal reasoning. Raz's point was that distinction and analogy could not proceed on arbitrary similarities and dissimilarities. Rationales of cases would determine which letters are relevant and which irrelevant. But Raz was unable to provide a formal account. In AI and Law, Kevin Ashley improved the model. Not just any letter could be added to the rule of the case or omitted from it. The fundamental change is this: each letter (each feature) is annotated inherently pro-plaintiff or pro-defendant. For any decision, each feature can be annotated with a + or - indicating that the feature supports the decision or supports the opposite decision. In Raz's example, [P] a+, b+, c-, d+, e-, g- / A, B, C --> X might be the annotation if a, b, and d are inherently X-evincing and c, e, and g are inherently evincing of not-X. With the information thus supplemented, distinction and analogizing can be constrained. A new case which seeks to distinguish, must do so by deleting one of the +-annotated features. Raz's novel case, [N] a1, b1, c1, not-e1, f1 could not be distinguished from P on the grounds of dissimilarity at e, since e is a negatively-annotated feature. According to Ashley, changing P's fact e to not-e can only improve the decision for X. Likewise, the rule of the case, P, can be qualified by an additional +-annotated feature, e.g., d+, or D: A, B, C --> X can become A, B, C, D --> X. Such a transformation does not occur by adding a feature negatively annotated, such as e-, or E. Of N2, Raz's second novel case, [N2] a2, b2, not-c2, d2, not-e2 subsumption under the rule of the case, P, is permitted. A negatively-annotated feature, C, has been removed from the prior case (since N exhibits C and N2 exhibits not-C). If A, B, and C together provide a rule for X, and C is inherently evincing of not-X, then, to Ashley, A and B suffice for X. One weakness of Ashley's analysis is that it assumes the annotation of features to be independent of the context provided by the other co-occurring features. With a formal model of argumentation, a more detailed record of the case can be made. The case does not only record which features (which letters) were relevant to the prior decision, [P] a, b, c, d, e, g / A, B, C --> X and what was the relation between the decision and each feature, [P] a+, b+, c-, d+, e-, g- / A, B, C --> X. The record includes the relevant arguments and their disputation trees. For example, here is a complete record in which four arguments relate to the decision, X, based on facts a, b, c, d, e, and g. Here, the symbol "--<" might be read "because of the reason(s)". [P] X --< A B A --< a B --< b not-X --< C C --< c not-C --< D D --< d not-X --< E G E --< e G --< g. The four arguments can be numbered, Arg1, Arg2, Arg3, and Arg4: [Arg1] X --< A B A --< a B --< b [Arg2] not-X --< C C --< c [Arg3] not-C --< D D --< d [Arg4] not-X --< E G E --< e G --< g. The single disputation tree in this dispute is the structure: Arg3 Arg2 Arg1 Arg4. There is a natural separation of features based on the arguments in which they are used. Some features (some letters) occur in arguments that support the decision; some occur in arguments that were used in opposition; some were used in both. This separation will allow a reconstruction of Ashley's considerations. It will no longer be necessary to assume that a feature is inherently pro-plaintiff or pro-defendant for whole classes of judicial decision-making. IV Our intuition is that an analogy to this case which makes reference to a sharing of the feature D, must also make reference to a sharing of the feature C. The feature D is relevant to the reasoning of the deciding only because D counters the effect of the feature C. Meanwhile, it is too simple to say whether D is relevant or not. It is relevant only insofar as C is relevant. For simplicity, assume as Raz apparently did that arguments have three levels. At the top level is the (putative) conclusion of the argument. At the middle level are the legal abstractions, concepts, or terms which, when conjoined, permit the conclusion (defeasibly of course). At the lowest level are evidentiary claims. We do not now define which analogies to the case are allowed and which are disallowed. Legal interpretation is hard to constrain. Instead, we simply identify three important ways that analogies to cases can be weakened: that is, three ways that the argument of the prior case can be elided. Legal reasoning may permit many kinds of interpretation, and we can sometimes identify the pattern of analogy being used in the interpretation. The best analogy reproduces all of the evidentiary features of the prior case. One kind of weakening of the analogy trades legal abstractions for factual situations. Thus, Raz was moved to write his rules of the case in terms of capital letters, A, and B, instead of a and b. In analogy, facts are mediated by concepts; other fact situations exemplifying the same legal concepts must admit the same argument. There are still, however, some dangers in this kind of weakening: different fact situations present different opportunities for counterargument and might fail to reproduce the same opportunities for reinstatement; more importantly, different fact situations permit Raz-like distinctions as the rule of the precedent case is reformulated. We refer to this kind of analogy-weakening as eliding particulars, or particulars-eliding. Another kind of weakening cuts short the disputation. Instead of finding sufficient similarity to make all of the same arguments that were made in the precedent case, certain lines of argument from the prior case can simply be omitted. The similarities between cases that would have served only to allow these lines of argument can be elided. Attacks that were made unsuccessfully in the prior case just cannot be made in the novel case. We refer to this kind of analogy-weakening as rebuttal-eliding. The third kind of weakening generalizes a single argument by omitting (or weakening) a legal concept from the middle level of some argument. This corresponds to a reinterpretation of a rule used by an argument. It also corresponds to using a different rule and trying to adopt, for the argument with the new rule, the success of the prior argument with the older, more particular rule. We refer to this kind of analogy-weakening as prerequisite-eliding. Each kind of eliding is a potential "rule of the case." In the example, the best analogy can be made with the novel case [N3] a, b, c, d, e, g although it might be debated that the best case should be [N4] a, A, b, B, c, C, d, D, e, E, g, G; where the same facts have entailed the same legal concepts. Particulars-eliding corresponds to dropping, for example, a, in exchange for a2, a different exemplification of A. [N5] a2, A, b, c, d, e, g. The main weakness of this kind of analogy is the opportunity for distinction. If a exemplifies A1 and a2 does not, then the adversary could provide a reinterpretation of argument Arg1: from [Arg1] X --< A B A --< a B --< b to [Arg1a] X --< A B A1 --< a B --< b. If the proper symbolization of the precedent case P really involves argument Arg1a, which might be supported by the language of the opinion, then the novel case of a2 can be distinguished because it fails to exemplify A1. Two lesser problems may occur with particulars-eliding analogies. First, a2 may introduce new opportunities for counterargument, such as not-X --< A2 A2 --< a2. These opportunities presumably can be seized by the adversary, after the particulars-eliding analogy has been made. Second, a2 may not exemplify some other concept which a did, and that concept might be used further down the disputation chain. It might be, for example, that [Arg2a] not-X --< C C --< c d [Arg3a] not-C --< D D --< d a; where Arg3a was the response to counterargument, Arg2a, in the precedent. Arg3a is available to those cases exemplifying a, but not to those exemplifying a2. Thus, an analogy based on a2 (and A, b, c, and d) would be weaker than an analogy based on a (and A, b, c, and d). Rebuttal-eliding corresponds to dropping, for example, all of the facts associated with Arg2 and Arg3: c and d. [N6] a, b, e, g. We require that if the features of some argument are omitted, then so must be the features of the dialectical subtree of that argument (unless the latter features appear somewhere else in the arguing of the case). Ashley would sometimes permit d without c; we would not. Suppose, as Ashley requires, that d has a generally positive effect on deciding X. Then it does so, in the case of N6, through some not yet represented, auxiliary argument, not through a strengthened analogy to P. As for Ashley, N7 is a legitimately weakened analogy to P: [N7] a, b, e because it lacks an essential feature, g, which was used (in Arg4) to attack an argument (Arg1) which supported the decision of the case. Prerequisite-eliding corresponds to dropping, for example, a whole branch of argument Arg1. It is like using [Arg1b] X --< A A --< a instead of [Arg1] X --< A B A --< a B --< b. Thus, the novel case [N8] a, c, d, e, g or even simply [N9] a might be subsumed under P by eliding prerequisites. Usually Arg1b is too weak an analogy, and a different mediating concept is substituted: [Arg1c] X --< A B2 A --< a B2 --< b; where B is a subclass of B2. In such cases, the judicial opinion is being reinterpreted to permit the extension from B to B2. It may well be that the rule in question is seminal with this case, P. Whether it should be rendered A B >-- X or A B2 >-- X is a fair question. The possibility of representing the rule of the case in various ways is a normal part of legal reasoning. V The rule of the case, or the rules of the case, are not as useful as the record of the case. Although a more detailed data structure makes it no longer possible to speak of a single rule of the case, it does contribute to a finer analysis and understanding of the case's rationale. The rationale of the case is exactly the record of disputation: the tree or trees of relevant arguments. The text of the opinion identifies the relevant features, but it also determines which combinations of features make sense for future judgements. Without a theory of argument, we must return to Raz's flat analysis that Raz himself deplored. Ashley was able to augment the formal model of the case and improve the analysis because he was thinking of argumentation. By actually modeling the arguments and their relations, yet better an analysis can be had. The logic of argument is still a new idea. It will certainly take time before the logicians make room for it in their mathematical world. What is equally certain to this author is quite a bit more optimistic. By the time the mathematicians understand the legitimacy of argument, philosophy of law will already have made substantial use of it.