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Summary "Formalism about Legal Reasoning" refers to the use of (often logic-based) formalisms to shed light on legal reasoning.  "Formal Models of Legal Reasoning," in contrast refers to the use of such formalisms to model actual legal reasoning.   Some works fit into both categories.  The former is also apt for discussions of the limits of the latter.  Both categories apply best to modeling or shedding light on judicial reasoning, or on the analysis of legal texts (be they statutes, constitutions [written or not], regulations, or exegeses of these), but are  less applicable to modeling or shedding light on the legislative or regulatory processes which produce these.
Key works Gardner 1987 is a general key work, while Gordon 1995 introduces dialogical models and dialogue logic, more generally, and discusses the issue of epistemic logic in formal models of law.  The seminal book Toulmin 2003 indirectly contributed far more to formal modeling using logic than the author intended, because for each objection Toulmin raised to the use of formal logic as a model, subsequent authors came along and found means of answering them within logic(s).
Introductions Hamfelt 1995 is a wonderful introduction to formal modeling of law, using classical logic as a framework.  The issue of epistemic logic in formal models of law is introduced extensively in Hage 2003.
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  1. M. Abraham, D. M. Gabbay & U. Schild (2011). Obligations and Prohibitions in Talmudic Deontic Logic. Artificial Intelligence and Law 19 (2-3):117-148.
    This paper examines the deontic logic of the Talmud. We shall find, by looking at examples, that at first approximation we need deontic logic with several connectives: O T A Talmudic obligation F T A Talmudic prohibition F D A Standard deontic prohibition O D A Standard deontic obligation. In classical logic one would have expected that deontic obligation O D is definable by $O_DA \equiv F_D\neg A$ and that O T and F T are connected by $O_TA \equiv F_T\neg (...)
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  2. Carlos Alarcon-Cabrera (1998). Von Wright's Deontic Logics and "Contrary-to-Duty Imperatives&Quot;. Ratio Juris 11 (1):67-79.
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  3. Carlos E. Alchourron (1996). On Law and Logic. Ratio Juris 9 (4):331-348.
  4. Robert Alexy (2000). Henry Prakken (1997), Logical Tools for Modelling Legal Argument. A Study of Defeasible Reasoning in Law. Argumentation 14 (1):65-72.
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  5. Katie Atkinson (2012). Introduction to Special Issue on Modelling Popov V. Hayashi. Artificial Intelligence and Law 20 (1):1-14.
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  6. Katie Atkinson (2008). Introduction to Special Issue on Modelling Legal Cases. Artificial Intelligence and Law 16 (4):329-331.
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  7. Katie Atkinson & Trevor J. M. Bench-Capon (2007). Practical Reasoning as Presumptive Argumentation Using Action-Based Alternating Transition Systems. Artificial Intelligence 171 (10-15):855-874.
    In this paper we describe an approach to practical reasoning, reasoning about what it is best for a particular agent to do in a given situation, based on presumptive justifications of action through the instantiation of an argument scheme, which is then subject to examination through a series of critical questions. We identify three particular aspects of practical reasoning which distinguish it from theoretical reasoning. We next provide an argument scheme and an associated set of critical questions which is able (...)
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  8. T. J. M. Bench-Capon (2012). Representing Popov V Hayashi with Dimensions and Factors. Artificial Intelligence and Law 20 (1):15-35.
    Modelling reasoning with legal cases has been a central concern of AI and Law since the 1980s. The approach which represents cases as factors and dimensions has been a central part of that work. In this paper I consider how several varieties of the approach can be applied to the interesting case of Popov v Hayashi. After briefly reviewing some of the key landmarks of the approach, the case is represented in terms of factors and dimensions, and further explored using (...)
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  9. T. J. M. Bench-Capon, T. Geldard & P. H. Leng (2000). A Method for the Computational Modelling of Dialectical Argument with Dialogue Games. Artificial Intelligence and Law 8 (2-3):233-254.
    In this paper we describe a method for the specification of computationalmodels of argument using dialogue games. The method, which consists ofsupplying a set of semantic definitions for the performatives making upthe game, together with a state transition diagram, is described in full.Its use is illustrated by some examples of varying complexity, includingtwo complete specifications of particular dialogue games, Mackenzie's DC,and the authors' own TDG. The latter is also illustrated by a fully workedexample illustrating all the features of the game.
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  10. Trevor Bench-Capon (2004). Book Review: Bram Roth, Case-Based Reasoning in the Law: A Formal Theory of Reasoning by Case Comparison. Ph. D. Thesis, the University of Maastricht, 2003. 181 Pp. [REVIEW] Artificial Intelligence and Law 12 (3):227-229.
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  11. Trevor Bench-Capon (1997). Argument in Artificial Intelligence and Law. Artificial Intelligence and Law 5 (4):249-261.
    In this paper I shall discuss the notion of argument, and the importanceof argument in AI and Law. I shall distinguish four areas where argument hasbeen applied: in modelling legal reasoning based on cases; in thepresentation and explanation of results from a rule based legal informationsystem; in the resolution of normative conflict and problems ofnon-monotonicity; and as a basis for dialogue games to support the modellingof the process of argument. The study of argument is held to offer prospectsof real progress (...)
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  12. Trevor J. M. Bench-Capon & Giovanni Sartor (2003). A Model of Legal Reasoning with Cases Incorporating Theories and Values. Artificial Intelligence 150 (1-2):97-143.
    Reasoning with cases has been a primary focus of those working in AI and law who have attempted to model legal reasoning. In this paper we put forward a formal model of reasoning with cases which captures many of the insights from that previous work. We begin by stating our view of reasoning with cases as a process of constructing, evaluating and applying a theory. Central to our model is a view of the relationship between cases, rules based on cases, (...)
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  13. Trevor Bench-Capon & Henry Prakken (2010). Using Argument Schemes for Hypothetical Reasoning in Law. Artificial Intelligence and Law 18 (2):153-174.
    This paper studies the use of hypothetical and value-based reasoning in US Supreme-Court cases concerning the United States Fourth Amendment. Drawing upon formal AI & Law models of legal argument a semi-formal reconstruction is given of parts of the Carney case, which has been studied previously in AI & law research on case-based reasoning. As part of the reconstruction, a semi-formal proposal is made for extending the formal AI & Law models with forms of metalevel reasoning in several argument schemes. (...)
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  14. Floris Bex, Henry Prakken, Chris Reed & Douglas Walton (2003). Towards a Formal Account of Reasoning About Evidence: Argumentation Schemes and Generalisations. [REVIEW] Artificial Intelligence and Law 11 (2-3):125-165.
    This paper studies the modelling of legal reasoning about evidence within general theories of defeasible reasoning and argumentation. In particular, Wigmore's method for charting evidence and its use by modern legal evidence scholars is studied in order to give a formal underpinning in terms of logics for defeasible argumentation. Two notions turn out to be crucial, viz. argumentation schemes and empirical generalisations.
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  15. L. Karl Branting (1993). A Computational Model of Ratio Decidendi. Artificial Intelligence and Law 2 (1):1-31.
    This paper proposes a model ofratio decidendi as a justification structure consisting of a series of reasoning steps, some of which relate abstract predicates to other abstract predicates and some of which relate abstract predicates to specific facts. This model satisfies an important set of characteristics ofratio decidendi identified from the jurisprudential literature. In particular, the model shows how the theory under which a case is decided controls its precedential effect. By contrast, a purely exemplar-based model ofratio decidendi fails to (...)
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  16. Eugenio Bulygin (2008). What Can One Expect From Logic in the Law? (Not Everything, but More Than Something: A Reply to Susan Haack). Ratio Juris 21 (1):150-156.
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  17. Eugenio Bulygin (2003). Review of Jaap Hage's Law and Defeasibility. [REVIEW] Artificial Intelligence and Law 11 (2-3):245-250.
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  18. Michael Clark (1997). Review of P. Wahlgren, Automation of Legal Reasoning. [REVIEW] Information and Communications Technology Law 6.
  19. Michael Clark (1992). Review of Kevin Ashley, Modelling Legal Argument. [REVIEW] Law, Computing and Artificial Intelligence 1 (1).
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  20. Stefania Costantini & Gaetano Aurelio Lanzarone (1995). Explanation-Based Interpretation of Open-Textured Concepts in Logical Models of Legislation. Artificial Intelligence and Law 3 (3):191-208.
    In this paper we discuss a view of the Machine Learning technique called Explanation-Based Learning (EBL) or Explanation-Based Generalization (EBG) as a process for the interpretation of vague concepts in logic-based models of law.The open-textured nature of legal terms is a well-known open problem in the building of knowledge-based legal systems. EBG is a technique which creates generalizations of given examples on the basis of background domain knowledge. We relate these two topics by considering EBG''s domain knowledge as corresponding to (...)
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  21. Phan Minh Dung & Phan Minh Thang (2009). Modular Argumentation for Modelling Legal Doctrines in Common Law of Contract. Artificial Intelligence and Law 17 (3):167-182.
    To create a programming environment for contract dispute resolution, we propose an extension of assumption-based argumentation into modular assumption-based argumentation in which different modules of argumentation representing different knowledge bases for reasoning about beliefs and facts and for representation and reasoning with the legal doctrines could be built and assembled together. A distinct novel feature of modular argumentation in compare with other modular logic-based systems like Prolog is that it allows references to different semantics in the same module at the (...)
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  22. Paul E. Dunne (2005). A Value-Based Argument Model of Convention Degradation. Artificial Intelligence and Law 13 (1):153-188.
    The analysis of how social conventions emerge and become established is rightly viewed as a significant study of great relevance to models of legal and social systems. Such conventions, however, do not operate in a monotonic fashion, i.e. the fact that a convention is recognised and complied with at some instant is no guarantee it will continue to be so indefinitely. In total rules and protocols may evolve, with or without the consent of individual members of the society, even to (...)
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  23. Norman Fenton, Martin Neil & David A. Lagnado (2013). A General Structure for Legal Arguments About Evidence Using Bayesian Networks. Cognitive Science 37 (1):61-102.
    A Bayesian network (BN) is a graphical model of uncertainty that is especially well suited to legal arguments. It enables us to visualize and model dependencies between different hypotheses and pieces of evidence and to calculate the revised probability beliefs about all uncertain factors when any piece of new evidence is presented. Although BNs have been widely discussed and recently used in the context of legal arguments, there is no systematic, repeatable method for modeling legal arguments as BNs. Hence, where (...)
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  24. Kathleen Freeman & Arthur M. Farley (1996). A Model of Argumentation and its Application to Legal Reasoning. Artificial Intelligence and Law 4 (3-4):163-197.
    We present a computational model of dialectical argumentation that could serve as a basis for legal reasoning. The legal domain is an instance of a domain in which knowledge is incomplete, uncertain, and inconsistent. Argumentation is well suited for reasoning in such weak theory domains. We model argument both as information structure, i.e., argument units connecting claims with supporting data, and as dialectical process, i.e., an alternating series of moves by opposing sides. Our model includes burden of proof as a (...)
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  25. Joseph S. Fulda (2010). The Logic of “Asked and Answered!”: The Case of the Traffic Light. Ratio Juris 23 (2):282-287.
    Uses erotetic logic to model the courtroom objection "Asked and Answered!".
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  26. Joseph S. Fulda (2000). The Logic of “Improper Cross”. Artificial Intelligence and Law 8 (4):337-341.
    Uses erotetic logic to model the courtroom objection "Improper Cross!". -/- Readers downloading the article should also please download the erratum et corrigendum, which is locally available.
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  27. Joseph S. Fulda (1989). The Logic of the Whole Truth. Rutgers Computer and Technology Law Journal 15 (2):435-446.
    Note: The author holds the copyright, and there was no agreement, express or implied, not to use a facsimile PDF. -/- Using erotetic logic, the paper defines the "the whole truth" in a manner consistent with U.S. Supreme Court precedent. It cannot mean "the whole story," as witnesses in an adversary system are permitted /only/ to answer the questions put to them, nor are they permitted to speculate, add irrelevant material, etc. Nor can it mean not to add an admixture (...)
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  28. Joseph S. Fulda (1988). The Logic of Expert Judging Systems and the Rights of the Accused. AI and Society 2 (3):266-269.
    Deals with the problem of enthymemes in expert systems designed to model legal reasoning; suggests that interactivity is crucial.
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  29. Roberto García, Rosa Gil & Jaime Delgado (2007). A Web Ontologies Framework for Digital Rights Management. Artificial Intelligence and Law 15 (2):137-154.
    In order to improve the management of copyright in the Internet, known as Digital Rights Management, there is the need for a shared language for copyright representation. Current approaches are based on purely syntactic solutions, i.e. a grammar that defines a rights expression language. These languages are difficult to put into practise due to the lack of explicit semantics that facilitate its implementation. Moreover, they are simple from the legal point of view because they are intended just to model the (...)
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  30. Simone Gittelson, Alex Biedermann, Silvia Bozza & Franco Taroni (2013). Modeling the Forensic Two-Trace Problem with Bayesian Networks. Artificial Intelligence and Law 21 (2):221-252.
    The forensic two-trace problem is a perplexing inference problem introduced by Evett (J Forensic Sci Soc 27:375–381, 1987). Different possible ways of wording the competing pair of propositions (i.e., one proposition advanced by the prosecution and one proposition advanced by the defence) led to different quantifications of the value of the evidence (Meester and Sjerps in Biometrics 59:727–732, 2003). Here, we re-examine this scenario with the aim of clarifying the interrelationships that exist between the different solutions, and in this way, (...)
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  31. Thomas F. Gordon (1993). The Pleadings Game. Artificial Intelligence and Law 2 (4):239-292.
    The Pleadings Game is a normative formalization and computational model of civil pleading, founded in Roberty Alexy''s discourse theory of legal argumentation. The consequences of arguments and counterarguments are modelled using Geffner and Pearl''s nonmonotonic logic,conditional entailment. Discourse in focussed using the concepts of issue and relevance. Conflicts between arguments can be resolved by arguing about the validity and priority of rules, at any level. The computational model is fully implemented and has been tested using examples from Article Nine of (...)
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  32. Thomas F. Gordon, Henry Prakken & Douglas N. Walton (2007). The Carneades Model of Argument and Burden of Proof. Artificial Intelligence 171 (10-15):875-896.
    We present a formal, mathematical model of argument structure and evaluation, taking seriously the procedural and dialogical aspects of argumentation. The model applies proof standards to determine the acceptability of statements on an issue-by-issue basis. The model uses different types of premises (ordinary premises, assumptions and exceptions) and information about the dialectical status of statements (stated, questioned, accepted or rejected) to allow the burden of proof to be allocated to the proponent or the respondent, as appropriate, for each premise separately. (...)
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  33. Thomas F. Gordon & Douglas Walton (2012). A Carneades Reconstruction of Popov V Hayashi. Artificial Intelligence and Law 20 (1):37-56.
    Carneades is an open source argument mapping application and a programming library for building argumentation support tools. In this paper, Carneades’ support for argument reconstruction, evaluation and visualization is illustrated by modeling most of the factual and legal arguments in Popov v Hayashi.
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  34. Jaap Hage (2003). Law and Defeasibility. Artificial Intelligence and Law 11 (2-3):221-243.
    The paper consists of three parts. In the first part five kinds of defeasibility are distinguished that is ontological, conceptual, epistemic, justification and logical defeasibility. In the second part it is argued that from these, justification defeat is the phenomenon that plays a role in legal reasoning. In the third part, the view is defended that non-monotonic logics are not necessary to model justification defeat, but that they are so to speak the natural way to model this phenomenon.
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  35. Jaap Hage (1996). A Theory of Legal Reasoning and a Logic to Match. Artificial Intelligence and Law 4 (3-4):199-273.
    This paper describes a model of legal reasoning and a logic for reasoning with rules, principles and goals that is especially suited to this model of legal reasoning. The paper consists of three parts. The first part describes a model of legal reasoning based on a two-layered view of the law. The first layer consists of principles and goals that express fundamental ideas of a legal system. The second layer contains legal rules which in a sense summarise the outcome of (...)
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  36. Jaap Hage & Aleksander Peczenik (2000). Law, Morals and Defeasibility. Ratio Juris 13 (3):305-325.
  37. Andreas Hamfelt (1995). Formalizing Multiple Interpretation of Legal Knowledge. Artificial Intelligence and Law 3 (4):221-265.
    A representation methodology for knowledge allowing multiple interpretations is described. It is based on the following conception of legal knowledge and its open texture. Since indeterminate, legal knowledge must be adapted to fit the circumstances of the cases to which it is applied. Whether a certain adaptation is lawful or not is measured by metaknowledge. But as this too is indeterminate, its adaptation to the case must be measured by metametaknowledge, etc. This hierarchical model of law is quite well-established and (...)
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  38. Jörg Hansen (2006). Deontic Logics for Prioritized Imperatives. Artificial Intelligence and Law 14 (1-2):1-34.
    When a conflict of duties arises, a resolution is often sought by use of an ordering of priority or importance. This paper examines how such a conflict resolution works, compares mechanisms that have been proposed in the literature, and gives preference to one developed by Brewka and Nebel. I distinguish between two cases – that some conflicts may remain unresolved, and that a priority ordering can be determined that resolves all – and provide semantics and axiomatic systems for accordingly defined (...)
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  39. Jürgen Hollatz (1999). Analogy Making in Legal Reasoning with Neural Networks and Fuzzy Logic. Artificial Intelligence and Law 7 (2-3):289-301.
    Analogy making from examples is a central task in intelligent system behavior. A lot of real world problems involve analogy making and generalization. Research investigates these questions by building computer models of human thinking concepts. These concepts can be divided into high level approaches as used in cognitive science and low level models as used in neural networks. Applications range over the spectrum of recognition, categorization and analogy reasoning. A major part of legal reasoning could be formally interpreted as an (...)
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  40. John F. Horty (2001). Argument Construction and Reinstatement in Logics for Defeasible Reasoning. Artificial Intelligence and Law 9 (1):1-28.
    This paper points out some problems with two recent logical systems – one due to Prakken and Sartor, the other due to Kowalski and Toni – designedfor the representation of defeasible arguments in general, but with a specialemphasis on legal reasoning.
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  41. Nguyen Duy Hung, Phan Minh Thang & Phan Minh Dung (2011). Modular Argumentation for Modelling Legal Doctrines of Performance Relief. Argument and Computation 1 (1):47-69.
    We present an argument-based formalism of contract dispute resolution following a modern view that the court would resolve a contract dispute by enforcing an interpretation of contract that reasonably represents the mutual intention of contract parties. Legal doctrines provide principles, rules and guidelines for the court to objectively arrive at such an interpretation. In this paper, we establish the appropriateness of the formalism by applying it to resolve disputes about performance relief with the legal doctrines of impossibility and frustration of (...)
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  42. Dan Hunter (1999). Out of Their Minds: Legal Theory in Neural Networks. [REVIEW] Artificial Intelligence and Law 7 (2-3):129-151.
    This paper examines the use of connectionism (neural networks) in modelling legal reasoning. I discuss how the implementations of neural networks have failed to account for legal theoretical perspectives on adjudication. I criticise the use of neural networks in law, not because connectionism is inherently unsuitable in law, but rather because it has been done so poorly to date. The paper reviews a number of legal theories which provide a grounding for the use of neural networks in law. It then (...)
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  43. Andrew J. I. Jones & Xavier Parent (2008). Normative-Informational Positions: A Modal-Logical Approach. [REVIEW] Artificial Intelligence and Law 16 (1):7-23.
    This paper is a preliminary investigation into the application of the formal-logical theory of normative positions to the characterisation of normative-informational positions, pertaining to rules that are meant to regulate the supply of information. First, we present the proposed framework. Next, we identify the kinds of nuances and distinctions that can be articulated in such a logical framework. Finally, we show how such nuances can arise in specific regulations. Reference is made to Data Protection Law and Contract Law, among others. (...)
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  44. Andrew J. I. Jones & Marek Sergot (1992). Deontic Logic in the Representation of Law: Towards a Methodology. [REVIEW] Artificial Intelligence and Law 1 (1):45-64.
    There seems to be no clear consensus in the existing literature about the role of deontic logic in legal knowledge representation — in large part, we argue, because of an apparent misunderstanding of what deontic logic is, and a misplaced preoccupation with the surface formulation of legislative texts. Our aim in this paper is to indicate, first, which aspects of legal reasoning are addressed by deontic logic, and then to sketch out the beginnings of a methodology for its use in (...)
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  45. Audun Jøsang & Viggo A. Bondi (2001). Legal Reasoning with Subjective Logic. Artificial Intelligence and Law 8 (4):289-315.
    Judges and jurors must make decisions in an environment of ignoranceand uncertainty for example by hearing statements of possibly unreliable ordishonest witnesses, assessing possibly doubtful or irrelevantevidence, and enduring attempts by the opponents to manipulate thejudge''s and the jurors'' perceptions and feelings. Three importantaspects of decision making in this environment are the quantificationof sufficient proof, the weighing of pieces of evidence, and therelevancy of evidence. This paper proposes a mathematical frameworkfor dealing with the two first aspects, namely the quantification ofproof (...)
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  46. Zsófia Kacsuk (2011). The Mathematics of Patent Claim Analysis. Artificial Intelligence and Law 19 (4):263-289.
    In patent law most of the crucial legal questions such as patentability and infringement are linked to the patent claims. The European Patent Office regards patent claims as a set of independent features which are examined separately in a more or less formal way. The author has found that this approach allows for developing a simple mathematical model which treats patent claim features as logical statements and patent claims as compound statements wherein the individual statements are connected by logical connectives. (...)
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  47. Tokuyasu Kakuta, Makoto Haraguchi & Yoshiaki Okubo (1997). A Goal-Dependent Abstraction for Legal Reasoning by Analogy. Artificial Intelligence and Law 5 (1-2):97-118.
    This paper presents a new algorithm to find an appropriate similarityunder which we apply legal rules analogically. Since there may exist a lotof similarities between the premises of rule and a case in inquiry, we haveto select an appropriate similarity that is relevant to both thelegal rule and a top goal of our legal reasoning. For this purpose, a newcriterion to distinguish the appropriate similarities from the others isproposed and tested. The criterion is based on Goal-DependentAbstraction (GDA) to select a (...)
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  48. Harm Kloosterhuis (2000). Analogy Argumentation in Law: A Dialectical Perspective. [REVIEW] Artificial Intelligence and Law 8 (2-3):173-187.
    In this paper I investigate the similarities betweenthe dialectical procedure in the pragma-dialecticaltheory and dialectical procedures in AI and Law. I dothis by focusing on one specific type of reasoning inlaw: analogy argumentation. I will argue that analogyargumentation is not only a heuristic forfinding new premises, but also a part of thejustification of legal decisions. The relevantcriteria for the evaluation of analogy argumentationare not to be found at the logical level of inference,but at the procedural level of the discussion. I (...)
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  49. Jacky Legrand (1999). Some Guidelines for Fuzzy Sets Application in Legal Reasoning. Artificial Intelligence and Law 7 (2-3):235-257.
    As an introduction to our work, we emphasize the parallel interpretation of abstract tools and the concepts of undetermined and vague information. Imprecision, uncertainty and their relationships are inspected. Suitable interpretations of the fuzzy sets theory are applied to legal phenomena in an attempt to clearly circumscribe the possible applications of the theory. The fundamental notion of reference sets is examined in detail, hence highlighting their importance. A systematic and combinatorial classification of the relevant subsets of the legal field is (...)
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  50. Hubert Lehmann (1990). Legal Concepts in a Natural Language Based Expert System. Ratio Juris 3 (2):245-253.
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