Showing papers on "Abductive reasoning published in 2007"

Abduction: the logic of discovery of Grounded Theory

Abstract: This essay is a contribution to the question whether grounded theory methodology (in the variant of STRAUSS & CORBIN) contains an abductive research logic as developed in the work of Charles Sanders PEIRCE. After going through the works of STRAUSS and CORBIN I answer the question with a resounding yes. But it does not only contain the logic of abductive reasoning but also that of qualitative induction. URN: urn:nbn:de:0114-fqs1001135

Inductive reasoning: Experimental, developmental and computational approaches

Abstract: Preface Aidan Feeney and Evan Heit 1. What is induction and why study it? Evan Heit 2. The development of inductive reasoning Brett K. Hayes 3. Interpreting asymmetries of projection in children's inductive reasoning Douglas Medin and Sandra Waxman 4. Property generalization as causal reasoning Bob Rehder 5. Availability in category-based induction Patrick Shafto, John Coley and Anna Vitkin 6. From similarity to chance Sergey Blok, Daniel Osherson and Douglas Medin 7. Theory-based Bayesian models of inductive reasoning Joshua Tenenbaum, Charles Spence and Patrick Shafto 8. Use of single or multiple categories in category-based induction Gregory Murphy and Brian Ross 9. Abductive inference: From philosophical analysis to neutral mechanisms Paul Thagard 10. Mathematical induction and induction in mathematics Lance Rips and Jennifer Asmuth 11. Induction, deduction, and argument strength in human reasoning and argumentation Mike Oaksford and Ulrike Hahn 12. Individual differences, dual processes, and induction Aidan Feeney 13. Taxonomising induction Steve Sloman.

Sense-making software for crime investigation: how to combine stories and arguments?

Abstract: Sense-making software for crime investigation should be based on a model of reasoning about evidence that is both natural and rationally well-founded. A formal model is proposed that combines artificial intelligence formalisms for abductive inference to the best explanation and for defeasible argumentation. Stories about what might have happened in a case are represented as causal networks and possible hypotheses can be inferred by abductive reasoning. Links between stories and the available evidence are expressed with evidential generalizations that express how observations can be inferred from evidential sources with defeasible argumentation. It is argued that this approach unifies two well-known accounts of reasoning about evidence, namely, anchored narratives theory and new evidence theory. After the reasoning model is defined, a design is presented for sense-making software that allows crime investigators to visualize their thinking about a case in terms of the reasoning model.

FreeEnCal: A Forward Reasoning Engine with General-Purpose

Abstract: A forward reasoning engine is an indispensable component in many advanced knowledge-based systems with purposes of creation, discovery, or prediction. This paper presents a forward reasoning engine with general-purpose, named "FreeEnCal", which can interpret and perform inference rules defined and given by its users, draw fragments of various classical and/or non-classical logic systems formalized as different formal systems, draw empirical theorems of various formal theories constructed based on various logic systems, and perform deductive, inductive, and abductive reasoning automatically. FreeEnCal can be used as a ready-made forward reasoning engine serving as a core and fundamental component in various advanced knowledge-based systems as well as an alone forward reasoning engine with general-purpose. The paper presents our basic ideas to design and implement FreeEnCal, facilities provided by FreeEnCal, and some applications of FreeEnCal.

Inductive Reasoning: Abductive Inference: From Philosophical Analysis to Neural Mechanisms

Abstract: WHAT IS ABDUCTION? In the 1890s, the great American philosopher C. S. Peirce (1931–1958) used the term “abduction” to refer to a kind of inference that involves the generation and evaluation of explanatory hypotheses. This term is much less familiar today than “deduction,” which applies to inference from premises to a conclusion that has to be true if the premises are true. And it is much less familiar than “induction,” which sometimes refers broadly to any kind of inference that introduces uncertainty, and sometimes refers narrowly to inference from examples to rules, which I will call “inductive generalization.” Abduction is clearly a kind of induction in the broad sense, in that the generation of explanatory hypotheses is fraught with uncertainty. For example, if the sky suddenly turns dark outside my window, I may hypothesize that there is a solar eclipse, but many other explanations are possible, such as the arrival of an intense storm or even a huge spaceship. Despite its inherent riskiness, abductive inference is an essential part of human mental life. When scientists produce theories that explain their data, they are engaging in abductive inference. For example, psychological theories about mental representations and processing are the result of abductions spurred by the need to explain the results of psychological experiments. In everyday life, abductive inference is ubiquitous, for example when people generate hypotheses to explain the behavior of others, as when I infer that my son is in a bad mood to explain a curt response to a question.

Formalising argumentative story-based analysis of evidence

Abstract: In the present paper, we provide a formalised version of a merged argumentative and story-based approach towards the analysis of evidence. As an application, we are able to show how our approach sheds new light on inference to the best explanation with case evidence. More specifically, it will be clarified how the events in a case story that are considered to be proven abductively explain the otherwise unproven events of the case story. We compare our approach with existing AI work on modelling legal reasoning with evidence.

Abducing Coreference by Model Construction

Abstract: In this paper, we argue that the resolution of anaphoric expressions in an utterance is essentially an abductive task following [12] who use a weighted abduction scheme on horn clauses to deal with reference. We give a semantic representation for utterances containing anaphora that enables us to compute possible antecedents by abductive inference. We extend the disjunctive model construction procedure of hyper tableaux [3, 14] with a clause transformation turning the abductive task into a model generation problem and show the completeness of this transformation with respect to the computation of abductive explanations. This abductive inference is applied to the resolution of anaphoric expressions in our general model constructing framework for incremental discourse representation which we argue to be useful for computing information updates from natural language utterances.

Reasoning with conditionals

Abstract: This paper reviews the psychological investigation of reasoning with conditionals, putting an emphasis on recent work. In the first part, a few methodological remarks are presented. In the second part, the main theories of deductive reasoning (mental rules, mental models, and the probabilistic approach) are considered in turn; their content is summarised and the semantics they assume for if and the way they explain formal conditional reasoning are discussed, in particular in the light of experimental work on the probability of conditionals. The last part presents the recent shift of interest towards the study of conditional reasoning in context, that is, with large knowledge bases and uncertain premises.

Inference to the Only Explanation

Abstract: In the above loveliness is a qualitative character that is distinct from a quasi-quantitative assessment of likeliness or probability. Indeed, the point of IBE is that loveliness is a guide to likeliness. In chapter 5 Lipton presents a case study, Semmelweis's investigation of puerperal (or childbed) fever, which permits a comparison with Hempel's discussion of the same case in support of his hypothetico-deductive account of confirmation. One interesting feature of Lipton's presentation is that loveliness is not discussed. This is because, as Lipton says, Semmelweis converted his problem into the question of the only explanation of the contrastive facts (p. 90). In what follows I want to explore the prospects for regarding Lipton's framework as permitting an account of abductive inference in which Inference to the Only Explanation plays a central role. It might be thought that Lipton himself, as just reported, has given at least an example of Inference to the Only Explanation. But, as we

Model-Based Reasoning in Science, Technology, and Medicine

Abstract: Abduction, Problem Solving, and Practical Reasoning.- Animal Abduction.- Communicative Gestures Facilitate Problem Solving for Both Communicators and Recipients.- The Concept of Fallacy is Empty.- Abductive Reasoning, Information, and Mechanical Systems.- Automated Abduction in Scientific Discovery.- Abduction, Medical Semeiotics and Semioethics.- Abduction and Modeling in Biosemiotics and Sociosemiotics.- Reason out Emergence from Cellular Automata Modeling.- Belief Ascription and De Re Communication.- Multiagent-Based Simulation in Biology.- Mathematics through Diagrams: Microscopes in Non-Standard and Smooth Analysis.- Models, Mental Models, Representations, and Medical Reasoning.- Cognition, Environment and the Collapse of Civilizations.- Cognitive Aspects of Tacit Knowledge and Cultural Diversity.- The Functional-Analogical Explanation in Chinese Science and Technology.- Model-Based Reasoning and Diagnosis in Traditional Chinese Medicine (TCM).- Model-Based Reasoning in Cognitive Science.- An Examination of Model-Based Reasoning in Science and Medicine in India.- Ontology, Artefacts, and Models of Reasoning.- The Wondering Angels of the Fractal Art.- Logical and Computational Aspects of Model-Based Reasoning.- Polynomizing: Logic Inference in Polynomial Format and the Legacy of Boole.- Abductive Inference and Iterated Conditionals.- Peircean Pragmatic Truth and da Costa's Quasi-Truth.- Sliding Mode Motion Control Strategies for Rigid Robot Manipulators.- Model-Based Chemical Compound Formulation.- Model-Based Reasoning for Self-Repair of Autonomous Mobile Robots.- Application of Bayesian Inference to Automatic Semantic Annotation of Videos.- An Algebraic Approach to Model-Based Diagnosis.- CYBERNARD: A Computational Reconstruction of Claude Bernard's Scientific Discoveries.- Do Computational Models of Reading Need a Bit of Semantics?.

Abductive Reasoning: Challenges Ahead

Abstract: The motivation behind the collection of papers presented in this THEORIA forum on Abductive reasoning is my book Abductive Reasoning: Logical Investigations into the Processes of Discovery and Explanation. These contributions raise fundamental questions. One of them concerns the conjectural character of abduction. The choice of a logical framework for abduction is also discussed in detail, both its inferential aspect and search strategies. Abduction is also analyzed as inference to the best explanation, as well as a process of epistemic change, both of which chal- lenge the argument-like format of abduction. Finally, the psychological question of whether humans reason abduc- tively according to the models proposed is also addressed. I offer a brief summary of my book and then comment on and respond to several challenges that were posed to my work by the contributors to this issue.

Abductive reasoning in neural-symbolic systems

Abstract: Abduction is or subsumes a process of inference. It entertains possible hypotheses and it chooses hypotheses for further scrutiny. There is a large literature on various aspects of non-symbolic, subconscious abduction. There is also a very active research community working on the symbolic (logical) characterisation of abduction, which typically treats it as a form of hypothetico-deductive reasoning. In this paper we start to bridge the gap between the symbolic and sub-symbolic approaches to abduction. We are interested in benefiting from developments made by each community. In particular, we are interested in the ability of non-symbolic systems (neural networks) to learn from experience using efficient algorithms and to perform massively parallel computations of alternative abductive explanations. At the same time, we would like to benefit from the rigour and semantic clarity of symbolic logic. We present two approaches to dealing with abduction in neural networks. One of them uses Connectionist Modal Logic and a translation of Horn clauses into modal clauses to come up with a neural network ensemble that computes abductive explanations in a top-down fashion. The other combines neural-symbolic systems and abductive logic programming and proposes a neural architecture which performs a more systematic, bottom-up computation of alternative abductive explanations. Both approaches employ standard neural network architectures which are already known to be highly effective in practical learning applications. Differently from previous work in the area, our aim is to promote the integration of reasoning and learning in a way that the neural network provides the machinery for cognitive computation, inductive learning and hypothetical reasoning, while logic provides the rigour and explanation capability to the systems, facilitating the interaction with the outside world. Although it is left as future work to determine whether the structure of one of the proposed approaches is more amenable to learning than the other, we hope to have contributed to the development of the area by approaching it from the perspective of symbolic and sub-symbolic integration.

Abduction and Inference to the Best Explanation

Abstract: Aliseda's Abductive Reasoning is focused on the logical problem of abduction. My paper, in contrast, deals with the epistemic problems raised by this sort of inference. I analyze the relation between abduction and inference to the best explanation (IBE). Firstly a heuristic and a normative interpretation of IBE are distin- guished. The epistemic problem is particularly pressing for the latter interpretation, since it is devoid of content without specific epistemic criteria for separating acceptable explanations from those which are not. Then I discuss two different normative interpretations of IBE. I. Niiniliuoto favours a "probabilistic- confirmational" translation of explanatory merit while S. Psillos thinks that the insight of IBE is lost in a pure probabilistic format. My conclusion is that Aliseda's theory of abduction fits better with a heuristic ac- count of IBE.

Model-Based Reasoning Methods within an Ambient Intelligent Agent Model

Abstract: Ambient agents react on humans on the basis of their information obtained by sensoring and their knowledge about human functioning. Appropriate types of reactions depend on in how far an ambient agent understands the human. On the one hand, such an understanding requires that the agent has knowledge to a certain depth about the human’s physiological and mental processes in the form of an explicitly represented model of the causal and dynamic relations describing these processes. On the other hand, given such a model representation, the agent needs reasoning methods to derive conclusions from the model and the information available by sensoring. This paper presents a number of such model-based reasoning methods. They have been formally specified in an executable temporal format, which allows for simulation of reasoning traces and automated verification in a dedicated software environment. A number of such simulation experiments and their formal analysis are described.

Automated Abduction in Scientific Discovery

Abstract: The role of abduction in the philosophy of science has been well studied in recent years and has led to a deeper understanding of many formal and pragmatic issues [1, 2, 3, 4, 5]. This paper is written from the point of view that real applications are now needed to help consolidate what has been learned so far and to inspire new developments. With an emphasis on computational mechanisms, it examines the abductive machinery used for generating hypotheses in a recent Robot Scientist project [6] and shows how techniques from Abductive Logic Programming [7] offer superior reasoning capabilities needed in more advanced practical applications. Two classes of abductive proof procedures are identified and compared in a case study. Backward-chaining logic programming methods are shown to outperform theorem proving approaches based on the use of contrapositive reasoning.

Logic of Sherlock Holmes in Technology Enhanced Learning

Abstract: Abduction is a method of reasoning that people use under uncertainty in a context in order to come up with new ideas. The use of abduction in this exploratory study is twofold: (i) abduction is a cross-disciplinary analytic tool that can be used to explain certain key aspects of human-computer interaction in advanced Information Society Technology (IST) environments; (ii) abduction is probably the central inferential mechanism at work when learners learn or in general make sense of things in an IST or mobile context. Consequently, abduction illuminates the special epistemological circumstances of IST enhanced learning, in particular when the learning materials and the learning environment have been arranged in accordance with constructivist pedagogical guidelines. A study of abductive reasoning will help us better understand IST enhanced learning and IST user behaviour as well as give us some valuable hints to the design of human-computer interaction in general.

Ignorance and semantic tableaux: Aliseda on abduction

Abstract: This is an examination of similarities and differences between two recent models of abductive reasoning. The one is developed in Atocha Aliseda's Abductive Reasoning: Logical Investigations into the Processes of Discovery and Evaluation (2006). The other is advanced by Dov Gabbay and the present author in their The Reach of Abduction: Insight and Trial (2005). A principal difference between the two approaches is that in the Gabbay- Woods model, but not in the Aliseda model, abductive inference is ignorance-preserving. A further differ- ence is that Aliseda reconstructs the abduction relation in a semantic tableaux environment, whereas the Woods-Gabbay model, while less systematic, is more general. Of particular note is the connection between abduction and legal reasoning.

Peirce on the role ofpoietic creation in mathematical reasoning

Abstract: C.S. Peirce defines mathematics in two ways: first as \"the science which draws necessary conclusions,\" and second as \"the study of what is true of hypothetical states of things\" (CP 4.227–244). Given the dual definition, Peirce notes, a question arises: Should we exclude the work of poietic hypothesis-making from the domain of pure mathematical reasoning? (CP 4.238). This paper examines Peirce's answer to the question. Some commentators hold that for Peirce the framing of mathematical hypotheses requires poietic genius but is not scientific work. I propose, to the contrary, that although Peirce occasionally seems to exclude the poietic creation of hypotheses altogether from pure mathematical reasoning, Peirce's position is rather that the creation of mathematical hypotheses is poietic, but it is not merely poietic, and accordingly, that hypothesis-framing is part of mathematical reasoning that involves an element of poiesis but is not merely poietic either. Scientific considerations also inhere in the process of hypothesis-making, without excluding the poietic element. In the end, I propose that hypothesis-making in mathematics stands between artistic and scientific poietic creativity with respect to imaginative freedom from logical and actual constraints upon reasoning.

A consequence finding approach for full clausal abduction

Abstract: Abductive inference has long been associated with the logic of scientific discovery and automated abduction is now being used in real scientific tasks. But few methods can exploit the full potential of clausal logic and abduce non-ground explanations with indefinite answers. This paper shows how the consequence finding method of Skip Ordered Linear (SOL) resolution can overcome the limitations of existing systems by proposing a method that is sound and complete for finding minimal abductive solutions under a variety of pruning mechanisms. Its utility is shown with an example based on metabolic network modelling.

Modern Logic and its Role in the Study of Knowledge

Abstract: NOWLEDGE IS at the heart of intelligent behaviour. The ability to obtain, manipulate and communicate knowledge, in explicit form, is what distinguishes humans from other animals. This suggests that any study of intelligent behaviour, theoretical or experimental, would have the same starting point, namely a Science of Knowledge, which studies the basic forms of knowledge, its acquisition, and its processing. Yet there does not seem to exist such a unified and mutually agreed science of knowledge. In ancient times philosophy, the ‘love of knowledge’, would aim to fulfil this role of the Mother of all Sciences, but philosophy has since long lost its central place and has mostly fragmented into specialised sciences such as physics, biology, and mathematics. Computer science, a relatively young branch on the tree of knowledge, has some aspirations to be the science of knowledge, but is currently at best a loosely connected collection of engineering technologies and abstract mathematical theory. (In fact, scholars of more established disciplines such as physics or chemistry often hesitate to call computer science a science at all, because its design-oriented approach does not fit in well with the doctrines of experimental sciences.) Artificial intelligence – the discipline studying fruitful connections between intelligent behaviour and computers – would be another contender, but has been accused of overstating its claims, having unclear goals, and applying sloppy methodology. In this chapter I argue that logic, in its widest sense, is – or at least, should be perceived as – the science of knowledge. This would be an unsurprising statement for a 19 century logician, who would study the kind of inductive reasoning involved in experimental sciences as eagerly as he would investigate the kind of reasoning that is employed in mathematical proofs. However, in the last century logic seems to have developed into a relatively specialised and not seldomly obscure branch of mathematics. This is all the more paradoxical since the first half of the 20 century has often been called ‘the Golden Age of logic’. Following the pioneering work of Gottlob Frege, who developed a forerunner of predicate logic called Begriffschrift (‘concept language’) in 1893, Russell and Whitehead published their three-volume Principia Mathematica between 1910 and 1913, in which they re-established the foundations of pure mathematics in logical terms. Whereas Kurt Gödel dealt a severe blow to the ambitions of logicians when he demonstrated that any logical system powerful enough to include natural numbers is also necessarily incomplete (i.e., the logical system allows the formulation of true statements which are demonstrably unprovable within the system), this didn’t stop logicians like Alonzo Church to develop ever more powerful logical systems (e.g., combinator logic and higher-order logic). Furthermore, Alfred Tarski invented what I consider one of the most important contributions of modern logic, namely the notion of an independent semantics.

Abduction through Semantic Tableaux versus Abduction through Goal-Directed Proofs

Abstract: In this paper, we present a goal-directed proof procedure for abductive reasoning. This procedure will be compared with Aliseda's approach based on semantic tableaux. We begin with some comments on Alis- eda's algorithms for computing conjunctive abductions and show that they do not entirely live up to their aims. Next we give a concise account of goal-directed proofs and we show that abductive explanations are a natural spin-off of these proofs. Finally, we show that the goal-directed procedure solves the problems we encountered in Aliseda's algorithms.

The Activation of Hypotheses during Abductive Reasoning

Abstract: The Activation of Hypotheses during Abductive Reasoning Martin R. K. Baumann (martin.baumann@phil.tu-chemnitz.de) Katja Mehlhorn (katja.mehlhorn@phil.tu-chemnitz.de) Franziska Bocklisch (franziska.bocklisch@phil.tu-chemnitz.de) Chemnitz University of Technology, Department of Psychology, Wilhelm-Raabe-Str. 43, 09107 Chemnitz, Germany observations. Following the above example, in most cases a patient complains not only about one symptom, such as a headache, but about a set of observations that could be a headache, sickness, and fever. Each of these symptoms can be caused by many different diseases. The physician’s task is to find the best explanation for the whole symptoms set. And, despite the complexity of the problem, the physician solves the problem in most cases quickly and accurately. How is this accomplished? Johnson and Krems (2001) suggested on the basis of their results on abductive reasoning that people use initial observations to construct a preliminary explanation for these observations. Succeeding observations are sequentially comprehended and integrated to generate a single current explanation for all observations seen so far. If an observation can be comprehended in different ways, that is, if there exist alternative elementary explanations for this new observation, the current explanation is used to decide between these alternatives. Only those elementary explanations for the new observations are considered as relevant that are compatible with the current explanation. Thus, the current explanation acts as an explanatory context for the comprehension and explanation of new observations. It reduces the complexity of the abductive reasoning problem as not all possible elementary explanations for a new observation are considered as relevant but only those that are compatible with the current explanation. Whereas Johnson and Krems’ model focuses on deliberate reasoning processes to describe the use of the current explanatory context, we assume that automatic comprehension processes based on spreading activation and constraint satisfaction also play a key role. It has been argued recently that both deliberate and automatic processes are involved in many reasoning tasks (Sloman, 1996) such as impression formation (Thagard & Kunda, 1998), hypothesis evaluation (Johnson, Zhang, & Wang, 1997), and medical diagnosis (Arocha & Patel, 1995). Thagard and Kunda explain how spreading activation processes can explain the effect of social stereotypes on the interpretation of behavior. Johnson, Zhang, and Wang show how automatic processes can provide information for the evaluation of hypotheses that is used subsequently in more deliberate processes to revise existing or generate new hypotheses in an abductive reasoning task. In our view these automatic processes also serve the function of making those elementary explanations of new observations highly available to the reasoner that have a high probability of being the relevant explanations in the Abstract Abductive reasoning, that is, finding an explanation for a set of observations, can be understood as a process of sequentially understanding and integrating new observations into a mental model about the current situation (Johnson & Krems, 2001; Josephson & Josephson, 1994). Whereas Johnson and Krems’ model focuses on conscious deliberate processes, it has been argued that automatic implicit processes also play an important role in abductive reasoning (e.g. Johnson, Zhang, & Wang, 1997). Adopting Kintsch`s (1998) construction-integration theory, we assume that automatic activation processes regulate the availability of possible explanations during the reasoning process. In our experiment, participants solved an artificial diagnosis task while the activation of explanatory hypotheses was measured. We found that explanatory hypotheses relevant in the current context for explaining a set of observations are kept in a more active state in memory than irrelevant or rejected hypotheses. Keywords: abductive reasoning; causal reasoning; automatic processes; explanations; activation. Introduction Generating a hypothesis to explain one or more observations is an essential part of many real world tasks. This kind of reasoning is called abductive reasoning (Josephson & Josephson, 1994). It is a vital subprocess, for example, in scientific discovery, medical diagnosis, software debugging, social attribution processes, and discourse comprehension. While explaining a given set of observations, the reasoner has often to decide between different alternative explanations to find the best explanation for the observations. We assume that both deliberate reasoning processes and automatic comprehension processes contribute to the generation of hypotheses (Johnson, Zhang, & Wang, 1997; Sloman, 1996). The goal of this paper is to examine how automatic comprehension processes constrain the consideration of hypotheses to the most plausible ones in the given context by making these hypotheses highly available to the reasoner and reducing the availability of implausible ones. Constructing an explanatory hypothesis can be a quite complex task. First, in many cases there is more than one possible explanation for a given observation. For example, headache is a common symptom of many diseases and is associated with many different causes. Second, the task is often not to explain one observation but a set of observations where each observation can be explained with more than one explanation. In such a case, a combination of elementary hypotheses has to be found that best explains all

Argumentation, stories and generalizations: a comment

Abstract: The underlying theory of the software program described in ‘Sense-Making Software for Criminal Investigation’ (Bex et al., 2007) complements modified Wigmorean analysis (MWA). Both adopt a qualitative rather than a quantitative approach. MWA is broadly compatible with the kind of logic involved, including abductive inference to the best explanation and the idea of defeasible argumentation. Both approaches are mainly valuable as aids to thinking, especially constructing and evaluating arguments, rather than as methods of presenting them in order to persuade. Both approaches can be applied at different stages of criminal investigation (and more broadly of legal processes), but the specific device of Wigmore charts (one part of MWA) is more useful in hypothesis testing and discarding than in hypothesis formation, which typically requires imaginative reasoning. The Anchored Narratives of Crombag et al. and MWA have similar theoretical assumptions, except that MWA gives a radically different account about the relationship between stories, generalizations and argument. The proposed program has considerable promise, but before it can be of positive practical value in police investigation, more attention needs to be given not only to the obvious dangers of using stories and generalizations in this context but also about what positive guidance can be given to mitigate these dangers. There is, however, an unresolved tension between the simplifying tendencies of formalized computer programming and the tendency of MWA to emphasize the complexities of practical inferential reasoning and argumentation in legal contexts.

Abduction as a strategy for concept formation in mathematics: Cardano postulating a negative

Abstract: When dealing with abductive reasoning in scientific discovery, historical case studies are focused mostly on the physical sciences, as with the discoveries of Kepler, Galilei and Newton. We will present a case study of abductive reasoning in early algebra. Two new concepts introduced by Cardano in his Ars Magna, imaginary numbers and a negative solution to a linear problem, can be explained as a result of a process of abduction. We will show that the first appearance of these new concepts fits very well Peirce’s original description of abductive reasoning. Abduction may be regarded as one important strategy for the formation of new concepts in mathematics. Peirce on abduction: a recapitulation Peirce’s definition of abduction has been understood and explained in two different meanings, coined by Magnani (2001) as selective and creative abduction. Selective abduction is the process of finding the right explanatory hypothesis from a given set of possible explanations. A common example of selection is medical diagnosis. In contrast, creative abduction generates the (right) explanatory hypothesis. As we are discussing concept formation, the abductive reasoning in our case study is of the creative type. We propose to view the formation of a new concept in mathematics as the hypothesis abduced to explain an anomaly. Peirce does not discuss concept formation as such but comes very close when he describes abduction as “the only logical operation which introduces new ideas” (1958, 5, 171). The existence of an anomaly is crucial in the cases we have studied, and is expressed by Peirce in similar wordings. He describes abduction as a form of inference motivated by the observation of a “surprising fact” (1958, 5, 1889), reasoning initiated by “genuine doubt” and “genuine surprise” (1958, 5, 524), a motivation to break away from our habits “due to some novel experience” (ibid.). Abduction takes place when “we find ourselves confronted with some experience contrary to our expectations” (1958, 7, 36). We will show below that Peirce’s depiction of the mental state associated with abduction aptly describes the historical cases under discussion. A last relevant aspect of Peirce’s account of abduction is that the generated hypothesis must follow specific conditions to become the explanatory hypothesis: “Namely, the hypothesis cannot be admitted, even as a hypothesis, unless it be supposed that it would account for the facts or some of them” (1958, 5, 188-9). This necessitates some necessary connection between the novel hypothesis and the observed anomaly. The generated hypothesis delivers, at that time, the most adequate explanation for the “surprising fact”.

The Ontology of Legal Possibilities and Legal Potentialities

Abstract: Ontologies in a legal expert system must be processed to suit all possible user cases within the field of law of the system. From the logical premises of a deductive system of express rules of law, legal ontologies may be implied to encompass the combinatorial explosion of possible cases that may lack one or more of the express antecedents in the deductive rule system. Express ontologies in inductive and abductive premises that are associated with the deductive antecedents, may also be adjusted by implication to suit the combinatorial explosion of possible cases. Implied legal ontologies may be determined to suit the user’s case and its legal consequences. The method of this determination and the processing of express black letter law accordingly, is considered by reference to the supplementation of ontology by logic and the supplementation of logic by ontology, in the legal domain; three bases of this method are discussed: law-making power, prior analytics, and the pillars of truth in science and law. Firstly, law-making authority includes the power to determine the logical category of legal premises, and legal truth tables (c.f. Wittgenstein, 1918); law is laid down as legal ontologies with logic attributes or structures. Thus, three ontological posits of law-makers provide for the logical processing of legal information. Rules of law are Major deductive premises laid down, formally or informally, as conditional propositions which may be systematised for extended deductive reasoning. Material facts in a case are laid down as inductive instances that particularise or define antecedents in rules of law; they also may be used as Minor deductive premises to determine the outcome of the case. Reasons for rules are laid down as and for strong or weak abductive reasoning. Secondly, legal knowledge engineering requires prior analytics (cf. Aristotle, 1952, originally c.335 BC ) for the acquisition of the expertise; by prior analytics, premises are formalised and systematized for automation of their associated heuristics. Legal epistemology both determines and implements logical structures; through prior analytics it uses ontologies of legal possibilities and potentialities, to comprehensively predetermine premises for its three forms of legal logic: deduction, induction and abduction. Thirdly, Lord Chancellor Bacon’s (1620) reconstruction of legal epistemology as scientific method for expanding knowledge, systematizes the sources of truth in law and science. It is here developed as a method of prior analytics for constructing an ontology of legal possibilities and legal potentialities. Such ontological construction is essential for determining the heuristics of combinatorial explosion derived from express legal rules to meet the possible cases of users; while legal experts need only construct the relevant part of the combinatorial explosion, for a client’s case, an expert system must be capable of constructing any relevant part to suit a user’s case.

Selecting Implications in Fuzzy Abductive Problems

Abstract: Abductive reasoning is an explanatory process in which potential causes of an observation are unearthed. We have concentrated on the formal definition of fuzzy abduction as an inversion of the Generalised Modus Ponens given by Mellouli and Bouchon-Meunier. While studying this formalism we noticed that some observations could not be explained properly. Observations, in abductive reasoning, are made within the conclusion space of the considered rule. Their potential shape is therefore highly constrained by the implication operator used. We claim that, given a feasible observation and a set of rules, we can categorise the set of implications to be used. Since a given observation will match only part of the conclusions in the rule-set, we offer a categorisation of a rule system coherent with observed data

Inference of abduction theories for handling incompleteness in first-order learning

Abstract: In real-life domains, learning systems often have to deal with various kinds of imperfections in data such as noise, incompleteness and inexactness. This problem seriously affects the knowledge discovery process, specifically in the case of traditional Machine Learning approaches that exploit simple or constrained knowledge representations and are based on single inference mechanisms. Indeed, this limits their capability of discovering fundamental knowledge in those situations. In order to broaden the investigation and the applicability of machine learning schemes in such particular situations, it is necessary to move on to more expressive representations which require more complex inference mechanisms. However, the applicability of such new and complex inference mechanisms, such as abductive reasoning, strongly relies on a deep background knowledge about the specific application domain. This work aims at automatically discovering the meta-knowledge needed to abduction inference strategy to complete the incoming information in order to handle cases of missing knowledge.

A Causal Theory of Abduction

Abstract: The article provides a uniform representation of abductive reasoning in the logical framework of causal inference relations. The representation covers in a single framework not only traditional, ‘classical’ forms of abduction, but also abductive reasoning in diagnosis, theories of actions and change, and abductive logic programming.