Dov Gabbay

Inductive Logic is number ten in the 11-volume Handbook of the History of Logic. While there are many examples were a science split from philosophy and became autonomous (such as physics with Newton and biology with Darwin), and while there are, perhaps, topics that are of exclusively philosophical interest, inductive logic — as this handbook attests — is a research field where philosophers and scientists fruitfully and constructively interact. This handbook covers the rich history of scientific turning points in Inductive Logic, including probability theory and decision theory. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration.

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, including Mediaeval and Renaissance Logic and Logic: A History of its Central. In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality. Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world. Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value. Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas.

Inductive Logic is number ten in the 11-volume Handbook of the History of Logic. While there are many examples were a science split from philosophy and became autonomous (such as physics with Newton and biology with Darwin), and while there are, perhaps, topics that are of exclusively philosophical interest, inductive logic — as this handbook attests — is a research field where philosophers and scientists fruitfully and constructively interact. This handbook covers the rich history of scientific turning points in Inductive Logic, including probability theory and decision theory. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration.

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.

Formal nonmonotonic systems try to model the phenomenon that common sense reasoners are able to “jump” in their reasoning from assumptions Δ to conclusions C without their being any deductive chain from Δ to C. Such jumps are done by various mechanisms which are strongly dependent on context and knowledge of how the actual world functions. Our aim is to motivate these jump rules as inference rules designed to optimise survival in an environment with scant resources of effort and time. We begin with a general discussion and quickly move to Section 3 where we introduce five resource principles. We show that these principles lead to some well known nonmonotonic systems such as Nute’s defeasible logic. We also give several examples of practical reasoning situations to illustrate our principles.

Much of cognitive science seeks to provide principled descriptions of various kinds and aspects of rational behaviour, especially in beings like us or AI simulacra of beings like us. For the most part, these investigators presuppose an unarticulated common sense appreciation of the rationality that such behaviour consists in. On those occasions when they undertake to bring the relevant norms to the surface and to give an account of that to which they owe their legitimacy, these investigators tend to favour one or other of three approaches to the normativity question. They are the analyticity or truth-in-a-model approach; the pluralism approach; and the reffective equilibrium approach.All three of these approaches to the normativity question are seriously flawed, never mind that the first two have some substantial provenance among logicians and the third has enjoyed a flourishing philosophical career.Against these views, we propose a strong version of what might be called normatively immanent descriptivism. We attempt to elucidate its virtues and to deal with what appears to be its most central vulnerability, embodied in the plain fact that actual human behaviour is sometimes irrational

Based on the premise that what is relevant, consistent, or true may change from context to context, a formal framework of relevance and context is proposed in which • contexts are mathematical entities • each context has its own language with relevant implication • the languages of distinct contexts are connected by embeddings • inter-context deduction is supported by bridge rules • databases are sets of formulae tagged with deductive histories and the contexts they belong to • abduction and revision are supported by a notion of consistency of formulae and sets of formulae which are relative to a context, and which can, in turn, be seen as constituents of agendas

There are several areas in applied logic where deletion from databases is involved in one way or another:Belief contraction Triggers of the form ‘If condition then remove A’, which are extensively used in database management systemsResource considerations as in relevance and linear logics, where addition or removal of resource can affect provabilityFree logic and the like, where existence and non-existence of individuals affects quantification.All of these areas have certain logical difficulties relating to the removal of elements. These difficulties are normally overcome by working on the meta-level. This paper offers a comprehensive model in which the various forms of removal are treated on the object level, i.e., as connectives of the object language

One of our purposes here is to expose something of the elementary logical structure of abductive reasoning, and to do so in a way that helps orient theorists to the various tasks that a logic of abduction should concern itself with. We are mindful of criticisms that have been levelled against the very idea of a logic of abduction; so we think it prudent to proceed with a certain diffidence. That our own account of abduction is itself abductive is methodological expression of this diffidence. A second objective is to test our conception of abduction's logical structure against some of the more promising going accounts of abductive reasoning. We offer our various suggestions in a benignly advisory way. The primary targets of our advice is ourselves, meant as guides to work we have yet to complete or, in some instances, start. It is possible that our colleagues in the abduction research communities will find our counsel to be of some interest. But we repeat that our first concern is to try to get ourselves straight about what a logic of abduction should encompass

The present collection of essays honours John Woods on the occasion of his eightieth birthday from contributors who wish to pay homage to this remarkable researcher whom they see not only as a scholar of prodigious energy and insight, but as a friend, colleague, collaborator, or former teacher. All of the essays touch upon topics Woods has taken a direct or indirect interest in, ranging from technical problems of mathematical logic and applications of formal methods through philosophical logic, philosophy of language, epistemology and history of ancient and modern logic. The sections themselves of the book cover relevance, paradoxes, inconsistency, fallacies, abduction, fictions, logic naturalized, law, argumentation and narratives, and many other topics: and represent the main research areas which have profited from Woods' reflections.

This paper is intended first for the formal argumentation community (see https://comma.csc.liv.ac.uk/). This community develops logics and systems modelling argumentation and dialogues. The community is in search of major applications areas for their models. One such application area e.g. is Law. The message of this paper is that there is another major application area for formal argumentation. There is an international community of sex offender therapist that is well established and well funded, and their therapy methods use (methods that can be modelled by) formal argumentation and logic. This community presents a natural application area for formal argumentation. We thus describe in this paper how the sex offender therapists work, to give the formal argumentation researcher a view of this application area. What is especially important about this application area is that in order to model it and learn from it, the formal argumentation community have to evolve their formal methods and adapt to this new application. Part of this enhancement is to modify and import certain methods from other areas of Logic e.g. from Non-Monotonic logic. The members of the formal argumentation community are not familiar, on average, with other areas of logic, and so we also describe in this paper, what we need from neighbouring logics. This makes this paper of interest also to sex offender therapist as well. They may be already familiar with their own practices, but the additional logics described will be of interest to them.

There is a generic way to add any new feature to a system. It involves identifying the basic units which build up the system and introducing the new feature to each of these basic units. In the case where the system is argumentation and the feature is probabilistic we have the following. The basic units are: the nature of the arguments involved; the membership relation in the set S of arguments; the attack relation; and the choice of extensions. Generically to add a new aspect to an argumentation network \ can be done by adding this feature to each component. This is a brute-force method and may yield a non-intuitive or meaningful result. A better way is to meaningfully translate the object system into another target system which does have the aspect required and then let the target system endow the aspect on the initial system. In our case we translate argumentation into classical propositional logic and get probabilistic argumentation from the translation. Of course what we get depends on how we translate. In fact, in this paper we introduce probabilistic semantics to abstract argumentation theory based on the equational approach to argumentation networks. We then compare our semantics with existing proposals in the literature including the approaches by M. Thimm and by A. Hunter. Our methodology in general is discussed in the conclusion

Given an argumentation network with initial values to the arguments, we look for algorithms which can yield extensions compatible with such initial values. We find that the best way of tackling this problem is to offer an iteration formula that takes the initial values and the attack relation and iterates a sequence of intermediate values that eventually converges leading to an extension. The properties surrounding the application of the iteration formula and its connection with other numerical and non-numerical techniques proposed by others are thoroughly investigated in this paper