The 'Humean thesis', formulated by Hannes Leitgeb, seeks to coordinate the plain beliefs of an ideal agent with its degrees of belief by requiring that they should jointly satisfy a certain equivalence condition expressing the idea of stability under conditionalization. We draw attention to a feature of the proposed condition, namely the scarcity of probability functions that can enter into it, which appears to impede it from playing the desired bridging role.
An extended post-publication version, last updated , is available here.
Readers are advised to make use of this version. Work on the subject, and furher material may be integrated into the text later. We report on progress and an unsolved problem in our attempt to obtain a clear rationale for relevance logic via semantic decomposition trees. Suitable decomposition rules, constrained by a natural parity condition, generate a set of directly acceptable formulae that contains all axioms of the well-known system R, is closed under substitution and conjunction, satisfies the letter-sharing condition, but is not closed under detachment.
To extend it, a natural recusion is built into the procedure for constructing decomposition trees. The resulting set of acceptable formulae has many attractive features, but it remains an open question whether it continues to satisfy the crucial letter-sharing condition. A short version, giving only the essential ideas, is scheduled to appear, perhaps in , in a volume Logic in Question. Copies available from the author on request. This text is expository. Springer Reviews results that have been obtained about bounds on the loss of probability occasioned by applying classically sound, but probabilistically unsound, Horn rules for inference relations.
It uses only elementary finite probability theory without appealing to linear algebra, and also provides some new results, in the same spirit, on non-Horn rules. Pages of Sven Ove Hansson ed.
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Series: Outstanding Contributions to Logic. Springer, An extended post-publication text, last updated , is available here. It extends results, adds comments, corrects minor errors, and hopefully is much more reader-friendly. I continue to work on the subject, adding new material to the text.
Investigates introduction and elimination rules focusing on the general questions of the existence, for a given truth-functional connective, of at least one such rule that it satisfies, and the uniqueness of a connective with respect to the set of all of them. An outline of the main results without proofs appeared as " Advice to the relevantist policeman " in The Logica Yearbook ed. Relevance logic is ordinarily seen as a subsystem of classical logic under the translation that replaces arrows by horseshoes. If, however, we consider the arrow as an additional connective alongside the horseshoe and other classical connectives, another perspective emerges.
Relevance logic, specifically the system R, may be seen as the output of a conservative extension of classical consequence into the language with arrow. We seek a better understanding of why an evaluation system devised by Tor Sandqvist yields full classical logic, despite its inferential character, by providing and analysing a direct proof of the fact using a suitable maximality construction. We reflect on lessons that the lottery and preface paradoxes provide for the logic of uncertain inference.
One of these lessons is the unreliability of the rule of conjunction of conclusions in such contexts, whether probabilistic or qualitative. Another lesson is the possibility of rational inconsistent belief. This leads us to discuss criteria for deciding when an inconsistent set of beliefs may nevertheless reasonably be retained. Journal of Philosophical Logic , A preliminary version, less complete than the final one, was published in Hykel Hosni and Franco Montagna eds , Probability, Uncertainty and Rationality. Applied Logic 7 December The concept of relevance between classical propositional formulae, defined in terms of letter-sharing, began to take on a fresh life in the late s when it was reconsidered in the context of the logic of belief change.
This paper develops and extends these ideas in a systematic way. ISBN: Reviews the connections between different kinds of nonmonotonic logic and the general idea of varying degrees of belief. Summarizes the leading ideas of the book B3. An explanation, without proofs, of the results of 64 , focussing on the application to the logic of belief change and discussing the significance of the formal results.
Journal of Symbolic Logic 72 September Shows that although AGM partial meet contraction and revision fail Parikh's criterion of relevance, the criterion is respected if we apply the operations to a normalized version of a belief set, obtained by applying the finest splitting theorem to it. Studia Logica 86 September Charts the ways in which closure properties of consequence relations for uncertain inference take on different forms according to whether the relations are generated in a quantitative or a qualitative manner.
A discussion of the connections and differences between completeness and representation theorems in logic, with examples drawn from classical and modal logic, the logic of friendliness, and nonmonotonic reasoning. Beziau Basel: Birkhauser Verlag, Defines and examines a notion of logical friendliness, a broadening of the familiar notion of classical consequence. Also reviews familiar notions and operations with which friendliness makes contact, providing a new light in which they may be seen.
Essays in Honour of Dov Gabbay, vol 2 , ed S. Artemov, H. Barringer, A. Garcez, L. Lamb, J. Gabbay and F. Guenthner Amsterdam, Springer, , pages An extended overview of the essentials of nonmonotonic logic.
Human Nonmonotonic Reasoning: the Importance of Seeing the Logical Strength of Arguments
Based on the perspectives developed in the papers 53 , Systems of natural deduction transform logically pure second-level derivations into first-level ones, introducing annotations and procedural steps but losing logical purity. Can we obtain logical purity without reconstructing everything in terms of sequent-calculi?
Chapter 6 pp in P. Springer Verlag, An overview of the basic ideas of nonmonotonic inference, written for a readership of economists with some background in probability but little if any in logic. Chapter 13 pp in P. An expository paper, describing some of the different kinds of conditionals that appear in ordinary discourse, and ways in which logicians have sought to model them.
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Volume Defeasible inheritance systems and reactive diagrams Dov M Gabbay. Oxford Academic.
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Issue Section:. You do not currently have access to this article. Download all figures. Sign in. You could not be signed in. Nonmonotonic logics were created as an abstraction of some types of common sense reasoning, analogous to the way classical logic serves to formalize ideal reasoning about mathematical objects. These logics are nonmonotonic in the sense that enlarging the set of axioms does not necessarily imply an enlargement of the set of formulas deducible from these axioms.