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David Makinson

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6 papers
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6

FLAP Journal 2025 Journal Article

Boole Between the Equations

  • David Makinson

Two questions are investigated. Why is it that, despite his general expla- nations, Boole’s logical operations can be difficult to pin down when he solves specific problems? Why did his late manuscript attempt to get rid of division by zero fall short of its goal? It is suggested that the former difficulty arises from his segmentation of solution-routines into stages, with the identity of the operations differing from stage to stage; while the latter failure stems from his continuing confinement to strictly equational reasoning. 1 Two questions Why is it that despite Boole’s general explanations, it is so difficult to pin down even his basic operations of multiplication, addition and subtraction when reading his solutions to logical problems? Why is it that when, in a late manuscript only now being published, he finally got down to the task of trying to eliminate the controverted operation of division from his procedures, he did not fully succeed? These are the central questions of the present paper. In outline, the conclusions are as follows: • For Boole, solving a logical problem consisted essentially in running a routine. The routines have stages, and the identities of even the basic operations of multiplication, addition and subtraction are not fixed, but vary according to the stage of the routine in which they appear. These stages and their opera- tions are analysed in section 3, revealing also several different ways of reading Boole’s celebrated {0, 1} principle. The author thanks David Waszek for generously making available the edited text of Boole’s manuscript ‘The nature of thought’ before its publication and general discussions of Boole’s work. Thanks also to Alex Citkin, Antonielly Garcia Rodrigues and again David Waszek for helpful information, comments and corrections on drafts.

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AIJ Journal 1994 Journal Article

Nonmonotonic inference based on expectations

  • Peter Gärdenfors
  • David Makinson

We show how nonmonotonic inferences may elegantly be interpreted in terms of underlying expectations. The fundamental idea is that when we reason, we make use of not only the information that we firmly believe, but also expectations that guide our beliefs without quite being part of them. We propose two ways of modelling the expectations used in nonmonotonic reasoning: by expectation sets, equipped with selection functions, and by expectation relations. For each of these we prove representation theorems and establish relations with several other modellings in the area, including Poole systems and preferential models. We also show that by using the notion of expectation, one can unify the treatment of the theory of belief revision and that of nonmonotonic inference relations. This is accomplished by viewing the relation of “epistemic entrenchment” used in Gärdenfors [18] and Gärdenfors and Makinson [20] as a kind of expectation ordering. Thus we see belief revision and nonmonotonic reasoning as basically the same process, albeit used for two different purposes.

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AIJ Journal 1991 Journal Article

Floating conclusions and zombie paths: Two deep difficulties in the “directly skeptical” approach to defeasible inheritance nets

  • David Makinson
  • Karl Schlechta

We discuss two difficulties in the “directly skeptical” approach to inference in defeasible inheritance nets, as developed by Horty, Thomason and Touretzky. We suggest that as a result of the general architecture of the approach, it is intrinsically unable to deal with a phenomenon of “floating conclusions”, and has great difficulty in accommodating a phenomenon of “zombi paths”. The conclusion drawn is that the directly skeptical approach cannot hope to do the work of an approach via the family of all extensions.

Details DOI
v2026.09.13