Logical problems with nonmonotonicity
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A few years ago, believing that human thinking is nonmonotonic, I tried to reconstruct a nonmonotonic reasoning by application of two monotonic procedures. I called them “step forward” and “step backward” (see ). The first procedure is just a consequence operation responsible for an extension of the set of beliefs. The second one, defined on the base of the logic of falsehood reconstructed for the given logic of truthfulness, is responsible for a reduction of the set of beliefs. Both procedures taken together were successfully verified by using so-called AGM (see ), postulates for expansion, contraction and revision formulated by Alchourrón, Gärdenfors and Makinson (e.g. ). Reasoning composed of the mutual application of both procedures seemed to be quite natural for modeling our thinking. At that time, I supposed that it should be nonmonotonic but I was wrong. It turned out impossible to satisfy a definition of the nonmonotonic inference by reasoning composed both steps. To understand why this is impossible, I began to analyze how nonmonotonicity is obtainable in some well-known cases in the literature. I analyzed the problem from two points of view: (1) non-formal examples for nonmonotonicity and (2) formal constructions of nonmonotonic operations/relations. The result of those investigations was astonishing: none of the considered by me cases of nonmonotonicity belonging to point (1) and almost none belonging to (2) satisfies the definition of nonmonotonic inference. Arguments against the nonmonotonic character of well-known examples for nonmonotonicity of human thinking are more precisely presented in . I present them below an abbreviated version of them.
- Alchourron, C., P. Gärdenfors and D. Makinson, “On the logic of theory change: contraction functions and their associated revision functions”, Theoria, 48 (1985): 4–37. DOI: 10.1111/j.1755-2567.1982.tb00480.x
- Gabbay, Dov M., C.J. Hogger and J.A. Robinson, Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 3, Clarendon Press, Oxford, 1994.
- Ginsberg, Matthew L., “AI and nonmonotonic reasoning”, pages 1–33 in .
- Łukowski, Piotr, “A formalisation of the “step forward — step backward” reasoning”, Anales del Seminario de Historia de la Filosofia, 18 (2001): 109–124.
- Łukowski, Piotr, “The procedures for belief revision”, pages 249–268 in Towards Mathematical Philosophy, Trends in Logic 28, Springer, 2009.
- Łukowski, Piotr, “Is human reasoning really nonmonotonic?”, Logic and Logical Philosophy, 22 (2013): 63–73. DOI: 10.12775/LLP.2013.004
- Poole, David, “Default logic”, pages 189–215 in .
- Makinson, David, “General patterns in nonmonotonic reasoning”, pages 35–110 in .
- Makinson, David, Bridgges from Classical to Nonmonotonic Logic, Text In Computing, Vol. 5, published by King’s College, 2005.
- Shoham, Y., Reasoning About Change, MIT Press, Cambridge, USA 1988.
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