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Thus from Corollary A.1 we obtain that |α ∧ ¬γ| w < w |α ∧ γ| w or X 1 ∪ X 2 < w Y 1 ∪ Y 2 -
Given that we initially established that C +WCA (α ∧ β) → γ ⊃ α → (γ ∨ (α ∧ ¬β)), substitution of logical equivalents yields C +WCA (α ∧ β) → γ ⊃ α → (γ ∨ ¬β), which is the same as C +WCA (α ∧ β) -
An instance of WCA is (α ∧ β) → γ ⊃ ((α ∧ β) ∨ (α ∧ ¬β)) → (γ ∨ (α ∧ ¬β)) which is (α ∧ β) -
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⊃ ((α ∨ (α ∧ ¬β ∧ γ)) → (β ∨ (α ∧ ¬β ∧ γ))). Simplifying (via CEA) yields -
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- Publisher
- Springer Journals
- Copyright
- Copyright © 2007 by Springer Science+Business Media B.V.
- Subject
- Computer Science; Complexity; Computer Science, general ; Mathematics, general; Artificial Intelligence (incl. Robotics)
- ISSN
- 1012-2443
- eISSN
- 1573-7470
- DOI
- 10.1007/s10472-007-9044-7
- Publisher site
- See Article on Publisher Site
Abstract
In nonmonotonic reasoning, a default conditional α→β has most often been informally interpreted as a defeasible version of a classical conditional, usually the material conditional. There is however an alternative interpretation, in which a default is regarded essentially as a rule, leading from premises to conclusion. In this paper, we present a family of logics, based on this alternative interpretation. A general semantic framework under this rule-based interpretation is developed, and associated proof theories for a family of weak conditional logics is specified. Nonmonotonic inference is easily defined in these logics. Interestingly, the logics presented here are weaker than the commonly-accepted base conditional approach for defeasible reasoning. However, this approach resolves problems that have been associated with previous approaches.
Journal
Annals of Mathematics and Artificial Intelligence – Springer Journals
Published: May 24, 2007