Adaptively applying modus ponens in conditional logics of normality
Summary (2 min read)
3. Modus Ponens in Conditional Logics of Normality
- In this section I will informally motivate and outline the main idea behind the modeling of a defeasible MP in this paper.
- Informally speaking, specificity occurs if a more specific argument overrides a more general one.
- In their case abnormalities are of the formA.
- There are two adaptive strategies that specify what it exactly means that a condition of a line is “unsafe”.
- Then, in Section 5, the adaptive logics for conditionally applying MP will be defined.
4. Adaptive logics
- An adaptive logicAL in standard format is a triple consisting of (i) a lower limit logic , which is a reflexive, transitive, monotonic, and compact logic that has a characteristic semantics and containsCL (classical logic), (ii) a set of abnormalitiesΩ, characterized by a (possibly restricted) logical form, and (iii) an adaptive strategy.
- Formulating an adaptive logic in the standard format provides the logic with all of the important meta-theoretic features, such as soundness and completeness (as is shown in (Batens, 2007)).
- The proof dynamics is governed by a markingdefinition for proof lines.
- Also for applications ofRC conditions are carried forward, as it was the case for RU.
- For the reliability strategy only models are considered whose abnormal part is a subset of the set of unreliable formulas.
5. Applying Modus Ponens Conditionally
- These are propositions that are excepted by the information given in the premises.
- 12 Instead of trying to have the final word on the discussion I want to point out that, as the example shows, the minimal abnormality strategy detaches floating conclusions, while the more skeptical reliability strategy rejects them.
- It is entailed byPmin. 13 The situation is slightly different inDRpx: besidesbn−1 andbn alsobn−2 ∨ ¬bn−2 isRp-derivable from the premises.
- Thus, in the given example their logic handles the transitive relations between defaults better than these systems, since (with both strategies)d is derivable following argumenta c d.
- This example illustrates a more complex case of specificity.
6. Discussion
- Also in comparison with other systems from the literature.
- Some advantages of the adaptive approach Adaptive logics offer a very generic framework enabling defeasible MP for conditional logics of normality since they can be applied to anyconditional lower limit logic as long as it is reflexive, transitive, monotonic and compact.
- What is derivable by classical logic from these maximal contingent extensions corresponds to the factual consequences the authors draw via default reasoning.
7. Conclusion
- In this paper an adaptive logic approach to Modus Ponens for conditional logics of normality was presented.
- By meansof benchmark examples it was demonstrated that the adaptive systems deal with specificity and conflicting 23.
- Adaptively Applying Modus Ponens 23 arguments in an intuitive way.
- Acknowledgements Research for this paper was supported by the Research Fund ofGhent University by means of Research Project 01G01907.
- I thank Joke Meheus and Du ja Šešelja and the three anonymous reviewers for valuable comments to aformer version of this paper.
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Cites background from "Adaptively applying modus ponens in..."
...For some illustrations of how to do deal with some well-known cases of specificity within the adaptive logics framework, see Straßer (2011, 2012)....
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References
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Additional excerpts
...Adding the following Rational Monotonicity principle to the core properties yields logic R (see Lehmann & Magidor, 1992):5 ‘ ððA↝CÞ ^ ðAO:BÞÞ ððA ^ BÞ↝CÞ ðRMÞ The core properties are not without criticism....
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...Some weakening or variants of Rational Monotonicity have been proposed: e.g. ‘ ððA↝BÞ ^ ððA ^ CÞO:BÞÞ ððA ^ CÞ↝BÞ (IRR) in the context of Description Logic by Giordano et al. (2009) or in the context of conditional deontic logics ‘ ððA↝BÞ ^ ðAO: ðB ^ CÞÞÞ ððA ^ CÞ↝BÞ (WRM) by Goble (2004)....
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35 citations
"Adaptively applying modus ponens in..." refers background in this paper
...Furthermore, the influential work in Kraus et al. (1990) is greatly generalised in Arieli and Avron (2000) by their plausible nonmonotonic consequence relations, and in Friedman and Halpern (1996) by their plausibility measures....
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33 citations
"Adaptively applying modus ponens in..." refers methods in this paper
...Furthermore, similar techniques as presented here for defeasible MP in the context of default reasoning can be applied to conditional deontic logics (see Straßer, 2010)....
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32 citations
"Adaptively applying modus ponens in..." refers methods in this paper
...Another rule-based approach is e.g. presented by Dung and Son (2001)....
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