# Adaptively applying modus ponens in conditional logics of normality

TL;DR: This paper presents an adaptive logic enhancement of conditional logics of normality that allows for defeasible applications of Modus Ponens to conditionals, and is enriched by the ability to perform default inferencing.

Abstract: This paper presents an adaptive logic enhancement of conditional logics of normality that allows for defeasible applications of Modus Ponens to conditionals. In addition to the possibilities these logics already offer in terms of reasoning about conditionals, this way they are enriched by the ability to perform default inferencing. The idea is to apply Modus Ponens defeasibly to a conditional and a fact on the condition that it is ‘safe' to do so concerning the factual and conditional knowledge at hand. It is for instance not safe if the given information describes exceptional circumstances: although birds usually fly, penguins are exceptional to this rule. The two adaptive standard strategies are shown to correspond to different intuitions, a skeptical and a credulous reasoning type, which manifest themselves in the handling of so-called floating conclusions.

## 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

...Some scholars voice worries (see e.g. Bonevac, 2003; Jeffry & Renee, 1994; Koons, 2009; Wobcke, 1995)....

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...In recent years the traditional formalisms of default reasoning such as presented in the landmark articles on default logic (Reiter, 1980), on circumscription (Mc-Carthy, 1980), and on autoepistemic logic (Moore, 1984) have been criticised and alternative conditional approaches have been developed....

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...compared to other systems which are able to model default inferencing such as Delgrande’s (1988), Lamarre’s (1993), or Geffner and Pearl’s (1992)....

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...Hence, the authors only offer a computational approximation in terms of an assumption-based truth maintenance-like system (see Kleer, 1987)....

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### "Adaptively applying modus ponens in..." refers background in this paper

...For instance their relevance for belief revision has been investigated in Wobcke (1995), Boutilier (1994b), and Booth (2001)....

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...Most of the research in this domain has been in the vein of the following influential conditional logics: Stalnaker (1968) and Lewis (2000) who offer an ontic interpretation of the conditional, Adams (1975) who introduces probabilities in the discussion, and Gärdenfors’ (1978) belief revision…...

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