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Adaptively Applying Modus Ponens in Conditional Logics of Normality

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Adaptive Logics for Defeasible Reasoning

Part of the book series: Trends in Logic ((TREN,volume 38))

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Abstract

This chapter 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 A ? B and a fact A on the assumption 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.

A previous version of this chapter was published in the Journal of Applied Non-Classical Logic under the same title [1]

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Notes

  1. 1.

    A floating conclusion is a proposition that can be reached by two conflicting and equally strong arguments (see our discussion in Sect. 2.5).

  2. 2.

    In some conditional logics of normality \(\leadsto \) is not primitive. For instance in [9] it is defined by making use of Kripkean modal logic. There the core properties are shown to be equivalent to an extension of the modal logic \({\mathbf {S4}}\). See [21] for a comparative study of various semantic systems for the core properties such as the preferential structures of [11], the \(\epsilon \)-semantics of [23], the possibilistic structures of [24] and \(\kappa \)-rankings of [25, 26].

  3. 3.

    We will use the name convention that is associated with conditional logics of normality (see [27, 28]) and not the one associated with nonmonotonic consequence relations which is used e.g. in [11].

  4. 4.

    The proofs are fairly standard and can be found e.g. in [11].

  5. 5.

    I adopt the names \({\mathbf {P}}\) and \({\mathbf {R}}\) for these logics from [19]. Although these are the same names as used for the systems in the pioneering KLM paper [11], the reader may be warned: the approach in terms of conditional logics differs from the KLM perspective which deals with rules of inference rather than with axioms. Also, strictly speaking, Rational Monotonicity as defined in [11] is a rule of inference whereas (RM) as defined above is an axiomatic counterpart to it.

  6. 6.

    Some weakening or variants of Rational Monotonicity have been proposed: e.g. \(\vdash _{{\mathbf {}}} ((A\leadsto B) \wedge ((A\wedge C) \not \leadsto \lnot B)) \supset ((A\wedge C) \leadsto B)\) (IRR) in the context of Description Logic by Giordano et al. [34] or in the context of conditional deontic logics \(\vdash _{{\mathbf {}}} ((A\leadsto B) \wedge (A\not \leadsto \lnot (B\wedge C))) \supset ((A\wedge C)\leadsto B)\) (WRM) by Goble in [35].

  7. 7.

    More precisely, Lexicographic Closure strengthens Rational Closure for all defaults with antecedents that have a finite rank: if \(A\) has finite rank and \(A\leadsto B\) is in the rational closure of \(D\), then \(A\leadsto B\) is in the lexicographic closure of \(D\).

  8. 8.

    The name (Inh) indicates that the property of being exceptional is inherited along \(\leadsto \)-paths.

  9. 9.

    A more precise notion of what it means that a condition is “unsafe” will be given in the next section by means of a marking definition.

  10. 10.

    Preemption plays an important role in the research on inheritance networks (see [48]).

  11. 11.

    While Ginsberg [49], and Makinson and Schlechta [50] argue for the acceptance, Horty [51] argues against it.

  12. 12.

    Note that in case we do not add \(a\not \leadsto \bot \) to our premises, \({\mathbf {P_{\min }}}\) is rather rigorous and also entails \(a \leadsto \bot \).

  13. 13.

    The reason is as follows. Suppose first that \(b_{n-1} \not \leadsto \lnot b_{n-2}\). In this case by means of (RM) and since \(b_{n-1} \leadsto b_n\) also \((b_{n-1} \wedge b_{n-2}) \leadsto b_n\). By (RT) and since \(b_{n-2} \leadsto b_{n-1}\), \(b_{n-2} \leadsto b_n\). But then since \(a\), \(a\leadsto \lnot b_n\) and \(a \leadsto \dots \leadsto b_{n-2} \leadsto b_n\), by (SpeG), \({\bullet }b_{n-2}\). Now suppose \(b_{n-1} \leadsto \lnot b_{n-2}\). Since \({\bullet }b_{n-1}\) we get \({\bullet }\lnot b_{n-2}\) by (Inh). Altogether, \({\bullet }b_{n-2} \vee {\bullet }\lnot b_{n-2}\). Note that this argument does not hold in \({\mathbf {Pp}}\) since it makes essentially use of (RM).

  14. 14.

    Note that \({\mathbf {P_{\min }}}\) entails \(a\leadsto d\) and moreover \(a\leadsto \bot \) (in case we do not manually add \(a \not \leadsto \bot \) to the premises, see also footnote 12).

  15. 15.

    The reason is as follows. Suppose \(\lnot {\bullet }c\). Suppose (i) \(b \leadsto \lnot a\). Since also \(a\), \(a\leadsto b\) and \(a\leadsto a\) (by (ID)) we get \({\bullet }b\) by (Spe2). Assume \(b \not \leadsto \lnot c\). By (RM), \((b\wedge c) \leadsto \lnot a\). Since also \(a\), \(a\leadsto a\) and \(a \leadsto (b \wedge c)\) (by (CC), \(a \leadsto b\) and \(a \leadsto c\)) we have \({\bullet }(b\wedge c)\) by (Spe2). By (Inh) and (CI) also \({\bullet }c\),—a contradiction. Hence, \(b \leadsto \lnot c\). By (Inh) \({\bullet }\lnot c\). Now suppose (ii) \(b \not \leadsto \lnot a\). Since \(b\leadsto \lnot e\) by (RM) \((a\wedge b)\leadsto \lnot e\). Since \(a \leadsto b\) by (RT) \(a\leadsto \lnot e\). By the latter, \(a\), \(\lnot {\bullet }c\) and \(a\leadsto c\) we have \(c \not \leadsto e\) due to (Spe2). By \(a\), \(a\leadsto \lnot e\) and \(a\leadsto c\leadsto d \leadsto e\) we have \({\bullet }d\) due to (SpeG). Assume \(d\not \leadsto \lnot c\). Then by (RM) \((c\wedge d)\leadsto e\) and by (RT), \(c \leadsto e\),—a contradiction. Hence \(d \leadsto \lnot c\) and by (Inh), \({\bullet }\lnot c\). Altogether we get \({\bullet }c \vee {\bullet }\lnot c\).

  16. 16.

    It is not entailed by \({\mathbf {P_{\min }}}\) in case we add \((p \wedge s \wedge r) \not \leadsto \bot \).

  17. 17.

    As will been shown, for instance, for Rational Closure in the next chapter.

  18. 18.

    As discussed in Sect. 6.2, ALs also employ semantic selections on the models of the \({\mathbf {LLL}}\).

  19. 19.

    Delgrande introduces in fact two equivalent proposals in this paper. The other one, which I do not discuss above, is based on forming maximal consistent extensions of the conditional knowledge base at hand (in contrast to the maximal consistent extensions of the factual knowledge which I discuss here). Note, however, that a similar criticism applies to both approaches.

  20. 20.

    Computing Conditional Entailment is a pretty complex and challenging task. Hence, the authors only offer a computational approximation in terms of an assumption-based truth maintenance-like system (see [57]).

  21. 21.

    The proofs can be found in the Appendix.

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Acknowledgments

I thank Joke Meheus, Dunja Šešelja, and the anonymous reviewers of the Journal of Applied Non-Classical Logic for valuable comments concerning a previous version of this chapter.

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    Christian Straßer

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Straßer, C. (2014). Adaptively Applying Modus Ponens in Conditional Logics of Normality. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_6

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