Abstract
In this section sequential combinations of adaptive logics are studied under a generic perspective. We will provide meta-theoretic insights and dynamic proof theories.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
Diderik Batens remarks that Peter Verdée was the first to notice this problem.
- 3.
We prove a slightly more generic version than we need for Lemma 3.1.2 where we only make use of the case \(\varGamma \subseteq {{\mathcal {W}}}\). However, the gained generality will be useful in the next Section.
- 4.
See Lemma 2.4.1.
- 5.
Frederik Van De Putte was the first to come up with a very similar counter-example concerning the lack of soundness. Let \(\varGamma ' = \{A_i^1 \vee A_j^1 \mid i,j \in \mathbb {N}, i\ne j\} \cup \{B \vee A_i^1 \vee A_i^2 \mid i \in \mathbb {N}\}\) and take \(\mathbf{AL_1^m}\) and \(\mathbf{AL_2^m}\) from above. It can be shown that \(B\) is a syntactic consequence but not a semantic one with respect to the semantics defined above.
- 6.
The rather technical and lengthy proof can be found in Appendix B.1.
- 7.
We use R as a metavariable for a generic inference rule.
- 8.
The formula on line 9 is not a Dab \(_1\)-formula, since it contains the abnormality \({}!^2r\) which is not a member of \(\varOmega _1\).
- 9.
In case some \({\mathbf{x_i}} = {\mathbf{m}}\) this definition also makes reference to the \(i\)-marking for minimal abnormality which we define in Sect. 3.3.4.
- 10.
In [5] we present another sequential proof theory where this is not necessary. However, the price to pay is that the marking definition for minimal abnormality is more complicated.
- 11.
Usually the superscript \(^n\) is used in order to indicate the normal selections strategy. However, since we use the subscript \(_n\) in order to indicate the \(n\)-th logic in our sequence and in order to avoid needless ambiguities, we use \(^\mathrm{ns}\) for the normal selections strategy in this section.
- 12.
Note that in Sect. 2.8 we present several variants of a marking definition for normal selections. We do not spell out the corresponding variants in this section, but the necessary adjustments are obvious.
References
Batens, D.: Adaptive Logics and Dynamic Proofs. A Study in the Dynamics of Reasoning (201x)
Putte, F.V.D.: Hierarchic adaptive logics. Logic J. IGPL 20(1), 45–72 (2012)
Putte, F.V.D., Straßer, C.: Three formats of prioritized adaptive logics: a comparative study. Logic J. IGPL 2(21), 127–159 (2013)
Putte, F.V.D., Straßer, C.: Extending the standard format of adaptive logics to the prioritized case. Logique et Analyse 55(220), 601–641 (2012)
Straßer, C., Putte, F.V.D.: Proof theories for superpositions of adaptive logics (201x). Logique et Analyse (Forthcoming)
Batens, D.: A general characterization of adaptive logics. Logique et Analyse. 173–175, 45–68 (2001)
Acknowledgments
The research in this chapter is inspired by the work on combinations of ALs by Diderik Batens and by Frederik Van De Putte. I also thank both of them for valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Straßer, C. (2014). Sequential Combinations of ALs. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-00792-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00791-5
Online ISBN: 978-3-319-00792-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)