Abstract
In this chapter we generalize the standard format of adaptive logics and thereby introduce an interesting larger class of adaptive logics that can be characterized in a simple and intuitive way. We demonstrate that the new format overcomes the two shortcomings of the standard format. On the one hand, logics with both qualitative and quantitative rationales can be expressed in it. On the other hand, the format is expressive enough to allow for the handling of priorities in various ways. We show that many adaptive logics that have been considered in the literature fall within this larger class -for instance adaptive logics in the standard format, adaptive logics with counting strategies, lexicographic adaptive logics-and that the characterization of this class offers many possibilities to formulate new logics. One of the advantages of the format studied in this chapter is that a lot of meta-theory comes for free for any logic formulated in it. We show that adaptive logics formulated in it are always sound and complete. Furthermore, many of the meta-theoretic properties that are usually associated with the standard format (such as cumulativity, fixed point property, (strong) reassurance, etc.) also hold for rich subclasses of logics formulated in the new format.
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.
The idea of selecting a certain set of models and then to define a semantic consequence relation on the basis of this selection is an integral part of many formal systems. Variants of it can be found in e.g., Shoham [8, 9], McCarthy [10], Schlechta [11], etc. Lindström [12] and Makinson [13] offer systematic overviews.
- 2.
In this chapter we will use \(\prec \) to denote a strict partial order. Of course, one can easily define the corresponding non-strict \(\preceq \) by \(a \preceq b\) iff \(a\prec b\) or \(a = b\). Hence, this is a purely conventional choice.
- 3.
It can easily be shown that \({\mathbf {AL_{\min _\subset }^\circ }}\) represents the universal Rescher-Manor consequence relation: \(A\) is derivable from all maximally consistent subsets of \(\varGamma \) iff \(\varGamma ^\circ \vdash _{\mathbf{AL}_{\min _\subset }^\circ } A\). See [15].
- 4.
- 5.
Of course, for finite sets \(\varphi \subset \psi \) implies \(|\varphi | < |\psi |\). However, for infinite sets the comparison by means of the cardinality does not allow to prefer \(\varphi \) to \(\psi \) in case of \(\varphi \subset \psi \), although the latter clearly indicates that \(\varphi \) is “better” (“less abnormal”) than \(\psi \).
- 6.
In the Appendix (Corollary C.2.1) we prove that \(\min _\subset ^\cup \bigl (\mathsf{Ab}_{{\mathbf {LLL}}}^{\varGamma }\bigr ) = \mathsf{Ab}_{{\mathbf {LLL}}}^{\varGamma '}\) where \(\varGamma ' = \varGamma \cup (\varOmega \setminus \bigcup \varSigma (\varGamma ))^{\,\,\check{\lnot }}\). This shows that the models in \(\min _\subset ^\cup \bigl (\mathsf{Ab}_{{\mathbf {LLL}}}^{\varGamma }\bigr )\) are indeed exactly the models which do not validate any abnormalities which are not part of any minimal \(\mathsf{Dab}\)-consequence.
- 7.
In [15] we show that the AL \({\mathbf{A}{{\mathbf {CL_\circ }}}^\mathbf{r }}\) in standard format that is characterized by the triple \({\langle {{\mathbf {CL_\circ }}}, {\varOmega _{\circ }},}\) reliability\({\rangle }\) represents the free Rescher-Manor consequence relation: \(A\) is a free Rescher-Manor consequence from \(\varGamma \) iff \({\varGamma ^{\circ }} \Vdash _\mathbf{ACL_{\circ }{}^\mathbf{r}}A\).
- 8.
Of course, we could define a logic on the basis of \({\mathbf {CL_\circ }}\) that realizes the same idea. Instead of e.g., \(\circ _3 A\) we could use \({\circ }{\circ }{\circ } A\) in order to express that \(A\) is stated at time point \(3\).
- 9.
\(\langle X, \prec \rangle \) is smooth iff for all \(x \in X\) there is a \(y \in \min _\prec (X)\) such that \(y \preceq x\).
- 10.
- 11.
- 12.
This is not the same as simply defining another partial order \(\prec _\mathsf{co}'\) by “\(\mathrm{Ab}(M) \prec _\mathsf{co}' \mathrm{Ab}(M')\) iff \(\mathrm{Ab}(M) \prec _\mathsf{co}\mathrm{Ab}(M')\) or \(\mathrm{Ab}(M) \subset \mathrm{Ab}(M')\)” and then to use \(\varPsi _{\prec _\mathsf{co}'}\). Note that since \({\subset } \subseteq {\prec _\mathsf{co}}\), also \({\prec _\mathsf{co}'} = {\prec _\mathsf{co}}\).
- 13.
We prove that \(\min _{\prec _c}\bigl (\mathsf{Ab}_{{\mathbf {LLL}}}^{\varGamma }\bigr )\) (where \({\mathbf {LLL}}\) qualifies as a lower limit logic such as our \({\mathbf {CL_\circ ^\star }}\)) is non-empty for \({\mathbf {LLL}}\)-non-trivial \(\varGamma \) in Sect. 5.8.2 (see Fact 5.8.2 in combination with Lemma 5.3.2).
- 14.
Where \(n \in \mathbb {N}^\infty \), we define \({\infty }/{n} =_\mathrm{df} \infty \). We show later that \(\min _\subset (\mathsf{Ab}_{{\mathbf {LLL}}}^{\varGamma })\) is non-empty for \({\mathbf {LLL}}\)-non-trivial \(\varGamma \) (see Lemma 5.3.2 and Theorem 5.5.4): hence we do not have to worry about division by zero.
- 15.
We still focus on the semantic aspect of ALs in order not to open more doors than necessary at this point of the discussion. However, this should not distract from the fact that all these semantic features have a syntactic counter-part. We will investigate also the syntax of ALs beginning with the next section.
- 16.
In order to reduce notational clutter, we will in the following not distinguish between \({\mathbf {LLL}}\) and \({\mathbf {LLL^+}}\) and always write \({\mathbf {LLL}}\) in order to denote either. (Compare the discussion in Sect. 2.7). The context will always disambiguate this.
- 17.
We will in the remainder skip the reference to \({\mathbf {LLL}}\) whenever the context disambiguates.
- 18.
\(X\) is a \(\prec \) -lower set of \(Y\) iff for all \(x \in X\) and all \(y\in Y\), if \(y \prec x\) then \(y \in X\).
- 19.
The reader may for the moment think of \(\varLambda _s^\varGamma \) as denoting the set \(\varLambda (\varXi _{s}(\varGamma ))\). However, we will have to make a slight adjustment below (see Definition 5.3.5).
- 20.
For the sake of simplicity, we disregard in this discussion “checked connectives” (see Sect. 2.7).
- 21.
Note that we cannot introduce new minimal \(\mathsf{Dab}\)-formulas in the proof.
- 22.
Note that T1–T3 are indeed applicable since \(\varXi ^\mathrm{sat}_s(\varGamma ) , \min _\subset ^\cup (\varXi _s(\varGamma )) \in \varUpsilon \) as we show in Lemma C.2.1 in the Appendix.
- 23.
In [12] these criteria were first proposed for preferential semantics.
- 24.
In Theorem 5.5.4 and Fact 5.8.2 we show that \(\langle X, \subset \rangle \) and \(\langle X, \prec _c\rangle \) are smooth for all \(X \in \varUpsilon \) which ensures T3. In view of the definition of \(\prec _c\), \(\min _{\prec _c}(X) \subseteq \min _\subset (X)\) which ensures T1. T2 is evident.
- 25.
See Fact C.3.1.
- 26.
This and the following lemma are proven in Appendix A.
- 27.
This is shown in the Appendix: Fact C.3.4 and Fact C.3.11.
- 28.
See Fact C.3.1 in Appendix C.
- 29.
This is proven in Appendix C.4.
- 30.
See Theorem C.4.3.
- 31.
Or equivalently: \(\varGamma \Vdash _{{\mathbf {AL_{\varvec{\Lambda }}^n}}} A\) iff there is an \(M \in {\mathcal {M}}_{{\mathbf {AL_{\varvec{\Lambda }}}}}\bigl (\varGamma \bigr )\) such that for all \(M' \in {\mathcal {M}}_{{\mathbf {LLL}}}\bigl (\varGamma \bigr )\) for which \(\mathrm{Ab}(M) = \mathrm{Ab}(M')\), \(M' \models A\).
- 32.
Note that \(\min _\subset (\varXi (\varGamma ))\) is just another way of writing \(\varPhi (\varGamma )\).
- 33.
In the Appendix we show that \(\varPsi _\prec \) and \(\varPsi _{[\prec _1, \ldots , \prec _{n}]}\) are threshold functions (see Fact C.3.8).
- 34.
See Appendix: \(\varPsi _\prec \) satisfies both DI \(_\prec \) (Fact C.3.3) and RA \(_\prec \) (Fact C.3.4), and hence by Fact 5.5.1 also CT and CM. \(\varPsi _{[\prec _1, \ldots , \prec _{n}]}\) satisfies CT and CM by Fact C.3.12.
- 35.
By Fact C.3.3, \(\varPsi _\prec \) satisfies DI \(_\prec \) and hence by Fact 5.6.1 also SIMP. By Fact C.3.13, \(\varPsi _{[\prec _1, \ldots , \prec _{n}]}\) satisfies DI \(_{\prec _n}\) and hence by Fact 5.6.1 also SIMP.
- 36.
T1 and T2 are trivially satisfied, for T3 see the discussion in Sect. 5.5.3.
- 37.
The fact that this logic indeed characterizes the reliability strategy is proven in Theorem C.2.1 in the Appendix.
- 38.
For this and other relationships among the criteria see Fig. 5.3. For the proofs see Fact C.3.1 in the Appendix.
- 39.
See Lemma C.1.1 in the Appendix.
References
Primiero, G., Meheus, J.: Majority merging by adaptive counting. Synthese 165, 203–223 (2008)
Primiero, G., Meheus, J.: Quasi-merging and pure-arbitration on information for the family of adaptive logics ADM. In: Proceeding of the Workshop on Logic and Intelligent Interaction, ESSLLI, pp. 21–30. Hamburg (2008)
Primiero, G., Meheus, J.: Adaptive arbitration by variant counting on commuatative bases with weights. In: Proceedings of the Fusion 2008 Conference, Köln, 1–3 July, pp. 1374–1380. ISBN 978-3-00-024883-2 (2008)
Putte, F.V.D., Straßer, C.: Extending the standard format of adaptive logics to the prioritized case. Logique at Analyse 55(220), 601–641 (2012)
Putte, F.V.D., Straßer, C.: A logic for prioritized normative reasoning. J. Logic Comput. 23(3), 568–583 (2013)
Putte, F.V.D.: Generic formats for prioritized adaptive logics. with applications in deontic logic, abduction and belief revision. Ph.D. thesis, Ghent University (March 28th, 2012)
Verdée, P.: Combining a rich paraconsistent negation with classical negation by means of an infinitely valued logic (in preparation)
Shoham, Y.: A semantical approach to nonmonotonic logics. In: Ginsberg, M.L. (ed.) Readings in Non-Monotonic Reasoning, pp. 227–249. Morgan Kaufmann, Los Altos (1987)
Shoham, Y.: Reasoning about change: time and causation from the standpoint of artificial intelligence. Cambridge, MIT Press (1988)
McCarthy, J.: Circumscription—a form of non-monotonic reasoning. Artif. Intell. 13, 27–29 (1980)
Schlechta, K.: Coherent Systems. Elsevier, Amsterdam (2004)
Lindström, S.: A semantic approach to nonmonotonic reasoning: inference operations and choice. Uppsala Prints and Reprints in, Philosophy, no 10, Department of Philosophy, University of Uppsala (1994)
Makinson, D.: General patterns in nonmonotonic reasoning. In: Handbook of Logic in Artificial Intelligence and Logic Programming, vol. III, pp. 35–110. Clarendon Press, Oxford (1994)
Rescher, N., Manor, R.: On inference from inconsistent premises. Theor. Decis. 1, 179–217 (1970)
Meheus, J., Straßer, C., Verdée, P.: Which Style of Reasoning to Choose in the Face of Conflicting Information? J. Logic. Comput. (forthcoming)
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)
Batens, D.: Minimally abnormal models in some adaptive logics. Synthese 125, 5–18 (2000)
Bossu, G., Siegel, P.: Saturation, nonmonotonic reasoning and the closed-world assumption. Artif. Intell. 25, 13–63 (1985)
Goldzsmidt, M., Morris, P., Pearl, J.: Amaximumentropy approach to nonmonotonic reasoning. IEEE Trans. Pattern Anal. Mach. Intell. 15(3), 220–232 (1993)
Acknowledgments
The results presented in this chapter are the product of joint research with Frederik Van De Putte.
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). Generalizing the Standard Format. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-00792-2_5
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)