Institute for Web Science and Technologies · Universität Koblenz - Landau
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Seminar "Measuring Inconsistency"

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Winter Term 2019 / 2020

Inconsistency is a ubiquitous phenomenon whenever knowledge is compiled is some formal language. The notion of inconsistency refers (usually) to multiple pieces of information and represents a conflict between those, i.e., they cannot hold at the same time. The two statements "It is sunny outside" and "It is not sunny outside" represent inconsistent information and in order to draw meaningful conclusions from a knowledge base containing these statements, this conflict has to be consolidated somehow. The field of Inconsistency Measurement provides an analytical perspective on the issue of inconsistency. Its aim is to quantitatively assess the severity of inconsistency in order to both guide automatic reasoning mechanisms and to help human modellers in identifying issues and compare different alternative formalisations.

Consider the following two knowledge bases K1 and K2 formalised in classical propositional logic modelling some information about the weather:

K1 = {sunny, -sunny, hot, -hot}
K2 = {-hot, sunny, sunny => hot,humid }

Both K1 and K2 are classically inconsistent, i.e., there is no interpretation satisfying any of them. But looking closer into the structure of the knowledge bases one can identify differences in the severity of the inconsistency. In K1 there are two apparent contradictions, i.e., {sunny,-sunny} and {hot,-hot} are directly conflicting formulas. In K2, the conflict is a bit more hidden. Here, three formulas are necessary to produce a contradiction ({-hot, sunny, sunny => hot}). Moreover, there is one formula in K2 (humid), which is not participating in any conflict and one could still infer meaningful information from this by relying on e.g. paraconsistent reasoning techniques. In conclusion, one should regard K1 as more inconsistent than K2. Inconsistency measures aim at formalising this intuition.

Seminar topics

  1. Measuring inconsistency using probability theory [11]
  2. Measuring inconsistency using distances [6]
  3. Measuring inconsistency using minimal inconsistent sets I [8]
  4. Measuring inconsistency using minimal inconsistent sets II [10]
  5. Measuring inconsistency using maximal consistent sets [1]
  6. Measuring inconsistency using forgetting [2]
  7. Measuring inconsistency using many-valued logics [13]
  8. Measuring inconsistency using prime implicates [9]
  9. Measuring inconsistency in argumentation graphs [7]
  10. Measuring inconsistency in first-order logic [5]
  11. Measuring inconsistency in description logic [12]
  12. Measuring inconsistency in probabilistic logics [3]
  13. Measuring inconsistency in data bases [4]
  14. Measuring inconsistency in non-monotonic logics [14]


  1. Meriem Ammoura, Yakoub Salhi, Brahim Oukacha, and Badran Raddaoui. On an MCS-based inconsistency measure. International Journal of Approximate Reasoning, 80:443–459, 2017.
  2. Philippe Besnard. Forgetting-based inconsistency measure. In Proceedings of the 10th International Conference on Scalable Uncertainty Management (SUM’16), pages 331–337, 2016.
  3. Glauber De Bona, Marcelo Finger, Nico Potyka, and Matthias Thimm. Inconsistency measurement in probabilistic logic. In John Grant and Maria Vanina Martinez, editors, Measuring Inconsistency in Information, volume 73 of Studies in Logic. College Publications, February 2018.
  4. Hendrik Decker and Sanjay Misra. Database inconsistency measures and their applications. In Proceedings of the 23rd International Conference on Information and Software Technologies (ICIST 2017), 2017.
  5. John Grant and Anthony Hunter. Analysing inconsistent first-order knowledgebases. Artificial Intelligence, 172(8-9):1064–1093, May 2008.
  6. John Grant and Anthony Hunter. Analysing inconsistent information using distance-based measures. International Journal of Approximate Reasoning, 89:3–26, 2017.
  7. Anthony Hunter. Measuring inconsistency in argument graphs. Technical report, University College London, 2017.
  8. Anthony Hunter and Sebastien Konieczny. Measuring inconsistency through minimal inconsistent sets. In Gerhard Brewka and Jerome Lang, editors, Proceedings of the Eleventh International Conference on Principles of Knowledge Representation and Reasoning (KR’2008), pages 358–366, Sydney, Australia, September 2008. AAAI Press, Menlo Park, California.
  9. Said Jabbour, Yue Ma, Badran Raddaoui, and Lakhdar Sais. Quantifying conflicts in propositional logic through prime implicates. International Journal of Approximate Reasoning, 2017.
  10. Said Jabbour and Lakhdar Sais. Exploiting MUS structure to measure inconsistency of knowledge bases. In Proceedings of the 22nd European Conference on Artificial Intelligence (ECAI’16), pages 991–998, 2016.
  11. Kevin M. Knight. Measuring inconsistency. Journal of Philosophical Logic, 31:77–98, 2001.
  12. Yue Ma, Guilin Qi, Pascal Hitzler, and Zuoquan Lin. Measuring inconsistency for description logics based on paraconsistent semantics. In Proceedings of the 9th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU ’07, pages 30–41, Berlin, Heidelberg, 2007. Springer-Verlag.
  13. Matthias Thimm. Measuring inconsistency with many-valued logics. International Journal of Approximate Reasoning, 86:1–23, July 2017.
  14. Markus Ulbricht, Matthias Thimm, and Gerhard Brewka. Measuring strong inconsistency. In Proceedings of the 32nd AAAI Conference on Artificial Intelligence (AAAI’18), pages 1989– 1996. AAAI Press, February 2018.


Preparatory meeting: July 15th, 2019, 15:15 (room F.522)

Seminar dates: Mondays 10:15-11:45, winter term 2019/20, (room F.330)


In order to participate in the seminar a participation in the preparatory meeting on July 15th, 2019, 15:15 (room F.522) is necessary. If you are interested in attending the seminar, please inform Matthias Thimm in advance by sending an informal e-mail.

Participants are expected to give a presentation (approx. 30 minutes) on one of the above topics. Following the seminar, a paper on the seminar topic (approx. 12 pages) is to be submitted.

Please adhere to the following general guidelines when preparing the presentation and the paper: pdf


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