Abstract
We present a logic for reasoning about the interplay between belief, evidence and trust in a multi-agent setting. We call this logic DL-BET which stands for “Dynamic Logic of Belief, Evidence and Trust”. According to DL-BET, if the amount of evidence in support a given fact \(\varphi \) and the ratio of evidence in support of \(\varphi \) to the total amount of evidence in support of either \(\varphi \) or its negation are sufficient then, as a consequence, one should be willing to believe \(\varphi \). We provide a sound and complete axiomatization for the logic and illustrate its expressive power with the aid of a concrete example.
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
Similar content being viewed by others
Notes
- 1.
As we will show in Sect. 3, formula \(\lnot \mathsf { T}_{i {, } j } \bot \) is valid in the logic DL-BET. Thus, if \(\mathsf { T}_{i {, } j }\) was a normal modal operator, \( \lnot (\mathsf { T}_{i {, } j } \varphi \wedge \mathsf { T}_{i {, } j } \lnot \varphi )\) would have been valid, which is highly counter-intuitive.
- 2.
Note that this ratio can be conceived as the probability that \(\varphi \) is true, computed of the basis of the number of evidence supporting \(\varphi \).
- 3.
Here we take the term “envisaged” to be synonymous of the term “imagined”. Clearly, there are situations that one can imagine that she considers impossible. For example, a person can imagine a situation in which she is the president of French republic and, at the same time, considers this situation impossible.
- 4.
The proof can be found in the extended version of this paper [28].
References
Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symbo. Logic 50, 510–530 (1985)
Artemov, S.N.: The logic of justification. Rev. Symb. Logic 1(4), 477–513 (2001)
Ayer, A.J.: Probability and Evidence. Columbia University Press, New York City (1972)
Baltag, A., Liu, F., Smets, S.: Reason-based belief revision in social networks. In: Slides, KNAW-Workshp on the Logical Dynamics of Information, Agency and Interaction, Amsterdam (2014)
Baltag, A., Moss, L.S., Solecki, S.: The logic of common knowledge, public announcements, and private suspicions. In: Gilboa, I. (ed.) Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 1998), pp. 43–56 (1998)
Baltag, A., Renne, B., Smets, S.: The logic of justified belief, explicit knowledge, and conclusive evidence. Ann. Pure Appl. Logic 165(1), 49–81 (2014)
Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Wooldridge, M., Bonanno, G., van der Hoek, W. (eds.) Logic and the Foundations of Game and Decision Theory. Texts in Logic and Games, vol. 3. Amsterdam University Press, Amsterdam (2008)
van Benthem, J.: Dynamic logic for belief revision. J. Appl. Non Class. Logic 17, 129–156 (2007)
van Benthem, J.: Logical Dynamics of Information and Interaction. Cambridge University Press, Cambridge (2011)
van Benthem, J., Fernández-Duque, D., Pacuit, E.: Evidence logic: a new look at neighborhood structures. In: Bolander, T., Braüner, T., Ghilardi, S., Moss, L. (eds.) Proceedings of Advances in Modal Logic, vol. 9, pp. 97–118. King’s College Press, London (2012)
van Benthem, J., Pacuit, E.: Dynamic logics of evidence-based beliefs. Studia Logica 99(1–3), 61–92 (2011)
Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)
Christoff, Z.: Dynamic logics of networks. Information Flow and the Spread of Opinion. Ph.D. thesis, ILLC, University of Amsterdam (2016)
Dastani, M., Herzig, A., Hulstijn, J., van der Torre, L.: Inferring trust. In: Leite, J., Torroni, P. (eds.) CLIMA 2004. LNCS, vol. 3487, pp. 144–160. Springer, Heidelberg (2005). doi:10.1007/11533092_9
Dietrich, F., List, C.: Judgment aggregation by quota rules: majority voting generalized. J. Theor. Polit. 19(4), 391–424 (2007)
Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Berlin (2007)
Dubois, D., Liu, W., Ma, J., Prade, H.: The basic principles of uncertain information fusion. An organised review of merging rules in different representation frameworks. Inf. Fusion 32, 12–39 (2016)
Dubois, D., Prade, H.: Representation and combination of uncertainty with belief functions and possibility measures. Comput. Intell. 4(3), 244264 (1988)
Falcone, R., Castelfranchi, C.: Social trust: a cognitive approach. In: Tan, Y.-H., Castelfranchi, C. (eds.) Trust and Deception in Virtual Societies, pp. 55–90. Springer, Netherlands (2001). doi:10.1007/978-94-017-3614-5_3. Chap. 1
Friedman, J.: Suspended judgment. Philos. Stud. 162(2), 165–181 (2013)
Grandi, U., Lorini, E., Perrussel, L.: Propositional opinion diffusion. In: Proceedings of the 14th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2015), pp. 989–997. ACM Press (2015)
Hunter, A., Booth, R.: Trust-sensitive belief revision. In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015), pp. 3062–3068. AAAI Press (2015)
Jøsang, A.: A logic for uncertain probabilities. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 9(3), 279–311 (2016)
Jøsang, A.: Interpretation of fusion and hyper opinions in subjective logic. In: 15th International Conference on Information Fusion, Singapore, pp. 1225–1232 (2017)
Keynes, J.M.: A Treatise on Probability. The Collected Writings, vol. 8. Macmillan, Hampshire (1973)
Kraus, S., Lehmann, D.J.: Knowledge, belief and time. Theoret. Comput. Sci. 58, 155–174 (1988)
Liau, C.-J.: Belief, information acquisition, and trust in multi-agent systems: a modal logic formulation. Artif. Intell. 149(1), 31–60 (2003)
Liu, F., Lorini, E.: Reasoning about belief, evidence and trust in a multi-agent setting (extended version). Technical report, Institut de Recherche en Informatique de Toulouse (IRIT) (2017)
Liu, F., Seligman, J., Girard, P.: Logical dynamics of belief change in the community. Synthese 191(11), 2403–2431 (2014)
Lorini, E.: A minimal logic for interactive epistemology. Synthese 193(3), 725–755 (2016)
Lorini, E., Jiang, G., Perrussel, L.: Trust-based belief change. In: Proceedings of the 21st European Conference on Artificial Intelligence (ECAI 2014), pp. 549–554. IOS Press (2014)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Singh, M.: Trust as dependence: a logical approach. In: Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2011), pp. 863–870. ACM (2011)
Xue, Y., Parikh, R.: Strategic belief updates through influence in a community. Stud. Logic 8, 124–143 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Liu, F., Lorini, E. (2017). Reasoning About Belief, Evidence and Trust in a Multi-agent Setting. In: An, B., Bazzan, A., Leite, J., Villata, S., van der Torre, L. (eds) PRIMA 2017: Principles and Practice of Multi-Agent Systems. PRIMA 2017. Lecture Notes in Computer Science(), vol 10621. Springer, Cham. https://doi.org/10.1007/978-3-319-69131-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-69131-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-69130-5
Online ISBN: 978-3-319-69131-2
eBook Packages: Computer ScienceComputer Science (R0)