# Reasoning About Belief, Evidence and Trust in a Multi-agent Setting

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## 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.

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## Notes

1. 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. 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. 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. 4.

The proof can be found in the extended version of this paper [28].

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Correspondence to Emiliano Lorini .

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### 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

• DOI: https://doi.org/10.1007/978-3-319-69131-2_5

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