Belief as Willingness to Bet

We investigate modal logics of high probability having two unary modal operators: an operator $K$ expressing probabilistic certainty and an operator $B$ expressing probability exceeding a fixed rational threshold $c\geq\frac 12$. Identifying knowledge with the former and belief with the latter, we may think of $c$ as the agent's betting threshold, which leads to the motto "belief is willingness to bet." The logic $\mathsf{KB.5}$ for $c=\frac 12$ has an $\mathsf{S5}$ $K$ modality along with a sub-normal $B$ modality that extends the minimal modal logic $\mathsf{EMND45}$ by way of four schemes relating $K$ and $B$, one of which is a complex scheme arising out of a theorem due to Scott. Lenzen was the first to use Scott's theorem to show that a version of this logic is sound and complete for the probability interpretation. We reformulate Lenzen's results and present them here in a modern and accessible form. In addition, we introduce a new epistemic neighborhood semantics that will be more familiar to modern modal logicians. Using Scott's theorem, we provide the Lenzen-derivative properties that must be imposed on finite epistemic neighborhood models so as to guarantee the existence of a probability measure respecting the neighborhood function in the appropriate way for threshold $c=\frac 12$. This yields a link between probabilistic and modal neighborhood semantics that we hope will be of use in future work on modal logics of qualitative probability. We leave open the question of which properties must be imposed on finite epistemic neighborhood models so as to guarantee existence of an appropriate probability measure for thresholds $c\neq\frac 12$.Comment: Removed date from v1 to avoid confusion on citation/reference, otherwise identical to v

AGREEING TO DISAGREE: A SURVEY

Aumann (1976) put forward a formal definition of common knowledge and used it to prove that two ""like minded"" individuals cannot ""agree to disagree"" in the following sense. If they start from a common prior and update the probability of an event E (using Bayes'' rule) on the basis of private information, then it cannot be common knowledge between them that individual 1 assigns probability p to E and individual 2 assigns probability q to E with p ¹ q. In other words, if their posteriors of event E are common knowledge then they must coincide. Aumann''s Agreement Theorem has given rise to a large literature which we review in this paper. The results are classified according to whether they are probabilistic (Bayesian) or qualitative. Particular attention is paid to the issue of how to interpret the notion of Harsanyi consistency as a (local) property of belief hierarchies.

Empirical Evaluation of Abstract Argumentation: Supporting the Need for Bipolar and Probabilistic Approaches

In dialogical argumentation it is often assumed that the involved parties always correctly identify the intended statements posited by each other, realize all of the associated relations, conform to the three acceptability states (accepted, rejected, undecided), adjust their views when new and correct information comes in, and that a framework handling only attack relations is sufficient to represent their opinions. Although it is natural to make these assumptions as a starting point for further research, removing them or even acknowledging that such removal should happen is more challenging for some of these concepts than for others. Probabilistic argumentation is one of the approaches that can be harnessed for more accurate user modelling. The epistemic approach allows us to represent how much a given argument is believed by a given person, offering us the possibility to express more than just three agreement states. It is equipped with a wide range of postulates, including those that do not make any restrictions concerning how initial arguments should be viewed, thus potentially being more adequate for handling beliefs of the people that have not fully disclosed their opinions in comparison to Dung's semantics. The constellation approach can be used to represent the views of different people concerning the structure of the framework we are dealing with, including cases in which not all relations are acknowledged or when they are seen differently than intended. Finally, bipolar argumentation frameworks can be used to express both positive and negative relations between arguments. In this paper we describe the results of an experiment in which participants judged dialogues in terms of agreement and structure. We compare our findings with the aforementioned assumptions as well as with the constellation and epistemic approaches to probabilistic argumentation and bipolar argumentation

Limit-Agreeing To Disagree

We reconsider Aumann' s seminal impossibility theorem that agents cannot agree to disagree in a topologically extended epistemic model. In such a framework, a possibility result on agreeing to disagree actually ensues. More precisely, agents with a common prior belief satisfying limit knowledge instead of common knowledge of their posterior beliefs may have distinct posterior beliefs. Since limit knowledge is defined as the limit of iterated mutual knowledge, agents can thus be said to limit-agree to disagree. Besides, an example is provided in which limit knowledge coincides with Rubinstein's (1989) notion of almost common knowledge, and the agents have almost common knowledge of posteriors yet distinct posterior beliefs. More generally, an epistemic-topological foundation for almost common knowledge is thus provided