We consider multi-agent systems where agents actions and beliefs are determined aleatorically, or "by the throw of dice". This system consists of possible worlds that assign distributions to independent random variables, and agents who assign probabilities to these possible worlds. We present a novel syntax and semantics for such system, and show that they generalise Modal Logic. We also give a sound and complete calculus for reasoning in the base semantics, and a sound calculus for the full modal semantics, that we conjecture to be complete. Finally we discuss some application to reasoning about game playing agents.
More general foundational work on reasoning probabilistically was done by de Finetti @cite_3 who established an epistemic notion of probability based on what an agent would consider to be a rational wager (the Dutch book argument). In @cite_21 , Milne incorporates these ideas into the logic of conditional events. Stalnaker @cite_9 has also considered conditional events and has presented conditional logic @cite_17 . Here, conditional refers to the interpretation of one proposition being contingent on another, although this is not quantified nor assigned a probability.
@cite_9 @cite_21 @cite_3 @cite_17
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Both in science and in practical affairs we reason by combining facts only inconclusively supported by evidence. Building on an abstract understanding of this process of combination, this book constructs a new theory of epistemic probability. The theory draws on the work of A. P. Dempster but diverges from Depster's viewpoint by identifying his "lower probabilities" as epistemic probabilities and taking his rule for combining "upper and lower probabilities" as fundamental. The book opens with a critique of the well-known Bayesian theory of epistemic probability. It then proceeds to develop an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called "monotone of order of infinity." and Dempster's rule for combining such set functions. This rule, together with the idea of "weights of evidence," leads to both an extensive new theory and a better understanding of the Bayesian theory. The book concludes with a brief treatment of statistical inference and a discussion of the limitations of epistemic probability. Appendices contain mathematical proofs, which are relatively elementary and seldom depend on mathematics more advanced that the binomial theorem. This article begins by outlining some of the history--beginning with brief remarks of Quine's-of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it. Combining computational models of argumentation with probability theory has recently gained increasing attention, in particular with respect to abstract argumentation frameworks. Approaches following this idea can be categorised into the constellations and the epistemic approach. While the former considers probability functions on the subgraphs of abstract argumentation frameworks, the latter uses probability theory to represent degrees of belief in arguments, given a fixed framework. In this paper, we investigate the case where probability functions are given on the extensions of abstract argumentation frameworks. This generalises classical semantics in a straightforward fashion and we show that our approach also complies with many postulates for epistemic probabilistic argumentation. We propose a new graded semantics for abstract argumentation frameworks that is based on the constellations approach to probabilistic argumentation. Given an abstract argumentation framework, our approach assigns uniform probability to all arguments and then ranks arguments according to the probability of acceptance wrt. some classical semantics. Albeit relying on a simple idea this approach (1) is based on the solid theoretical foundations of probability theory, and (2) complies with many rationality postulates proposed for graded semantics. We also investigate an application of our approach for inconsistency measurement in argumentation frameworks and show that the measure induced by the probabilistic graded semantics also complies with the basic rationality postulates from that area.