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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A conditional sentence expresses a proposition which is a function of two other propositions, yet not one which is a truth function of those propositions. I may know the truth values of “Willie Mays played in the American League” and “Willie Mays hit four hundred” without knowing whether or not Mays, would have hit four hundred if he had played in the American League. This fact has tended to puzzle, displease, or delight philosophers, and many have felt that it is a fact that calls for some comment or explanation. It has given rise to a number of philosophical problems; I shall discuss three of these. 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. Part 7 A preliminary survey: heads and tails - preliminary considerations heads and tails - the random process laws of "large numbers" the "central limit theorem". Part 8 Random processes with independent increments: the case of asymptotic normality the Wiener-Levy process behaviour and asymptotic behaviour ruin problems ballot problems. Part 9 An introduction to other types of stochastic process: Markov processes stationary processes. Part 10 Problems in higher dimensions: second-order characteristics and the normal distribution the discrete case the continuous case the case of spherical symmetry. Part 11 Inductive reasoning, statistical inference: the basic formulation and preliminary clarifications the case of independence and the case of dependence exchangeability. Part 12 Mathematical statistics: the scope and limits of the treatment the likelihood principle and sufficient statistics a Bayesian approach to "estimation" and "hypothesis testing" the connections with decision theory.