Comp. by: Prabhu Stage: Proof Chapter No.: 7 Title Name: Douven Date:9/9/20 Time:20:55:20 Page Number: 128   Rational Belief in Lottery- and Preface-Situations Impossibility Results and Possible Solutions Gerhard Schurz Introduction: Rational Belief – Qualitative and Quantitative Rational agents represent their beliefs about the world in two rather different ways. One way is by means of qualitative beliefs, or yes-or-no beliefs: Here a proposition p is either believed, or disbelieved (i.e., ¬p is believed), or belief about p is suspended. Thereby, with “rational belief that p” it is meant that the proposition p is believed to be true, or accepted as being true, whereby the system of beliefs satisfies certain rationality conditions. Standard rationality conditions for qualitative belief (or acceptance) are consistency and logical closure. The canonical reasons for these conditions are that rational agents strive for true beliefs, that only consistent beliefs can be true, and that truth is closed under logical inference.  The second way of representing beliefs is quantitative, by means of degrees of belief. The standard rationality condition here is that degrees of belief should satisfy the axioms of probability. A canonical reason for this requirement is that if rational degrees of beliefs are measured in terms of fair betting quotients, sure losses can be avoided if and only if (iff ) degrees of belief obey the axioms. To express the rationality conditions in a formally precise way, we use the following notation: Small letters a, (a i ), b,... denote arbitrary sentences of an assumed formal language L having at least the expressive power of (classical) prepositional logic with ¬, ^, as propositional connectives. Capital letters A, B, ... denote sets of sentences.  The full logical closure requirement applies to ideal agents that are logically omniscient. For real agents, the closure requirement can be satisfied only by those inferences that are “simple enough” to be mastered by the agent. However, the problems of the logical closure condition will arise for very simple inferences that are mastered by typical real agents: namely, applications of the rule of conjunction.  In: Igor Douven (ed.), Lotteries, Knowledge, and Rational Belief: Essays on the Lottery Paradox, Cambridge: Cambridge University Press 2021, 128-146.