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'

real'

'

world, we require agent descriptions to incor- porate some elements of uncertainty. Note that, in this paper, we will not consider multi-agent settings, in which agents have uncertain predictions about other agents'

uncertainty. Here we use a possible worlds semantics (also known as Kripke structures, or models [17]) as the seman- tical basis that characterises modal logics of knowledge and belief, and as a means of expressing uncertainty with respect to the true state of the world. Given a situation of a system, one could draw a map of states considered possible and, consequently, be able to determine what is believed in that situation. In this context, a set of possible worlds would represent the doxastic possibilities. In other words, by having worlds that are named '

'

possible'

'

, an agent expresses its '

'

doubts'

'

about which is the '

'

real'

'

situation, i.e., its uncertainty about the true state of the world. The more worlds an agent considers possible, the more uncertain it is, and the less it believes. This is what makes possible worlds a qualitative measure of an agent'

s uncertainty [12]. The most popular choices for modelling informational attitudes such as beliefs, involve annotating the agent with a KD45-like logic [8,19]. However, when using logics such as KD45 in its standard form, an agent cannot distinguish situations in which the evidence for a certain fact is '

equally distributed'

over its alterna- tives, from situations in which there is only one, almost negligible, counterexample to the '

fact'

. Probabilistic logics (cf. [21]) and probabilistic modal logics [12] are a way to address this. In particular, probabilistic logics of knowledge and belief [13,6,24] aim at removing the limitations implied by classical epistemic and doxastic logic. In epistemic logic the formalization is restricted to sentences such as '

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agent knows u'

'

or '

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agent does not know u'

'

, in which no quanti?cation of the agent'

s certainty is possible. We present a logic that builds upon the natural framework of Kripke models, while allowing us to reason about uncertainty. For us it is both important and interesting to capture, and express, the notion of degrees of uncertainty within the agent itself. This means, intuitively, that we want to express statements like: '

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agent i believes that the probability of statement b being true is greater than x'

'

. In this sense, the agent can have more (or less) con?dence in certain facts. More speci?cally, we introduce the PFKD45 Logic which extends, in some aspects, the system PFD given in [24] (which in turn was inspired by the logic from [9]). We propose not only a complete axiomatization for the logic, but also a decision procedure that permits us to verify the satis?ability of PFKD45-formulae. We claim that PFKD45 represents a good candidate for representing and reasoning about uncertainty within computational agents, especially because, contrary to many logical approaches to probabilistic reasoning, it is conceptually simple and logically compact―we come back to this in the conclud- ing section. This paper is organised as follows. In Section

2 we present a description of the language, including basic de?nitions and a complete axiomatization. In the subsequent section we provide the semantics and establish some meta-logical properti........