A key issue when reasoning with default rules is how to order them so as to derive plausible conclusions according to the more specific rules applicable to the situation under concern, to make sure that default rules are not systematically inhibited by more general rules, and to cope with the problem of irrelevance of facts with respect to exceptions. Pearl's system Z enables us to rank-order default rules. In this paper we show how to encode such a rank-ordered set of defaults in possibilistic logic. We can thus take advantage of the deductive machinery available in possibilistic logic. We point out that the notion of inconsistency tolerant inference in possibilistic logic corresponds to the bold inference ; 1 in system Z. We also show how to express defaults by means of qualitative possibility relations. Improvements to the ordering provided by system Z are also proposed.

. This short paper relates the conditional object-based and possibility theorybased approaches for reasoning with conditional statements pervaded with exceptions, to other methods in nonmonotonic reasoning which have been independently proposed: namely, Lehmann's preferential and rational closure entailments which obey normative postulates, the infinitesimal probability approach, and the conditional (modal) logics-based approach. All these methods are shown to be equivalent with respect to their capabilities for reasoning with conditional knowledge although they are based on different modeling frameworks. It thus provides a unified understanding of nonmonotonic consequence relations. More particularly, conditional objects, a purely qualitative counterpart to conditional probabilities, offer a very simple semantics, based on a 3-valued calculus, for the preferential entailment, while in the purely ordinal setting of possibility theory both the preferential and the rational closure entai...

The links between Shoham's preference logic and possibilistic logic, a numerical logic of uncertainty based on Zadeh's possibility measures, are investigated. Starting from a fuzzy set of preferential interpretations of a propositional theory, we prove that the notion of preferential entailment is closely related to a previously introduced notion of conditional possibility. Conditional possibility is then shown to possess all properties (originally stated by Gabbay) of a well-behaved non-monotonic consequence relation. We obtain the possibilistic counterpart of Adams ' e-semantics of conditional probabilities which is the basis of the probabilistic model of non-monotonic logic proposed by Geffner and Pearl. Lastly we prove that our notion of possibilistic entailment is the one at work in possibilistic logic, a logic that handles uncertain propositional formulas, where uncertainty is modelled by degrees of necessity, and where partial inconsistency is allowed. Considering the formerly established close links between Gardenfors'epistemic entrenchment and necessity measures, what this paper proposes is a new way of relating belief revision and non-monotonic inference, namely via possibility theory. 1

This paper is an investigation of the relationship between conditional objects obtained as a qualitative counterpart to conditional probabilities, and nonmonotonic reasoning. Roughly speaking, a conditional object can be seen as a generic rule which allows us to get a conclusion provided that the available information exactly corresponds to the "context" part of the conditional object. This gives freedom for possibly retracting previous conclusions when the available information becomes more specific. Viewed as an inference rule expressing a contextual belief, the conditional object is shown to possess all properties of a well-behaved nonmonotonic consequence relation when a suitable choice of connectives and deduction operation is made. Using previous results from Adams' conditional probabilistic logic, a logic of conditional objects is proposed. Its axioms and inference rules are those of preferential reasoning logic of Lehmann and colleagues. But the semantics relies on a three-valu...

Cox's well-known theorem justifying the use of probability is shown not to hold infinite domains. The counterexample also suggests that Cox's assumptions are insu cient to prove the result even in infinite domains. The same counterexample is used to disprove a result of Fine on comparative conditional probability.

It is shown that the notion of conditional possibility can be consistently introduced in possibility theory, in very much the same way as conditional expectations and probabilities are defined in the measure- and integral-theoretic treatment of probability theory. I write down possibilistic integral equations which are formal counterparts of the integral equations used to define conditional expectations and probabilities, and use their solutions to define conditional possibilities. In all, three types of conditional possibilities, with special cases, are introduced and studied. I explain why, like conditional expectations, conditional possibilities are not uniquely defined, but can only be determined up to almost everywhere equality, and I assess the consequences of this nondeterminacy. I also show that this approach solves a number of consistency problems, extant in the literature. INDEX TERMS: Possibility integral, integral equation, conditional possibility. 1 CONDITIONAL POSSIBILITY: A SURVEY This is the second in a series of three papers on the measure- and integral-theoretic aspects of possibility theory. Here I specifically deal with conditional possibility. I shall make ample use of the results, definitions and notational conventions, given in the first paper of this series, which

Cox's Theorem provides a theoretical basis for using probability theory as a general logic of plausible inference. The theorem states that any system for plausible reasoning that satisfies certain qualitative requirements intended to ensure consistency with classical deductive logic and correspondence with commonsense reasoning is isomorphic to probability theory. However, the requirements used to obtain this result have been the subject of much debate. We review Cox's Theorem, discussing its requirements, the intuition and reasoning behind these, and the most important objections, and finish with an abbreviated proof of the theorem.

In this article we propose a qualitative (ordinal) counterpart for the Partially Observable Markov Decision Processes model (POMDP) in which the uncertainty, as well as the preferences of the agent, are modeled by possibility distributions. This qualitative counterpart of the POMDP model relies on a possibilistic theory of decision under uncertainty, recently developed. One advantage of such a qualitative framework is its ability to escape from the classical obstacle of stochastic POMDPs, in which even with a finite state space, the obtained belief state space of the POMDP is infinite. Instead, in the possibilistic framework even if exponentially larger than the state space, the belief state space remains finite. 1

The assumptions needed to prove Cox's Theorem are discussed and examined. Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and, arguably, not natural. I recently wrote a paper (Halpern, 1999) casting doubt on how compelling a justification for probability is provided by Cox's celebrated theorem (Cox, 1946). I have received (what seems to me, at least) a surprising amount of response to that article. Here I attempt to clarify the degree to which I think Cox's theorem can be salvaged and respond to a glaring inaccuracy on my part pointed out by Snow (1998). (Fortunately, it is an inaccuracy that has no affect on either the correctness or the interpretation of the results of my paper.) I have tried to write this note with enough detail so that it can be read independently of (Halpern, 1999), but I encourage the reader to consult (Halpern, 1999), as well as the two major sources it is based on (Cox, 1946; Paris, 1994), ...

Cox’s theorem states that, under certain assumptions, any measure of belief is isomorphic to a probability measure. This theorem, although intended as a justification of the subjectivist interpretation of probability theory, is sometimes presented as an argument for more controversial theses. Of particular interest is the thesis that the only coherent means of representing uncertainty is via the probability calculus. In this paper I examine the logical assumptions of Cox’s theorem and I show how these impinge on the philosophical conclusions thought to be supported by the theorem. I show that the more controversial thesis is not supported by Cox’s theorem.