Precise Propagation of Upper and Lower Probability Bounds in System P

In this paper we consider the inference rules of System P in the framework of coherent imprecise probabilistic assessments. Exploiting our algorithms, we propagate the lower and upper probability bounds associated with the conditional assertions of a given knowledge base, automatically obtaining the precise probability bounds for the derived conclusions of the inference rules. This allows a more flexible and realistic use of System P in default reasoning and provides an exact illustration of the degradation of the inference rules when interpreted in probabilistic terms. We also examine the disjunctive Weak Rational Monotony of System P+ proposed by Adams in his extended probability logic.Comment: 8 pages -8th Intl. Workshop on Non-Monotonic Reasoning NMR'2000, April 9-11, Breckenridge, Colorad

Probabilistic entailment in the setting of coherence: The role of quasi conjunction and inclusion relation

In this paper, by adopting a coherence-based probabilistic approach to default reasoning, we focus the study on the logical operation of quasi conjunction and the Goodman-Nguyen inclusion relation for conditional events. We recall that quasi conjunction is a basic notion for defining consistency of conditional knowledge bases. By deepening some results given in a previous paper we show that, given any finite family of conditional events F and any nonempty subset S of F, the family F p-entails the quasi conjunction C(S); then, given any conditional event E|H, we analyze the equivalence between p-entailment of E|H from F and p-entailment of E|H from C(S), where S is some nonempty subset of F. We also illustrate some alternative theorems related with p-consistency and p-entailment. Finally, we deepen the study of the connections between the notions of p-entailment and inclusion relation by introducing for a pair (F,E|H) the (possibly empty) class K of the subsets S of F such that C(S) implies E|H. We show that the class K satisfies many properties; in particular K is additive and has a greatest element which can be determined by applying a suitable algorithm

Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects

We make a probabilistic analysis related to some inference rules which play an important role in nonmonotonic reasoning. In a coherence-based setting, we study the extensions of a probability assessment defined on $n$ conditional events to their quasi conjunction, and by exploiting duality, to their quasi disjunction. The lower and upper bounds coincide with some well known t-norms and t-conorms: minimum, product, Lukasiewicz, and Hamacher t-norms and their dual t-conorms. On this basis we obtain Quasi And and Quasi Or rules. These are rules for which any finite family of conditional events p-entails the associated quasi conjunction and quasi disjunction. We examine some cases of logical dependencies, and we study the relations among coherence, inclusion for conditional events, and p-entailment. We also consider the Or rule, where quasi conjunction and quasi disjunction of premises coincide with the conclusion. We analyze further aspects of quasi conjunction and quasi disjunction, by computing probabilistic bounds on premises from bounds on conclusions. Finally, we consider biconditional events, and we introduce the notion of an $n$-conditional event. Then we give a probabilistic interpretation for a generalized Loop rule. In an appendix we provide explicit expressions for the Hamacher t-norm and t-conorm in the unitary hypercube

Numerical Representations of Acceptance

Accepting a proposition means that our confidence in this proposition is strictly greater than the confidence in its negation. This paper investigates the subclass of uncertainty measures, expressing confidence, that capture the idea of acceptance, what we call acceptance functions. Due to the monotonicity property of confidence measures, the acceptance of a proposition entails the acceptance of any of its logical consequences. In agreement with the idea that a belief set (in the sense of Gardenfors) must be closed under logical consequence, it is also required that the separate acceptance o two propositions entail the acceptance of their conjunction. Necessity (and possibility) measures agree with this view of acceptance while probability and belief functions generally do not. General properties of acceptance functions are estabilished. The motivation behind this work is the investigation of a setting for belief revision more general than the one proposed by Alchourron, Gardenfors and Makinson, in connection with the notion of conditioning.Comment: Appears in Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence (UAI1995

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

The efficiency of the non-profit enterprise: constitutional ideology, conformist preferences and reputation

According to one thesis the non profit enterprise (in short NPE) is able to attract ideological entrepreneurs and workers (Rose-Ackerman 1996). In fact I prove that without the ideological element, a simple game between the entrepreneur, worker and beneficiary is condemned to an opportunistic equilibrium, beneficial to the internal members of the organization but detrimental to the beneficiary. Thus the NPE does not better than its for profit counterpart. In my model ideologues, both entrepreneurs and workers, share a principle of justice seen as the constitutional ideology of the NPE, agreed upon in an hypothetical ex ante bargaining game. The constitutional principle provides an independent source of motivation (a source of utility) of the players, in so far as they believe in the reciprocity of conformity to the ideology by all the participants. I call this conformity-based utility “ideological”, and I see it as the representation of a preference for expected conformity to the given constitutional principle. The philosophical underpinnings of this reform of the players' utility functions in worked out by distinguishing two concepts of preferences of the Self: consequentialist preferences and conformist preferences. The latter are preferences for those actions that are part of states of affairs described in terms of interdependent actions conforming to an abstract norm or principle, which become effective once the preferences' holder does expect that the other players do they part in that state of affairs and they do expect that himself do his part in the same state of affairs. What result is that a player's ideological utility depends on the expectation of deontological modes of behaviour followed by all the participants, himself included. On this basis it is possible to overcome personal incentives to embrace opportunistic behaviour, so that the proper Non-profit Enterprise emerges. It is proved that in the “social enterprise game” amongst the member of the organisation there exists an organisational equilibrium minimising transaction costs to the beneficiaries. At last, this equilibrium rests on the emergence of an expectations system of reciprocal conformity to the constitutional ideology. As the existence - not even the selection – of the internal organizational equilibrium rests heavily on the existence of the appropriate system of reciprocal expectation, the problem of how we can justify the emergence of the appropriate system of beliefs must be underlined. Here is where the explicit moral codes of the NPE enters the scene. I see the code of ethics as the building block for deriving a reputation equilibrium between the NPE as a whole and its external stakeholders within a repeated game, whose stage-game is the typical game of trust played under incomplete knowledge and unforeseen contingencies. At last the conformist-motivation model and the reputation model under unforeseen contingencies are shown to play together in a mutually supporting explanation of the efficiency of the NPE.