Conclusion

Abstract

In this book, a profound and extensive investigation of how to handle uncertain knowledge both in qualitative and in quantitative settings was presented. The unifying framework announced in its title is provided by conditionals. Observing conditional structures and preserving conditional beliefs turned out to be of crucial importance when reasoning under uncertainty and changing epistemic knowledge bases.

The approach to conditionals developed and used here is quite different from the logical one usually taken. Rather we featured a dynamic view on conditionals as actors on worlds, shifting them appropriately to establish conditional beliefs. In this context, an essential notion was that of a conditional valuation function allowing us to conceive conditional reasoning actually as extending propositional reasoning by a new dimension. Conditional valuation functions abstract from the concrete representation of epistemic attitutes (e.g. by probability distributions or by ordinal conditional functions) and provide the formal framework to reveal fundamental patterns of (quantified) conditional reasoning. In particular, the idea of preserving conditional beliefs under change operations could be put in precise formal terms, capturing interactions of high complexity between atoms and relating them to the (sets of) conditionals inducing change. It was shown that this principle of conditional preservation not only has a crucial meaning for quantitative conditional reasoning, but also covers corresponding ideas and approaches in qualitative settings. Therefore, it can be considered to formalize an important paradigm in general epistemic resoning.

Despite the complicatedness of the underlying theory, revisions of probability distributions and ordinal conditional functions preserving conditional beliefs turned out to follow a strikingly simple conditional design. This approach was pursued within the probabilistic framework to elaborate further conditions such a revision should reasonably satisfy. We found that crucial topics in this area were the realization of a functional concept depending on conditional as well as on numerical structures, the independence of syntactical representation of probabilistic knowledge and the coherent handling of iterated revisions (logical coherence). Together with the principle of conditional preservation, the corresponding three postulates were apt to distinguish a single probabilistic revision operator, namely revision by optimizing entropy (ME-revision). Thus we characterized this powerful technique of incorporating new probabilistic information by focussing on the consistent handling of conditionals, establishing the ME-methods as a most important tool to carry out quantified uncertain reasoning. We classified ME-reasoning according to the formal framework of nonmonotonic reasoning, proving it to satisfy crucial demands in this area. Moreover, we showed how to use the formal representation of the ME-distribution to derive some inference patterns, that is, how to calculate ME-probabilities in some simple, but illustrative examples.