The book can also be used as an excellent text for graduate-level courses in AI, operations research, or applied probability.

Many have claimed that unspecific evidence sometimes demands unsharp, indeterminate, imprecise, vague, or interval-valued probabilities. Against this, a variant of the diachronic Dutch Book argument shows that perfectly rational agents always have perfectly sharp probabilities.

We examine the closure conditions of the probabilistic consequence relation of Hawthorne and Makinson, specifically the outstanding question of completeness in terms of Horn rules, of their proposed (finite) set of rules O. We show that on the contrary no such finite set of Horn rules exists, though we are able to specify an infinite set which is complete.

We investigate a lattice of conditional logics described by a Kripke type semantics, which was suggested by Chellas and Segerberg – Chellas–Segerberg (CS) semantics – plus 30 further principles. We (i) present a non-trivial frame-based completeness result, (ii) a translation procedure which gives one corresponding trivial frame conditions for arbitrary formula schemata, and (iii) non-trivial frame conditions in CS semantics which correspond to the 30 principles.

Conditional structures lie at the heart of the sciences, humanities, and everyday reasoning. It is hence not surprising that conditional logics – logics specifically designed to account for natural language conditionals – are an active and interdisciplinary area. The present book gives a formal and a philosophical account of indicative and counterfactual conditionals in terms of Chellas-Segerberg semantics. For that purpose a range of topics are discussed such as Bennett’s arguments against truth value based semantics for indicative conditionals.

This paper consists of some lecture notes in which conditional logic is treated as an extension of modal logic. Completeness and filtration theorems are provided for some basis systems.

Systems of logico-probabilistic reasoning characterize inference from conditional assertions that express high conditional probabilities. In this paper we investigate four prominent LP systems, the systems _O, P_, _Z_, and _QC_. These systems differ in the number of inferences they licence _. LP systems that license more inferences enjoy the possible reward of deriving more true and informative conclusions, but with this possible reward comes the risk of drawing more false or uninformative conclusions. In the first part of the paper, we (...) present the four systems and extend each of them by theorems that allow one to compute almost-tight lower-probability-bounds for the conclusion of an inference, given lower-probability-bounds for its premises. In the second part of the paper, we investigate by means of computer simulations which of the four systems provides the best balance of reward versus risk. Our results suggest that system _Z_ offers the best balance. (shrink)

We examine the closure conditions of the probabilistic consequence relation of Hawthorne and Makinson, specifically the outstanding question of completeness in terms of Horn rules, of their proposed (finite) set of rules O. We show that on the contrary no such finite set of Horn rules exists, though we are able to specify an infinite set which is complete.

We chart the ways in which closure properties of consequence relations for uncertain inference take on different forms according to whether the relations are generated in a quantitative or a qualitative manner. Among the main themes are: the identification of watershed conditions between probabilistically and qualitatively sound rules; failsafe and classicality transforms of qualitatively sound rules; non-Horn conditions satisfied by probabilistic consequence; representation and completeness problems; and threshold-sensitive conditions such as `preface' and `lottery' rules.

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, →, in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, 'C → B' holds just in case P[B | C] ≥ (...) r. Thus, each conditional in a given family behaves like conditional probability above some specific support level. (shrink)

Using formal systems to represent and reason about uncertainty.

This paper presents a formalism that combines useful properties of both logic and probabilities. Like logic, the formalism admits qualitative sentences and provides symbolic machinery for deriving deductively closed beliefs and, like probability, it permits us to express if-then rules with different levels of firmness and to retract beliefs in response to changing observations. Rules are interpreted as order-of-magnitude approximations of conditional probabilities which impose constraints over the rankings of worlds. Inferences are supported by a unique priority ordering on rules (...) which is syntactically derived from the knowledge base. This ordering accounts for rule interactions, respects specificity considerations and facilitates the construction of coherent states of beliefs. Practical algorithms are developed and analyzed for testing consistency, computing rule ordering, and answering queries. Imprecise observations are incorporated using qualitative versions of Jeffrey's rule and Bayesian updating, with the result that coherent belief revision is embodied naturally and tractably. Finally, causal rules are interpreted as imposing Markovian conditions that further constrain world rankings to reflect the modularity of causal organizations. These constraints are shown to facilitate reasoning about causal projections, explanations, actions and change. (shrink)

THE INDICATIVE CONDITIONAL. A PROBABILISTIC CRITERION OF SOUNDNESS FOR DEDUCTIVE INFERENCES Our objective in this section is to establish a prima facie case ...

The standard use of the propositional calculus ('P.C.?) in analyzing the validity of inferences involving conditionals leads to fallacies, and the problem is to determine where P.C. may be ?safely? used. An alternative analysis of criteria of reasonableness of inferences in terms of conditions of justification rather than truth of statements is proposed. It is argued, under certain restrictions, that P. C. may be safely used, except in inferences whose conclusions are conditionals whose antecedents are incompatible with the premises in (...) the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn. (shrink)

Wolfgang Spohn presents the first full account of the dynamic laws of belief, by means of ranking theory. This book is his long-awaited presentation of ranking theory and its ramifications.

This monograph provides a new account of justified inference as a cognitive process. In contrast to the prevailing tradition in epistemology, the focus is on low-level inferences, i.e., those inferences that we are usually not consciously aware of and that we share with the cat nearby which infers that the bird which she sees picking grains from the dirt, is able to fly. Presumably, such inferences are not generated by explicit logical reasoning, but logical methods can be used to describe (...) and analyze such inferences. Part 1 gives a purely system-theoretic explication of belief and inference. Part 2 adds a reliabilist theory of justification for inference, with a qualitative notion of reliability being employed. Part 3 recalls and extends various systems of deductive and nonmonotonic logic and thereby explains the semantics of absolute and high reliability. In Part 4 it is proven that qualitative neural networks are able to draw justified deductive and nonmonotonic inferences on the basis of distributed representations. This is derived from a soundness/completeness theorem with regard to cognitive semantics of nonmonotonic reasoning. The appendix extends the theory both logically and ontologically, and relates it to A. Goldman's reliability account of justified belief. This text will be of interest to epistemologists and logicians, to all computer scientists who work on nonmonotonic reasoning and neural networks, and to cognitive scientists. (shrink)

Kyburg goes half-way towards objective Bayesianism. He accepts that frequencies constrain rational belief to an interval but stops short of isolating an optimal degree of belief within this interval. I examine the case for going the whole hog.