Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system (...) acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone. (shrink)

This book, formed as a series of essays in honour of Professor Carl Baudenbacher, addresses the very art of judicial reasoning, and features contributions from many of the foremost current or former national, supranational, or international judges. This unique volume is intended first and foremost for legal scholars, but its approachable style makes it readily accessible for students and for those with a general interest in the application of the law and justice in today's multi-layered world. The collection of essays (...) is rather more philosophical and reflective as opposed to doctrinal. Each contribution focuses on the nature and operation of justice, the independence of the judiciary, and on judicial style primarily from the perspective of the judges themselves. The book provides perspectives on what it means to be accountable and independent as a judge, the role of language and languages in the quest for justice, while other contributions acquaint readers with the some of the structures of courts themselves, or indeed question for whom judgments are written. Each chapter has been written by a presiding judge, or head of an institution and the book is divided into three parts: - Part I Art and Method - Part II Justice and the Judiciary - Part III Reasoning and Language(s). (shrink)