Conditional Belief

This chapter first presents the static and dynamic rules of ranking theory. Then it shows how ranking theory solves the problem of iterated belief revisions.

Why Should I?

This chapter first answers the question of why conditional beliefs should obey the axioms and update rule of ranking theory. This includes a defense of the conditional theory of conditional belief that characterizes conditional belief in terms of belief and counterfactuals. Then the instrumentalist view of rationality, or normativity, underlying this answer is discussed. The chapter concludes with a discussion of conditional obligation and conditional belief.

Belief First

This chapter first discusses which agents this book focuses on and which ends they are assumed to have. Then it briefly describes how this relates to conditional belief and belief revision.

Belief and Counterfactuals

This book is the first of two volumes on belief and counterfactuals. It consists of six of a total of eleven chapters. The first volume is concerned primarily with questions in epistemology and is expository in parts. Among other theories, it provides an accessible introduction to belief revision and ranking theory. Ranking theory specifies how conditional beliefs should behave. It does not tell us why they should do so nor what they are. This book fills these two gaps. The consistency argument tells us why conditional beliefs should obey the laws of ranking theory by showing them to be the means to attaining the end of holding true and informative beliefs. The conditional theory of conditional belief tells us what conditional beliefs are by specifying their nature in terms of non-conditional belief and counterfactuals. In addition, the book contains several novel arguments, accounts, and applications. These include an argument for the thesis that there are only hypothetical imperatives and no categorical imperatives; an account of the instrumentalist understanding of normativity, or rationality, according to which one ought to take the means to one’s ends; as well as solutions to the problems of conceptual belief change, logical learning, and learning conditionals. A distinctive feature of the book is its unifying methodological approach: means-end philosophy. Means-end philosophy takes serious that philosophy is a normative discipline, and that philosophical problems are entangled with each other. It also explains the importance of logic to philosophy, without being a technical theory itself.

System Z for Conditional Belief Bases with Positive and Negative Information

In non-monotonic reasoning, conditional belief bases mostly contain positive information in the form of standard conditionals. However, in practice we are often confronted with negative information, stating that a conditional does \emph{not} hold, i.e. we need a suitable approach for reasoning over belief bases $\Delta$ with positive and negative information. In this paper, we investigate the interaction of positive and negative information in a conditional belief base and establish a property for partitions of $\Delta$ that is equivalent to consistency. Based on this property, we develop a non-trivial extension of system Z for mixed conditional belief bases and provide an algorithm to compute this partition.

Selection Strategies for Inductive Reasoning From Conditional Belief Bases and for Belief Change Respecting the Principle of Conditional Preservation

Given a belief base ∆ consisting of a set of conditionals,there are many different ways an agent may inductivelycomplete the knowledge represented by ∆ to a completeepistemic state; two well-known approaches are given by systemP and system Z, and also each ranking model of ∆ induces afull inference relation. C-representations are special rankingmodels that obey the principle of conditional indifference.Inductive reasoning using c-representations can be done withrespect to all c-representations, with respect to a subclass of,e.g., minimal c-representations, or with respect to singlec-representations. In this paper, we present and investigateselection strategies for determining single c-representations tobe used for inductive reasoning from belief bases. We developaxioms for specifying characteristics of selection strategies.We illustrate which desirable properties, like syntaxsplitting, are ensured by the axioms, and develop constructionsfor obtaining selection strategies satisfying the axioms.Furthermore, we also present and study the extension of selectionstrategies to c-revisions that follow the principle ofconditional preservation and that have been employed successfullyin various belief change settings.

A Qualitative Theory of Cognitive Attitudes and their Change

Abstarct We present a general logical framework for reasoning about agents’ cognitive attitudes of both epistemic type and motivational type. We show that it allows us to express a variety of relevant concepts for qualitative decision theory including the concepts of knowledge, belief, strong belief, conditional belief, desire, conditional desire, strong desire, and preference. We also present two extensions of the logic, one by the notion of choice and the other by dynamic operators for belief change and desire change, and we apply the former to the analysis of single-stage games under incomplete information. We provide sound and complete axiomatizations for the basic logic and for its two extensions.

Counterfactuals and Probability

Dorothy Edgington has been a resolute defender of an NTV account of conditionals, according to which a conditional does not express a proposition that makes a categorical claim about the world, but instead make a qualified claim, or express a conditional belief, qualified by or conditional on the proposition expressed by the antecedent. Unlike some philosophers who defend an NTV view for indicative conditionals, but not for subjunctive or counterfactual conditionals, Edgington argues for the more radical thesis that both kinds of conditionals should be given a non-propositional analysis. This chapter considers Edgington’s NTV account of subjunctive conditionals, the role of objective probability in the account, and its relation to the possible-worlds propositional analysis of subjunctive conditionals.

Syntax Splitting = Relevance + Independence: New Postulates for Nonmonotonic Reasoning From Conditional Belief Bases

Syntax splitting, first introduced by Parikh in 1999, is a natural and desirable property of KR systems. Syntax splitting combines two aspects: it requires that the outcome of a certain epistemic operation should only depend on relevant parts of the underlying knowledge base, where relevance is given a syntactic interpretation (relevance). It also requires that strengthening antecedents by irrelevant information should have no influence on the obtained conclusions (independence). In the context of belief revision the study of syntax splitting already proved useful and led to numerous new insights. In this paper we analyse syntax splitting in a different setting, namely nonmonotonic reasoning based on conditional knowledge bases. More precisely, we analyse inductive inference operators which, like system P, system Z, or the more recent c-inference, generate an inference relation from a conditional knowledge base. We axiomatize the two aforementioned aspects of syntax splitting, relevance and independence, as properties of such inductive inference operators. Our main results show that system P and system Z, whilst satisfying relevance, fail to satisfy independence. C-inference, in contrast, turns out to satisfy both relevance and independence and thus fully complies with syntax splitting.