Joseph Y. Halpern

Recent work in psychology and experimental philosophy has shown that judgments of actual causation are often influenced by consideration of defaults, typicality, and normality. A number of philosophers and computer scientists have also suggested that an appeal to such factors can help deal with problems facing existing accounts of actual causation. This article develops a flexible formal framework for incorporating defaults, typicality, and normality into an account of actual causation. The resulting account takes actual causation to be both graded and comparative. We then show how our account would handle a number of standard cases. 1 Introduction2 Causal Models3 The HP Definition of Actual Causation4 The Problem of Isomorphism5 Defaults, Typicality, and Normality6 Extended Causal Models7 Examples7.1 Omissions7.2 Knobe effects7.3 Causes versus background conditions7.4 Bogus prevention7.5 Causal chains7.6 Legal doctrines of intervening causes7.7 Pre-emption and short circuits8 Conclusion

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kii [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable

What is an inference rule? This question does not have a unique answer. One usually finds two distinct standard answers in the literature; validity inference $(\sigma \vdash_\mathrm{v} \varphi$ if for every substitution $\tau$, the validity of $\tau \lbrack\sigma\rbrack$ entails the validity of $\tau\lbrack\varphi\rbrack)$, and truth inference $(\sigma \vdash_\mathrm{t} \varphi$ if for every substitution $\tau$, the truth of $\tau\lbrack\sigma\rbrack$ entails the truth of $\tau\lbrack\varphi\rbrack)$. In this paper we introduce a general semantic framework that allows us to investigate the notion of inference more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and first-order logic

The question of whether knowledge is definable in terms of belief, which has played an important role in epistemology for the last 50 years, is studied here in the framework of epistemic and doxastic logics. Three notions of definability are considered: explicit definability, implicit definability, and reducibility, where explicit definability is equivalent to the combination of implicit definability and reducibility. It is shown that if knowledge satisfies any set of axioms contained in S5, then it cannot be explicitly defined in terms of belief. S5 knowledge can be implicitly defined by belief, but not reduced to it. On the other hand, S4.4 knowledge and weaker notions of knowledge cannot be implicitly defined by belief, but can be reduced to it by defining knowledge as true belief. It is also shown that S5 knowledge cannot be reduced to belief and justification, provided that there are no axioms that involve both belief and justification

The appropriateness of S5 as a logic of knowledge has been attacked at some length in the philosophical literature. Here one particular attack based on the interplay between knowledge and belief is considered: Suppose that knowledge satisfies S5, belief satisfies KD45, and both the entailment property (knowledge implies belief) and positive certainty (if the agent believes something, she believes she knows it) hold. Then it can be shown that belief reduces to knowledge: it is impossible to have false beliefs. While the entailment property has typically been viewed as perhaps the least controversial of these assumptions, an argument is presented that it can plausibly be viewed as the culprit. More precisely, it is shown that this attack fails if we weaken the entailment property so that it applies only to objective (nonmodal) formulas, rather than to arbitrary formulas. Since the standard arguments in favor of the entailment property are typically given only for objective formulas, this observation suggests that care must be taken in applying intuitions that seem reasonable in the case of objective formulas to arbitrary formulas

We propose new definitions of (causal) explanation, using structural equations to model counterfactuals. The definition is based on the notion of actual cause, as defined and motivated in a companion article. Essentially, an explanation is a fact that is not known for certain but, if found to be true, would constitute an actual cause of the fact to be explained, regardless of the agent's initial uncertainty. We show that the definition handles well a number of problematic examples from the literature

We examine carefully the rationale underlying the approaches to belief change taken in the literature, and highlight what we view as methodological problems. We argue that to study belief change carefully, we must be quite explicit about the ontology or scenario underlying the belief change process. This is something that has been missing in previous work, with its focus on postulates. Our analysis shows that we must pay particular attention to two issues that have often been taken for granted: the first is how we model the agent's epistemic state. (Do we use a set of beliefs, or a richer structure, such as an ordering on worlds? And if we use a set of beliefs, in what language are these beliefs are expressed?) We show that even postulates that have been called beyond controversy are unreasonable when the agent's beliefs include beliefs about her own epistemic state as well as the external world. The second is the status of observations. (Are observations known to be true, or just believed? In the latter case, how firm is the belief?) Issues regarding the status of observations arise particularly when we consider iterated belief revision, and we must confront the possibility of revising by and then by ¬.

We propose a generalization of expected utility that we call generalized EU (GEU), where a decision maker’s beliefs are represented by plausibility measures and the decision maker’s tastes are represented by general (i.e., not necessarily real-valued) utility functions. We show that every agent, “rational” or not, can be modeled as a GEU maximizer. We then show that we can customize GEU by selectively imposing just the constraints we want. In particular, we show how each of Savage’s postulates corresponds to constraints on GEU.

Judea Pearl (2000) was the first to propose a definition of actual causation using causal models. A number of authors have suggested that an adequate account of actual causation must appeal not only to causal structure but also to considerations of normality. In Halpern and Hitchcock (2011), we offer a definition of actual causation using extended causal models, which include information about both causal structure and normality. Extended causal models are potentially very complex. In this study, we show how it is possible to achieve a compact representation of extended causal models

Three notions of definability in multimodal logic are considered. Two are analogous to the notions of explicit definability and implicit definability introduced by Beth in the context of first-order logic. However, while by Beth’s theorem the two types of definability are equivalent for first-order logic, such an equivalence does not hold for multimodal logics. A third notion of definability, reducibility, is introduced; it is shown that in multimodal logics, explicit definability is equivalent to the combination of implicit definability and reducibility. The three notions of definability are characterized semantically using (modal) algebras. The use of algebras, rather than frames, is shown to be necessary for these characterizations

ABSTRACT For changing opinion, represented by an assignment of probabilities to propositions, the criterion proposed is motivated by the requirement that the assignment should have, and maintain, the possibility of matching in some appropriate sense statistical proportions in a population. This ‘tracking’ criterion implies limitations on policies for updating in response to a wide range of types of new input. Satisfying the criterion is shown equivalent to the principle that the prior must be a convex combination of the possible posteriors. Furthermore, this is equivalent to the requirement that prior expected values must fall inside the range spanned by possible posterior expected values. The tracking criterion is liberal; it allows for, but does not require, a policy such as Bayesian conditionalization, and can be offered as a general constraint on policies for managing opinion over time. Examples are given of non-Bayesian policies, both ones that satisfy and ones that violate the criterion. _1_ Introduction _2_ Alternative Updating Policies _3_ Modelling the Situation for Normal Updating _4_ Tracking: A Criterion for Updating Policies _5_ Tracking: Precise Formulation and Relation to Convexity _6_ The Spanning Criterion _7_ Non-Bayesian Policies that Satisfy the Spanning and Tracking Criteria _8_ Policies that Violate the Spanning and Tracking Criteria Appendix Appendix

We consider the issue of what an agent or a processor needs to know in order to know that its messages are true. This may be viewed as a first step to a general theory of cooperative communication in distributed systems. An honest message is one that is known to be true when it is sent (or said). If every message that is sent is honest, then of course every message that is sent is true. Various weaker considerations than honesty are investigated with the property that provided every message sent satisfies the condition, then every message sent is true

Galles & Pearl (l998) claimed that s [possible-worlds] framework.s framework. Recursive models are shown to correspond precisely to a subclass of (possible-world) counterfactual structures. On the other hand, a slight generalization of recursive models, models where all equations have unique solutions, is shown to be incomparable in expressive power to counterfactual structures, despite the fact that the Galles and Pearl arguments should apply to them as well. The problem with the Galles and Pearl argument is identified: an axiom that they viewed as irrelevant, because it involved disjunction (which was not in their language), is not irrelevant at all

There have been two major lines of research aimed at capturing resource-bounded players in game theory. The first, initiated by Rubinstein (), charges an agent for doing costly computation; the second, initiated by Neyman (), does not charge for computation, but limits the computation that agents can do, typically by modeling agents as finite automata. We review recent work on applying both approaches in the context of decision theory. For the first approach, we take the objects of choice in a decision problem to be Turing machines, and charge players for the “complexity” of the Turing machine chosen (e.g., its running time). This approach can be used to explain well-known phenomena like first-impression-matters biases (i.e., people tend to put more weight on evidence they hear early on) and belief polarization (two people with different prior beliefs, hearing the same evidence, can end up with diametrically opposed conclusions) as the outcomes of quite rational decisions. For the second approach, we model people as finite automata, and provide a simple algorithm that, on a problem that captures a number of settings of interest, provably performs optimally as the number of states in the automaton increases

An agent often has a number of hypotheses, and must choose among them based on observations, or outcomes of experiments. Each of these observations can be viewed as providing evidence for or against various hypotheses. All the attempts to formalize this intuition up to now have assumed that associated with each hypothesis h there is a likelihood function  μ h , which is a probability measure that intuitively describes how likely each observation is, conditional on h being the correct hypothesis. We consider an extension of this framework where there is uncertainty as to which of a number of likelihood functions is appropriate, and discuss how one formal approach to defining evidence, which views evidence as a function from priors to posteriors, can be generalized to accommodate this uncertainty.

A teacher announced to his pupils that on exactly one of the days of the following school week (Monday through Friday) he would give them a test. But it would be a surprise test; on the evening before the test they would not know that the test would take place the next day. One of the brighter students in the class then argued that the teacher could never give them the test. "It can't be Friday," she said, "since in that case we'll expect it on Thurday evening. But then it can't be Thursday, since having already eliminated Friday we'll know Wednesday evening that it has to be Thursday. And by similar reasoning we can also eliminate Wednesday, Tuesday, and Monday. So there can't be a test!" The students were somewhat baffled by the situation. The teacher was well-known to be truthful, so if he said there would be a test, then it was safe to assume that there would be one. On the other hand, he also said that the test would be a surprise. But it seemed that whenever he gave the test, it wouldn't be a surprise. Well, the teacher gave the test on Tuesday, and, sure enough, the students were surprised.

There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the indistinguishability relation is typically taken to be an equivalence relation (and thus transitive). It is shown that if the uncertainty perception and the question of when an agent reports that two things are indistinguishable are both carefully modeled, the problems disappear, and indistinguishability can indeed be taken to be an equivalence relation. Moreover, this model also suggests a logic of vagueness that seems to solve many of the problems related to vagueness discussed in the philosophical literature. In particular, it is shown here how the logic can handle the sorites paradox

In 1960, E. P. Wigner, a joint winner of the 1963 Nobel Prize for Physics, published a paper titled On the Unreasonable Effectiveness of Mathematics in the Natural Sciences [61]. This paper can be construed as an examination and affirmation of Galileo's tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of mathematics in accurately describing physical phenomena. Wigner viewed these examples as illustrations of what he called the empirical law of epistemology, which asserts that the mathematical formulation of the laws of nature is both appropriate and accurate, and that mathematics is actually the correct language for formulating the laws of nature. At the same time, Wigner pointed out that the reasons for the success of mathematics in the natural sciences are not completely understood; in fact, he went as far as asserting that “… the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.”