We propose new definitions of 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.

Using formal systems to represent and reason about uncertainty.

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. (shrink)

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. (shrink)

We propose a new definition of actual causes, using structural equations to model counterfactuals. We show that the definition yields a plausible and elegant account of causation that handles well examples which have caused problems for other definitions and resolves major difficulties in the traditional account.

Reasoning About Knowledge is the first book to provide a general discussion of approaches to reasoning about knowledge and its applications to distributed ...

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 algebras. The use of algebras, rather than frames, is shown to be necessary for these characterizations. (shrink)

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.

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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.”. (shrink)

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. (shrink)

Natural conditions are provided that are sufficient to ensure that causality as defined by approaches that use counterfactual dependence and structural equations will be transitive.

We consider a setting where a decision maker’s uncertainty is represented by a set of probability measures, rather than a single measure. Measure-by-measure updating of such a set of measures upon acquiring new information is well known to suffer from problems. To deal with these problems, we propose using weighted sets of probabilities: a representation where each measure is associated with a weight, which denotes its significance. We describe a natural approach to updating in such a situation and a natural (...) approach to determining the weights. We then show how this representation can be used in decision making, by modifying a standard approach to decision making—minimizing expected regret—to obtain minimax weighted expected regret. We provide an axiomatization that characterizes preferences induced by MWER both in the static and dynamic case. (shrink)

I show how thinking in terms of the protocol used can help clarify problems related to anthropic reasoning and self-location, such as the Doomsday Argument and the Sleeping Beauty Problem.

The menu-dependent nature of regret-minimization creates subtleties when it is applied to dynamic decision problems. It is not clear whether forgone opportunities should be included in the menu. We explain commonly observed behavioral patterns as minimizing regret when forgone opportunities are present. If forgone opportunities are included, we can characterize when a form of dynamic consistency is guaranteed.

We give a simple proof characterizing the complexity of Presburger arithmetic augmented with additional predicates. We show that Presburger arithmetic with additional predicates is Π 1 1 complete. Adding one unary predicate is enough to get Π 1 1 hardness, while adding more predicates (of any arity) does not make the complexity any worse.

Sufficient conditions are given under which ratifiable acts exist.

doi: 10.1017/S1755020308090084, Published by Cambridge University Press 31 March 2009 in Volume 1, Number 4 of The Review of Symbolic Logic . On page 541, in the 4 th paragraph, in line 7, an error occurred. The sentence should correctly read: “For all worlds w , if there is more than one grain of sand in the pile in w , then there is still more than one grain of sand after removing one grain of sand.”.

Chateauneuf and Faro axiomatize a weighted version of maxmin expected utility over acts with nonnegative utilities, where weights are represented by a confidence function. We argue that their representation is only one of many possible, and we axiomatize a more natural form of maxmin weighted expected utility. We also provide stronger uniqueness results.

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. (shrink)

We show that a 0–1 law holds for propositional modal logic, both for structure validity and frame validity. In the case of structure validity, the result follows easily from the well-known 0–1 law for first-order logic. However, our proof gives considerably more information. It leads to an elegant axiomatization for almost-sure structure validity and to sharper complexity bounds. Since frame validity can be reduced to a Π11 formula, the 0–1 law for frame validity helps delineate when 0–1 laws exist for (...) second-order logics. (shrink)

We show that a 0–1 law holds for propositional modal logic, both for structure validity and frame validity. In the case of structure validity, the result follows easily from the well-known 0–1 law for first-order logic. However, our proof gives considerably more information. It leads to an elegant axiomatization for almost-sure structure validity and to sharper complexity bounds. Since frame validity can be reduced to a Π11 formula, the 0–1 law for frame validity helps delineate when 0–1 laws exist for (...) second-order logics. (shrink)