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. (shrink)

Review of Joseph Y. Halpern (ed.), Theoretical Aspects of Reasoning About Knowledge: Proceedings of the 1986 Conference (Los Altos, CA: Morgan Kaufmann, 1986),.

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.

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)

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)

This collection of essays is a Festschrift for Ernest W. Adams, and is based on a symposium that was held in his honor in 1993. As the title suggests, most of the essays focus on probability and the logic of conditionals, and the relationship between them; they draw their inspiration from Adams’s seminal work on the subject. As a computer scientist, I was struck by just how much the topics discussed play a major role in much recent work in computer (...) science, and how relevant much recent work in computer science is to some of the issues discussed here. (shrink)

This collection of essays is a Festschrift for Ernest W. Adams, and is based on a symposium that was held in his honor in 1993. As the title suggests, most of the essays focus on probability and the logic of conditionals, and the relationship between them; they draw their inspiration from Adams’s seminal work on the subject. As a computer scientist, I was struck by just how much the topics discussed play a major role in much recent work in computer (...) science, and how relevant much recent work in computer science is to some of the issues discussed here. (shrink)

We present the logic K/2 which is a logic with classical implication and only the left part of classical negation.We show that it is possible to define a classical negation into K/2 and that the classical proposition logic K can be translated into this apparently weaker logic.We use concepts from model-theory in order to characterized rigorously this translation and to understand this paradox. Finally we point out that K/2 appears, following Haack's distinction, both as a deviation and an extension of (...) K. (shrink)

The common liberal understanding of reproductive autonomy – characterized by free choice and a principle of non-interference – serves as a useful way to analyse the normative appeal of having certain choices open to people in the reproductive realm, especially for issues like abortion rights. However, this liberal reading of reproductive autonomy only offers us a limited ethical understanding of what is at stake in many kinds of reproductive choices, particularly when it comes to different uses of reproductive technologies and (...) third-party reproduction. This is because the liberal framework does not fully capture who benefits from which reproductive options, the extent of the risks and harms involved in various reproductive interventions, and the reasons for why people are driven to make certain reproductive choices. (shrink)

Literature on testimonial injustice and ways that perpetrators might combat it have flourished since Miranda Fricker’s ground-breaking work on testimonial injustice. Less attention has been given, however, to the role of bystanders. In this paper, I examine the accountability that bystanders may have for their omissions to redress testimonial injustice. I argue that bystander accountability applies in cases where it is opportune for bystanders to intervene, and if they are also sufficiently equipped and able to redress the testimonial injustice. Moreover, (...) I recommend that we move beyond virtue responsibilism for ameliorative thinking about testimonial injustice. (shrink)

Bio-heteronormative conceptions of the family have long reinforced a nuclear ideal of the family as a heterosexual marriage, with children who are the genetic progeny of that union. This ideal, however, has also long been resisted in light of recent social developments, exhibited through the increased incidence and acceptance of step-families, donor-conceived families, and so forth. Although to this end some might claim that the bio-heteronormative ideal is not necessary for a social unit to count as a family, a more (...) systematic conceptualization of the family—the kind of family that matters morally—is relatively underexplored in the philosophical literature. This paper makes a start at developing and defending an account of the family that is normatively attractive and in line with the growing prevalence of non-conventional families and methods of family-formation. Our account, which we call a constitutive-affirmative model of the family, takes the family to be constituted by an ongoing process of relevant affective and affirmative relations between the putative family members. (shrink)

It was natural and inevitable that his two Aethiopias, in the eastern and western extremities of the world, should be identified with the countries of the two dark-skinned peoples in the Far East and the Far West of the Ancient World: India and Mauretania. There was the difficulty that the real Aethiopia was in Africa, neither in the Far East nor in the Far West. Serious writers on geography tried to reconcile Homer and the geographical facts. I wish to excerpt (...) from Strabo three theories concerning Aethiopia and the Aethiopians which will then be used to elucidate passages in Latin literature. (shrink)

It was natural and inevitable that his two Aethiopias, in the eastern and western extremities of the world, should be identified with the countries of the two dark-skinned peoples in the Far East and the Far West of the Ancient World: India and Mauretania.There was the difficulty that the real Aethiopia was in Africa, neither in the Far East nor in the Far West. Serious writers on geography tried to reconcile Homer and the geographical facts.I wish to excerpt from Strabo (...) three theories concerning Aethiopia and the Aethiopians which will then be used to elucidate passages in Latin literature. (shrink)