Online research in philosophy

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. (shrink)

Department of Computer Science, University of California, Los Angeles, Los Angeles, CA 90095, USA judea{at}cs.ucla.edu' + u + '@' + d + ''//--> 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. Introduction Causal models: a review 2.1 Causal models 2.2 Syntax and semantics (...) The definition of cause Examples A more refined definition Discussion AAppendix: Some Technical Issues A.1 The active causal process A.2 A closer look at AC2(b) A.3 Causality with infinitely many variables A.4 Causality in nonrecursive models. (shrink)

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 ¬. (shrink)

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)

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 entialment 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)

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 τ, the validity of τ [σ] entails the validity of τ[φ]), and truth inference $(\sigma \vdash_\mathrm{t} \varphi$ if for every substitution τ, the truth of τ[σ] entails the truth of τ[φ]). 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. (shrink)

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.

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. (shrink)

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. (shrink)