If an agent believes that the probability of E being true is 1/2, should she accept a bet on E at even odds or better? Yes, but only given certain conditions. This paper is about what those conditions are. In particular, we think that there is a condition that has been overlooked so far in the literature. We discovered it in response to a paper by Hitchcock (2004) in which he argues for the 1/3 answer to the Sleeping Beauty problem. (...) Hitchcock argues that this credence follows from calculating her fair betting odds, plus the assumption that Sleeping Beauty’s credences should track her fair betting odds. We will show that this last assumption is false. Sleeping Beauty’s credences should not follow her fair betting odds due to a peculiar feature of her epistemic situation. (shrink)

The aim of this paper is to argue for a revised and precisified version of the infamous Verifiability Criterion for the meaningfulness of declarative sentences. The argument is based on independently plausible premises concerning probabilistic confirmation and meaning as context-change potential, it is shown to be logically valid, and its ramifications for potential applications of the criterion are being discussed. Although the paper is not historical but systematic, the criterion thus vindicated will resemble the original one(s) in some important ways. (...) At the same time, it will also be more modest insofar as meaningfulness will turn out to be relativized linguistically and probabilistically, and different choices of the linguistic and probabilistic parameters may lead to different verdicts on meaningfulness. (shrink)

The article recapitulates what logic is about traditionally and works out two roles it has been playing in philosophy: the role of an instrument and of a philosophical discipline in its own right. Using Tarski’s philosophical-logical work as case study, it develops a logical reconstructionist methodology of philosophical logic that extends and refines Rudolf Carnap’s account of explication and rational reconstruction. The methodology overlaps with, but also partially diverges from, contemporary anti-exceptionalism about logic.

Pure mathematical truths are commonly thought to be metaphysically necessary. Assuming the truth of pure mathematics as currently pursued, and presupposing that set theory serves as a foundation of pure mathematics, this article aims to provide a metaphysical explanation of why pure mathematics is metaphysically necessary.

In this chapter we will deal with “mechanizing” induction, i.e. with ways in which theoretical computer science approaches inductive generalization. In the field of Machine Learning, algorithms for induction are developed. Depending on the form of the available data, the nature of these algorithms may be very different. Some of them combine geometric and statistical ideas, while others use classical reasoning based on logical formalism. However, we are not so much interested in the algorithms themselves, but more on the philosophical (...) and theoretical foundations they share. Thus in the first of two parts, we will examine different approaches and inductive assumptions in two particular learning settings. While many machine learning algorithms work well on a lot of tasks, the interpretation of the learned hypothesis is often difficult. Thus, while e.g. an algorithm surprisingly is able to determine the gender of the author of a given text with about 80 percent accuracy [Argamon and Shimoni, 2003], for a human it takes some extra effort to understand on the basis of which criteria the algorithm is able to do so. With that respect the advantage of approaches using logic are obvious: If the output hypothesis is a formula of predicate logic, it is easy to interpret. However, if decision trees or algorithms from the area of inductive logic programming are based purely on classical logic, they suffer from the fact that most universal statements do not hold for exceptional cases, and classical logic does not offer any convenient way of representing statements which are meant to hold in the “normal case”. Thus, in the second part we will focus on approaches for Nonmonotonic Reasoning that try to handle this problem. Both Machine Learning and Nonmonotonic Reasoning have been anticipated partially by work in philosophy of science and philosophical logic. At the same time, recent developments in theoretical computer science are expected to trig- ger further progress in philosophical theories of inference, confirmation, theory revision, learning, and the semantic and pragmatics of conditionals. We hope this survey will contribute to this kind of progress by building bridges between computational, logical, and philosophical accounts of induction. (shrink)

In this paper we describe and interpret the formal machinery of abstraction processes in which the domain of abstracta is a subset of the domain of objects from which is abstracted.