Here you can get a sneak preview at some of my drafts and unpublished papers.
Work in progress
"Real and hyperreal possibility: infinitesimal probabilities in branching time structures"
Abstract We investigate how branching time (BT) structures can be combined with probability theory. In particular, we consider assigning infinitesimal probabilities -- available in non-Archimedean probability theory -- to individual histories. We illustrate the proposal by applying it to an infinite sequence of coin tosses. We also demonstrate how the approach works in light of the problems of future contingents and historical counterfactuals. We introduce the concept of 'hyperreal possibility' as a new modal notion between 'impossibility' and 'real possibility'.
"Hyperreals and Their Applications"
Abstract Hyperreal numbers are an extension of the real numbers, which contain infinitesimals and infinite numbers. This document is prepared as a handout for two tutorial sessions on "Hyperreals and their applications", presented at the Formal Epistemology Workshop 2012 (May 29-June 2) in Munich. It is set up as an annotated bibliography about hyperreals. It does not aim to be exhaustive or to be formally precise; instead, its goal is to direct the reader to relevant sources in the literature on this fascinating topic. The document consists of two parts: sections 1-3 introduce NSA from different perspectives and sections 4-9 discuss applications, with an emphasis on topics that may be of interest to formal epistemologists and to philosophers of mathematics or science.
"The Lockean Thesis and Stratified Belief: Rational Beliefs for Real People?"
Abstract Traditional decision theory has a normative goal and models agents with highly idealized properties. More recent decision-theoretic models take into account basic descriptive aspects, although they remain normative in scope. In economical decision theory, considering human cognitive limitations has led to the concept of bounded rationality. In epistemology, similar considerations have led to the realization that in order to reason about probabilistic information, humans may need to extract full beliefs from it first. The Lockean Thesis aims to inform us how to achieve this extraction in a rational way. In formal epistemology, the Lockean Thesis is usually modelled in terms of a probability threshold. In this article, we discuss an alternative formal model, called Stratified Belief. Instead of postulating a threshold, this model is better understood in terms of an approximation of probability values, where the coarseness of the approximation is context-dependent.
The main goal of this article is to investigate whether the Lockean Thesis and Stratified Belief are psychologically plausible models. We review relevant findings from psychological experiments on mental number representation and processing of probabilistic information. There is indeed psychological evidence that humans rely on an approximative representation of continuous quantities, but there remains a gap between the formal and the experimental sides. On the one hand, the Lockean Thesis and Stratified Belief do not cover the dimension of utility; on the other hand, there are no experiments on probability perception without utility or other confounding factors. This is the first time that the model of Stratified Belief is discussed in relation to the psychological literature, so this is an explorative article rather than an exhaustive exposition.