Leon Horsten

Fair infinite lotteries

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.

Formal Methods in the Philosophy of Science

In this article, we reflect on the use of formal methods in the philosophy of science. These are taken to comprise not just methods from logic broadly conceived, but also from other formal disciplines such as probability theory, game theory, and graph theory. We explain how formal modelling in the philosophy of science can shed light on difficult problems in this domain.

Norms for Theories of Reflexive Truth

In the past two decades we have witnessed a shift to axiomatic theories of truth. But in this tradition there has been a proliferation of truth theories. In this article we carry out a meta-theoretical reflection on the conditions that we should want axiomatic truth theories to satisfy.

Term Models for Abstraction Principles

Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the method is poorly behaved unless the background second-order logic is predicative.

One Hundred Years of Semantic Paradox

This article contains an overview of the main problems, themes and theories relating to the semantic paradoxes in the twentieth century. From this historical overview I tentatively draw some lessons about the way in which the field may evolve in the next decade.

Infinitesimal probabilities (epub ahead of print)

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. 1u2003Introduction 2u2003The Limits of Classical Probability Theory u2003u20032.1u2003Classical probability functions u2003u20032.2u2003Limitations u2003u20032.3u2003Infinitesimals to the rescue? 3u2003NAP Theory u2003u20033.1u2003First four axioms of NAP u2003u20033.2u2003Continuity and conditional probability u2003u20033.3u2003The final axiom of NAP u2003u20033.4u2003Infinite sums u2003u20033.5u2003Definition of NAP functions via infinite sums u2003u20033.6u2003Relation to numerosity theory 4u2003Objections and Replies u2003u20034.1u2003Cantor and the Archimedean property u2003u20034.2u2003Ticket missing from an infinite lottery u2003u20034.3u2003Williamson’s infinite sequence of coin tosses u2003u20034.4u2003Point sets on a circle u2003u20034.5u2003Easwaran and Pruss 5u2003Dividends u2003u20035.1u2003Measure and utility u2003u20035.2u2003Regularity and uniformity u2003u20035.3u2003Credence and chance u2003u20035.4u2003Conditional probability 6u2003General Considerations u2003u20036.1u2003Non-uniqueness u2003u20036.2u2003Invariance Appendixu2003 1u2003Introduction 2u2003The Limits of Classical Probability Theory u2003u20032.1u2003Classical probability functions u2003u20032.2u2003Limitations u2003u20032.3u2003Infinitesimals to the rescue? u2003u20032.1u2003Classical probability functions u2003u20032.2u2003Limitations u2003u20032.3u2003Infinitesimals to the rescue? 3u2003NAP Theory u2003u20033.1u2003First four axioms of NAP u2003u20033.2u2003Continuity and conditional probability u2003u20033.3u2003The final axiom of NAP u2003u20033.4u2003Infinite sums u2003u20033.5u2003Definition of NAP functions via infinite sums u2003u20033.6u2003Relation to numerosity theory u2003u20033.1u2003First four axioms of NAP u2003u20033.2u2003Continuity and conditional probability u2003u20033.3u2003The final axiom of NAP u2003u20033.4u2003Infinite sums u2003u20033.5u2003Definition of NAP functions via infinite sums u2003u20033.6u2003Relation to numerosity theory 4u2003Objections and Replies u2003u20034.1u2003Cantor and the Archimedean property u2003u20034.2u2003Ticket missing from an infinite lottery u2003u20034.3u2003Williamson’s infinite sequence of coin tosses u2003u20034.4u2003Point sets on a circle u2003u20034.5u2003Easwaran and Pruss u2003u20034.1u2003Cantor and the Archimedean property u2003u20034.2u2003Ticket missing from an infinite lottery u2003u20034.3u2003Williamson’s infinite sequence of coin tosses u2003u20034.4u2003Point sets on a circle u2003u20034.5u2003Easwaran and Pruss 5u2003Dividends u2003u20035.1u2003Measure and utility u2003u20035.2u2003Regularity and uniformity u2003u20035.3u2003Credence and chance u2003u20035.4u2003Conditional probability u2003u20035.1u2003Measure and utility u2003u20035.2u2003Regularity and uniformity u2003u20035.3u2003Credence and chance u2003u20035.4u2003Conditional probability 6u2003General Considerations u2003u20036.1u2003Non-uniqueness u2003u20036.2u2003Invariance u2003u20036.1u2003Non-uniqueness u2003u20036.2u2003Invariance Appendixu2003

Absolute Infinity in Class Theory and in Theology

In this article we investigate similarities between the role that ineffability of Absolute Infinity plays in class theory and in theology.