Internal Laws of Probability, Generalized Likelihoods and Lewis’ Infinitesimal Chances–A Response to Adam Elga

The rejection of an infinitesimal solution to the zero-fit problem by A. Elga ([2004]) does not seem to appreciate the opportunities provided by the use of internal finitelyadditive probability measures. Indeed, internal laws of probability can be used to find a satisfactory infinitesimal answer to many zero-fit problems, not only to the one suggested by Elga, but also to the Markov chain (that is, discrete and memory-less) models of reality. Moreover, the generalization of likelihoods that Elga has in mind is not as hopeless as it appears to be in his article. In fact, for many practically important examples, through the use of likelihoods one can succeed in circumventing the zero-fit problem. 1 The Zero-fit Problem on Infinite State Spaces 2 Elga’s Critique of the Infinitesimal Approach to the Zero-fit Problem 3 Two Examples for Infinitesimal Solutions to the Zero-fit Problem 4 Mathematical Modelling in Nonstandard Universes? 5 Are Nonstandard Models Unnatural? 6 Likelihoods and Densities A Internal Probability Measures and the Loeb Measure Construction B The (Countable) Coin Tossing Sequence Revisited C Solution to the Zero-fit Problem for a Finite-state Model without Memory D An Additional Note on ‘Integrating over Densities’ E Well-defined Continuous Versions of Density Functions 1 The Zero-fit Problem on Infinite State Spaces In general, the zero-fit problem arises when two candidates for a probabilistic law of nature assign probability (‘fit’) zero to an observation and it is therefore not possible to prefer one over the other. This is the case, for instance, on The Author (2007). Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. doi:10.1093/bjps/axl028 For Permissions, please email: [email protected] Advance Access published