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 problemby A. Elga (2004]) does not seem to appreciate the opportunitiesprovided by the use of internal finitely-additive probabilitymeasures. Indeed, internal laws of probability can be used tofind a satisfactory infinitesimal answer to many zero-fit problems,not only to the one suggested by Elga, but also to the Markovchain (that is, discrete and memory-less) models of reality.Moreover, the generalization of likelihoods that Elga has inmind is not as hopeless as it appears to be in his article.In fact, for many practically important examples, through theuse of likelihoods one can succeed in circumventing the zero-fitproblem.

1 The Zero-fit Problem on Infinite State Spaces

2Elga's Critique of the Infinitesimal Approach to the Zero-fitProblem

3 Two Examples for Infinitesimal Solutions to theZero-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 Solutionto the Zero-fit Problem for a Finite-state Modelwithout Memory

D An Additional Note on ‘Integrating over Densities’

E Well-defined Continuous Versions of Density Functions