Kian Salimkhani

In this paper, I discuss the relationship between the method of arbitrary functions and single-case probabilities, and offer an interpretation of these probabilities by reference to physical symmetries of chance experiments. The method of arbitrary functions attempts to explain outcome frequencies through the underlying deterministic dynamics of chance experiments and the constant proportions of these dynamics that lead to different outcomes in repeated chance experiments. However, the method of arbitrary functions alone appears to provide no comprehensive interpretation of deterministic chance since it relies on distributions of initial conditions that, in turn, require interpretation. Moreover, even without reference to distributions of initial conditions, the method of arbitrary functions may not be able to provide genuine single-case probabilities since the constant proportions appear to become relevant for outcomes only in repeated chance experiments. In this paper, I propose an explanation of the constant proportions of dynamics via the physical properties of single chance experiments. By doing so, I attempt to develop the method of arbitrary functions into a comprehensive interpretation of probability that does not rely on the distribution of initial conditions and covers genuine single-case probabilities. Specifically, I show how the constant proportions result from what I will call _the relevant physical symmetries of chance experiments_, i.e., from a regular constitution of chance devices with respect to the dynamically realized symmetry transformations in single chance experiments.

Broadly speaking, there are three views on whether Leibniz’s Principle of the Identity of Indiscernibles (PII) is violated in the case of similar particles. According to the earliest view, PII is always violated (call this the no discernibility view); according to the more recent weak discernibility view, PII is at least valid in a weak sense. No and weak discernibility have been referred to as orthodoxy. Steven French has argued that although PII is violated, similar particles can still be regarded as individuals, or, alternatively, as non-individuals: French famously concluded therefore that metaphysics is underdetermined by physics. Call this thesis orthodox underdetermination. Most recently, some authors have turned against orthodoxy by arguing that PII is valid in more than a weak sense – call this the new discernibility view, also referred to as heterodoxy. Since heterodoxy is backed up by physical considerations, metaphysics now seems to be determined by physics: physics indicates that PII is valid. In this paper, I argue that with respect to entangled states, there are two ways to establish PII’s validity, which yield two different ontological interpretations of entanglement. Therefore, a form of underdetermination returns within the heterodox framework. I argue that heterodox underdetermination deserves some attention, because the two ontological interpretations might yield different explanations of the violation of Bell inequalities.