Peter Vranas

Why does classical equilibrium statistical mechanics work? Malament and Zabell (1980) noticed that, for ergodic dynamical systems, the unique absolutely continuous invariant probability measure is the microcanonical. Earman and Rédei (1996) replied that systems of interest are very probably not ergodic, so that absolutely continuous invariant probability measures very distant from the microcanonical exist. In response I define the generalized properties of epsilon-ergodicity and epsilon-continuity, I review computational evidence indicating that systems of interest are epsilon-ergodic, I adapt Malament and Zabell’s defense of absolute continuity to support epsilon-continuity, and I prove that, for epsilon-ergodic systems, every epsilon-continuous invariant probability measure is very close to the microcanonical.

[1] You have a crystal ball. Unfortunately, it’s defective. Rather than predicting the future, it gives you the chances of future events. Is it then of any use? It certainly seems so. You may not know for sure whether the stock market will crash next week; but if you know for sure that it has an 80% chance of crashing, then you should be 80% confident that it will—and you should plan accordingly. More generally, given that the chance of a proposition A is x%, your conditional credence in A should be x%. This is a chance-credence principle: a principle relating chance (objective probability) with credence (subjective probability, degree of belief). Let’s call it the Minimal Principle (MP).

I outline six components of a comprehensive proposal for overhauling the foundations of deontic logic. (1) Actions and prescriptions are temporally indexed; more precisely, they attach to nodes of a tree in a branching time structure. (2) Actions are (modeled as) sets of branches and can be coarse- or fine-grained depending on whether or not they have proper subsets which are also actions. (3) Prescriptions have satisfaction and violation sets; these are sets of branches which may—but need not—be or include actions. (4) Prescriptive propositions, which state that an action is obligatory/permitted/forbidden according to a given prescription, are defined by relating the action with the satisfaction and violation sets of the prescription. (5) Conditional prescriptions can—but need not—be derived from unconditional or even from other conditional ones. (6) Thick prescriptions, in contrast to thin ones, prescribe or proscribe actions with varying intensities, and can have embedded subprescriptions (some of which are negative, namely “contraryto-duty”). Most of the above components are inspired by the literature, but their combination is novel.

David Lewis (1980) proposed the Principal Principle (PP) and a “reformulation” which later on he called ‘OP’(Old Principle). Reacting to his belief that these principles run into trouble, Lewis (1994) concluded that they should be replaced with the New Principle (NP). This conclusion left Lewis uneasy, because he thought that an inverse form of NP is “quite messy”, whereas an inverse form of OP, namely the simple and intuitive PP, is “the key to our concept of chance”. I argue that, even if OP should be discarded, PP need not be. Moreover, far from being messy, an inverse form of NP is a simple and intuitive Conditional Principle (CP). Finally, both PP and CP are special cases of a General Principle (GP); it follows that so are PP and NP, which are thus compatible rather than competing.