Peter Vranas

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.

I was quite excited when I first read Restall and Russell’s (2010) paper. For two reasons. First, because the paper provides rigorous formulations and formal proofs of implication barrier theses, namely “theses [which] deny that one can derive sentences of one type from sentences of another”. Second (and primarily), because the paper proves a general theorem, the Barrier Construction Theorem, which unifies implication barrier theses concerning four topics: generality, necessity, time, and normativity. After thinking about the paper, I am satisfied with its treatment of the first three topics, namely generality, necessity, and time. But I am not satisfied with its treatment of normativity, so my comments are exclusively on that topic.

Suppose that a sign at the entrance of a hotel reads: “Don’t enter these premises unless you are accompanied by a registered guest”. You see someone who is about to enter, and you tell her: “Don’t enter these premises if you are an unaccompanied registered guest”. She asks why, and you reply: “It follows from what the sign says”. It seems that you made a valid inference from an imperative premise to an imperative conclusion. But it also seems that imperatives cannot be true or false, so what does it mean to say that your inference is valid? It cannot mean that the truth of its premise guarantees the truth of its conclusion. One is thus faced with what is known as “Jørgensen’s dilemma” (Ross 1941: 55-6): it seems that imperative logic cannot exist because logic deals only with entities that, unlike imperatives, can be true or false, but it also seems that imperative logic must exist. It must exist not only because inferences with imperatives can be valid, but also because imperatives (like “Enter” and “Don’t enter”) can be inconsistent with each other, and also because one can apply logical operations to imperatives: “Don’t enter” is the negation of “Enter”, and “Sing or dance” is the disjunction of “Sing” and “Dance”. A standard reaction to this dilemma consists in basing imperative logic on analogues of truth and falsity. For example, the imperative “Don’t enter” is satisfied if you don’t enter and is violated if you enter, and one might say that an inference from an imperative premise to an imperative conclusion is valid exactly if the satisfaction (rather than the truth) of the premise guarantees the satisfaction of the conclusion. But before getting into the details, more needs to be said on what exactly imperatives are

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.