Online research in philosophy

David Lewis, Michael Thau, and Ned Hall have recently argued that the Principal Principle—an inferential rule underlying much of our reasoning about probability—is inadequate in certain respects, and that something called the ‘New Principle’ ought to take its place. This paper argues that the Principle Principal need not be discarded. On the contrary, Lewis et al. can get everything they need—including the New Principle—from the intuitions and inferential habits that inspire the Principal Principle itself, while avoiding the problems that originally caused them to abandon that principle.

There are at least three core principles that define the chance role: (1) the Principal Principle, (2) the Basic Chance Principle, and (3) the Humean Principle. These principles seem mutually incompatible. At least, no extant account of chance meets more than one of them. I offer an account of chance which meets all three: L*-chance. So the good news is that L*-chance meets (1)–(3). The bad news is that L*-chance turns out unlawful and unstable.

Faced with the paradox of undermining futures, Humeans have resigned themselves to accounts of chance that severely conflict with our intuitions. However, such resignation is premature: The problem is Humean supervenience (HS), not Humeanism. This paper develops a projectivist Humeanism on which chance claims are understood as normative, rather than fact stating. Rationality constraints on the cotenability of norms and factual claims ground a factual-normative worlds semantics that, in addition to solving the Frege-Geach problem, delivers the intuitive set of possibilia for each chance law. Hence, the account does not entail HS, and the paradox does not arise. A confirmation theory is developed, and the 'principal principle' is justified.

The Principal Principle (PP) says that, for any proposition A, given any admissible evidence and the proposition that the chance of A is x%, one's conditional credence in A should be x%. Humean Supervenience (HS) claims that, among possible worlds like ours, no two differ without differing in the spacetime-point-by-spacetime-point arrangement of local properties. David Lewis (1986b, 1994a) has argued that PP contradicts HS, and the validity of his argument has been endorsed by Bigelow et al. (1993), Thau (1994), Hall (1994), Strevens (1995), Ismael (1996), Hoefer (1997), and Black (1998). Against this consensus, I argue that PP might not contradict HS: Lewis's argument is invalid, and every attempt -- within a broad class of attempts -- to amend the argument fails.

David Lewis's influential work on the epistemology and metaphysics of objective chance has convinced many philosophers of the central importance of the following two claims: First, it is a serious cost of reductionist positions about chance (such as that occupied by Lewis) that they are, apparently, forced to modify the Principal Principle--the central principle relating objective chance to rational subjective probability--in order to avoid contradiction. Second, it is a perhaps more serious cost of the rival non-reductionist position that, unlike reductionism, it can give no coherent explanation for why the Principal Principle should hold. I argue that both of these claims are fundamentally mistaken.

David Lewis's influential work on the epistemology and metaphysics of objective chance has convinced many philosophers of the central importance of the following two claims: First, it is a serious cost of reductionist positions about chance (such as that occupied by Lewis) that they are, apparently, forced to modify the Principal Principle--the central principle relating objective chance to rational subjective probability--in order to avoid contradiction. Second, it is a perhaps more serious cost of the rival non-reductionist position that, unlike reductionism, it can give no coherent explanation for why the Principal Principle should hold. I argue that both of these claims are fundamentally mistaken.

Lewis’ Principal Principle (PP) aims at clarifying the connection between chance (i.e. objective probability) and credence (i.e. subjective probability). It is generally assumed that the chance that an event will occur does not depend on our credence in the occurrence of that event. Nevertheless, chances constrain our credence, and Lewis’ PP is an attempt to capture this connection.

one's subjective probability for a proposition should conform to one's beliefs about that proposition's objective chance of coming true. David Lewis has argued (i) that this principle provides the defining role for chance; (ii) that it conflicts with his reductionist thesis of Humean supervenience, and so must be replaced by an amended version that avoids the conflict; hence (iii) that nothing perfectly deserves the name ‘chance’, although something can come close enough by playing the role picked out by the amended principle. We show that in fact there must be ‘chances’ that perfectly play what Lewis takes to be the defining role. But this is not the happy conclusion it might seem, since these ‘chances’ behave too strangely to deserve the name. The lesson is simple: much more than the Principal Principle—more to the point, much more than the connection between chance and credence—informs our understanding of objective chance. 1 Introduction 2 Preliminaries 3 Undermining futures and the New Principle 4 The Old Principle rescued? 5 The New Bug 6 Conclusion.

[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).

A chance-credence norm states how an agent's credences in propositions concerning objective chances ought to relate to her credences in other propositions. The most famous such norm is the Principal Principle (PP), due to David Lewis. However, Lewis noticed that PP is inconsistent with many accounts of chance that attempt to reduce chance facts to non-modal facts. Those who defend such accounts of chance have offered two alternative chance-credence norms, both of which are consistent with reductionism about chance: the first is the New Principle (NP), formulated by Ned Hall and Michael Thau and grudgingly accepted by Lewis; and the second is Jenann Ismael's General Recipe (IP). However, while NP and IP are each consistent with the sort of reductionism that conflicts with PP, they are incompatible with one another. Thus, the question arises: Which should the reductionist favour? In this paper, I argue that the consequences of IP when coupled with a reductionist account are unacceptable, so we must accept NP. I conclude by considering two responses to my arguments.