A Puzzle about Objective Chance and Causation

Suppose that this is how a given casino’s 10-cent slot machine works: it has a random number generator which produces a  string of numbers between 1 and 1000, given a seed value.  Pulls of the lever are put into correspondence, chronologically, with this randomly-generated string.  If a lever pull matches a certain designated number, say, 222, then that lever pull gets a payout of $90.  Here’s a proposition about these slot machines:

A) The objective chance that the slot machine pays out, on any given pull, is 1/1000.

It’s true that, if we were to know the value of the seed and the nature of the random number generator, then we could figure out precisely when the machine will pay out.  But, given determinism, precisely the same thing is true of any coin flip or die roll.  Were we to know the precise microphysical initial conditions of the coin flip and the laws of nature, we could figure out whether the coin will land heads or tails.  This is no obstacle to there being an objective chance associated with an event – it only tells us that a precise specification of the microphysical initial conditions is inadmissible information.  Similarly, the seed-value and the method of random number generation is inadmissible information when it comes to the slot machine.  But this doesn’t mean that there isn’t an objective chance that the slot machine pays out, on any given pull.

Here’s another proposition:

B) The objective chance that any given roll of a fair, six-faced die lands 1-up is 1/6.

This should be beyond reproach.

Finally, consider this proposition:

C)  If there is a robust causal law to the effect that events of type A cause all and only events of type B — so that every A event leads to a B event, and no B event is caused by anything other than an A event — then the objective chance of an A event occurring is equal to the objective chance of a B event occurring.

Besides being intuitively plausible, I take (C) to be one of the central claims underlying the Bayes-Net approach to testing causal hypotheses.  When we model causation and objective chance in the way specified by Pearl’s and Spirtes et. al.’s causal models, we allow the causal laws codified in the structural equations to induce a probability function over the endogenous variables.  If (C) were false, then this would be illegitimate.

The Puzzle is that (A), (B), and (C) are inconsistent, as the following story demonstrates.

Suppose that the casino owners want to know the seed value for their slot machine.  They want, that is, inadmissible information that will let them calculate, ahead of time, what their bottom line will look like after a certain number of pulls of the slot machine.  However, while protective of their bottom line, they aren’t unscrupulous.  They don’t want to plant the seed, they just want to know what it is.  So, here’s what they do:  they produce 6 randomly-selected seed values, using standard techniques (clipping three numbers from the end of a 10-digit decimal expansion of an arbitrarily selected time, e.g.).  Then, they roll a die to determine which of these seed values will go into the slot machine.

Suppose that it’s true that, if the first seed is selected, then the slot machine will pay out on the 1001st pull of the lever.  If any of the other seeds are selected, then the slot machine will not pay out on the 1001st pull.  Then, there is a robust causal law asserting the following:  The slot machine will pay out on the 1001st pull if and only if the die landed 1-up.

If (B) is true, then the objective chance of the die landing 1-up is 1/6.  But then, if (C) is true, then the objective chance of the machine paying out on the 1001st pull is 1/6 — since there is a robust causal law saying that the die lands 1-up if and only if the 1001st pull pays out.  By (C), the objective chance of the cause must be equal to the objective chance of the effect.  So the objective chance of the machine paying out on the 1001st pull must be 1/6.  But this contradicts (A), which says that the objective chance of the machine paying out on any given pull is 1/1000, not 1/6.

It’s true, of course, that both the die roll and the causal law involves all sorts of inadmissible information.  But inadmissible information is only relevant to the question of what our credence should be.  The puzzle, as I’ve formulated it, has absolutely nothing to do with credence.  It has to do only with the objective chance function, and the connection between the objective chances of various events which are related by robust causal laws.

Tooley on Laws of Nature and Counterfactual Support

So I was looking over Dan’s reconstruction of Tooley’s argument below, and I’m still somewhat worried about its validity.

Dan mentioned that I thought there might be some funny business with premises (3) and (10) (on his first reconstruction):

(3) It is a nomological truth that all salt, when in water, dissolves.

(10) It is true that if this piece of salt were in water and were not dissolving, it would not be in the vicinity of a piece of gold.

Dan’s certainly right that there’s no straight-up contradiction here (like there would be if (10) was stating a material conditional).  However, something still feels very odd about the line of argumentation being taken.  What originally bothered me was the fact that, in (10), we are using a putative law to support a counterfactual about what would happen in a situation in which that law (or, rather, the law from which it is derived) is violated.  And I’m not convinced that any law can support a counterfactual like this.

This is a problem because Tooley is trying to draw a distinction between two classes of nomological truths (the laws proper and the logical consequences of laws).  He is arguing that the second class cannot support certain counterfactuals which the first class can – and that, therefore, they must be treated differently.  However, if he demonstrates this by pointing to a certain class of counterfactuals which are not only problematic for the second class, but for the first class as well, then he’s failed to draw the distinction.

So, I’ve been convinced by Dan that Tooley’s argument demonstrates that nomological truths like (L) all salt, when in water and near gold, dissolves in water. have difficulty supporting certain counterfactuals.  However, I haven’t been convinced that laws proper don’t face the very same difficulty.  On my understanding of things, if I can show that a plain jane law faces similar difficulties with the same kind of counterfactual, then I will have undermined Tooley’s distinction.

Tooley’s Laws of Nature and Counterfactual Support

In “The Nature of Laws”, Michael Tooley argues that some proper subclass of the nomological truths are laws of nature, since laws should support counterfactuals and not all nomological truths do that.

He says, “If one says that all nomological statements support counterfactuals, and that it is a nomological truth that all salt when both in water and near gold dissolves, one will be forced to accept [that if this piece of salt were in water and were not dissolving, it would not be in the vicinity of a piece of gold], whereas it is clear that there is good reason not to accept [that].”  (Last line in first paragraph of section 3.)

philosopher makes mistake

In “How to Define Theoretical Terms” (1970), David Lewis says the following. Take a theory T that introduced a new term ‘t’. Replace ‘t’ in T with an appropriate variable to form an open sentence R`. Lewis now claims that ‘t’ is correctly defined as follow:

t = the unique x such that R`

Note the uniqueness requirement. If there are multiple realizations of R` (that is, variable assignments that satisfy R`) differing in what they assign to x, then ‘t’ is denotationless. Van Fraassen (1997) argues that, provided only that T is consistent and has an infinite model, such will always be the case.

The Road to Ramseyan Humility

I’ve mentioned to a few of you around here that I think it is Lewis’ solution to “Putnam’s Paradox” that lands him in “Ramseyan Humility”. Here’s a sketch of why. I apologize for assuming some background familiarity with both “Putnam’s Paradox” and “Ramseyan Humility”, but I’m trying to keep things punchy.

Consider Lewis’ argument for Humility. Lewis’ assumes that T is realized by fundamental properties, which (because fundamental properties are plausibly wholly distinct from the roles they realize) allows Lewis to run the permutation argument to establish multiple possible realizations of T. But why does Lewis assume that T is realized by fundamental properties? There are actually several questions here. First, why does Lewis think that T is even prima face the sort of thing that is realized—that is, why does Lewis think that the Ramsey sentence approach to the content of T is the right one? Second, why does Lewis think that T is realized (rather than holding a non-representational interpretation of the Ramsey sentence as Sklar does)? Third, assuming that T is realized by something, why does Lewis think that T is realized by (metaphysically robust) natural properties [rather than, say, (metaphysically thin) classes]? And, finally, fourth, assuming that T is realized by natural properties, why does Lewis think that T is realized by completely natural properties, i.e., fundamental properties (rather than, say, natural “enough” properties).

Although there are four questions here, I think Lewis’ answer to the first three will be roughly the same: realism. His answer to the fourth question, however, can be found in RH where he makes a particular observation about scientific progress. Let’s take these questions in order.