Salmon, in Causality and Explanation, suggests that causal processes are demarcated from pseudo-processes by their ability to transmit marks – causal processes can transmit marks; pseudo-processes can’t. About mark transmission: “A mark that has been introduced into a process by means of a single intervention at a point A is transmitted to point B if and only if it occurs at B and at all stages of the process between A and B without additional interventions.” (CaE, p. 197)

Here’s a simple example: There’s a rotating spotlight in the center of a circular room which casts a spot of light on the wall. The light ray traveling from the spotlight to the wall is a causal process; interpose a red filter in the beam near its source and the spot on the wall will be red. The spot of light moving around the wall is a pseudo-process; no interposing of a red filter (or intervention of any sort) can make the spot maintain its redness (or retain a mark of any sort) as it moves on.

We can ask this question, though: How does the process make the mark appear elsewhere within it? (CaE, p. 197) Salmon thinks the answer is ‘astonishingly simple’: it doesn’t (not in any deep sense, anyway). The transmission of a mark from point A in a causal process to point B in the same process just is the fact that it appears at each point between A and B without further interactions.(CaE, p. 197)

I don’t think this is right. It doesn’t get mark transmission right in close by possible worlds (maybe even in our world). Consider a world w. In w there’s a particle a. a can have properties P, Q, and R. Choose any time t. The probability that a will be P at t is 1/3. Likewise for Q and R. There’s another particle b in w. If b strikes a, a will be P, but only during the strike. Suppose b strikes a at t1. b then ricochets and barrells into space. a, then, is P at t1. Suppose also that, by chance, a is P at t2 and at all times between t1 and t2. It seems that, by Salmon’s criteria, a mark (i.e. P) is transmitted from t1 to t2 along the a’s-travels-process. But clearly it’s not. Mark transmission seems to not be as simple as Salmon takes it to be.

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6 Responses to An “At-At” Theory of Causal Influence

You say the probability of a being P at t for any time t is 1/3 and you also say that a is P at each of the time points between t1 and t2.

First, there is the worry over whether it even makes sense to say that the probability of a particle’s being P at t (a time point!) is 1/3. Doesn’t QM merely talk about the probabilities of particles having properties during INTERVALS (rather than at time points)?

Second, it’s not clear that it makes sense to suppose both that the probability of a’s being P at t for any time t is 1/3 AND that a is P at each of (I assume) infinitely (!) many times t within a given interval. It seems that in any world where the latter is the case, it is NOT the case that the probability of a’s being P at ANY time t is 1/3, for if it were, the probability of a’s being P at SOME time during the interval would be 1.

(You may be able to solve these problems by going to a world where there are finitely many time points within a given interval of time—but then there’s the worry over whether there is such a world.)

I was pretty sloppy in setting up my case, so you were right to worry as you did. (I hope I’m a little less sloppy here, but no guarantees.)

Re 1: I don’t know a great deal about QM. (When I read introductions to QM, they usually roll over me like warm breezes, and I’m back to being nearly as ignorant as I was before I began.) You’re right about QM talking only about the probabilities of particles having momentum and position during intervals (though I don’t know whether that generalizes to all properties – if the math says so, then it’s so): “… although there are no exact, or point-position or momentum eigenstates, there are certainly eigenstates of ‘position (momentum) restricted to a finite range’.” (Particles and Paradoxes, p. 31) I’m doubtful though that e.g. a momentum eigenstate’s not being square-integrable, so that QM can’t deliver probabilities regarding a particle’s having an exact momentum, really calls into question the INTERPRETABILITY of a statement like: the probability that a will have momentum x at t is 1/3. Does there not being QM translations of such statements really show (somehow) that they don’t make sense (simpliciter, not just in QM)? Isn’t there a metaphysically possible world governed by QM*, with analogous statements definable in QM*? If so, then stipulate w to be one of those worlds.

Suppose that I’m wrong though. Suppose “The probability that a will have momentum x at t is 1/3” really doesn’t make good sense. My example works just as well with intervals. I’ll show how while dealing with 2.

Re 2: Let me do this with possible worlds (e.g. if, in w, at t, p(P)=1/2, then the probability, at t, that a will be P in [tm,tn] is 1/2). Choose any time interval [tm,tn]. Let P={a-is-P-in-[tm,tn]-worlds}, Q={a-is-Q-in-[tm,tn]-worlds}, and R={a-is-R-in-[tm,tn]-worlds}. Also, let B={b-strikes-a-in-[tm,tn]-worlds}. Let F be the smallest sigma-algebra that contains {P,Q,R,B}. Choose any PRIOR probability distribution p on F. I’m stipulating that in w: p(P|(w*\B))=p(Q|(w*\B))=p(R|(w*\B))=1/3 (where w* is the set of all possible worlds). Now things run as before: If b strikes a, a will be P, but only during the strike. b strikes a at t1. b then ricochets and barrells into space. a is P at t1. By chance, a is P at t2 and at all times between t1 and t2. It seems that, by Salmon’s criteria, a mark (i.e. P) is transmitted from t1 to t2 along the a’s-travels-process. But again, clearly it’s not.

A point of information: *Why* is it clear that a mark was not transmitted in this case? Why is this a problem for Salmon’s account of causal processes, and not *just* an interesting result when the account is extended to chancy processes?

Also: Perhaps, for cases involving chancy processes, should Salmon count events which result from their residual, “background” chance as events due to intervention? (By “residual, background chance” what I have in mind is something similar to what proponents of counterfactual analyses of causal processes typically employ to deal with these cases.)

Here is a better response: Add a chance-raising condition for chancy cases. So a mark that has been introduced into a chancy process by means of a single intervention at a point A is transmitted to point B if and only if it occurs at B and at all stages C of the process between A and B without additional interventions, and p(B)

Okay, so my post was cut off. Here is the last paragraph again:

Here is a better response: Add a chance-raising condition for chancy cases. So a mark that has been introduced into a chancy process by means of a single intervention at a point A is transmitted to point B if and only if it occurs at B and at all stages C of the process between A and B without additional interventions, and prob(B)is less than prob(B|A) and prob (C|A). Something like this will give the intuitively correct judgment about the case, plus we’re not violating the reductionistic spirit of Salmon’s account.

What exactly is the difference between “we get an interesting result when we apply the theory to X” and “we get a counterexample when we apply the theory to X”?

Also, you write:

“Something like this will give the intuitevely correct judgment about the case, plus we’re not violationing the reductionistic spririt of Salman’s account.”

I’m not sure the latter is the case, since your conditional probabilities are, as of yet, unreduced.

Regarding your first question: The thought is that sometimes an intuitive judgment should force us to revise or abandon a theory, but other times a theory should for us to revise or abandon an intuitive judgment. (Of course, there’s no decision procedure that would be able to tell you, exactly, when one should do one rather than the other. But it does not therefore follow that this is not a useful distinction for philosophical purposes.)

Regarding your second question: You’re correct that I failed to provide a fully revealing analysis of conditional probabilities that makes no ineliminable reference to causal concepts. But then again, it was just a blog post. And of course, whether one can provide such an analysis is an open philosophical question. (Salmon, at least at one point in his career, thought so.) My point was simply that *if* you can provide such an analysis, then it seems like one can avoid the troubling case.

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Here’s a couple undercooked thoughts.

You say the probability of a being P at t for any time t is 1/3 and you also say that a is P at each of the time points between t1 and t2.

First, there is the worry over whether it even makes sense to say that the probability of a particle’s being P at t (a time point!) is 1/3. Doesn’t QM merely talk about the probabilities of particles having properties during INTERVALS (rather than at time points)?

Second, it’s not clear that it makes sense to suppose both that the probability of a’s being P at t for any time t is 1/3 AND that a is P at each of (I assume) infinitely (!) many times t within a given interval. It seems that in any world where the latter is the case, it is NOT the case that the probability of a’s being P at ANY time t is 1/3, for if it were, the probability of a’s being P at SOME time during the interval would be 1.

(You may be able to solve these problems by going to a world where there are finitely many time points within a given interval of time—but then there’s the worry over whether there is such a world.)

I was pretty sloppy in setting up my case, so you were right to worry as you did. (I hope I’m a little less sloppy here, but no guarantees.)

Re 1: I don’t know a great deal about QM. (When I read introductions to QM, they usually roll over me like warm breezes, and I’m back to being nearly as ignorant as I was before I began.) You’re right about QM talking only about the probabilities of particles having momentum and position during intervals (though I don’t know whether that generalizes to all properties – if the math says so, then it’s so): “… although there are no exact, or point-position or momentum eigenstates, there are certainly eigenstates of ‘position (momentum) restricted to a finite range’.” (Particles and Paradoxes, p. 31) I’m doubtful though that e.g. a momentum eigenstate’s not being square-integrable, so that QM can’t deliver probabilities regarding a particle’s having an exact momentum, really calls into question the INTERPRETABILITY of a statement like: the probability that a will have momentum x at t is 1/3. Does there not being QM translations of such statements really show (somehow) that they don’t make sense (simpliciter, not just in QM)? Isn’t there a metaphysically possible world governed by QM*, with analogous statements definable in QM*? If so, then stipulate w to be one of those worlds.

Suppose that I’m wrong though. Suppose “The probability that a will have momentum x at t is 1/3” really doesn’t make good sense. My example works just as well with intervals. I’ll show how while dealing with 2.

Re 2: Let me do this with possible worlds (e.g. if, in w, at t, p(P)=1/2, then the probability, at t, that a will be P in [tm,tn] is 1/2). Choose any time interval [tm,tn]. Let P={a-is-P-in-[tm,tn]-worlds}, Q={a-is-Q-in-[tm,tn]-worlds}, and R={a-is-R-in-[tm,tn]-worlds}. Also, let B={b-strikes-a-in-[tm,tn]-worlds}. Let F be the smallest sigma-algebra that contains {P,Q,R,B}. Choose any PRIOR probability distribution p on F. I’m stipulating that in w: p(P|(w*\B))=p(Q|(w*\B))=p(R|(w*\B))=1/3 (where w* is the set of all possible worlds). Now things run as before: If b strikes a, a will be P, but only during the strike. b strikes a at t1. b then ricochets and barrells into space. a is P at t1. By chance, a is P at t2 and at all times between t1 and t2. It seems that, by Salmon’s criteria, a mark (i.e. P) is transmitted from t1 to t2 along the a’s-travels-process. But again, clearly it’s not.

A point of information: *Why* is it clear that a mark was not transmitted in this case? Why is this a problem for Salmon’s account of causal processes, and not *just* an interesting result when the account is extended to chancy processes?

Also: Perhaps, for cases involving chancy processes, should Salmon count events which result from their residual, “background” chance as events due to intervention? (By “residual, background chance” what I have in mind is something similar to what proponents of counterfactual analyses of causal processes typically employ to deal with these cases.)

Here is a better response: Add a chance-raising condition for chancy cases. So a mark that has been introduced into a chancy process by means of a single intervention at a point A is transmitted to point B if and only if it occurs at B and at all stages C of the process between A and B without additional interventions, and p(B)

Okay, so my post was cut off. Here is the last paragraph again:

Here is a better response: Add a chance-raising condition for chancy cases. So a mark that has been introduced into a chancy process by means of a single intervention at a point A is transmitted to point B if and only if it occurs at B and at all stages C of the process between A and B without additional interventions, and prob(B)is less than prob(B|A) and prob (C|A). Something like this will give the intuitively correct judgment about the case, plus we’re not violating the reductionistic spirit of Salmon’s account.

Alex,

What exactly is the difference between “we get an interesting result when we apply the theory to X” and “we get a counterexample when we apply the theory to X”?

Also, you write:

“Something like this will give the intuitevely correct judgment about the case, plus we’re not violationing the reductionistic spririt of Salman’s account.”

I’m not sure the latter is the case, since your conditional probabilities are, as of yet, unreduced.

Dustin,

Regarding your first question: The thought is that sometimes an intuitive judgment should force us to revise or abandon a theory, but other times a theory should for us to revise or abandon an intuitive judgment. (Of course, there’s no decision procedure that would be able to tell you, exactly, when one should do one rather than the other. But it does not therefore follow that this is not a useful distinction for philosophical purposes.)

Regarding your second question: You’re correct that I failed to provide a fully revealing analysis of conditional probabilities that makes no ineliminable reference to causal concepts. But then again, it was just a blog post. And of course, whether one can provide such an analysis is an open philosophical question. (Salmon, at least at one point in his career, thought so.) My point was simply that *if* you can provide such an analysis, then it seems like one can avoid the troubling case.