Responses by Tim McGrew to questions I raised in an e-mail concerning the multiplication rule:
In order for an argument to be a ʽgoodʼ one, the confidence we have in the truth of the conclusion must be greater than .5, otherwise we canʼt claim to know the conclusion (I think this is uncontroversial).
The trouble is that "good argument" is not sufficiently well defined for us to say whether it must meet this criterion. An argument that nails down the probability of the conclusion at .51 -- no higher and no lower -- meets the criterion you named; but is it really, for just that reason, a "better" argument than one that shows that the conclusion has a probability of at least .4 and perhaps much more?
In order for an argument to be a ʽgoodʼ one, the confidence we have in the truth of the conclusion must be greater than .5, otherwise we canʼt claim to know the conclusion (I think this is uncontroversial).
The trouble is that "good argument" is not sufficiently well defined for us to say whether it must meet this criterion. An argument that nails down the probability of the conclusion at .51 -- no higher and no lower -- meets the criterion you named; but is it really, for just that reason, a "better" argument than one that shows that the conclusion has a probability of at least .4 and perhaps much more?
But then, as I expected, certain atheists (like Stephen Law) have been applying the ʽmultiplication ruleʼ to arguments for the existence of God. As you probably know, Glenn Peoples defends a version of the moral argument which has 5 premises. Law pointed out that even if each premise had a probability of .8 that the conclusion was only .33 likely to be true.
As I pointed out in a few comments on Glenn's website around the turn of the year, the multiplication rule (1) applies only when we are assured of independence, and (2) sets a lower bound only, not an upper bound.
I forwarded this to an old professor of mine who has a similar background in philosophy compared to you, and he agreed with Law, and even said that he thinks that if more people were sensitive to this ʽmultiplication ruleʼ in the philosophy of religion that they would realize that it is sort of a waste of effort.
I am sorry to have to say that your old professor is simply mistaken. And it's a good thing too, even for views that I am sure he would approve of. After all, why should a skeptical argument about the multiplication of probabilities apply only to arguments for the existence of God? How about to theories in cosmology, or geology, or biology? Yikes!
Needless to say, I am a little disturbed by this. I think I have an idea of how you might want to respond to this initially because I suspect that you are the DIMA that responded to Law on Glennʼs blog.
I commented as "Tim." However, if I recall, Dima did state the key point more or less correctly before I arrived.
Even if you arenʼt DIMA, I would like to know that even if it is true that the .33 above is only the lower bound of the premises whereas the conclusion could have an upper bound up to 1, are we stuck with not knowing where in this range Glennʼs argument (or any argument for that matter) lies in this range?
Only if this is the only argument we are employing. If it is one among many -- as it should be if we have a cumulative case for the existence of God -- then it can be simply one step along the way. A number of such arguments taken together may raise the probability of their common conclusion as close to 1 as you please.
It seems like this ʽmultiplication ruleʼ lands us in skepticism if all we can say is, well, I know that the probability of the conclusion being true is somewhere between .33 and 1, but I have no clue where in that range it lands.
Remember that "I have no clue" applies only if the moral argument is the only argument you've got.
Cheers!
Tim
BRAD BOWEN FROM SECULAR OUTPOST SAYS:
If two events or states of affairs are independent, then the probability that both will occur is equal
to the multiplication of the probabilities of those two events. If p is an event (or state of affairs) that is independent of an event (or state of affairs) q, then:
P(p & q) = P(p) x P(q)
But if p and q are dependent events, then the probability formula is a bit different:
P(p & q) = P(p) x P(q/p)
Suppose p is 'getting heads on coin toss 1' and q is 'getting heads on coin toss 2'. Assuming that the outcome of coin toss
1 has no influence on the outcome of coin toss 2, we can conclude that p and q are independent events, and use the first, simpler formula above:
P(p & q) = .5 x .5 = .25
But if p is “It will rain in Seattle today” and q is “The streets in Seattle will get wet today”, then the truth or falsehood of
p has an obvious influence on the truth or falsehood of q, namely if p is true, then it is virtually certain that q will also be true. Since these are dependent events, we must use the second, more complex formula to calculate the probability of the conjunction of the two claims:P(p & q) = P(p) x P(q/p)
Since it is virtually certain that q is the case given the assumption that p is the case, P(q/p) is approximately equal to 1.0, so the probability that both p and q are true is about the same as the probability that p is true.
In arguments where two or three premises work together to support the conclusion, each of the two or three premises must be true in order for the argument to work and provide rational support for the conclusion. We can thus determine the probability that all two or three premises are true by using one of the above probability formulas. But we should use the simpler probability formula (where the probability of each event/premise is multiplied with the probability of the other premise(s) only if the events/premises are independent.
As I pointed out in a few comments on Glenn's website around the turn of the year, the multiplication rule (1) applies only when we are assured of independence, and (2) sets a lower bound only, not an upper bound.
I forwarded this to an old professor of mine who has a similar background in philosophy compared to you, and he agreed with Law, and even said that he thinks that if more people were sensitive to this ʽmultiplication ruleʼ in the philosophy of religion that they would realize that it is sort of a waste of effort.
I am sorry to have to say that your old professor is simply mistaken. And it's a good thing too, even for views that I am sure he would approve of. After all, why should a skeptical argument about the multiplication of probabilities apply only to arguments for the existence of God? How about to theories in cosmology, or geology, or biology? Yikes!
Needless to say, I am a little disturbed by this. I think I have an idea of how you might want to respond to this initially because I suspect that you are the DIMA that responded to Law on Glennʼs blog.
I commented as "Tim." However, if I recall, Dima did state the key point more or less correctly before I arrived.
Even if you arenʼt DIMA, I would like to know that even if it is true that the .33 above is only the lower bound of the premises whereas the conclusion could have an upper bound up to 1, are we stuck with not knowing where in this range Glennʼs argument (or any argument for that matter) lies in this range?
Only if this is the only argument we are employing. If it is one among many -- as it should be if we have a cumulative case for the existence of God -- then it can be simply one step along the way. A number of such arguments taken together may raise the probability of their common conclusion as close to 1 as you please.
It seems like this ʽmultiplication ruleʼ lands us in skepticism if all we can say is, well, I know that the probability of the conclusion being true is somewhere between .33 and 1, but I have no clue where in that range it lands.
Remember that "I have no clue" applies only if the moral argument is the only argument you've got.
Cheers!
Tim
BRAD BOWEN FROM SECULAR OUTPOST SAYS:
If two events or states of affairs are independent, then the probability that both will occur is equal
to the multiplication of the probabilities of those two events. If p is an event (or state of affairs) that is independent of an event (or state of affairs) q, then:
P(p & q) = P(p) x P(q)
But if p and q are dependent events, then the probability formula is a bit different:
P(p & q) = P(p) x P(q/p)
Suppose p is 'getting heads on coin toss 1' and q is 'getting heads on coin toss 2'. Assuming that the outcome of coin toss
1 has no influence on the outcome of coin toss 2, we can conclude that p and q are independent events, and use the first, simpler formula above:
P(p & q) = .5 x .5 = .25
But if p is “It will rain in Seattle today” and q is “The streets in Seattle will get wet today”, then the truth or falsehood of
p has an obvious influence on the truth or falsehood of q, namely if p is true, then it is virtually certain that q will also be true. Since these are dependent events, we must use the second, more complex formula to calculate the probability of the conjunction of the two claims:P(p & q) = P(p) x P(q/p)
Since it is virtually certain that q is the case given the assumption that p is the case, P(q/p) is approximately equal to 1.0, so the probability that both p and q are true is about the same as the probability that p is true.
In arguments where two or three premises work together to support the conclusion, each of the two or three premises must be true in order for the argument to work and provide rational support for the conclusion. We can thus determine the probability that all two or three premises are true by using one of the above probability formulas. But we should use the simpler probability formula (where the probability of each event/premise is multiplied with the probability of the other premise(s) only if the events/premises are independent.
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