Deductive Arguments And Multiplying Probabilities

[Moot]

So this is something that's been addressed quite a lot on this forum but I've never understood the objections to it. Long ago SolElQoy tried explaining this to me and others but I came away none the wiser. So here's another attempt at getting to the bottom of this.

The idea is that the probability of the premises become multiplied in a deductive argument. This means that if an argument has two premises that have a .6 probability of being true, the conclusion only has a .36 probability of being true.

Let's take the newer version of the KCA (forgive me if I'm not formulating it exactly right):

1. If the Universe began to exist, it had a cause.
2. The Universe began to exist.
C. Therefore, the Universe had a cause.

Now let's say that we give each premise a .6 confidence level. How likely is the conclusion to be true? It seems pretty obvious to me that the answer is .36. There are three possibilities and here's how I'd put the probabilities:

1. The Universe began to exist and had a cause = .36
2. The Universe began to exist without a cause = .24
3. The Universe did not begin to exist = .4

This seems right to me, but Dr. Craig (and others I'm sure) has a standard that says a good argument simply needs to have premises that are more plausible than their negation. But that seems to contradict the above.  Somethings gotta give! It's been suggested that the answer lies in the difference between probability and plausibility but I can't understand why that would be. Even if plausibility and probability are different things, I can't see why a plausibility estimate couldn't be roughly translated into a probability estimate. It seems that we do that all the time. Here's another example that has nothing to do with God:

1. If my neighbour's dog is in the park, it's wearing a dog-sweater.
2. My neighbour's dog is in the park.
C. Therefore, my neighbour's dog is wearing a dog-sweater.

Edit: My neighbours dog only wears a sweater if it's in the park (it's not really the same structure otherwise).

Let's say again that we give each premise a .6 confidence level. Is anyone claiming that conclusion is more likely than not in that scenario? If yes, why? Where did I go wrong? If no, does the same reasoning apply to the KCA or is there a relevant difference?

When I've brought this up in the past I've had some accusations of dishonesty and irrationality thrown at me so let me just make it very clear that my reasoning doesn't only apply to arguments for theism. I'm also certainly not claiming to be pwning Craig. I'm guessing there's a good reason for his standard, I'm just not sure what it is. Also, just to show that this isn't an objection the RF atheists have dreamed up, here's Christian philosopher/apologist Calum Miller on Craigs KCA. I get that Miller isn't regarded as an authority on this, but he seems well respected among theists:

"Again, for completeness, I should briefly detail one or two final concerns. These are to do with the framework of the argument as a whole. The first is that in general, I find many deductive arguments unsatisfactory. It is not always clear how the plausibility of the premises is supposed to relate to the plausibility of the conclusion. To secure a conclusion which, given our evidence, is more probable than not, the conjunction of the premises of a deductive argument must have a probability of greater than 0.5. But the fact that each premise is more likely than not, given our evidence, is certainly not sufficient to establish this.

This raises a further problem, something of a “dwindling probabilities” argument: we can only guarantee a probability of the conclusion insofar as the conjunction of the argument’s premises is probable. But if each premise is only somewhat more probable than not, the probability of the conclusion we can derive drops off pretty quickly. Suppose we think that P(P1) = P(P2|P1) = 0.7: then P(P1 & P2) = 0.49. Let P3 be the premise that if the universe has a cause, the cause of the universe is personal. Suppose P(P3|P1 & P2) = 0.7. Then P(P1 & P2 & P3) = 0.343.  Of course, some might question these probabilities: but it is clear that the more uncertain steps there are, the more the probability we can guarantee for the conclusion decreases. Since a lot of steps in the KCA are far from certain, and since some seem only slightly more probable than their negations, this presents a serious problem for the KCA."

Sorry for the long post. Looking forward to responses.

[Johan Biemans (jbiemans)]

While I disagree with the more probable than not criteria, what they are saying is not that you should keep the initial probability.  They are saying something like:

If premise A is more probable than not, then you should assign premise A with a 100% probability. (in other words, call it true).  In other words, it seems like they are asking you to accept something as true, when it is only more likely true than not, rather than just accepting it as more likely true than not.  The only reason for this would be because of something is true, you cannot rationally reject it, but if something is more likely true than not, you can still rationally reject it on the basis that it is possibly false.

[Moot]

While I disagree with the more probable than not criteria, what they are saying is not that you should keep the initial probability.  They are saying something like:

If premise A is more probable than not, then you should assign premise A with a 100% probability. (in other words, call it true).  In other words, it seems like they are asking you to accept something as true, when it is only more likely true than not, rather than just accepting it as more likely true than not.  The only reason for this would be because of something is true, you cannot rationally reject it, but if something is more likely true than not, you can still rationally reject it on the basis that it is possibly false.

Hmm... Thanks.

This still makes no sense to me. Why in the world would I regard it as 100% true if I only think it's more probable than not. Take my dog example. Wouldn't it be downright irrational of me to believe that the conclusion was most likely true?

[hatsoff]

1.  You only multiply probabilities for independent events. This doesn't undermine your point but let's just make sure we're being precise.

2. It's occassionally reasonable to be confident in mutually inconsistent statements but in practice this doesn't happen with small numbers of statements---like the kca's three.

[Moot]

1.  You only multiply probabilities for independent events. This doesn't undermine your point but let's just make sure we're being precise.

2. It's occassionally reasonable to be confident in mutually inconsistent statements but in practice this doesn't happen with small numbers of statements---like the kca's three.

Thanks. I agree with 1, but I'm not quite sure I understand what you mean by 2. The KCA doesn't involve any mutually inconsistent statements, does it?

Also, did I (in your opinion) get the probabilities right here (given .6 probability for the premises):

1. The Universe began to exist and had a cause = .36
2. The Universe began to exist without a cause = .24
3. The Universe did not begin to exist = .4

If no, why? If yes, doesn't this undermine Craig's standard for a good argument?

[Language-Gamer]

1.  You only multiply probabilities for independent events. This doesn't undermine your point but let's just make sure we're being precise.

2. It's occassionally reasonable to be confident in mutually inconsistent statements but in practice this doesn't happen with small numbers of statements---like the kca's three.

I agree with moot that (2) is very deep. Maybe you can expand on it more.

[aleph naught]

The norms of inference cannot be reduced to rules of probability. Another reason to think that there are objective normative properties. (#thereforemoralrealism).

[kurros]

The norms of inference cannot be reduced to rules of probability.

Why not?

[Language-Gamer]

Grabbing popcorn for this ultimate showdown of our greatest minds. Maybe this can be turned into a debate on Bayesian epistemology!

[Moot]

The norms of inference cannot be reduced to rules of probability. Another reason to think that there are objective normative properties. (#thereforemoralrealism).

Could you expand on this? I don't see why I wouldn't assign a confidence level of (for example) .6 to the question "did the Universe begin to exist?".

[aleph naught]

The norms of inference cannot be reduced to rules of probability.

Why not?

Because of exactly cases like this, where it's reasonable to accept the conclusion of an inference as roughly the same degree that it's reasonable to accept the premises. The probability decays as we draw more inferences, but the reasonableness doesn't seem to.

[Pragmatic]

The norms of inference cannot be reduced to rules of probability.

Why not?

Because of exactly cases like this, where it's reasonable to accept the conclusion of an inference as roughly the same degree that it's reasonable to accept the premises. The probability decays as we draw more inferences, but the reasonableness doesn't seem to.

Well, the OP would disagree with the last sentence. I'm inclined to disagree as well.

[kurros]

2. It's occassionally reasonable to be confident in mutually inconsistent statements but in practice this doesn't happen with small numbers of statements---like the kca's three.

In what sense? I don't think they can both have greater probability than 0.5 each if they are mutually exclusive. See this proof:

1. Suppose P(X|Y) = P(Y|X) = 0  (mutually exclusive)
2. Suppose P(Y) > 0.5

3. P(Y) = P(Y|X)P(X) + P(Y|~X)P(~X) = P(Y|~X)P(~X)            : from 1.
4. P(Y|~X)P(~X) > 0.5                   : from 3 + 2
5. P(~X) > 0.5 / P(Y|~X)            : rearrange 4
6. P(~X) > 0.5                           : because P(Y|~X) <= 1
7. 1 - P(X) > 0.5
8. P(X) < 0.5
QED.

Also because being mutually exclusive means that P(X or Y) = P(X) + P(Y), and therefore both P(X) and P(Y) cannot be greater than 0.5 without P(X) + P(Y) being greater than 1 and violating the probability rules.

[kurros]

The norms of inference cannot be reduced to rules of probability.

Why not?

Because of exactly cases like this, where it's reasonable to accept the conclusion of an inference as roughly the same degree that it's reasonable to accept the premises. The probability decays as we draw more inferences, but the reasonableness doesn't seem to.

I would say this exact case is behaving perfectly sensibly according to a probabilistic interpretation.

[Moot]

The norms of inference cannot be reduced to rules of probability.

Why not?

Because of exactly cases like this, where it's reasonable to accept the conclusion of an inference as roughly the same degree that it's reasonable to accept the premises. The probability decays as we draw more inferences, but the reasonableness doesn't seem to.

Thanks, but I really can't understand why. It doesn't seem reasonable to me to accept the conclusion (given a .6 confidence level in the premises). Why should I regard it as reasonable?

Also, would you say the same for my dog example?