FTA part 8: Clarification on the multiverse vs. necessity (part of the Aron series)

Greg says,
I am sorry that my lengthy response wasn't very clear.  Let me see if we can get on the same page.  (Of course, it's not essential that we agree, right?  Just essential that we understand the other person's arguments.  I really appreciate that you're taking the time to do so with mine!)

So, there are a couple of misunderstandings here.  Let's try to tackle them one by one.  The first one is that I don't understand your model.  At the beginning of your last message, you said, "our universe is just one out of a pool of infinity conceivable universes."  Later, you said, "There would be no need for me to resort to a multiverse."  So do you believe in the existence of multiple universes, or not?

My guess is not, because you said "conceivable".  But since I don't know, let's explore both options.

Option 1: First, let's assume there is just one universe.  Yes, it is one universe out of an infinity of conceivable universes, but none of these other universes is real.  Since there is no explanation of how or why this particular universe is actualized over the other conceivable universes, any discussion of probability is useless anyway, normalizability problem or not.  Do you agree? (I think so because at the end of your last message, you said, "the concept of probability is simply not meaningful in this context".  Unless I misunderstand what you meant by that statement.)  And if so, then you are stuck with an apparently finely-tuned universe with no explanation for it.  The universe just IS, and that's the way it is.  Pretty strange, right?

Option 2: Now let's assume there really is a real multiverse ensemble.  In that case, there really is some physical mechanism that generates universes, so this real, physical mechanism has real probability distributions for how it generates all the constants and initial conditions.  This is what I was saying last time, and if so, then again, you end up with either a very small probability of our universe being the way that it is, or a finely-tuned mechanism.  Again, non-theists like the first (very small probability) because you can conceivably defeat that by N large.  A finely-tuned mechanism would be problematic because then you have fine tuning at the most basic property of all of physical reality, and you can't explain that.

I really like one of the papers you pointed me to by Colyvan et al.  In particular, I liked it when they said, "After all, if they [meaning the constants] could not have been different, the probability of the universe being just as we find it is 1, and no fine tuning has occurred. But what is the modality invoked here? Logical possibility? Conceptual possibility? Physical possibility? This is rarely spelled out in the usual presentations of the argument." (p. 326)

This is what I was saying, and I think you would agree.  They go on to discuss the problem with using logical possibility as the modality, precisely because it runs into the normalizability problem, as you pointed out.  What they are missing here is that, if there is no real mechanism that "decides" which constants to pick from, there is no point in talking about probability anyway, again, normalizability problem or not.  We end up with the universe just IS.

Do you see a third option besides either these other universes are real, or they are not?  Or do you see my characterization of the first option as flawed?  (I think this is where more discussion will occur, but I'd like to hear what you have to say about it before I ramble about this on and on.  And on and on...as I tend to do.

Another misunderstanding I think we had is related to what I just laid out as our two Options.  In particular, you quoted me as saying, "everything's equally impossible or our current value is necessary."  That in a nutshell is what I was saying our two options were.  But I got that from the Colyvan paper: "The fine tuning argument, on its most plausible interpretation, hence not only shows that life-permitting universes are improbable, but, arguably, that they are impossible!" (p. 327)  Juxtapose that statement with, "Physical possibility (construed as consistency with the laws of physics and physical constants as we find them) however, restricts the range too much for the proponent of the fine tuning argument, leaving the actual values as the only possible ones, and hence setting the probability at 1!" (p. 329, original emphases removed)

Another misunderstanding is how you then go on to characterize the normalizability problem: "each possible universe is either equally impossible, or they all have a small nonzero probability. They can't be impossible, because then the probabilities don't add up to 1, and they can't have a nonzero probability, because then the probabilities add up to infinity."  The either/or statement you lead off with is not true.  (Before I go on, I do think you characterized the normalizability problem accurately, but I don't think its conditions are met in reality.) Of course there are probability distributions with an infinite domain that are normalizable.  We just don't know what the correct probability distribution to use is.  But again, either there is a real mechanism that generates these universes, in which case there is a real distribution so it is really normalizable; or there is not, in which case it is futile to talk about any probability distribution because there is nothing to draw from.

Final misunderstanding: "If I'm right that the concept of probability is simply not meaningful in this context, then this dissolves the mystery. There would be no need for me to resort to a multiverse or necessity."  Yeah, totally, you may be right that the concept of probability is not meaningful.  That's what I was trying to say in my previous message (and is captured in Option 1 of this message).  And, in which case, you would not need to resort to a multiverse because you have already assumed there is not one.  But then you are stuck with necessity, because the universe just IS.

OK, so those are the critical parts where I either misunderstood you, or where I think you misunderstood me.

Also, let me end with this: this discussion is awesome and I hope you don't get too frustrated at how long I take to respond.  You keep up the good work with cordially asking questions and rebutting Christians' arguments.  I know a lot of atheists (and Christians too) that just want to have their say.  Maybe that's you too, but you're hiding it really well, which means that's not you.

What I should have said in all of that was simply, I think you characterized the normalizability problem correctly but I just don't think it's relevant. Because either we're dealing with a real mechanism (which then must be normalizable by virtue of its being real) or not, in which case a discussion of probability is futile. What do you think?

Take care!

PS: I just took a look back at my previous message, and I even used "Option 1" and "Option 2" in that message too. I forgot, and I guess it's just what I think so strongly that it came out twice.  Shame on me, because it looks like that means I didn't actually explain anything new this time.  Let me know if that's true.

PPS: What I should have said in all of that was simply, I think you characterized the normalizability problem correctly but I just don't think it's relevant. Because either we're dealing with a real mechanism (which then must be normalizable by virtue of its being real) or not, in which case a discussion of probability is futile. What do you think?

[See summary page of this discussion, with links to all the posts, here.]

FTA part 7: Aron asks clarifying questions (part of the Aron series)

Aron wrote:
Thanks for the response!  I want to make sure I understand you, and that you understand me.  My objection was this: our universe is just one out of a pool of infinity conceivable universes.  P(FT/~G) is the probability of picking a universe like ours at random.  Probabilities only make sense if they add to one (ex/ for a die 1/6x6=1).  So for the FTA to work, we need to be able to assign each possible universe a probability that allows them to add up to 1.  But this is impossible because if each universe has a probability of 0, they all add up to 0.  And if each universe is given a small nonzero probability, they add up to infinity.  Since the probabilities can't add up to 1, it is meaningless to talk about probabilities here.   The objection is that our intuitions have led us to extend to concept of "probability" far beyond the context in which it is applicable.

The way you characterized the normalizability objection is like this: "everything's equally impossible or our current value is necessary."  I'm not sure this is what I was getting at.  Instead, it should say "each possible universe is either equally impossible, or they all have a small nonzero probability. They can't be impossible, because then the probabilities don't add up to 1, and they can't have a nonzero probability, because then the probabilities add up to infinity."

If I'm right that the concept of probability is simply not meaningful in this context, then this dissolves the mystery.  There would be no need for me to resort to a multiverse or necessity.

In sum, I'm not quite sure exactly what your objection was to the normalizability problem.

[See summary page of this discussion, with links to all the posts, here.]

FTA part 6: Defying intuition, the multiverse, or necessity (part of the Aron series)

Greg says,
Yes, the normalization problem does seem to come up, doesn't it?  But the more I think about it, the more I think it's a cover-up.  Here's what I mean.  Just like the initial objection you raised about degree of fine tuning not translating into a rigorous probability, in this case this is just another layer of subtlety, but the conclusion is the same.  In going deeper with this, we are essentially just pushing it back another layer.

Another way to think about it is, the first level of fine-tuning is very intuitive, and speaks easily to the common person.  "Wow!  Look how finely-tuned these constants are!  This argues for intention in the make-up of the universe."  This is the intuitive conclusion, and sometimes intuition is right.

On the other hand, sometimes intuition is wrong.  For someone who wants to contest this conclusion (and please don't consider that turn of the phrase to mean I think the challenger of the FTA is disingenuous...we need to think deeply about it), there is always a way to get out of it.  There's always a door to exit for the skeptic.  But every time you exit the door, you end up in another room that is smaller and more difficult to exit.  A smaller door is there to exit the next room, and still smaller.  Pretty soon you'll need one of Alice's mushrooms to get out of the door, it's so small.  How deep does the rabbit hole go?

Did that sound pompous?  Sorry, I thought of that word picture last night and I really liked it.    In any case, my point is that the more one plays the skeptic to deny what intuition is telling us, the harder you have to work and the more you have to deny precious bits of reality.

OK, now that I've played it up so much, do I actually have an argument?  (Hee-hee, I hope so.  We'll see if you like it or not.)

So let's start with the point I made last time.  Either the small life-permissible interval of G is improbable, or the probability distribution from which we are drawing G must be itself finely-tuned (aka, atypical).  That makes intuitive sense.  It's a bit harder to understand than the basic "G must be within one part in 10^60, therefore God did it," but it's still pretty intuitive.  The rebuttal to that is we have no reason to consider any particular sort of probability measure.  Indeed, the normalizability problem destroys fine-tuning: either everything's equally impossible, or our current value is necessary (P = 1).  What method do we have to restrict the probability distribution to some intermediate shape?

So, we drop-kick intuition and need to go one level deeper.  (Remember how I asked, "How deep does the rabbit hole go?"  I have a feeling to get to the bottom of this conversation, we'll eventually have to discuss properly basic beliefs and brains-in-a-vat.  It's a steep price to pay to be constantly skeptical of the intuitive power of the FTA.)  If we really want to have a probability distribution to draw from, we need a mechanism. Here our discussion will bifurcate into two plausible solutions.

(Before I do that, can I mention an aside here?  Initially, I presented the FTA as a rigorous Bayesian-type proof.  Recall you challenged my ability to say P(FT | ~G) is super-low.  Now I just want to recall the point that these probabilities in Bayesian arguments are *epistemic*.  Meaning, they're "what are the odds of that happening?"-type probabilities.  This is the reason why the Bayesian argument goes through, because most will understand the fine-tuning of the constants and conditions of the universe and of earth and intuitively agree that P(FT | ~G) is low, even if it can't be proven rigorously.)

OK, back to the bifurcation.  There are now two naturalistic options (to avoid God): (1) either the universe is alone (and necessary), or (2) it is one of an ensemble of universes, commonly called the multiverse (which then itself is necessary).

Option 1: if the universe is necessary and alone, then all the constants and conditions could not have been other than what they are.  In that sense, all of these probabilities would be unity.  How could it have been any other way if the universe itself is necessary?  But if that is the case, we again are stuck with asking why it had to have been this way.  What is it about the universe and necessity that made it so that life could possibly exist?  Especially when it seems like there are so many other ways it could have been that would have precluded life.  Again, we are now not only stuck with asking "Why is there something rather than nothing?" (since the universe has no explanation for its existence, it would seem rather odd that it would be the necessary entity), but also with "Why is the universe the way that it is" (since it being just the way it is permitted intelligent life to develop within it to ask these questions).

Now, one way you could answer these questions is flippantly.  Dr. Krauss is a famous example of this, with his, "'Why' questions are silly."  But I don't regard you as thinking that.  So then why do you think there is something rather than nothing?  Why do you think the universe is the way it is?  Remember, without God and thus without intention, there is no explanation for these facts.

Option 2: if the universe is one of many universes in the multiverse, then plausibly this could explain how the perceived fine-tuning arose.  Returning to the normalizability problem, the main issue I have with it is, if there really is a natural mechanism that "chooses" values of constants for the universe, then it cannot have the normalizability problem.  This is because it must have a real probability distribution, not this hypothetical/philosophical/no-logical-restriction type distribution.  So the existence of the multiverse then solves the normalizability problem: either the probability distribution is typical, and thus our universe is rare, or the universe-generating mechanism itself has a finely-tuned probability distribution to produce a bunch of universes like ours.  In the second case, our universe is not rare (they're all like ours), but the fine-tuning is in the multiverse itself.

Skeptics rather like the first case: our universe is rare, but the number of universes (probabilistic resources) is so high that one such as ours is bound to have been generated randomly.

I could go on and on about the multiverse, so let me leave it with this so you can respond before I go off the deep end: I don't see Option 1 as viable.  For the skeptic, the universe just can't be the necessary entity. It raises too many questions about sufficient explanation.  So, Option 2, the multiverse, must be the fall back if you want to escape through the ever-shrinking skeptical door.  Further more, the second case of Option 2 (the multiverse itself is finely-tuned) cannot be the case for the skeptic, as this would be identical in ontology to Option 1.  Therefore, as I see it, the skeptic must choose the first case of Option 2: the multiverse generates widely-varying random universes, one of which is the lucky one (ours).

Is that where you think you'd go with this?

[See summary page of this discussion, with links to all the posts, here.]

FTA part 5: Aron introduces the normalizability problem (part of the Aron series)

Aron says:
If I grant a uniform distribution for the sake of argument, then the probability of G being "just right" is the ratio: (life permitting values/possible values).  As far as I know (and I'm no expert) there's nothing in modern physics that restricts the range of possible values.  Robin Collins, for example says, "The value of G, for instance, conceivably could have been any number between 0 and infinity" http://home.messiah.edu/%7Ercollins/Fine-tuning/FINETLAY.HTM).

So we can either say that the range of physically possible values is infinite, or we can say that we simply have no idea what the range is.  The second option kills the fine tuning argument, so you should prefer the first option.

Here is my problem: the axiom of normalizability requires that the probabilities of all the possibilities add up to 1.  If there are infinity possible values, and each is given the same super small non-zero probability, this adds up to infinity.  If, instead, we give each possibility a zero probability, it adds up to 0.  Either way, we can't normalize the probability space, so we can't meaningfully talk about probabilities in this context.  P(FT/~G) is not low; it just doesn't even make sense to ask for this number.  (This is the argument made by McGrew et al. here: http://philpapers.org/rec/MCGPAT.  The point was also independently made by Colyvan et al. here:
http://www.colyvan.com/papers/finetuning.pdf, and by Paul Davies in "The Mind of God.")

One solution to the normalizability problem is to drop the assumption of a uniform distribution.  A nonuniform distribution would allow us to normalize a space of infinite possibilities.  You took this approach and said something like this:  "I recognize that there are multiple possible distributions, and we don't know which distribution is correct.  But since the set of life favoring distributions is just a small set of the total number of possible distributions, the probability that the actual distribution favors life permitting values is still very low."

Notice that this approach does away with the assumption of a uniform distribution over the range of possible values, but then assumes a uniform distribution over the range of possible distributions.  While this proposal allows us to normalize the space of possible values, it simply recreates the normalizability problem, because now we are unable to normalize the space of possible distributions.  We are faced with an infinite number of possible distributions, and you seem to be asking that we lay a uniform distribution over this infinite range.  This is the normalizability problem all over again.
Another possible solution is to find a way to limit the range of possible values, but I don't think this works.  You seem to have done this by focusing on possible values for G between 0 and 2.  Why exactly did you restrict the range of possibilities this way?

[See summary page of this discussion, with links to all the posts, here.]

FTA part 4: Why fine-tuning means the universe is improbable (part of the Aron series)

Greg says:
OK, great, all that is good to know for me. Hopefully, it will help us avoid talking past each other.

I'll go ahead now and engage with your points from your previous post. First, I am not sure if we want to go down the path of discussing Spinoza. I don’t have much interest in it, and you didn’t seem to push it too hard, so we’ll put that out of mind, unless you want to bring it back up at some point.

But perhaps what I could say about it is to give you a bible verse that is germane to the topic of P(FT | G):

Isaiah 45:18 - he who created the heavens,
he is God;
he who fashioned and made the earth,
he founded it;
he did not create it to be empty,
but formed it to be inhabited

OK, on to the topic of P(FT | ~G). In vernacular, what is the probability that the finely-tuned aspects of the universe would arise naturalistically? This is the term in the fine-tuning argument that typically takes on values like 10^-60 or 10^-120, etc. It’s the term that I called “epsilon.”

In particular, you said that I needed to be careful with that term, and you are absolutely right. Like you said, if you really want to think deeply about this, it’s not really accurate to just take the 10^-60 number and say that’s the probability. Again, you are 100% correct. So why do so many people (including myself) do that? TBH, I think for most people, they probably don’t realize the subtlety. For me, it’s just so much easier to communicate the idea that way, and in the end, if you want to go deeper, the conclusion is the same anyway, because you are just pushing back the fine tuning one step. But the route is indeed more subtle.

BTW, please read the following as if I were discussing an exciting topic that I like to think about and on which I am interested in hearing your feedback, rather than some guns-blazing attack on the non-theistic worldview. Because the former is the way I mean it, rather than the latter.

To make matters concrete, let’s unpack the discussion by focusing on the example of big G, which is the constant that is often said to be finely-tuned to one part in 10^60. It is not necessarily the best example to take, since it has its flaws, but it’s certainly an easy one to discuss. As well, anything I say here can be easily generalized to discussion of other constants.

Now, without loss of generality (and for the sake of discussion), we are free to pick units such that G = 1, so that the life-permitting range for values of G is 1 - 10^-60 < G < 1 + 10^-60. This is a narrow range for sure, but it does not necessarily translate into a low probability, because that depends on the probability distribution from which we are randomly drawing values of G.

(And of course, we are now ranging into philosophical/metaphysical speculation...we have no known mechanism by which we may suppose the existence of a probability distribution, nor one from which a value of G could be “drawn”, but in the end I don’t think it matters. I think the argument makes intuitive sense and will apply to just about any theoretical/hypothetical mechanism that one could come up with.)

Let’s imagine for a second that the probability distribution from which we are drawing the value of G is uniform from zero to two. In that case, p(G) = 0.5 uniformly on that interval. Then P(1 - 10^-60 < G < 1 + 10^-60) does indeed equal 10^-60. In this “special” case, the range of life-permitting values really does equal the probability of getting a value within that range.

But what if the probability distribution were a normal distribution with mu = 1 and sigma = 10^-61? In other words, a really, really tight distribution around the desired value of G = 1. In that case, we are *virtually guaranteed* to have a value of G within the life permitting range.

However…(do you see where I’m going with this?)...the only way you may legitimately assume we have such a “special,” atypical probability distribution for G is if you admit there is fine-tuning in the probability distribution itself. How in the world would one, apart from an intelligent creator with a purpose in mind, possibly justify having a probability distribution that forces this otherwise seemingly serendipitous, life-permitting value of G? (And that’s just one finely-tuned parameter.)

In other words, in my estimation, by correctly pointing out that the narrow fine-tuning range does not equate directly to probability, you escape the fine-tuning at that level, only to encounter it at a deeper level. With the fine-tuning argument, not only do you have to answer the age-old question of, “Why is there something rather than nothing,” but also, “Why is the something (that be, rather than nothing) the way that it is?”

[See summary page of this discussion, with links to all the posts, here.]

FTA part 3: Aron answers my general questions (part of the Aron series)

Hi Greg!

1. Not a believer. I'm interested because I have friends who are and it's just generally interesting.

2. Kalam or fine tuning. They have the most intuitive appeal, and they implicate lots of philosophical issues.

3. Problem of evil or potential inconsistencies in the definition of theism.

4. Not a scientism-ist. These sorts of questions are inherently philosophical questions, not scientific ones.

[See summary page of this discussion, with links to all the posts, here.]

FTA part 2: General questions for Aron (part of the Aron Series)

Aron, how are you doing? I’m glad to hear from you again! I hope you continue to think deeply about these questions.

To be honest, I didn't realize that I was going against what Larry believed about using epistemic probabilities and such. At any rate, I am glad we came to some agreement about how TAG could possibly be used in an argument, even if we still don’t agree on whether certainty in TAG renders evidence for God useless.

Hey, but now that we've come to some sort of conclusion, it sounds like you want to switch gears and go more in-depth into the fine tuning? Would that be correct? I’m interested in doing that. I find the fine tuning argument fascinating, and it is definitely one of the arguments that first opened the door to my skeptical way of thinking to allow me to entertain the possibility that God exists.

But before we get too in-depth with this argument, and I see that you have put forth a couple challenges to it, I was wondering if you’d answer a couple of questions? The reason why is I don’t know you very well. (Maybe Larry does, but I don’t!)

First, I am assuming you are not a believer in Jesus in the classical Christian sense. Is that right? Did you ever at one point consider yourself to be a Christian?

Assuming your answer to the first question is, “No, I am not a believer,” what is your interest in Ratio Christi?

What is your favorite argument in favor of the existence of God?

What is your favorite argument against the existence of God?

A lot of atheists these days take a very hard scientistic stance, in which the only allowable evidence in discussions about whether God exists is evidence that can be tested scientifically. I am assuming that would not be your stance, given how into the TAG you got, but I just wanted to make sure.

Thanks a bunch for humoring me on answering these questions. The reason why I think they’re important is because knowing your background might help us avoid talking past each other. To be fair, I will give you my answers.

I am a believer, but I grew up atheist. My favorite argument in favor of the existence of God is...well, it’s hard to pin one down. I've switched back and forth over the years, but as I said, the fine tuning argument was one of the first that I heard and it really went along way to convincing me to go from being an atheist to theist. For this reason, I am really passionate about apologetics, which is why I am passionate about Ratio Christi.

My favorite argument for atheism is the problem of evil, as well as the horrific events in the OT as evidence against Yahweh being perfectly good; those two are very tough problems for the Christian.

I am clearly not a proponent of scientism, although I am a scientist.

[See summary page of this discussion, with links to all the posts, here.]

FTA part 1: Aron challenges that the probability if fine tuning is low (part of the Aron Series)

Aron wrote:
I like your cumulative case method because it looks like you're using Jeffrey conditioning. The probability of G in light of TAGs uncertain status=P(G/TAG)xP(TAG)+P(G/~TAG)xP(~TAG). The left side would be 1x.6, and the right side would be .4 multiplied by whatever you think P(G) is given all the other evidence. In this case, you can use TAG along with evidential arguments. That's why I think the Bayesian approach is best, bc it can let you do stuff that you couldn't do if you were offering TAG as a proof. I don't think Larry would be open to this approach though.

My initial argument was offered against someone who uses TAG as a deductive argument, so my point still holds in that context. I think you offer a good way to sidestep the problem. But even with the method you offer, my main point still stands: evidential arguments can only make a contribution if you're open to the possibility that TAG is wrong.

About fine tuning. I agree for the most part. There are a few ways I could push back. For what it's worth, there's a long tradition, going back to at least Spinoza, of arguing that P(FT/G) is either 0 or near 0. (Spinoza obviously didn't talk about fine tuning per se, but he argued that God wouldn't/couldn't create a universe bc he'd have no reason to do so, given his lack of wants/needs).

Also, be careful with P(FT/~G). The fine tuning data doesn't say this number is small, it says that the life permitting range of values is small. If someone says a constant is fine tuned to one part in 10 billion, they aren't saying that there is a 1/10 billion chance that the constant would have that value. Instead, they are just saying that if the value of the constant were changed by one ten billionth of a percent, then life couldn't exist. Claims about fine tuning are about the narrowness of the life permitting range of values a constant could take. Additional argumentation is needed to show that its improbable that a constant would take a value in that range. To say that P(FT/~G) is low, you need to make some philosophical assumptions, such as the adoption of the principle of indifference and the rejection of the axiom of countable additivity.

[See summary page of this discussion, with links to all the posts, here.]

Aron and the transcendental argument (part of the Aron Series)

Please see below for my discussion with Aron about the transcendental argument for God's existence (TAG).  I apologize for the abrupt beginning, but I jumped into the conversation in medias res, as it were.  And unfortunately I have no way of retrieving Aron's earliest points in the argument, including his formal argument points (1-8).

For links to the full series, see here.


=========

Greg Reeves wrote:
As a scientist/engineer and not a philosopher, I am also not the best qualified, but with that disclaimer... If Ron is right that the possibility of having evidence against God is necessary for the design argument to succeed, then yes indeed the design argument fails. And he would be absolutely right about the first line of argumentation.

But even if we grant him that needing the possibility of having evidence against God is required for successful evidential arguments against God, Argument 1 is a paradox, not a contradiction, because he is leaving out crucial qualifiers in statements (1) and (8).

When he says "the design argument succeeds" he should be saying "were logic possible without God, the design argument succeeds" (a true statement).

At the end, when he says "the design argument fails" it is instead "now that we have proven that God is necessary for logic to be possible, and the design argument rests on the operation of logic, then the design argument fails (in that it cannot not be true)". This last statement does not contradict the previous statement; they are different statements and he is guilty of equivocation in the word "fails".
In other words, the design argument goes through under certain premises, and does not under others. So what? What he has shown is that if TAG is true, the design argument fails, but only because God is proven rather than uncertain; while if TAG is false, then the design argument succeeds and therefore God is a high probability. In other words, he has shown that God is either a certainty or a highly probable being.

What about his second line of argumentation? Equivocation again. If he is granting TAG in the second argument, then not only is (1) true, but the laws of logic would not hold were God to not exist. Therefore, instead of (3) he should be saying "The fine tuning argument is a successful evidential argument for God if logic is possible without God (assumption)." Instead of (4) it would be "Therefore, God would exist even if logic were possible without God." Then (6) would become, "But there could be theoretical evidence against God, given the success of the fine tuning argument, were logic to be possible without God." And (7) would be a near-tautology: "Therefore, there is no being whose nature is the foundation of logic were logic possible without God." Then (8) Therefore, God would not exist were logic possible without God." Finally, (9) "Therefore, logic is impossible without God." Which is where you began anyway since he started by granting TAG.

We should all (myself included) be very careful about his hidden premises/equivocation.

What about his proof that starts out with "If there is a true statement that takes the form 'there is evidence for x', then it confirms theism?" In this proof, I think he is starting with the assumption that TAG is true, meaning the laws of logic depend on God. But this proof clearly has a problem with it...I am probably not getting my terms 100% correct, but I think Ron is conflating epistemic probability (what we think is true based on the evidence) from ontological probability (what actually IS). If we are uncertain that God exists, then based on evidence, we can epistemically put a probability on our belief (crudely). Such and such evidence favors the interpretation that God exists. Other evidence may favor atheism. But in an uncertain world you can truly have evidence that favors a proposition that is untrue. Therefore, having evidence for a proposition does not make that proposition true...all it does is make the statement "there is evidence for this proposition" true.

But if God really does exist, then atheism (here I am using it as the state of affairs in which no God exists) is ontologically untrue. No matter how much evidence you may say there is for atheism, if God does in fact exist that makes atheism untrue. So the way in which he wants "there is evidence for atheism" to mean atheism is true is just a false maneuver. So let's update his actual argument with this in mind:

1. If there is a true statement that takes the form “there is evidence for x”, then it confirms theism (because of TAG).
2. The statement “there is evidence for atheism” is a statement that takes the form “there is evidence for x”.
3. Therefore, if the statement “there is evidence for atheism” is true, then it confirms theism (which has already been proven because TAG was granted).
4. If the statement “there is evidence for atheism” is true, then it, in principle, challenges theism because it means evidence against theism exists.
5. Therefore, if the statement “there is evidence for atheism” is true, then it both confirms (100%) and challenges (makes you wonder about) theism.
6. If the truth of the statement “there is evidence for atheism” both confirms and challenges theism, then the conclusion that one may draw from the evidence for atheism (ie, that atheism is true) statement is necessarily false. In other words, even though there is evidence for atheism, that evidence does not go through.
7. Therefore, the statement “there is evidence for atheism” if true, may lead you to a false conclusion if you don't realize that such a logically true statement confirms theism (100%) under TAG.
8. Therefore, Premise 1 necessarily leads to the conclusion that while there can be evidence for atheism, atheism is still false.

So in summary, atheism can have evidence for it, but *if TAG is granted* then atheism is false, so evidence for atheism is incorrect. But if TAG is not granted, then you can marshal evidence all you want and we'll see which one stacks up better. Ultimately the existence of the laws of logic have a more comfortable fit in a theistic worldview rather than an atheistic one, and the atheist is left with the uncomfortable task of explaining their existence.

============

Aron wrote:
Greg, in your first post, what exactly do you mean with this paragraph: "In other wordsthe design argument goes through under certain premises, and does not under others. So what? What he has shown is that if TAG is true, the design argument fails, but only because God is proven rather than uncertain; while if TAG is false, then the design argument succeeds and therefore God is a high probability. In other words, he has shown that God is either a certainty or a highly probable being." My point was to show that, under the assumptions of TAG, the design argument fails, and under the assumptions of the design argument, TAG fails. Are you agreeing with me that the success of one entails the failure of the other?

I'm talking about epistemic probability here.

When I talk about evidence, I mean that some fact makes a hypothesis more epistemically likely than it would have been without it. E is evidence if P(H/E)>P(H). This condition will be met if P(E/H)>P(E/~H).

You're absolutely right that there can be evidence against a true hypothesis (i.e., evidence that lowers it's epistemic probability). For example, if I were framed for murder and the weapon was planted in my sock drawer, this would increase the epistemically probability in the detective's mind that I was guilty. This is despite the fact that, ontologically, the probability that I'm guilty is 0.

So there is no doubt that evidence can pull in different directions epistemically. Some evidence may suggest I'm the killer, and other evidence may suggest I'm innocent. However, no single piece of evidence can simultaneously confirm and challenge a hypothesis. This would mean that for that particular evidence E, P(E/H)>P(E/~H) and P(E/H)

P(H/E) and P(H)

So imagine some fact F raises the epistemic probability of atheism. This would mean P(F/atheism)>P(F/theism), and therefore P(atheism/F)>P(atheism). But, if we think that logic presupposes theism, then we should think every fact about the world increases the epistemic probability of theism to 100% (this is bc in order for there to be facts, laws like the law of identity and non contradiction must apply). This would mean that while F raises the epistemic probability of atheism, it must do the same for theism. Contradiction.

Thus, while it is true that we can have evidence against a true hypothesis, this is only true when that hypothesis is not the foundation of logic itself. If you want to admit the possibility of evidence against theism, you need to drop the premise that logic depends on God.


===================

Greg Reeves wrote:
Aron, sorry that I did not know you had responded to me.  I guess I didn't get any update saying so.  Like I said originally, I am a scientist and engineer, so you have to take what I say with a grain of salt about these matters.

What I meant in the first paragraph is that under some conditions, TAG goes through, and under others DAG (design) goes through.  Meaning, if you accept TAG, and you condition your DAG argument on TAG, then you end up with an inescapable 100% probability for God, because you have already conditioned your DAG argument on the background that God exists.  So, yes I do agree with you that the success of TAG entails the failure of DAG *only if you are correct that* and *only in the sense that* you must have the possibility of some evidence contrary to an argument to make the argument successful.  But I did say up front that I do not necessarily accept that. But again, even if I did, then DAG would only fail in that sense, but would not fail in the sense that God has been proven to not exist.  It seems to be only a technicality.

But I definitely could be wrong about that, so let's explore your suggestion about probabilities.  Let me make sure I am understanding you correctly.  You are concerned that if you have a piece of evidence that increases your epistemic probability for atheism, then under TAG it also simultaneously decreases the evidence for atheism (because any logical construct, if it exists, on TAG, proves theism).  You are then worried that if you accept TAG then you get a logical contradiction.  Therefore, TAG cannot be true.

Problem is, since we are dealing with epistemic probability, you have to be *very careful* because epistemic probability can be very tricky.  You can end up sneaking all sorts of stuff in the back door.  Here is where I think you are going wrong.  I will try to parse my answer in the context of your previous paragraph that started with "So imaging some fact F...":

So imagine some fact F raises the epistemic probability of atheism if you do not accept TAG. This would mean P(F/atheism)>P(F/theism), and therefore P(atheism/F)>P(atheism). But, if we think that logic presupposes theism, in other words, if we then condition our probabilities on TAG, then we should think every fact about the world increases the epistemic probability of theism to 100% (this is bc in order for there to be facts, laws like the law of identity and non contradiction must apply). This would mean P(atheism/F & TAG)

Remember, formally the law of non-contradiction says both A and ~A cannot hold at the same time, in the same way, and *under the same circumstances*.  Conditioning on TAG completely changes your circumstances.  So we are back to what I said in my original post.  It depends on your premises.  You change your premises and then of course your conclusions can change.  If logic were possible without God (ie, you don't condition on TAG), then you could use logic to try to prove atheism.  If indeed TAG is true, and you condition on it, then any piece of evidence that you previously used in support of atheism when you did not condition on TAG is no longer in support of atheism.  If you wish, I could write out a full Bayesian analysis on this, but I get the feeling that most readers of this forum would not benefit from it.


=============

Aron wrote:
I don't think we disagree about anything. Once you have come to accept TAG and incorporated it into your background knowledge, then any probability assessments you make will be conditioned on TAG. And you agree that if we condition on TAG, there could never be evidence against theism. But if this is the case, then there could never be evidence for theism either, so the design argument won't work. You seem to agree with this, but think it's "trivial." The reason I don't think it is trivial is that it forces people to make a choice. If you think TAG is true, then you can't think that biological complexity makes theism more likely than it would otherwise be. And if you think biological complexity lends support to theism - i.e. Pr(theism/biology)>Pr(theism) - then you can't have TAG in your background knowledge. I have seen people make cumulative cases that include both arguments, and I don't think this is an option. You can't simultaneously believe that both arguments are sound.

Greg, I wouldn't mind seeing your Bayesian analysis of Dawkins.


===========

Greg Reeves wrote:
Aron, I am glad we aren't disagreeing then, but I do want to take you to task a little bit because I was primarily responding to how you said these things were *contradictions*.  It is not a formal contradiction because you are either conditioning on TAG or not, and that changes your outcome.

But I like how you want to be very precise about it.  If I were to be building such a cumulative case, I probably wouldn't have caught that, but now that you point it out, I'll be careful.  But I think you can still do it...let me unpack what I mean by that.

First, if one is being as precise as you are, I still argue it is "trivial", because in that case, Pr(theism | biology) >= Pr(theism) --- note here the greater than or equal to rather than the strict greater than --- if you include TAG in your background.  That is because TAG means Pr(theism) = 1.  So Pr(theism | biology) = 1.  So it is a trivial result as to whether you "add" the argument from biological design to your background knowledge.  In either case, Pr(theism) = 1.

So how in the world would you build a cumulative case with TAG included?  Well, since we're talking about epistemic probability here (which is a measure of belief rather than frequency), then you could say "Let 'A' be the event that someone believes TAG is true with a 60% probability".  Could you not then condition on "A"?  Would then TAG not be part of a cumulative case?  I think it could be.

In that case, TAG is just another part of your toolbox in the cumulative case for God.  This is where I fall because I think it's a powerful argument, but no one will be 100% convinced on the basis of this argument alone.  On a frequentist approach, either TAG is true, or not; just like either God actually does exist or not.  But in terms of epistemic/Bayesian thinking, the question is rather, how convinced are you that this argument goes through? If you are 90% certain of TAG, then both Pr(theism | A & B) >= Pr(theism | B) and Pr(theism | A) > 90%, where A = TAG is 90% probable and B = background knowledge.

If you argue for some reason it cannot be part of a cumulative case, then we are faced with two possible choices: either you accept TAG and then arguing further about the existence of God is trivial since you accept the proof from TAG, or you reject TAG and then we go on with the rest of the cumulative case.  Either way, any other evidence/argument I mount in favor of theism is at least neutral: in the former case it's irrelevant and in the latter it strengthens the case for God.

Which camp do you fall under?  Do you accept TAG and therefore need no further convincing on the strength of that argument alone?  Or do you reject TAG and therefore are open to discussing the strength of the other myriad arguments for God's existence?

Again, if you are like me, then TAG is a powerful argument and if correct then God's existence is 100% certain.  But we don't know for sure it's correct, meaning that, in terms of epistemic probabilities, God's existence is not 100% certain. Therefore, we marshall other arguments.

Regarding the Bayesian analysis of Dawkins's statements, it is specifically about the case for the fine tuning of the universe.  The general Bayesian analysis goes like this:

Pr(G | FT) = P(FT | G)*P(G)/P(FT)

where G = God exists and FT = the fine tuning of the universe is instantiated.  As usual, we can split the denominator into two terms:

P(FT) = P(FT | G)*P(G) + P(FT | ~G)*(1 - P(G))

Now, the fine tuning argument says that P(FT | ~G) = epsilon (ie, small).  Usually numbers like 10^-120 or 10^-10^123 are thrown around, but the exact number is not important, just that it's small.

For the sake of simplicity, let me just say that P(FT | G) = 1 (ie, God would indeed make a universe in which advanced life is possible, and of course by necessity such a universe would be finely tuned as we observe).  We can keep this term around for precision but I think it's easier to discuss over facebook if I make this assumption.  The results are essentially the same either way unless you want to argue P(FT | G) = epsilon also, which I think would be hard to justify.

Anyway, this leaves us with:

P(G | FT) = P(G) / (P(G) + epsilon*(1 - P(G))

So the only thing left here "unknown" is our prior probability of the existence of God.  Now, you can see right away that if the fine tuning argument is correct in that epsilon is small, then only if you have an absurdly low prior for God existing can you escape the conclusion that P(G | FT) is close to 1.  Well, that's exactly what Dawkins does.  He says, "It doesn't matter how improbable our universe is; the probability that God exists is smaller."  That's a sneaky statement, but what it means is that P(G) = 0.  The only number that could be a probability that is smaller than *any other number* that is also a probability is zero.  By definition.

Mathematically, Dawkins's statement is: for every epsilon > 0, 0 <= P(G) < epsilon.  This is *mathematically identical* to saying P(G) = 0.  But if someone's prior for God is zero, then there is no reason to have any discussion.  Dawkins is essentially saying, "I don't care what the scientific evidence for fine tuning says, I will choose to believe that God does not exist."  That is not reason or rationality, that is blind belief.  Belief, as it were, in spite of the evidence.  :-P

But let's instead say that your prior is also super-small.  Let's say it's much smaller than epsilon even.  Then you are left with:

P(G | FT) = P(G)/epsilon

In other words, the most hardened skeptic, unless he is exhibiting blind faith that God does not exist (and thus P(G) = 0 for him), would objectively increase his epistemic belief that God exists by orders of magnitude once the evidence for the fine tuning of the universe were examined.  And if you have a set prior for God's existence (and not a moving target so that as more evidence comes in, you "conveniently" make your P(G) smaller), all you have to do is wait a while.  As we discover more about the universe, I predict that epsilon will get smaller and smaller.  If I am right, then eventually, epsilon will either shrink past your set P(G), or you will have to find some other reason to reject this argument.  Or else you are fooling yourself.

Now, don't get me wrong, you can attack what I've said in a number of places.  I did gloss over what one might think P(FT | G) is.  Also, you can argue that the universe is not in fact finely-tuned.  However, in the first case, as I've said, unless you unjustifiably put P(FT | G) ~ epsilon or less, then it doesn't really matter what P(FT | G) is.  In the second case, you would be going against hard scientific data, and thus must object out of a precommitment to a non-theistic philosophy rather than objectively examining the evidence.  I think that in either case you are in a weaker position than the theist.

I can go through a similar Bayesian argument for the resurrection.  I like that one too.

Series: a discussion with Aron from Maryland

Dear all,
Last year, I began an internet discussion with a non-theist named Aron from Maryland.  At the moment, this discussion is ongoing, and will be continuously updated here, so check back often.

We started out talking about the Transcendental Argument for God's existence (TAG), and in the end agreed how it could be used in a bigger cumulative case.

We have since transitioned to discussing the fine-tuning argument (FTA).  It's great discussion, and I hope others will follow along.

Transcript of our discussion about TAG (we start transitioning to discussing FTA at the end):
http://twobooksapproach.blogspot.com/2015/01/aron-and-transcendental-argument-part.html

FTA part 1: Aron challenges that the probability if fine tuning is low

A beginning to our discussion of the FTA, where Aron opens with acknowledging our general agreement of how to apply the TAG, and also suggests the probability that our universe would be the way it is (usually called "finely-tuned" in theistic circles) in not actually low:
http://twobooksapproach.blogspot.com/2015/01/fta-part-1-aron-challenges-that.html

FTA part 2: General questions for Aron

I answer with some general questions for Aron to make sure we're not talking past each other:
http://twobooksapproach.blogspot.com/2015/01/fta-part-2-general-questions-for-aron.html

FTA part 3: Aron answers my general questions

Aron then answers my general questions, so I think we're ready to start the discussion in earnest.
http://twobooksapproach.blogspot.com/2015/01/fta-part-3-aron-answers-my-general.html

FTA part 4: Why fine-tuning means the universe is improbable

I begin to lay out the fine-tuning argument and press why we can say the probability that the fine-tuning would occur without God is small.  Please note that Aron and I both agree that the constants of the universe are finely-tuned, in that they cannot vary by much before the universe is no longer life permitting.
http://twobooksapproach.blogspot.com/2015/01/fta-part-4-why-fine-tuning-means.html

FTA part 5: Aron introduces the normalizability problem

Aron responds by noting that since we are appealing to "possible universes" (which may not even exist), we have no idea what values of the constants of physics are more probable than others, so we must assume any value has equal probability to any other. However, since we have no way to restrict the ranges these values may take on, we have to allow for infinite ranges (i.e., zero to infinity).  Therefore, the probability distribution is not really a probability distribution.  In this statement, Aron is introducing the "non-normalizability" problem.  To strengthen his argument, Aron cites famed Christian apologist Dr. Timothy McGrew.  This is a big deal.
http://twobooksapproach.blogspot.com/2015/06/fta-part-5-aron-introduces.html

FTA part 6: Defying intuition, the multiverse, or necessity

I agree that the non-normalizability problem that Aron introduced in the previous post is a big deal.  Then I lay out how the intuition of the FTA argument is layered, and every time a skeptic denies the intuitive conclusion that fine-tuning leads to the conclusion that God most likely designed the universe, he or she must give up some more obvious conclusion for a more skeptical one.  This is my "How deep does the rabbit hole go?" story.  But, we do need to go deeper, so I note that there are two options.  Either the multiverse exists, in which case the probabilities are indeed normalizable (because there is a real mechanism generating universes), or our universe is all there is.  In the second case (which Aron wants to go with), even if the probabilities are meaningless (since there is only one universe), we still have a situation (fine-tuning) that demands an explanation.
http://twobooksapproach.blogspot.com/2015/06/fta-part-6-defying-intuition-multiverse.html

FTA part 7:  Aron asks clarifying questions

Aron reasserts that the non-normalizability problem, since it leads to the conclusion that probabilities are meaningless, also means that the fine-tuning argument just does not work.  He asks for more clarification and explanation on my part.

http://twobooksapproach.blogspot.com/2015/06/fta-part-7-aron-asks-clarifying.html

FTA part 8:  Clarification on the multiverse vs. necessity

Here I clarify what I said in my previous post.  In particular, I clarify that if Aron wants to go with a single universe, and claim that fine-tuning does not lead to small probabilities, this is the same as saying (1) the universe just IS, with no explanation for it, and none needed; and (2) the universe just IS FINELY-TUNED FOR LIFE, with no explanation for that either, and none needed.

http://twobooksapproach.blogspot.com/2015/06/fta-part-8-clarification-on-multiverse.html

Link: Richard Dawkins and Circular Reasoning

Over at The Cumulative Case, I've made a recent post about Richard Dawkins and his circular reasoning, which echoes a post from a while back here at the Two Books Approach.  In essence, I have attributed a statement to Dr. Dawkins that, if followed to its logical conclusion, means to deny that the fine-tuning of the universe implies the existence of God, is predicated on first assuming God cannot exist.  That is, the probability of God existing is identically zero.  Of course, this isn't logic or reasoning, it is simply blind faith.  Now, I don't suppose I'll know for sure if Dr. Dawkins really holds to this view, so for the benefit of the doubt, I will say that he does not.  Even so, I think it is interesting to unpack what the following statement means: "No matter how improbable this universe is by chance, the probability of God is even less."

An excerpt from the relevant post from The Cumulative Case:

If you start off assuming that God cannot exist, then no amount of evidence, no matter how strong, can budge you.  Dawkins is essentially saying, "I don't care what the scientific evidence for fine tuning says, I will choose to believe that God does not exist." That is not reason or rationality, that is blind belief. Belief, as it were, in spite of the evidence.

For the readers' convenience, I've pasted the relevant point from my old post (note the edit where I have made my statements more circumspect by inserting [many] where the word "all" used to be):

[T]he statement "no matter how improbable this universe is by chance, the probability of God is even less" is tantamount to saying "the probability of God existing is zero." Think about it. The only non-negative number that is guaranteed to be smaller than all positive numbers is zero. This is quite a strong statement. It goes far beyond saying God doesn't exist. It says that God cannot exist. In other words, Dawkins is using an assumption that God cannot exist to try to prove that God does not exist. It is a completely circular argument.

Here's another way to think about it, for the more math-oriented folks. In probability and statistics, the proof we're trying to make is something called a conditional probability. We see an improbable universe around us. What is the probability that God exists given we live in an improbable universe (ie, what's P(G|U))? Using Bayesian inference, we can easily come up with:

P(G|U) = P(G)/(epsilon + P(G)).

Here P(G) is the prior probability that God exists, and "epsilon" is the small chance that this universe came together by coincidence ([many] scientists would agree that epsilon is very small...something like 10^-50 or less). When doing Bayesian inference, you often have to bring in some a priori assumptions to assign prior probabilities (hence the name), so we have to guess at what P(G) is. But you never outright assume that P(G) is identically zero (or one). That would be the same as saying "no matter what our studies tell me, I will choose to believe X." (That's called blind faith.) Usually, when you don't know, you simply set your prior probabilities equal to 1/2. It's easy to see that P(G|U) (the probability that God exists given the universe we live in) would be extremely close to one for any reasonable choice of P(G). The only choice that makes P(G|U) small is P(G) = 0. Which is apparently what Dawkins wants to say.

Link: The Inevitable Consequence of An Atheistic Worldview

One of my favorite Christian authors is J. Warner Wallace (Cold Case Christianity), who gave a fantastic lecture here at NC State in December.  I was browsing his blog the other day and came across this piece where he describes an intriguing exchange between some skeptical readers of his blog.

In essence, one of the commenters came out and said in very plain words what the consequences are of an atheistic worldview.  The bottom line is there is no foundation for morality.  I recommend it as a quick read.

Skeptics don't think Christians think

I have just now come to the realization that, even though there are fantastic arguments in favor of the Christian worldview and of the truth of the bible, many skeptics still think that Christian apologists are charlatans.  That is, these skeptics think that our reasons for belief are grasping at straws, and that our belief comes first, and this faith blinds us to the mistakes we are making in our arguments.  Or worse, that we know our arguments are bad, but we keep advancing them in hopes to keep the faith alive.

This really hit home with me when I considered several bad arguments against Christianity from the starting assumption that all arguments in favor of Christianity were wrong.  (That is, Christians have zero reason for their belief).  Check these out (and these are just a few):

"You just believe because your parents told you to -- therefore, God doesn't exist."  

This is a terrible argument!  But if you start with the assumption that Christianity has no arguments in its favor, then you see why this is compelling.

"You just believe because you were born in Western society.  If you were born anywhere else, you would have adopted the prevalent religion of that location.  Therefore, all religions are false."

This argument is so empty it's hilarious.  First, even if it were true, it doesn't invalidate Christianity.  And second, it's self-refuting: you could apply this argument to atheism (like I do here in "Objection 1").  So why is it so convincing to so many people?  It's because there is the underlying assumption that religions (especially Christianity) have no basis for their belief!  So you can't apply it to atheism, because that has a basis for its belief (so the assumption goes).

Religions are a mind-virus (a meme).  They spread because of societal and cultural pressures.  Therefore, God doesn't exist.

Again, this argument doesn't hold water unless you're already assuming there are no reasons for belief in God.  And again, this argument can be turned against atheism...but the person putting forth the argument doesn't realize that because they don't realize their implicit (and incorrect) assumption: that atheism has evidence for it but religion does not.

Even in the multiverse, our universe is rare

Last time I ended with the cliffhanger, saying the multiverse (the idea there are a vast number of other universes out there) strengthens the design argument for God.  This discussion stems from the widespread realization, from scientific data, that our universe appears "designed".  The constants of physics and structure of the universe are finely-tuned: they must be just-right in order for life to exist.  Even non-theistic scientists acknowledge this.  (See here for a great website that has compiled many quotes to this effect.)  However, what if there are an infinite number of universes, each with their own random laws of physics?  By sheer numbers then, we should expect at least one bio-friendly universe to exist.

And the fact that we exist says that such a universe exists.

This last point is incredibly important.  It is called a "selection effect".  Even though it's rare that a bio-friendly universe exists, the idea is that it is not rare for observers to note they live in a bio-friendly universe.  The very fact that observers exist (and are living) shows they must be in such a universe.

But here's where it gets weird.  It turns out that, of all possible universes with observers, our universe is still rare.  Because of the extreme degree of fine-tuning necessary to make this universe bio-friendly, it is actually far more probable that a universe composed simply of a star and a host of planets (including the life-friendly one) just popped into existence (POOF!) out of the quantum vacuum.  I repeat: it is far, far more probable that an entire life-friendly solar system just popped into existence with no explanation.

It gets weirder.  Considering those types of universes are more probable, and that presumably in some of those universes, life will have evolved to the point of being technologically advanced, it's actually more probable that we are just in a Matrix-like computer simulation from these advanced life-forms.  In that manner, we'd be "intelligently designed" (but also our world would not be "real").

It gets weirder.  Even more probable than the solar-system-out-of-nothing scenario, by far the most common type of universe with an observer(s) is one in which a single brain pops into existence, looks around and notes the nothingness in which it sits, and then pops out of existence.  (This is called the "Boltzmann Brain".)

All of these then beg the question: why do we live in a universe that is so extremely finely-tuned (to such a degree that it is prohibitively rare, even in a multiverse), when it's far more likely that we wouldn't live in such a universe?  The clear-cut answer is that our universe did not arise by chance.  We are not just a lucky accident of the quantum vacuum churning out random universes.  Our universe was supernaturally designed by the One who has the power and care to undertake such a creative event.

The Multiverse: science or a cop-out?

Recently, over at townhall.com, columnist Dennis Prager wrote a piece on "Why Some Scientists Embrace the 'Multiverse'".  The article is a good read; in the first part, Prager describes the incredible scientific evidence for the design of the universe for the benefit of life.  The bio-friendliness of the universe is undeniable, and is admitted by scientists of all stripes.  But the "design" of the universe then begs the question: who designed it?  Or, how did it come to be this way?

However, Prager then transitions into a discussion of the multiverse (the idea that there are many, perhaps infinite, universes out there).  If true, the multiverse could get around the problem of the fine-tuning.  After all, if there are an infinite number of universes out there, all with random laws of physics, there would just have to be one in which the laws were "just right" for life to exist.  No matter how improbable it would be for you to hit that bullseye perfectly, given an infinite number of tries, it's bound to happen.  And that's the universe we live in.

But Prager implies the multiverse is solely proposed by scientists in order to get around the fine-tuning.  This is incorrect.  The multiverse hypothesis is a direct inference from the laws of physics.  If we've got the equations of our own universe right, there is a very good chance there are other universes out there.  Just because it's impossible to detect them doesn't mean they aren't there.  And that certainly doesn't mean the multiverse hypothesis is some concoction by atheists to get around the fine-tuning.

So if the multiverse exists, and I don't think we can discount it, where does that leave the fine-tuning argument?  It does seem to provide a nice refutation of fine-tuning pointing to God as designer.  Perhaps that's why it's so popular, and why it gets so much attention.  (And perhaps that's Prager's actual point: that the multiverse is popular with scientists, even though there is no direct evidence for it, because it seems to satisfy their worldview.)  But the reality is, the multiverse only makes the design argument for God stronger.

Is Dawkins spreading propaganda?

Well, despite the sensationalistic title of this post, I think the answer is probably "no, I wouldn't go that far as to say that."  But I have been amazed at some of the things he keeps on saying, held in tension with some things I know people talk to him about.  For example, he continues to deride all believers as people who do not think, yet he is acquainted with one of the greatest proponents for Christian thinking: fellow Oxford professor (and prolific author) John Lennox.

This all came together for me in an "aha!" moment when I read the first four sentences of this article about William Lane Craig (which by the way is a great article in and of itself and I recommend reading it; the first four sentences are almost irrelevant to the rest of the article) where the author depicts Dawkins as decrying the notion of giving a Christian apologist publicity.

This made me wonder: it is really about the search for truth and advancing science and reason for Dawkins?  Or is it about empty rhetoric, sound bytes, and attempting to control what people hear?  The irony of it all is that Dawkins (and now many others) blames the spread of religion on "memes".  Originally, memes didn't have to be funny images with catchy phrases; they can also be just what your parents taught you, or what your friends like, or what's cool at school.  Any cultural element that gets passed from one person to another could be a meme.  Yet, through the use of sound bytes, etc. (which appeal so much to this generation), it is actually atheism that is now propagating* by meme.  Very interesting.

* - By the way, the word "propaganda" has the same root as the word "propagate".  We now come full circle.

Why am I both a scientist and a Christian?

In this day, it's en vogue to say that if you think rationally, then you give up Christianity.  No doubt this seems true.  In fact, many churches also promote this view (either implicitly or explicitly).  The internet is filled with anecdotes about teenagers who came with the hard questions, yet their pastors/elders/"heroes" in the church just told them to have more faith.

Talk about a recipe for unbelief.

The sad thing is that this is a complete misunderstanding, promoted by both Christians and non-Christians alike.  Last time I professed that I am both a scientist and a Christian, but I didn't say why.  Here it is: Science was birthed out of a Christian world.  It borrows Christian ideas about the world as basic philosophical foundations.  And most importantly (at least for me): scientific, philosophical, and historical evidence undergirds, rather than erodes, the reasonableness of belief in the God of the bible.

But don't take my word for it.  Here are a few sites that would agree (and these are far from exhaustive; they are just the ones that I thought of off the top of my head):
www.reasons.org
www.str.org
www.reasonablefaith.org
www.godandscience.org
www.thepoachedegg.net

I am a scientist and a Christian

There is a false dichotomy that most people believe in without even thinking about it: that either science is true, or religion (i.e. Christianity) is true, but not both.  For example, according to reviews of "The Unbelievers," that movie implicitly assumes this false dilemma and rides that assumption throughout.

But that doesn't hold water.  I am a scientist and a Christian, and far from internally being at odds with myself, the scientific (and engineering) knowledge I have strengthens my belief in God.

What's interesting is that oftentimes, atheists maintain their lack of belief in any god not for scientific or logical reasons, but for non-rational (emotional) reasons.  Take the oft-quoted Aldous Huxley on Christian morals: "We objected to the morality because it interfered with our sexual freedom."

Or infamous atheist Thomas Nagel: "I want atheism to be true and am made uneasy by the fact that some of the most intelligent and well-informed people I know are religious believers. It isn’t just that I don’t believe in God and, naturally hope that I’m right in my belief. It’s that I hope that there is no God! I don’t want there to be a God; I don’t want the universe to be like that."

(Incidentally, these quotes support what I was saying a few days ago in regard to Objection 4.)

For more, see this post on the Christian Apologetics Alliance webpage.

A response to "Dear Believer, Why Do You Believe" part 2 of 2

Last time, I began a critique of the popular YouTube clip titled, "Dear Believer, Why Do You Believe".  The thing to keep in mind is that the main premise of this video is that no religious person has adequate reasons to believe.  If this premise were true, then the objections raised in this video actually hold some weight.  However, in that case, the objections would essentially be moot: if no religious person has good reasons for their belief, what is the point of raising more objections.

But instead, religious folk, and Christians in particular, have many reasons to believe!  If you follow my blog, you'll find some of those reasons.


As for the rest of my critique of the video, I have identified six common objections to religion in general, and Christianity in particular, and I offered a response to the first three last time.  Here I will go through the three remaining objections raised in this video.

The problem for the atheist who espouses the views in this video is there are not only many adequate reasons to believe in a god in general, and the Christian God in particular, but these reasons are also immensely compelling.  But even if they were only moderately compelling, the existence of such reasons would totally dismantle this video.  In a way, the narrator is underhandedly delivering a stinging insult to religious people worldwide, saying they live in the dark ages.

Objection 4: “Religions are just crutches to help us feel better.”

I wonder if religions aren’t just ancient constructs, in an attempt to explain unexplained phenomenon.  Though irrational in content, they are not irrational in their emergence.  But we no longer live in the dark; science is ablaze in our world, we no longer live in the cave!  We no longer require comforting stories that make us feel safe, comforted or valued.  Isn’t it time our faith matches our discoveries?  Our ideas our new perspective?  Greater awe in reality rather than in fantasy?

This has many problems, first and foremost that atheists may hold their lack of belief in any gods also as a crutch.  If there is no cosmic being, then who am I held accountable to for my immoral actions?  No one.  OK, not every atheist feels this way, but certainly some do.  In the same vein, not every Christian feels like God is a crutch.  Sometimes, I wish God's influence on my life would just go away so I could be autonomous.  I mean, who doesn't like the idea of being their own master?  Who wants to submit to someone in higher authority than themselves?

On the other hand, again, if there are reasons to believe in God, then crutch or not, it is something to consider.

Objection 5: “It's arrogant to think you have the right religion”

Isn’t it time to stop thinking that we are somehow the reason why this universe was made?  That our culture is somehow better than other cultures?  It’s time to learn how the universe really is, even if that deflates our conceits, and forces us to admit we do not have all the answers.  You must confront these fundamental questions.  

I wonder why this is arrogant?  I've posted on this before, but it bears repeating: ideas aren't arrogant.  If you follow the evidence where it leads, how is that arrogant?  Furthermore, taking the position that the narrator does, how is his position not arrogant? Saying believers live in the dark ages?  The narrator surely thinks he's in a position of privileged knowledge.

Objection 6: “It's arrogant to think we occupy a privileged place in the universe.”

Isn’t it time to stop thinking that we are somehow the reason why this universe was made? That our culture is somehow better than other cultures? It’s time to learn how the universe really is, even if that deflates our conceits, and forces us to admit we do not have all the answers.

Last point, but it's beginning to sound repetitive: this would be valid if there were no evidence that we do indeed occupy a privileged place in the universe (in terms of importance).  But there is evidence.  Perhaps there is still debate going on about that, but the scientific data are fairly clear: there are only a handful of places and times in the universe that intelligent life could exist (and maybe only one unique time and place).  If that is what the data say, then again: how is this arrogant?

Conclusion

Every point made in this video seems nice on the surface, especially if you are already indoctrinated to the ways of the new atheists.  Especially if you accept without question the assumption that science and reason are on one side, and faith (which is blind) on the other.  But just for a moment examine your assumption.  If there are good reasons to believe in Jesus as Lord, then nothing this video says makes any sense.  There are no salient points made.

A response to "Dear Believer, Why Do You Believe" part 1 of 2

The popular YouTube clip titled, "Dear Believer, Why Do You Believe" is an attempt to show the fallacy of religion and religious belief. The video is very well done, and has a narrator with a soothing voice calmly raising several objections to religion.  There seems to be real power in this video because the objections keep coming, keep piling up, and if the unwitting believer (or unbeliever) watches, it's really easy to be either bewildered by the objections, if the believer is not ready to answer them, or to say, "Yeah, that's so true!" if you are an unbeliever.

The problem is, this clip is filled with logical fallacies and arguments that are easily shown to be false.  In this blog post, I will go through the main objections to religion raised in this video (which also happen to be commonly used by unbelievers) and show how each one of them fails.

Summary of the video

Much of the video is devoted to the question of why a religious person believes.  The implication is that believers have no good reason to believe.  In fact, you only believe because your parents told you so.  The narrator asks, since there are many religions out there, which cannot all be right, then how would a person choose one religion over any other?  The narrator then claims that this means no religions are indeed correct.

The end of the video pays homage to modern science, saying that "now we know better".  The narrator does not blame ancient peoples for turning to fantasy to comfort themselves, but in this day and age, because of the advent of modern science, we should leave those superstitions behind.  The implication is that, any reasonable and clear thinking person will cut him- or herself free from the fairy-tale of religion and realize the "truth": there is no god.

The irony is that, if you are using reason to help form your basis of belief regarding the big questions, such as, "Who am I?", "Why are we here?", and "Is there a God?", you will soon see that this video has nothing logical to say about it.  In fact, most of the objections against religion raised by this video are self-refuting, a hallmark of poor reasoning.

In this series of this posts, I will go through six objections to religion in general (and Christianity in particular) portrayed by this video.  In each case, I will state the objection as I see it being raised by the video (either implicitly or explicitly), paraphrase (not direct quoting, although I will put it in quote blocks) the relevant parts of the video, and then respond to the objection(s).

Objection 1: “You’re only a Christian because you were born in America to Christian parents.”

Is the faith you practice the dominant one within your culture?  Aren't you suspicious that most people adopt the religion of the society in which they were born? Yet remain convinced they've found the one true faith?  Did you know that most people choose it not for reasons, but because they were born into it?  Can it be just an accident of geography?  Did you know nearly all religious devotees believe what they are taught to believe by their parents?

This is the main thrust of the video: religious people around the world have no reason to believe what they believe.  I actually have no idea whether or not the assertions made by the narrator here are true.  Is it really true that most people don't choose their religion for good reasons, but because their parents told them to?  Maybe so.  But this objection to religion commits two fallacies: (1) it is self-refuting (this is the death knell of any argument), and (2) it commits the genetic fallacy.

It is self-refuting because it cannot withstand its own scrutiny.  The implication is that since religious people only believe because they were born in a culture dominated by their chosen religion, then that religion cannot be true.  But the same can be true of unbelievers of any stripe.  You are only a postmodernist because you were born in late 20th-century/early 21st century America.  Or you are only an unbeliever because you were told so by your parents. Yes, I understand that many unbelievers in America (especially young people) grew up in the church, but since there is a trend of young people leaving the faith, I could just as easily say it's a cultural thing for young people to do (and not based on reasons).  Especially if these young people leaving the faith are spouting the same self-refuting objections found in this video.

This objection also commits the genetic fallacy, which says that because a belief's origin is suspect, the belief cannot be true.  But this is incorrect: just because someone's belief in God stems from their upbringing does not necessarily mean the belief is false.  It could be false, but you would have to bring a valid argument against it to show that, not a fallacious one such as this.

Objection 2: “How do you know you have it right?  Have you checked out all the other religions of the world? And if every member of faith feels just as strongly as you do, what are the odds you’re right?”

There are 2 dozen major religions.  Furthermore, did you know there are more than 45,000 denominations of Christianity alone, each claiming to understand ultimate truth better than the rest?  Each member of every faith is just as devoted and sincere and convicted as you? Did you know they also read infallible texts, have airtight apologetics, have experienced miracles, etc.?  Yet, since every religion is mutually exclusive, they cannot all be right, right?  If every member of every faith feels just as you do, what are the odds you’re right?  

In this little tight block of paraphrased-text from the video, the main common objection is that there are also other believers out there; how do you know they're not right, and you're not wrong?  The problem with this objection is clear: just because there are others out there who don't believe as I do, does not mean that what I believe is wrong.  The veracity of a religion, or of any other point of view, is not a popularity contest, in which a position must certainly be false if there are enough people who hold a different position.  This objection has three major problems with it: (1) it is self-refuting, (2) it is internally inconsistent, and (3) assumes the correct point of view is simply a popularity contest.

Before I go into those problems, I would like to commend the narrator for holding a position that is often unpopular among unbelievers, at least among the post-modernist types: that every (or nearly so) religion is mutually exclusive.  No, it is clear that not all roads lead to God.  After all, most of the time, the basic claims of a religion are in direct opposition to the basic claims of other religions.  For more, see this post by Eric Chabot in response to "Dear Believer..."

This objection is self-refuting again because it cannot stand up to its own scrutiny.  Atheism is also a set of beliefs that not everyone holds.  There are many out there whose religious beliefs are just as sincere and convicted as the beliefs held by the atheist. And certainly atheism contradicts most if not all religions.  They can't both (atheism and a given religion) be right, right?  So by this argument, atheism is itself subject to this popularity contest, and if that is the case, hands-down it will lose.  No, in order to advance atheism as a "better" choice than a given religion, you actually have to marshal evidence for it, not just say that there are too many religions out there to pick one for certain.

It is also internally inconsistent.  If we are to take the narrator at his word (and I think this is highly suspect), we note that he makes claims such as, "they all have infallible texts, air-tight apologetics, and have experienced miracles".  First, it is certainly untrue that multiple conflicting religions have air-tight apologetics.  Only one at best can have "air-tight" apologetics.  Furthermore, regarding the infallible texts and miracles, if that is true, that argues far more in favor of at least one religion being correct, or at least that the atheistic worldview is incorrect.  After all, on atheism, there are no infallible texts nor miracles experienced.  What the narrator is doing here is, in a backhanded way, saying that all religions claim to have these things, but they really do not.  If that is true, then yes, let's abandon religion.  But this is a bald assertion that needs to be supported by well-reasoned arguments. No such arguments are put forth here.  Instead, I would argue that Christianity in particular does have air-tight apologetics, making it the clear choice over all other religions, including atheism.  And if Christianity does indeed have air-tight apologetics, where does that leave this argument?  It leaves it lacking any apologetic reasoning of its own.

Finally, it is clear that the correct point of view is not the winner of a popularity contest.  Like I mentioned above, if that were true, then Christianity would be the winner.  On this view, the atheist should "reasonably" abandon his faith in atheism and turn towards the God of the bible.  In fact, this objection is at direct odds with his first objection in the video.  If "truth" is voted in, then Christianity would always be the winner, since people (according to the narrator) choose their faith based on what their parents (or some other meme) told them.  How then would a majority-dominant religion (and thereby the "correct" religion by the narrator's point of view implicit in this objection) ever be overturned?  It's correct because it's popular, and then it's passed on to the next generation as the majority religion by meme.  The final irony with this is that the narrator clearly does not believe that the best belief set is chosen by popularity.  If you watch the whole video, he espouses using reason, testing, and logical thinking to test which worldview is correct.  So, in the end, he answers his own question: "How do you know you have it right?"  The answer is: because of logic, reason, and testing the spirits.

Objection 3: “You're an atheist too, just for one less god than I am.”

I've been told my unbelief is guarantee of missing heaven and going to hell, but whose heaven/hell? Should I, just to be safe, accept God? But whose God?  Given so many options, what are the chances?  Might I be better off wagering on no God rather than the wrong God? What if you’re wrong?  What if not Jehovah, but Allah?  Or Wu-tan?  Or some other god on the other side of the planet you've never even heard of? 

Truth is, you already know what it’s like to be an atheist for all gods but your own.  The way you view them (other people) is the same way they view you.  Every devout Hindu, for example, has embraced his faith for the exact same reasons you've embraced yours. Yet you do not find his reasons compelling, nor do you lose sleep at night wondering whether you’ll wake up in his hell.  Given this, is it so hard to see why some of us just take our atheism one God further?

I am often amazed at this line of reasoning, which I have heard several times before.  It seems so inane to me, I am surprised that people still use it.  It's so baseless, yet frequent, that I often wonder whether there is some subtle point here that I am missing.  If so, I would like to have someone explain it to me more fully.  For now, I will simply critique it as it appears in the video.  This objection fails because the definition of atheism is belief in no god, not belief in one to the exclusion of other (possible) gods.  It also fails because there are actually very good reasons to believe in at least some general deity, which of course rules out atheism.

So, how could one who believes in at least one god (a "religious" person) at the same time not believe in any god (an atheist)?  The entire premise of this objection is absurd in the strictest sense of the word.  It is, by definition, contradictory.  However, I think it might be best to give the benefit of the doubt and simply assume the narrator was using a turn-of-phrase to really call into question how a believer knows that he has the "right" god.

In that case, the narrator is trying to argue that all religions have the same evidential basis: zero.  If that is the case, then he is right starting from Objection 1: we have no reason for choosing our own set of beliefs except that they have been thrust on us.  So then, how can we possibly look down on others in the world that also believe for no reason?

However, this objection falls flat on its face because it only holds water if there are no good reasons for belief in a particular god.  But it actually gets worse for the atheist: not only are there good reasons to believe in the God of the bible in particular, but there are also good reasons to believe in at least some personal, transcendent, omnipotent, omnibenevolent, omniscient god in general, including the cosmological, moral, teleological and arguments, as well as the argument from reason.  These would clearly rule out atheism as a foundation for belief, but not the major monotheistic religions of the world.

Do extraordinary claims require extraordinary evidence? (Part 2)

Summary

Last time I introduced the topic, claiming that miracles do not need miracle-level evidence to support them.  The reason: you cannot assume your conclusion before the argument begins!  That is the mistake that atheists make (perhaps not realizing it) when they state that extraordinary claims require extraordinary evidence.  But like I said last time, you cannot a priori assume that miracles cannot happen; if you do this, there is no reason to have the discussion. 

Below, I show this rather rigorously using Bayes' Theorem, in a combination with a very simple analysis of the prior probability of a miracle happening.  I also put forth the Resurrection as a concrete example.  Some skeptical readers may be disgruntled with some of the numbers I put to things.  But even so, keep in mind that I'm not after some mathematical proof  beyond a shadow of a doubt that a given miracle did occur. No, I am simply showing that the statement "extraordinary claims require extraordinary evidence" is false.  Indeed, you'll find that, even with very conservative assumptions, miracles can be plausibly supported by ordinary evidence.

But readers be forewarned: there is a lot of math below!

The mathematics of Bayes' Theorem

To put things mathematically, Bayes theorem tells you how to evaluate P(A | B): the probability of a proposition, "A", given (i.e., in light of) another proposition, "B".  To see this, note that the definition of such a conditional probability is:

P(A | B) = P(A & B)/P(B).

If we multiply through by "P(B)" then we get:

P(A | B)*P(B) = P(A & B).

Since "A" and "B" are interchangeable in the right hand side of this equation, you can also write:

P(A | B)*P(B) = P(A & B) = P(B | A)*P(A).

Moving P(B) back to the other side:

P(A | B) = P(B | A)*P(A)/P(B).

Let's put some concreteness to this.  Say "A" is the proposition that some miracle happened (for simplicity, call it "M" instead of "A").  Now say "B" is the quite ordinary evidence for said miracle (and call it "E" for evidence).  Now:

P(M | E) = P(E | M)*P(M)/P(E).

In other words, the probability that a miracle happened, given you have evidence for that miracle, is how well the miracle can explain the evidence, weighted by how likely you think the miracle and evidence should occur on their own.  These are called prior probabilities. That is, P(M) is, "What do you think is the probability of a miracle happening before you consider the evidence?"  And P(E) is, "What do you think the probability of the evidence happening is, without considering a miracle happened?"  Since the evidence is "ordinary", P(E) is probably not too small.

But it seems we're stuck with this nasty prior probability, P(M).  Surely the probability of a miracle happening is so small (by definition) that you must then conclude that P(M | E) is never going to be large enough to convince someone, right?  That is, unless P(E) is super-duper small.  Unless the evidence is also extraordinary.  Hence, the conclusion that to prove a miracle you need miraculous evidence.

This is where you have to question your presuppositions. This is where you have to take a second look at what goes into the prior, P(M).

Why you don't need miraculous evidence to prove a miracle.

The probability of any proposition, A, can be split up into two parts contingent on another proposition, B:

P(A) = P(A | B)*P(B) + P(A | ~B)*P(~B),

(where "|" means "given," and "~" is a negation).  In our case, what's P(M), the prior probability of a miracle, M?  It can be split up into two parts; the probability of a miracle happening given that God exists (G) and the probability of a miracle happening given God does not exist (~G):

P(M) = P(M | G)*P(G) + P(M | ~G)*P(~G).

This is mathematically true, and also philosophically sound, because if you are arguing about whether or not miracles are possible (and usually it's an argument between a theist and an atheist), you cannot a priori assume that God does not exist.  That is the basis of the whole argument.  If someone assumes the other person's position is impossible (probability zero), then there can be no discussion.  So you cannot assume P(G) = 0.  You can assume it is small, but not arbitrarily small (as Dawkins attempts to do).

So let's look at the terms that make up P(M).  I argue (and so would the atheist) that P(M | ~G) is essentially zero.  A miracle is not going to happen if God does not exist, because by definition it is an extraordinarily rare or impossible event.  So the second term disappears, and we're left with:

P(M) = P(M | G)*P(G).

I think we can successfully argue that P(M | G) (ie, the probability that the miracle in question would happen assuming God exists) is not vanishingly small. (If we are specifically saying G = God of the Bible, and M = the Resurrection, then I would say P(M | G) = 1.)  So the only way you end up with a zero prior for a miracle happening is if you assume that God cannot exist (ie, P(G) = 0).

Note that this is different from believing God does not exist.  Atheists believe God does not exist, but no self-respecting atheist believes God cannot exist.  If you ask someone whether they think P(G) = 0 and they say yes, the conversation is over.  That person believes that God cannot exist and therefore nothing you say can convince them.  You always must leave room for your beliefs to be falsified.

So P(G) > 0.  If you are talking about any God, then P(G) should be higher than 1/2, since more than half of all people believe in a god of one form or another (and if we're just basing our priors off of low-shelf statistics, which is what you normally do).  Along those lines, if you're talking about the God of the Bible, then P(G) is more like 1/3.  But even if you are speaking to a hardened atheist, such as Richard Dawkins, then P(G) is as high as 1 - 6.9/7 = 0.0143.  I would be willing to make that allowance.

You can evaluate P(M | G) in various ways.  If the god you are talking about has a holy text that claims the particular miracle in question (as in, the Bible and the Resurrection), then I would say P(M | G) = 1 (for all intents and purposes). On the other hand, if you are just attributing some random "miracle" to some random god, then it gets tricky.  For the sake of argument, let's say P(M | G) is a conservative one out of ten.  This means P(M) is rather high: one out of a thousand!  We take insurance policies out against probabilities more remote than that.  We play the lottery on the hope of winning big with probabilities far more remote than that.

If you are still disgruntled at my analysis of P(M), keep in mind that my argument for P(G) is essentially unassailable.  If you deny that argument, you are entering into the land of logical fallacy.  You may question my choice for P(M | G), however.  In that case, let's just leave P(M) equal to "some small number" p.  That way, we can see later what Bayes' Theorem does, and try to agree on a value for "p" later.  (Just remember: you can't make p = 0 a priori.)

Finishing the analysis

Now that we are armed with the knowledge that p can't be equal to zero, let's go ahead and complete our analysis using Bayes' Theorem.  We last left Bayes Theorem as:

P(M | E) = P(E | M)*p/P(E).

We can split P(E) up in the same way that we split up P(M), but this time we will condition it on "M":

P(E) = P(E | M)*P(M) + P(E | ~M)*P(~M),

or,

P(E) = P(E | M)*p + P(E | ~M)*(1 - p).

Putting this back into the equation for P(M | E):

P(M | E) = P(E | M)*p / (P(E | M)*p + P(E | ~M)*(1 - p)).

If we rearrange the P(E | M) factors:

P(M | E) = p / (p + (1-p)*P(E | ~M)/P(E | M)).

Now we see there are two operative variables here: "p" (the prior probability of a miracle; remember, this cannot equal zero), and the following ratio:

R = P(E | ~M)/P(E | M).

This last term is the explanatory power of the miracle.  How much more likely is it that the evidence would have happened if miracle did not occur over if the miracle had occurred?  If the miracle happening makes the evidence more likely to occur, then R < 1.  If the miracle happening makes the evidence less likely to occur, then R > 1.  Obviously, if you are marshalling evidence for a miracle, let's hope you've chosen something that favors the miracle happening, so then R < 1.  How much less than one, of course, depends on the particular miracle you are investigating and what evidence you have for it.

At any rate, this gives us a very simply formula for the probability of a miracle happening given the evidence at hand:

P(M | E) = p / (p + (1-p)*R).

To make things concrete, let's choose a specific example.

A concrete example: the Resurrection 

Let's now suppose the miracle, M, we're investigating is the Resurrection.  Of course Habermas and Licona have presented five "minimal facts" that most biblical scholars, skeptical and conservative, agree upon in regards to the events surrounding Jesus' death.  However, since this is an internet blog and not a scholarly work, and since that means there are many hyper-skeptics out there who might visit and comment on this blog, let's just stick to one piece of evidence: the conversion and martyrdom of the early church fathers.  And, let's just stick with two church fathers: Paul and James.

In this case, the probability P(M | E), what we wish to find, is the probability that the Resurrection occurred, given the (ordinary historical) evidence that Paul and James were both converted from hostility/skepticism and were both martyred for their beliefs (never renouncing the claim that they saw the risen Lord).

The probability p = P(M) = P(M | G)*P(G), where P(G) is the probability that the God of the Bible exists, and P(M | G) = 1.  So your prior p is essentially how likely you think it is that the God of the Bible exists.  Remember: the world's most famous atheist says this is about 0.01 (more or less. I admit that if he were interviewed, he might say the particular God of the Bible is even less likely to exist than any random god, but then why does he spend so much time focused on Christianity?).

So the question boils down to: what's R?  In words, R is the probability that Paul and James would die for their beliefs given the Resurrection did not happen, divided by the probability that the they would die for their beliefs given the Resurrection did happen.  It seems extremely, extremely, extremely unlikely to me that, if the Resurrection did not happen, then both Paul and James would be converted and go to their deaths proclaiming Christ is King.  But if it is even just 10 times more likely for the Resurrection to make more sense about either man dying for Jesus, then R = 0.01, and we get:

P(M | G) = 0.01 / (0.01 + 0.99*0.01) > 0.5

Now, you may not agree with me that Paul and James actually did convert and actually did die for their beliefs.  You may not even agree with me that the Resurrection makes more sense of them dying for their beliefs over the Resurrection not happening.  But what you are now forced to accept is that it does not take much in the way of "extraordinary" evidence for even a miracle to be plausible.  Remember, I took the position of the most die-hard atheist and a pitiful collection of historical evidence (compared to the wealth of historical evidence that we do have), and a conservative estimate of what that might mean, and I still am forced to arrive at the conclusion that the resurrection is more than 50% likely.

Conclusion

In these posts (including last time), I have shown, using careful (and conservative) mathematical and philosophical arguments that it does not take extraordinary evidence to back and extraordinary claim.  This is based on the recognition that you cannot assume your conclusion before you start. (That is the only way that you could ever arrive at the conclusion that extraordinary claims require extraordinary evidence.)  Once you give even a little ground to the possibility that God exists, and you must do so for intellectual honesty, you are forced to accept the fact that, given enough ordinary evidence, such as well-attested to historical facts, or personal testimony, that miracles actually can be plausible.  And while it is true that the evaluation of some of these probabilities are subjective, keep in mind this was not meant as a mathematical proof of any particular miracle, only a demonstration that, even with very conservative assumptions, miracles can be plausibly supported by ordinary evidence.