Bayes’ Theorem and the LCA (p.2)

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Bayes’ Theorem and the LCA

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Some Preliminaries

The theist can say that the reason why something exists rather than nothing is that God is a necessary being he exists by the necessity of his own nature. An atheist could believe the reason why there is something rather than nothing by being a Platonist, thinking that abstract objects like numbers genuinely exist as nonphysical entities independently of the mind. Yet (in my humble opinion at least) it seems more likely that numbers are more like ideas in the mind and have no more real existence then Sherlock Holmes. If there is no explanation for why something exists (including the necessity of a thing’s nature), then the contingent universe is all there is, and the contingent universe just exists inexplicably without an external cause.

Yet “If there is no explanation for why there is something rather than nothing, then the contingent universe just exists inexplicably” is true then “If the contingent universe has an explanation of its existence, then there is an explanation for why there is something rather than nothing.” To simplify it then I’ll use “Why is there something rather than nothing?” and “Why does the contingent universe exist?” more or less interchangeably, since in all likelihood they probably have the same answer: either (1) it exists inexplicably; or (2) the explanation is a transcendent personal entity that is metaphysically necessary.

Bayes’ Theorem and the LCA

Maverick Atheism doesn’t give any reason to prefer “there is no explanation for why the universe would exist” over “there is an explanation for why the universe would exist.” He seems to argue that “there is an explanation” has only a very slight advantage from the evidence of the universe’s existence. Is he right? Here Bayes’ theorem can be of some help. Suppose we use the following symbols:

• H is the hypothesis that the PSR holds true with respect to the universe (there is an explanation for why it exists; a sufficient reason for its existence).
• ¬H symbolizes that it is not the case that the PSR holds true with respect to the universe.
• E is the evidence of the universe existing.

Basically, ¬H says, “there is no reason for why the universe would exist” in the sense that if the universe does exist, it does so inexplicably with no external cause for its existence. One version of the Bayes’ theorem is the following:

Pr(H|E) =	Pr(H) × Pr(E|H) Pr(E)

What is the likelihood that the universe exists if there is a sufficient reason for its existence? That probability is 100%, so Pr(E|H) = 1. Suppose we have a true agnostic about whether there is an explanation for the universe’s existence (i.e. H and ¬H have equal prior probability) but hadn’t yet taken into account the evidential force E has for H. One might be tempted to set Pr(E) as 0.5, but then we’d get this:

Pr(H|E) =	0.5 × 1 0.5	= 1

As the old saying goes, if it sounds too good to be true, it probably is. There’s something wrong with the above equation; a little math shows that for Pr(E) to be 0.5 here Pr(¬H) must be 0, and that’s mathematically impossible if Pr(H) = 0.5. So what went wrong? Getting the right value for Pr(E) is trickier than one might think. We should be careful not to confuse Pr(E) with Pr(E|¬H); it may be true that in the absence of sufficient reason for the universe’s existence the universe is just as likely to exist as not, but that won’t necessarily hold true for Pr(E), because Pr(E) is equal to the following:

Pr(E) =

Pr(H) × Pr(E|H) + Pr(¬H) × Pr(E|¬H)

So by my lights, using the form of Bayes’ theorem where Pr(E) is on the bottom is not advisable here. With that in mind, another form of Bayes’ theorem is this:

Pr(H|E) =	Pr(H) × Pr(E|H) Pr(H) × Pr(E|H) + Pr(¬H) × Pr(E|¬H)

What about Pr(E|¬H)? We’re looking for the likelihood of E given just ¬H. Without an external cause of the universe to make it exist (and the universe lacking necessary existence) we can estimate the likelihood of it existing given just ¬H as 50%. Suppose one starts out as truly agnostic as to whether there is an explanation for the universe’s existence (i.e. both H and ¬H have a prior probability of 0.5) but had not yet considered the evidential force of the universe’s existence; call this person Al. Agnostic Al is using these probabilities:

• Pr(H) = 0.5
• Pr(E|H) = 1
• Pr(¬H) = 0.5
• Pr(E|¬H) = 0.5

And plugging those values into above formulation of Bayes’ theorem gives us this:

Pr(H|E) =	0.5 × 1 0.5 × 1 + 0.5 × 0.5	=	0.5 0.75	= 2⁄3 ≈ 0.67

So it seems that if one considered H and “there is no reason for why the universe would exist” to be equally likely but hadn’t yet taken into account the evidential force of the universe’s existence for H, one should assign H a probability of about 67% in the absence of further arguments. While that probability isn’t overwhelmingly huge, it’s not exactly negligible either. The existence of the universe would constitute some fairly significant evidence for H, in part because one advantage H has over “there is no reason for why the universe would exist” is that on H it is more likely that the universe would exist. So if in the absence of a sufficient reason “nothing at all exists” is just as likely as “something exists” then this provides a significant advantage for the hypothesis that there is some explanation for why there is something rather than nothing, at least for the aforementioned agnostic Al.

The mathematics is airtight; to reject it one would have to reject the input probabilities. I suspect Pr(E|H) would be pretty uncontroversial given the sense of “sufficient reason” being used here, but the other parts of the probability calculations are more vulnerable.

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Bayes’ Theorem and the LCA

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Bayes’ Theorem and the LCA

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A while back the Maverick Atheism blog wrote a rebuttal to the Leibnizian Cosmological Argument (LCA), that was in part a rebuttal to my own series on the Leibnizian cosmological argument (though it was also largely a rebuttal to Christian philosopher William Lane Craig, whose form of the LCA I borrowed).

A Brief Recap

For those who aren’t familiar with Bayes’ theorem, I recommend reading this quick introduction to Bayes’ theorem. To recap my series on the LCA a bit, here’s one of the versions I used (one I labeled LCA 1A):

1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
2. The universe exists.
3. If the universe does have an explanation for its existence, that explanation is God.
4. Therefore, the universe has an explanation of its existence (from 1 and 2).
5. Therefore, the explanation for the existence of the universe is God (from 3 and 4).

Premise 1 then is a version of the principle of sufficient reason. Another version of the LCA, one I labeled LCA 3, goes like this, where the contingent universe is the totality of all contingent things:

1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
2. The contingent universe exists.
3. If the contingent universe has an explanation for its existence, that explanation is God.
4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
5. Therefore, the explanation of the contingent universe is God (from 3 and 4).

I argued that the contingent universe could have failed to exist, and I also argued that the external cause of the universe would have to be an eternal, transcendent, metaphysically necessary, personal entity. Here’s how it worked: The external cause of the contingent universe could not itself be contingent (since then we wouldn’t have an explanation for the contingent universe) and so must be necessary. What is necessary is also eternal, since at no time and in no circumstances can metaphysically necessary entities fail to exist. Since the physical universe itself is contingent, the external cause would have to be nonphysical, and there are only two candidates in the metaphysical literature for nonphysical entities: abstract objects (like numbers) and unembodied minds (like God). But abstract objects can’t cause anything, so the only known viable explanation would be a nonphysical mind. We thus wind up with an eternal, transcendent, metaphysically necessary, personal entity that is the external cause of the universe. I also argued for a more modest conclusion: that the only known viable explanation for the contingent universe is eternal, transcendent, metaphysically necessary personal entity as the external cause of the universe—and that of course sounds suspiciously like theism, enough to make atheism rather implausible if we knew such an entity exists. Hence this toned down version:

1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or an external cause.
2. The contingent universe exists.
3. If the contingent universe has an explanation for its existence, that explanation is is an eternal, transcendent, metaphysically necessary, personal entity.
4. Therefore, the contingent universe has an explanation of its existence (from 1 and 2).
5. Therefore, the explanation of the contingent universe is an eternal, transcendent, metaphysically necessary, personal entity (from 3 and 4).

God is classically a metaphysically necessary being (i.e. he exists in all possible worlds) and the external cause of the physical universe. But if God is metaphysically necessary, his existence would also explain why there is something rather than nothing.

Maverick Atheism’s Rebuttal

Maverick Atheism attacked the first premise and argued that perhaps the universe does not have an explanation for its existence. The position he put forward:

So we can accept that ceteris paribus a worldview that explains e.g. why there is something rather than nothing is better than one that doesn’t, but given the plausibility of physical reality existing eternally without an external cause, the degree of evidential support this provides is rather small.

There is an element of plausibility in this; most of us can at least envisage the physical universe existing eternally with no external cause. But now consider the following scenario, borrowing largely from my first article on the Leibnizian cosmological argument.

Suppose we humans learned of an eternally existing monument at the center of the universe that says, “I, the Lord thy God, am the sustainer of the universe and have sustained it throughout all eternity” (if questions of different languages bother you, imagine further that it displays this message through a kind of mechanical telepathy such that anybody who looks at the monument sees the message in her own language).

Most of us can at least envisage the monument existing eternally and without an external cause for its existence. Still, somehow the monument existing eternally with no explanation for its existence sounds rather implausible here. Why?

One factor is the likelihood of there being a monument like this in the absence of a sufficient reason for its existence (like some intelligent entity being the external cause of the monument’s existence), and intuitively that likelihood is low. Where H is the hypothesis (of there being a sufficient reason) and E is the evidence (the monument’s existence), we might say that Pr(E|¬H) is very low. If you’ll recall my quick introduction to Bayes’ theorem, you’ll know that a low Pr(E|¬H) is a factor in making Pr(H|E) high.

Is the likelihood of there being something rather than nothing extremely low in the absence of a sufficient reason, an improbability akin to the hypothetical monument existing without a sufficient reason? Maybe not. Still, the question, “Why is there something rather than nothing?” is interesting precisely because (in the absence of a sufficient reason) it seems like it could have been the case that there existed nothing rather than something. Something very similar holds with the contingent universe, since the contingent universe is the totality of contingent things and thus “no contingent thing exists” is no less a real possibility than “nothing at all exists.” Similarly, “no physical thing exists” is no less a real possibility than “nothing at all exists.” In this article, I’ll examine the insights Bayes’ theorem has on this possibility of there being nothing rather than something.

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Internet Dating

Like many nerds, I’m a bit like the last dodo bird: very single with approximately equal probability of finding a suitable mate. Fortunately I have access to the internet, but I find it to be of limited use. Some online match-up sites I’ve used:

1. eHarmony.com
2. match.com
3. falseHope.com
4. tooDifficultToFindTheRightOne.com
5. bangYourHeadAgainstTheKeyboard.com