Reading an earnest and thought provoking editorial₁ from one James Wood, reviewing 'Letter To a Christian Nation' by Sam Harris. Though atheist himself, he admits a flagging patience with certain attitudes of atheists. I can concede that an atheist's superior and glib demeanor may be due to frustration and no small amount of pessimistic inference about the human condition, though I had to comment about a rebuttal he gives regarding Bertrand Russell's celestial teapot₂.

He claims that God, so much grander and more complex than a teapot, cannot be banished with such a simplistic comparison, when I would insist that God is actually much less believable than the teapot for that exact reason. I think Russell's teapot is due for an update which is more approachable and grounded. Here goes:

I claim that there is a discarded Coke can somewhere in the vastness of the Sahara, but I will brook absolutely no discussion about doubting my claim or investigating it for veracity. "Okay," you think, "I suppose I can assume that much to be true. Whatever this man's sources, the odds of a Coke can being somewhere in the desert must be considerable." But I then elaborate with claims that it's actually many, many cans, folded into glorious and artistically pleasing forms, and my obdurate refusal to discuss how it can be proved continues. At this point even the most generous theists would likely start getting annoyed with my odd behavior, yet at the very least what I'm asking you to believe isn't outside the realm of possibility. For all you know (though I refuse to allow you to check) there could be a folk art bazaar currently set up in the Sahara, so really it costs you very little to entertain my view.

And then I say that the cans have taken on beautiful, shimmering consciousness and are forming a society which hides from humanity, burying their chrome castles beneath the sand and moving their aluminum cities whenever we get too close to discovering them. "But..." you try to cut in. Before you can even begin to tell me what you find odd about my fantasy, I'm on the next detail. I claim that all of our major technological achievements of the last several hundred years are all thanks to the secret influence of the Shiny Can People.

Now you have countless legitimate doubts, but every time you try to tell me that, for starters, soda didn't even come in aluminum cans several hundred years ago, I insist that you weren't there so you can't be sure, and how could a mere burden of proof destroy the mighty empire of the Shiny Cans?

I like the utility of the can people because it doesn't start with an outlandish proposition, but if you stick around it gets absolutely ridiculous. Not only does that remind me more of how religion is actually sold, but it also serves to strengthen the original analogy of the teapot by reminding the curious mind that Russell's teapot is infinitely smaller and less complex than God, making it much less embarrassing to genuinely believe in since it would have so much more room to hide.

Odinn Celusta

1) http://www.newrepublic.com/article/the-celestial-teapot

2) http://en.wikipedia.org/wiki/Russell's_teapot

In this post (based on results from MIRI's recent workshop), I'll be looking at whether reflective theories of logical uncertainty (such as Paul's design) still suffer from Löb's theorem.

Theories of logical uncertainty are theories which can assign probability to logical statements. Reflective theories are theories which know something about themselves within themselves. In Paul's theory, there is an external P, in the meta language, which assigns probabilities to statements, an internal P, inside the theory, that computes probabilities of coded versions of the statements inside the language, and a reflection principle that relates these two P's to each other.

And Löb's theorem is the result that if a (sufficiently complex, classical) system can prove that "a proof of Q implies Q" (often abbreviated as □Q → Q), then it can prove Q. What would be the probabilistic analogue? Let's use □_aQ to mean P('Q')≥1-a (so that □₀Q is the same as the old □Q; see this post on why we can interchange probabilistic and provability notions). Then Löb's theorem in a probabilistic setting could:

Probabilistic Löb's theorem: for all a<1, if the system can prove □_aQ → Q, then the system can prove Q.

To understand this condition, we'll go through the proof of Löb's theorem in a probabilistic setting, and see if and when it breaks down. We'll conclude with an example to show that any decent reflective probability theory has to violate this theorem.

This post is a minor note, to go along with the post on the probabilistic Löb theorem. It simply seeks to justify why terms like "having probability 1" are used interchangeably with "provable" and why implications symbols "→" can be used in a probabilistic setting.

Take a system of classical logic, with a single rule of inference: modus ponens:

From A and A→B, deduce B.

Having a single rule of inference isn't much of a restriction, because you can replace other rules of inference ("from A₁,A₂,... and A_n, deduce B") with an axiom or axiom schema ("A₁∧A₂∧...∧A_n → B") and then use modus ponens on that axiom to get the other rule of inference.

In this logical system, I'm now going to make some purely syntactical changes - not changing the meaning of anything, just the way we write things. For any sentence A that doesn't contain an implication arrow →, replace

A with P(A)=1.

Similarly, replace any sentence of the type

A → B with P(B|A)=1.

This is recursive, so we replace

(A → B) → C with P(C | P(B|A)=1 )=1.

And instead of using modus ponens, we'll use a combined Bayesian inference and law of total probability:

From P(A)=1 and P(B|A)=1, deduce P(B)=1.