P
Perplexity
Guest
As per a request from a different thread, here is a general outline of one of the strongest arguments for atheism I know of.
Before getting into the argument, it’s necessary that I go over a little Bayes’ theorem first.
Bayes’ theorem is a method by which we can discern how probable a position is, given some piece of evidence.
The probability I’m speaking about is a numerical representation of your degree of confidence that a proposition is true or false. We represent our highest degree by 1.0, our epistemic indifference with 0.5 and our least degree with 0.0. This is a subjective thing, but that’s not a problem: we numerically represent subjective things like pain without worrying about the subjectivity. i.e., ‘How much pain do you feel on a scale of 1-10?’ etc.
So, the theorem asks us to discern which numbers we’ll represent our confidence with in certain propositions, it takes these numbers, plugs them into a formula, and calculates the probability we want, namely, how probable a position is given some piece of evidence.
I’ll explain which propositions the formula needs us to discern first, and some notation we’ll need:
H stands for [Hypothesis]. Hypothesis here just means the position we’re discerning the probability of. It could be ‘I like toast’ or ‘It’s cold’ etc.
Our background knowledge is all the things we know and believe, it’s represented as K, or sometimes B.
E is used to represent some proposed evidence. So, perhaps I take my pleasure derived from eating toast to be evidence [E] for the hypothesis [H] that ‘I like toast.’
E must always be ‘cut out from’ K, i.e., K stands for all your knowledge and beliefs minus whatever E is.
P stands for ‘the probability’ and | stands for ‘given.’
So, P(H|E&K) means ‘the probability of H given E and K.’ P(E|H&K) stands for the probability of E given H and K.
Now, on to the relevant probabilities.
There are the prior probabilities (or priors for short). These are how probable a hypothesis is given your background knowledge alone. They’re represented as ‘P(H|K).’ So, the probability that I ate recently [H], given that I feel full [a piece of knowledge in K] is quite high.
The posterior probability (or posterior for short) is how probable a hypothesis is given the proposed evidence and your background knowledge. It’s represented as P(H|E&K).
Finally, you’ve got the likelihood term: P(E|H&K) which is just how likely E is, given H and K. Does supposing H lead us to expect E? Or, would E be surprising given H?
Bayes’ theorem, then, is the following:
P(H|E&K) = [P(H|K) * P(E|H&K)] / [P(H|K) * P(E|H&K)] + [P(~H|K) * P(E|~H&K)]
Now, ‘[P(H|K) * P(E|H&K)] + [P(~H|K) * P(E|~H&K)]’ is P(E|K). So sometimes, the theorem simplifies and says: P(H|E&K) = P(H|K) * P(E|H&K) / P(E|K).
Here’s what it could look like in practice:
Suppose H = It’s cold, ~H = It’s not cold, and E = I’m shivering.
Suppose the following probabilities:
P(H|K)=0.8
P(~H|K)=0.2
P(E|H&K)=0.9
P(E|~H&K)=0.3
Bayes’ theorem takes these and plugs them in as follows:
P(H|E&K) = [0.8 * 0.9] / [0.8 * 0.9] + [0.2 * 0.3] = 0.92.
Before getting into the argument, it’s necessary that I go over a little Bayes’ theorem first.
Bayes’ theorem is a method by which we can discern how probable a position is, given some piece of evidence.
The probability I’m speaking about is a numerical representation of your degree of confidence that a proposition is true or false. We represent our highest degree by 1.0, our epistemic indifference with 0.5 and our least degree with 0.0. This is a subjective thing, but that’s not a problem: we numerically represent subjective things like pain without worrying about the subjectivity. i.e., ‘How much pain do you feel on a scale of 1-10?’ etc.
So, the theorem asks us to discern which numbers we’ll represent our confidence with in certain propositions, it takes these numbers, plugs them into a formula, and calculates the probability we want, namely, how probable a position is given some piece of evidence.
I’ll explain which propositions the formula needs us to discern first, and some notation we’ll need:
H stands for [Hypothesis]. Hypothesis here just means the position we’re discerning the probability of. It could be ‘I like toast’ or ‘It’s cold’ etc.
Our background knowledge is all the things we know and believe, it’s represented as K, or sometimes B.
E is used to represent some proposed evidence. So, perhaps I take my pleasure derived from eating toast to be evidence [E] for the hypothesis [H] that ‘I like toast.’
E must always be ‘cut out from’ K, i.e., K stands for all your knowledge and beliefs minus whatever E is.
P stands for ‘the probability’ and | stands for ‘given.’
So, P(H|E&K) means ‘the probability of H given E and K.’ P(E|H&K) stands for the probability of E given H and K.
Now, on to the relevant probabilities.
There are the prior probabilities (or priors for short). These are how probable a hypothesis is given your background knowledge alone. They’re represented as ‘P(H|K).’ So, the probability that I ate recently [H], given that I feel full [a piece of knowledge in K] is quite high.
The posterior probability (or posterior for short) is how probable a hypothesis is given the proposed evidence and your background knowledge. It’s represented as P(H|E&K).
Finally, you’ve got the likelihood term: P(E|H&K) which is just how likely E is, given H and K. Does supposing H lead us to expect E? Or, would E be surprising given H?
Bayes’ theorem, then, is the following:
P(H|E&K) = [P(H|K) * P(E|H&K)] / [P(H|K) * P(E|H&K)] + [P(~H|K) * P(E|~H&K)]
Now, ‘[P(H|K) * P(E|H&K)] + [P(~H|K) * P(E|~H&K)]’ is P(E|K). So sometimes, the theorem simplifies and says: P(H|E&K) = P(H|K) * P(E|H&K) / P(E|K).
Here’s what it could look like in practice:
Suppose H = It’s cold, ~H = It’s not cold, and E = I’m shivering.
Suppose the following probabilities:
P(H|K)=0.8
P(~H|K)=0.2
P(E|H&K)=0.9
P(E|~H&K)=0.3
Bayes’ theorem takes these and plugs them in as follows:
P(H|E&K) = [0.8 * 0.9] / [0.8 * 0.9] + [0.2 * 0.3] = 0.92.