I’ve recently been reading about a rationalist blogger who converted to Catholicism. She may have assigned subjective probabilities like: P(\text{Objective Morality}) = 0.9 \ P(\text{God}) = 0.05Then she may have introspected and come up with: P(\text{Objective Morality | God}) = 0.99 \ P(\text{Objective Morality | No God}) = 0.02 \text{ }We can calculate: P(\text{God|Objective Morality}) = \frac{P(\text{Objective Morality|God}) \times P(\text{God})}{P(\text{Objective Morality}) } = \frac{0.99 \times 0.05}{0.9} = 0.55 \ P(\text{God|No Objective Morality}) = \frac{P(\text{No Objective Morality|God}) \times P(\text{God})}{P(\text{No Objective Morality}) } = \frac{0.01 \times 0.05}{0.1} = 0.005Abbreviating Objective Morality by "OM", and "God" by "G", this state of affairs is inconsistent, because we intuitively see that: P(G) \neq P(G|OM) \times P(OM) + P(G|¬OM) \times P(¬OM)To resolve it, she could either increase her subjective probability of there being a God P(\text{God})= P(\text{G | OM }) \times P(\text{OM})
- P(\text{G | ¬OM }) \times P(\text{¬OM }) = 0.55 \times 0.9 + 0.005 \times 0.1 = 0.4955 \approx 0.5or she could reduce her probability of there being some kind of objective morality P(\text{OM})= P(\text{OM | G }) \times P(\text{G})
- P(\text{OM | ¬G }) \times P(\text{¬G }) = 0.99 \times 0.1 + 0.02 \times 0.95 = 0.118She could also reconsider P(God|Objective Morality) or P(Objective Morality|God). Anyways, I find myself very confused by this state of affairs. Is this a solved question? Is there a purely principled way of resolving this which only takes into account the 4 numbers P(OM), P(G), P(OM|G) and P(G|OM)? Is there a standard way of using some kind of metaprobabilities?
I would make explicit that her beliefs about her subjective probabilities are inaccurate observations of her implied underlying logically omniscient, consistent belief system. She can then assign each possible underlying consistent belief system a probability, and update that assignment once she realizes that some of the possible systems were not consistent. What this comes out to is that whether she should update her belief in God or Objective Morality comes down to which of her beliefs she is less certain about.
The 4 given probabilities are actually perfectly consistent within the equations you are using. It is provable that whatever 4 probabilities you use the equations will be consistent.
Therefore the question becomes "where did my maths go wrong?"
P(G|OM) = 0.055, not 0.55
I’m pretty confident that the only way probabilities can actually be inconsistent is if it is over constrained (e.g. in this case you define 5 relevant probabilities instead of 4). The whole point of having axioms is to prevent inconsistencies provided you stay inside them.
P.S. Good job on noticing your confusion!
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0.9 = P(Objective Morality) ≠ P(God) * P(Objective Morality | God) + P(No God) * P(Objective Morality | No God) = 0.05 * 0.99 + 0.95 * 0.02 = 0.0685. That’s inconsistent, right?
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Argh,you’re right,I didn’t check that one. P(OM) cancels on the P(G) equation so that one isn’t over constrained.
However for the equation for P(OM) 4 variables is over constrained, 3 is enough.