For the simple Raegan example we can let A=“cut federal support to local governments” and B=“Reagan will provide federal support for unwed mothers.”
We compare to just on it’s own and this is clearly lower. Now before I have asked you these questions you have some prior for and based on your entire life up till now, so in reality these priors are and . By asking you this question I change the resulting X to some X’ in a way that isn’t strictly based on the A and B laid out in the question, depending on the person this could alter the probabilities at play.
I.E. I ask you on its own, you reply with where (we combine the information of your life up till now with the fact you’ve just been asked this question). Now I could take the same person and instead have asked and you would get (or the equivalent reversal depending on how the person thinks about things) where . These no longer obey the same inequalities as before. Now we could ask if they should and dive into how priors should be updated but there is a genuine difference in both cases at this meta-level of “why is this person asking me this question” that means there is always some extra information conveyed. Depending on how you think priors should be updated the thinking could still be fallacious.
This universal prior idea I got from Jaynes’ book on probability in science.
For the simple Raegan example we can let A=“cut federal support to local governments” and B=“Reagan will provide federal support for unwed mothers.”
We compare to just
I.E. I ask you
This universal prior idea I got from Jaynes’ book on probability in science.