For example, if SBF evading jail would increase your confidence in the statement "The US justice system is wholly corrupt" then SBF being sentenced should decrease your confidence in it.
In your example, according to logic, if X implies Y, then if you don't have Y, you necessarily don't have X. If this were a logic exercise, then not "SBF goes to jail" necessarily implies not "the government is not corrupt."
However, in real life there's no connection between the two.
I imagine you’re remembering that P -> Q does not mean ~ P -> ~Q. That’s right, but you can absolutely get to ~Q -> ~P.
In conclusion, naysayers say he wont be convicted is imying and thus proves that the gov is corrupt. The top comment says he may get a sentence, meaning the government is not necessarily corrupt. Yaysayers say the gov is not corrupt and he will get a conviction iff he is guilty.
This is trivial, but difficult to formalize. Thanks for your correction.
- If the government is corrupted it does not matter if SBF is guilty, he will not go jail. - If the government is not corrupted and SBF is not guilty, he will not go to jail. - Only if the government is not corrupted and SBF is guilty, he will go to jail.
The problem is: There are more factors in life that just a corrupted government and guilt. There are jurys, capable lawyers, incapable DAs, loopholes, you name it.
So in truth we have "(corruption ∨ ¬guilty ∨ X) <-> ¬jail", with X being the unknown. Thus, if SBF does not go to jail, it could be true that the government is not corrupted, that SBF is guilty, but any of the other factors were at work.
I think this is what people are really arguing about: what will be causally relevant for the outcome. Mind you, even a conviction would not convict (ha!) people that the convernment is not corrupt. They'd rather say that somebody did not pay enough, other interests were at work, aliens, and so on.
The truth is that you cannot infer much based on a singular outcome if you do not have extremely good insight into the mechanics behind the outcome. Which is precisely why people rather update priors as a way to build up an evaluation based on statistics over a longer time frame. Quite ingenious, if you ask me.
As others have mentioned, X implies Y does in fact require ~Y implies ~X. I think your example is confusing because "the government is corrupt" means many different things, but you're using it in a rather specific way ("the government is protecting SBF"). The equivalence of `X implies Y` and `~Y implies ~X` is more manifest through the following example
"The government is protecting SBF, so SBF won't go to jail"
"SBF went to jail, so the government wasn't protecting him."You're modelling X and Y as propositions and you're correct about the inference of ~X and ~Y, but Bayesian updating is about degree of belief in those propositions, which your inference is not a claim about.