Perhaps the most bizarrely frustrating conversation I've had with a friend has to do with predicting something will occur with 50% probability. It is a waste of time to make such predictions – and my reasoning isn't solely mathematical (it doesn't change e.g. calculations of how well calibrated you are, though, which should raise red flags). It's philosophical.

Let's say you have a prediction, Glurbleflop. It's a totally opaque statement. What's it even mean? How could you even begin to guess how likely it is? Wouldn't your prior have to draw from your background assumption about random statements, e.g. random statements are only true 5% of the time?

This is truly and deeply wrong. Because boolean logic means I can construct a second statement, "Not Glurbleflop". Those two, together, are exhaustive. And since I can't see into Glurbleflop, I don't know if it's a "negative statement" – whatever that would mean – so it really means I should treat these two as equivalent. Or, more precisely, I can choose Glurbleflop between them without the loss of generality.

This is what's meant by saying a 50% prediction contains no information. You haven't even cracked open Glurbleflop and you're already there.

But I hear you exclaim, "I do so have information! It's just all the pro-Glurbleflop information balances precisely with the anti-Glurbleflop stuff! I know the statement, I can see the meaning, develop an understanding, and at the end of the day, it seems like a coin flip!"

But that information, as precious as it may be, is not contained in that 50% prediction. The details are discarded, all the intermediate conditional probabilities ignored, opposing reasoning canceled out when producing a simple percentage prediction. At the end of the day you're saying nothing. If you want to discuss that information, give me the conditional probabilities – if the anti- stuff isn't true or doesn't matter, predict a pro-, and vice versa. But to claim great wisdom and present ignorance is a big stinker.