I wrote a wall of text describing a problem I see in many/most RPGs, with an analysis of some of the typical options for addressing it and why they are insufficient, but then decided to skip most of the rationalization and just propose my solution. If y'all find it interesting enough to debate then we can expand on it.
Goal:
Avoid certainty in situations where characters would only have strong hunches, not absolute knowledge, in order to make subsequent decision-making more meaningful. For example, when trying to determine if a prisoner is lying, mechanically represent the fact that even strong indications of honesty/dishonesty might have other explanations. In other words, The player sees the die roll on the table, but the character is still guessing. The goal is to model that.
Proposal:
When using a skill to determine if a character knows something, but that "knowing" it would realistically mean believing, on a success the DM makes an additional secret roll. If this roll produces a 1, the DM gives the player the wrong answer. The die used starts at d4 and goes up 1 step for each 5 points above the DC that was rolled. If the original roll was a natural 20, the DM rolls 2 dice (of the appropriate type) and only lies if both come up 1.
The problem here is that
1. 5e characters are already clowns when it comes to their specializations, let alone things they are not proficient at. Having good stats and an easy DC, you already fail 20% of the time. Now you're saying that an additional ~20% of their successes will be failures, only worse, because they now don't know they failed. At this point I'd just be giving up on skills as a player.
2. Somehow characters who are poor at skills get less bad information.
Might I suggest instead doing the following:
1. Rolling the check in secret
2. If the check succeeds, give the player the correct information and an impression of how sure they are, based on how much the check succeeded by.
2. If the check fails but the d20 roll is below 5, give no information.
3. If the check fails and the d20 roll is above 5, give out incorrect information and base the confidence on the roll of the d20, with 20 being absolutely certain and 6 being just a hunch.
4. If the players have advantage, pick whichever roll succeeds OR the lowest roll. If they have disadvantage, pick whichever roll fails OR the highest roll.
The end result of this will be:
Experts will tend to have correct information, and will tend to be confident in it's correctness. They will be more confident when the correct information is something easy for them. When they are wrong, they will tend to have no information. They will almost never have incorrect information for an easy task, but they may still fail. They will only ever be totally wrong and absolutely confident in their answer at extremely difficult to impossible checks.
Those not suited to a task and with no training will often be wrong, but for easy information will be uncertain of it. As the difficulty of a task more outpaces their expertise, they will be more and more certain that wrong answers are correct.
Those completely unsuited to a task and with no training will often be wrong, and are more likely to think they are right.
Now, this might all look like a lot of mechanics to deal with, but honestly I think that the hardest part of any system like this is coming up with incorrect but believable information to hand out to each player in the first place.