Mathematics and memorization:
Under K=JTB:
If you were taught to memorize Planck's constant: "6.62607004 × 10-34 m2 kg / s" by a professor of physics, you would be justified in believing that it's true. But do you have any idea what it means? And if not, do you really know that 6.62607004 × 10-34 m2 kg / s?
With K=UTB (Understood True Belief) you would have to understand the mathematics/physics used in this equation in order to claim knowledge.
Consider the famous Gettier example:
Smith has applied for a job, but, it is claimed, has a justified belief that "Jones will get the job". He also has a justified belief that "Jones has 10 coins in his pocket." Smith therefore (justifiably) concludes (by the rule of the transitivity of identity) that "the man who will get the job has 10 coins in his pocket."
“In fact, Jones does not get the job. Instead, Smith does. However, as it happens, Smith (unknowingly and by sheer chance) also had 10 coins in his pocket. So his belief that "the man who will get the job has 10 coins in his pocket" was justified and true. But it does not appear to be knowledge.”
With K=JTB it could and has been said that Smith has knowledge. He’s justified in his belief and coincidentally correct. However can it be said that has understands why the truth is true? If he does, then he has knowledge. If he doesn’t then he does not.
The Gettier case is also flawed, in my opinion, because it asserts knowledge about the future. There is no point in time where the statement “Jones will get the job” is true. This is because by the time Jones get’s the job it would be meaningless to say “Jones will get the job” and that statement during that time will have no truth value. The sentences “Jones got the job” and “I believe Jones will get the job” both have truth values and can’t very well be rephrased as Gettier cases.
Knowledge via authority:
Consider someone who’s confronted by Pythagoras and Anaximander around 530 BC. Pythagoras tells this person the earth is round, Anaximander says it’s flat. The person ultimately sides with Pythagoras although he felt both sides seemed almost equally as likely. Does this person have knowledge?
K=UTB would say “no.” Because he doesn’t understand why the Earth is flat, he just believes something an authoritative figure claimed, even though he could have just as easily believed Anaximander.
Knowledge via assumption:
Consider a man named John who plans on attending a basketball game with his friend. He tells his friend that everyone on both teams will be wearing Nikes. He knows that based off statistics this is most likely the case, and so is entirely justified in his belief.
However at the last second he has a family emergency and has to rush back home. A basketball player, Kobe, however planned on wearing a pair of Reeboks to honor his late grandfather (who also wore Reeboks) but accidently left them at his house (mansion). He borrowed a pair of his teammates Nikes instead.
Although John’s assumption ended up being right and he was justified in believing it, it hardly seems like anything predictive or assumptive can be considered knowledge unless it's grounded in a priori necessity.
