What does TRUE mean in JUSTIFIED TRUE BELIEF?
We define knowledge as "justified true belief".
Now, my question is what does the term TRUE mean in the formal definition? Why not only "justified belief" is enough? If I have justification for one of my beliefs, why can't I say "I know it"? Why does the belief need to be true?
This is slightly tricky as not everyone uttering that may have the same conception of truth, but generally speaking I think the definition only makes sense for some external/correspondence notion of truth.
As a prototypical example (although modern philosophical epistemology has mostly abandoned this viewpoint), in "standard" epistemic logic an agent can only know facts that are true in some external sense, because Kax → x, i.e. if agent a knows x then x is true from the logic's (external observer) standpoint. In contrast, the agent may believe facts that are not true in that sense. In formal terms, Bax does not imply x. Almost universally in these (logic) settings Kax implies Bax but not vice-versa.
As it's usually noted in contemporary texts, epistemic logic has been far more successful in computer science. In a multi-processor, unified main-memory setting for example, there is no dispute as what it means that location (bit) x in main memory is true, whereas a processor may or may not know this, depending on when x was last written relative to when a processor read it.
To give you a translation of that idea in a less technical setting: if Alice hides a treasure in some spot in the woods, but Bob follows Alice without her noticing him and Bob digs up the treasure, then after that point we may say that Alice believes the treasure is still buried at that spot but [we know] it's not true that the treasure is still buried there. You could argue that from Alice's standpoint, she still "knows" that fact to be true, but in the approach taken in epistemic logic, the fact is not true anymore from an external/objective standpoint. So in that strict (logic) framing, you cannot express that she knows something that turns out to be false (from an objective/external standpoint), but only that she believes something like that.
In a broader philosophical context, this kind of simple/strict theory of truth is obviously problematic. E.g. in epistemic reliabilism:
a particular justified belief may be false; however, its method or mode of acquisition must in general lead to true convictions.
(Emphasis mine. Quote from V.F. Hendricks Mainstream and Formal Epistemology.)
For example (in a continuation of the Alice-like example), you know where you [last] put your various things (e.g. in your house), and [thus] normally expect to still find them there. So on a usually reliable basis, you can claim you know where your things are.
Seeing the other answer, I'll add here that the Gettier problem isn't that much related to the def of truth in JTB but with the issue of distinguishing justification from luck. (Various reliabilitists have claimed to have solved/addressed that too, but by additional means which differ from merely adopting a reliabilist standpoint, although it's hard to say that any consensus has been reached on that.)
It might be easier to think about this in terms of the meaning of the word ‘know’. In that case, the ‘true’ part of JTB amounts to the following claim:
If S knows that p, then p
For example, if I tell you that Ann knows that today is Sunday, you can deduce that today is Sunday.
Why isn't justified belief enough for knowledge? Suppose that Bob is about to go outside and he justifiably believes that it's not raining (say he checked the weather 5 minutes ago). But suppose that what he believes is false – it has just started raining. As he is about to go out, we can say: Bob doesn't know that it's raining. If what we say is correct, as it seems to be, then ‘S justifiably believes that p’ does not entail ‘S knows that p’.
By the way, the JTB theory has been refuted long ago. See for example the Gettier problem.
"Justified" and "belief" already hinted our knowledge innately dooms to have some nuanced subjective nature. Thus without the only remaining "true" requirement, there won't be any common objective ground to form our intersubjective sharable knowledge. After all, no matter what definition knowledge becomes, it's meant to be quite stable and sharable...
As Fizz correctly points out, this is an expression of the correspondence theory - and external realism - at the base of the JTB theory of knowledge: When we say that we do know something about the world, we essentially claim that it is actually true that this is the case in the world, not merely that we are justified in thinking that this is probably the case. And this claim is void if it turns out to be false, ie. we commonly would not say that someone knew about something if they made a false claim.
For example, it would be odd to say that Einstein knew that nuclear energy cannot possibly be used in 1934. He just did not know better than to make a false claim, no matter how well-justified it was in 1934, made by one of the leading scientists in the field no less.
As the corresponding SEP entry puts it:
Truth is a metaphysical, as opposed to epistemological, notion: truth is a matter of how things are, not how they can be shown to be. So when we say that only true things can be known, we’re not (yet) saying anything about how anyone can access the truth.
In other words: We can reason and build justified beliefs about the world as much as we like, if the world does not follow our reasoning, we can hardly say our claims were expressions of knowledge, since knowledge is about something beyond our own concepts, beliefs, and reasons. It is that which the T condition tries to capture.
The remaining epistemological problem, of course, is that we can, in turn, only be justified in our belief that something is true, and never be absolutely certain when it comes to empirical knowledge (as we know since Hume and Kant). Every piece of historical and contemporary knowledge can turn and for the most part has turned out to be false and, hence, not actual knowledge about what it claimed to be about, but a 'mere' representation / model/ conceptualisation of the object instead. That is why knowledge is, within JTB, to be understood as a claim about the world proper (from within our current representation /model /conceptualisation of this world) which has to be falsifiable since only if it can be tested and turn out to be false, it is more than dogmatism.
This absolute/metaphysical notion of truth is dubious when it comes to a more pragmatic take on the subject, see eg. Sellars contra Chisholm, or Putnam's critical work against naïve/external realism. These pragmatic objections led to people questioning metaphysics in general since there could be no philosophy beyond epistemology, but that's how it should be understood within the JTB theory of knowledge, which is, at its core, foundationalist as this answer of mine indicates (Chisholm is the knight champion of 20th century foundationalist epistemology).
Supposed I find a discarded card from a fortune-telling machine that says, "Your lucky lottery numbers are 1 2 3 17 23 42." I am an expert in probability and I believe, based on all available evidence, that lottery tickets are a bad investment, so I shouldn't buy one. Later, I check the lottery results and find that, with those numbers, I would have won.
My belief that the lottery ticket was a bad investment was justified, in the sense that on all logic and evidence pointed in that direction. But my belief turned out to be untrue.
In these circumstances it might be reasonable to say that I had a justified false belief that does not count as knowledge.
"Justified" and "true" are not the same thing. "Justified" means that it's reasonable to believe it, not that it's true. False beliefs can be justified under the correct circumstances, and it's even possible for true beliefs not to be justified.
A famous (and perhaps controversial) example of this is Democritus's proposal for the atom. When first proposed, it was a highly speculative idea (because he couldn't possibly verify it experimentally at that point); it obviously ended up being correct, but it was a long time before scientists were able to fully support the idea with evidence. That being said, it's much more reasonable to believe that everything's made up of atoms now than it was when the idea was first proposed (because we have much more evidence that it's true).
On the other hand, the fact that something ends up being false doesn't automatically mean that it was irrational for people to believe it at the time. There were all kinds of ideas throughout history that appeared to be supported by evidence but later ended up being false.
