I've seen it claimed that Theists and scientists alike work off Revelation, requiring things to be revealed. Forgive me if I'm leery of sentences that use a word once (especially a Capitalized One), and then essentially (re)define it in a later clause, but I've seen many trains driven through the logical holes created by such sentences.
The revelations (I'll use small 'r' here) that scientists require are of a rather particular and restricted nature which, because of its restrictions, benefits from a relevance not shared by personal, divine, transcendentally introspective, or Theistic Revelation (big 'R'). Big-R Revelation can be privately accessible to only those special few to whom it is Revealed. Big-R Revelation can be contingent on holding certain beliefs, conducting your life in certain ways, or somehow getting yourself into the appropriate relationship with the appropriate supernatural entities. Big-R Revelations can be incorrect (as you say: there can certainly be "false revelations"). There's a lot of things that are able to be big-R Revelations, and the more different things something can be, the less you can say about it.
Small-r revelation... the stuff which scientists depend upon to be revealed in the course of their inquiries... is terribly constrained by comparison. Small-r revelations cannot depend on your beliefs or supernatural relationships. Small-r revelations must be publicly accessible, in the sense that any criteria one must satisfy to observe the revelation are objectively verifiable ("you must have access to a 520nm laser" is ok, "you must be in a truly prayerful state of mind" is not).
There's a lot of things that aren't able to be small-r revelations, which means there's more that can be said about them. In particular, by virtue of their publicness, we can distinguish whether something is a true revelation or a false Revelation (i.e. "Yes, yes, we're sure that it has been Revealed to you that by mixing soap and ballpoint pens with a potato masher, you initiated Cold Fusion. Please forgive us for not funding your research after it turned out your discovery could not be repeated when anyone aside from yourself was able to observe the process.")
In that regard, at least, I agree with the point that "it is not impossible to distinguish between one revelation or the other as being true or false." That only holds for small-r revelations, though... the kind that, yes, scientists work off.
Big-R Revelation, though, the kind that scientists don't use (except in the same manner as daydreams, science fiction, and off-the-wall questions from their occasionally surprising mother-in-law: as inspiration for a direction of inquiry) and which a number of Theistic positions depend upon, does not have the benefit of having a way that one can distinguish between true ones and false ones.
Or, rather, there are Revealed ways to make such distinctions, and naturally those Revealed ways confirm themselves to be true Revelations, but they also are confirmed to be false Revelations by other such Revealed ways. If, contrary to what I suspect you believe, the ways which had been Revealed to you (or those whose authority you submit to) for distinguishing between true and false Revelations were actually false... how would you discern that? How could you discern that?
This also relates to moral relativism or "agreeing-to-disagree". Consider yourself fortunate if you haven't heard anyone say something like "well, there's what's true (Revealed) for me, and what's true (Revealed) for you, and they're not the same thing, so the best we can do is live and let live, and agree to disagree." Statements like that can only be based on big-R Revelations, which can differ from person to person, and without any non-Revealed means for anyone to discern the truth or falsity of a Revelation, you really are stuck treating them all on equal footing.
Well, except for things that have been Revealed to you, of course, because even though those may be false Revelations, it's been Revealed to you that they aren't. And, even though that may have been a false Revelation, it's been Revealed to you that it isn't. And, even though that may have been a false Revelation, it's been Revealed to you that it isn't. And, even though that may have been a false Revelation, it's been Revealed to you that it isn't. And even though it's clear that you never actually can get into the clear with this line of reasoning, most people will stop after finitely many steps and conclude that, yep, their Revelations really are true and are have been adequately confirmed.
Being stuck treating almost all Relevations on equal footing is where one becomes mired in the neighborhood moral relativism, since you're stuck dealing with the situation that, even if not all claims are true, most claims are equally true, and that's a bad place to be in and a hard one to get out of.
But that's the price of giving big-R Revelation a role in arbitrating truth.
Comments to the effect of "one can similarly point to Theists in regards to the truth of an immaterial reality", are not unwelcome comment, because it's a false similarity that runs afoul of the difference between little-r revelation and big-R Revelation.
Scientists, confined to little-r revelation, have a limited ability to disagree. Now, I'm not saying they don't disagree! I wouldn't deny some of the terrible and unfortunate rows in the history of science, nor would I sweep under the rug the incidents where conflicting positions benefited from a dialectic resolution that led to understanding superior to either initial point of view. That said, the publicness and commonality of little-r revelations does put a severe damper on the extent to which honest scientists (be they be Muslim or Infidel) can disagree.
Theists, on the other hand, manage to generate more and more disagreements as time goes on, thanks to their centering on big-R Revelations. Whether it's how the Catholic church isn't orthodox while the Orthodox church isn't catholic and the Mormons are Christians who aren't Christian (depends on which Christian Theists with sufficient training and experience you ask), or it's how the truth of immaterial reality that the sufficiently trained and experienced Hindus have a consensus on fails to match up with the truth of immaterial reality that the sufficiently trained and experienced Jews have a consensus on, it's decidedly not the case that there is consensus among all trained and experienced Theists as to the truth of immaterial realities.
So, mentions of consensus, Theists, and immaterial realness do well to highlight a corollary to my objection to big-R Revelation: while small informed consensus can be useful for approaching truth in the context of little-r revelation, it becomes useless in the context of big-R Revelation. Since, as I said before, searching for truth is important to me both in itself and as a necessary path to discerning "The Good", I'm loathe to see potentially useful truth-seeking tules rendered useless!
Shifting gears.
In bringing up trustworthyness, I admit that irrationality does not prevent someone from holding a true position: there's an... interesting... character in this area who is known for ranting down streets spouting general nonsense punctuated by the occasional exclamation of "...and therefore Lincoln is DEAD!" It's a rather disconcerting tirade but, all other things aside, I can't fault the correctness of his conclusion.
In his public proclamations, at least, the guy is certainly irrational. With the exception of loaning him money (you can't give him money unless you're ready to avoid him henceforth, he WILL remember you and eventually return anything you send his way), it's pretty safe to not consider him trustworthy... at least as far as communication goes. I'm not under the impression that his irrationality or the untrustworthiness of his reasoning are grounds to claim that Lincoln is still alive and well, and that neither age nor gunshot wounds have slowed him down much at all.
All that said, if he exclaimed about one of those American presidents that most Americans never learn about (let's say Wuggie, for example) and you were an American like most, I think you'd be wise to take the position: "I don't know who Wuggie is, whether Wuggie even exists or not, and given existence, whether Wuggie is dead or not... and a proclamation that Wuggie is real and really dead that comes from an irrational and factually untrustworthy person does not serve to persuade me of any such things."
In discussions about atheists accusing Theists of irrationality, it seems that sometimes the concerns is less with how Theists related to what they thought was true, and more with how atheists relate to what theists think is true. As such, while irrationality and untrustworthiness would not preclude a Theist from having randomly stumbled upon the truth, such conditions would preclude other people from taking them seriously. Not knowing whether they've actually got the truth or not, and knowing that a broken clock is more than twenty thousand times more likely to be wrong than to be right, if you're going to leap to any conclusion other than "I don't know", it's wiser to conclude that the irrational person is wrong, rather than right.
Changing gears again.
Stage magic such as sawing lovely assistants in half, which step far outside the constraints that reason imposes on truth, can be properly distinguished from the truth. For those not steeped in Revelations of the truth of ESP, mind-reading tricks can be distinguished from truth as well, despite the senses and intuition continuing to insist that there's not room for other explanations. Hidden-information legerdemain tricks, however, can only be properly distinguished from the truth by A> cracking the trick, or B> running it enough times in a controlled manner to demonstrate that the appropriate statistics are wrong. A> is obstructed by taking advantage of known unreliabilities in perceptions, and B> isn't relevant to this issue. Unless I've accidentally constructed a false dichotomy, this is a setting where, no, you can't properly distinguish the illusion from the truth.
Gear shift.
What is entailed by 'falsifiability'? 'The truth' is, unless you subscribe to irrationality, necessarily not falsifiable. There aren't just certain truths about the world which are not falsifiable: if you think you can find a truth which is falsifiable, I'll show you that what you've got isn't a truth.
No, it's not truths which anyone ought to try to falsify. That's a fool's errand. It's assertions of truth. It's claims of truth. There are certain claims which are not falsifiable. Lots of them, actually. Solipsism's a perennially annoying one. Claims that the world came into existence half a second ago, and everything burst into being in such a way that there appears to be a historical record and memories and, in general, a "past". Assertions that every time someone pats a kitten, four invisible intangible green monkeys who are powerless to interact with anything in any way spring into existence. These are claims about what is true, and they're not falsifiable.
Yes, of course the monkey thing could be true and, if that were the case, you couldn't prove otherwise. That's not the complaint, though. The issue is that we have no way (not even in principle) of knowing whether the monkey-claim is correct or not.
One can certainly say something doesn't have to be falsifiable in order to be true... and that's weaker than could be said. One can even say that something must not be falsifiable in order to be true, since what is true is (to those indulging in rationality) not false. If you want to say that a claim doesn't have to be falsifiable in order for the claim to be correct... yeah, you can have that one too. The negation of the kitten-monkey claim is equally unfalsifiable, and that doesn't mean that such monkeys do spontaneously manifest, any more than the original claim's unfalsifiability means they don't.
The problem is that whenever you have a position which depends on the truth or falseness of the content of a claim, and that claim's correctness with respect to the world cannot, even in principle, be established, then you're saddled with a position whose relevance to the world cannot, even in principle, be established. You can claim it's relevant until the cows come home, but there's no possible way to demonstrate that it actually is, because such a demonstration would serve to confirm or falsify the unfalsifiable claim, and that's not an option (it violates the assumption that the claim is unfalsifiable, and unless you're irrational, that's not a tenable situation.)
In other words, if unfalsifiable claims form part of a foundation, then everything that truly depends on them for support can have no impact on any aspect of the world, in any way, whatsoever. They cannot make anyone's lives better, or make anyone happier, or lead to more peace, or endow every young girl with a kitten that she wouldn't get anyhow, because if they did, those differences would serve to make the unfalsifiable claims falsifiable.
In short: Unfalsifiable claims can't help to justify a set of beliefs, because the truth or falsity of what they assert is necessarily irrelevant.
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