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If you are a patient person, you might follow these people to the park, snoop around for the invisible beast, try to open communication, maybe test for it scientifically (say you wait for rain and look for a place where rain is not hitting the ground directly). When you see no super-lizard, hear no telepathic voice, and find no evidence of an invisible creature ("it has no corporeal form for the rain to hit," the believers might explain), you will likely loose patience and stop wasting time with the giant invisible super-lizard assertion.
That is not to say that no such super-lizard exists - one may, just like an invisible pink unicorn might exist in your garage (and make itself completely incorporeal when the car pulls in), or Krishna might exist as told in the Hindu stories or Yahweh might exist as told in the Judeo-Christian stories.
However, if there is a lack of reasonable evidence supporting the claims of believers, and if all evidence leads you to think that the dogma, stories and holy books of organized religions (or silly super-lizard cults) are simply stories invented, twisted or evolved by often sincere believers, then it seems reasonable not to take these stories at their word.
Likely at the time the original authors of some holy book wrote about a given miracle, they were interpreting some natural event as having a supernatural cause (this being the common way of viewing many events at the time). Perhaps the tales of Krishna or Jesus or Buddha were based upon some actual historical teachers or cult leaders (whether charlatans or simply sincere believers whose words and actions were altered by the transcriber and altered as the story was passed on). At any rate, these things seem more plausible - given the lack of extraordinary evidence for extraordinary claims - than taking any or all religious dogma at face value.
Of course, this is not the same thing as disproving those stories, disproving that the Judeo-Christian god exists or that the Hindu gods exist. That would be a much taller order. Many people - believer and non-believer alike - point out that a negative existential claim cannot be proven. That is, you cannot prove that something does not exist. This is because it would appear to require searching the entire universe from end-to-end-to-end and showing that every place in the universe was void of that thing - a task entirely out of our hands.
It is relatively easier to prove a positive existential claim, to prove that something exists. For example, to prove that horses exist, you merely need to search until you find one. It might be an impossibly long search if there exists only one horse and it is in the last place in the universe you search, but on average this will not be the case, whereas disproving the existence of something always appears to require the impossible task of comprehensively surveying the entire universe.
However, things may not be that simple. If someone claims that the electromagnetic force exists everywhere in the universe, this is a positive existential claim. Its negative would be that the electromagnetic force does not exist everywhere in the universe. In this case, the positive claim can only be proven by an impossible all-inclusive search of the universe to make sure that there is not some pocket somewhere where the electromagnetic force does not exist. The negative claim, on the other hand, can be proven true by finding just one place where the force does not exist.
This example shows that it is not negative existential claims which are impossible to prove. Rather, the problem with proving propositions such as "unicorns do not exist" or "the Judeo-Christian god does not exist" is not that they are negative existential claims, but that the scope of the claim is so large as to be prohibitive of our testing it.
Thus, the common claim of the inability to disprove the existence of something is really just an example of the problem of induction (described briefly here). We can never be certain of any claim which says something holds at every place or time in the universe because we cannot comprehensively test such a claim. We can only make a limited test and infer conclusions based upon that.
The sun may have risen every morning of my life up until now, but it is entirely possible it will not rise tomorrow (due to an unexpected cosmic event like a black hole tearing open space and absorbing the sun and its light). We simply assume that the sun will rise tomorrow because if we did not make such assumptions we could not function - we would be crippled with the skepticism of possibility which forever haunts any epistemology.
It seems reasonable to accept a claim which might be false ("the sun will rise tomorrow") when the evidence available is completely in its favor and there is no evidence against it. Likewise, it seems reasonable to reject a claim which might be true ("Moses parted the Red Sea") when the evidence available is against it (e.g. all of our knowledge of the laws of hydrodynamics and the non-existence of magic powers). The point here is that a person can deny the literal truth of religious claims such as a deity's existence (considering those stories to be living mythology) without claiming to disprove the claims. This avoids the need for an impossible task like searching the entire universe before denying some assertion.
Of course, some non-believers might argue that many existential religious claims can indeed be disproved without needing the search the entire universe. Someone might say, for example, that the deity Yahweh of the Christian bible can be proven not to exist by logic alone because the idea is self-contradictory (an omnipotent, omniscient and benevolent deity appears inconsistent with the world around us, as argued here). There is no need to examine every nook and cranny of the universe to disprove a negative existential claim if one can show that the corresponding positive existential claim is logically impossible or contradictory.
However, even if a religious myth is in obvious contradiction or simply does not jive with history or physics, that could possibly be because our logic or history or science are imperfect and the myths are actually true. But to give this far-out possibility more than passing concern - to actually change our behavior based upon it - is to invite insanity, since a person would have to base their behavior on an infinity of possibilities, even apparent logical impossibilities, simply because we cannot get around the epistemic regress that leads to skeptical long-shot possibilities. How does a person choose how to act if every mutually-exclusive possibility is to be taken seriously?
Thus, if a person thinks that a religion's claims are human-made (either direct fabrications or else sincere misunderstandings based on an unscientific view of the world back then or else twisting of a story over hundreds of years), then it seems perfectly reasonable to assume (not claim to prove in some absolute sense) that those claims are not true.
If they happen to be true, it is a reasonable mistake since there simply was not sufficient evidence to make the long-shot possibility worth accepting. It seems unreasonable to accept something for which you do not think there is evidence. Granted, a person may be mistaken in any given case because their evidence is incomplete, so there is a need to be careful and open-minded, but it seems that at any given time a person's assumptions about the world should be based on the evidence so far. Judge the evidence of a given religious claim (e.g. "God exists") for yourself, but even if you cannot disprove the claim it still seems entirely reasonable to deny it.
Now there might exist some other thing which one might label "God", based on something other than the mythologies of organized religions. This certainly seems more plausible, but personally the plausibility is in direct proportion to how specific and sure one's beliefs about such a thing is. For as someone starts saying "God is this and this, and has done that and that, and told me such and such, and wants us all to act in this way", it seems to become the same as organized religions (or at least how they were in their infancy - it would remain to be seen whether this person attracts an audience and it becomes an organized religion).
On the other hand, claims that there is something bigger out there, or that something exists beyond or around or within us all - vague spiritual or mystic ideas (as opposed to hard and fast claims of truth) - appeal to me and seem reasonable to tentatively explore when the person considering them is a seeker rather than an evangelical. That is, if there is something worthy of being called "God" (though a term without so much baggage might be in order), I have trouble imagining that it would be something a few people could really come to know directly and fully (unless it was such that it 'revealed itself' in some undeniable way to us all). Any mystery that can be so easily solved does not seem worth calling "God" in the religion-free sense of the term.
And while we may not be able to disprove (or prove) any conception of "God" (or any conception of anything), I suggest it is reasonable to give the most consideration to those which do not seem based on made-up myths (claims that precisely this happened in exactly that way). Thus, in general, it is more worth contemplating "spiritual" ideas (to use another term loaded with baggage) which are put forth tentatively or speculatively by seekers rather than assertions made by those who have already found the Truthtm.
"I have seen several entirely sincere people who thought they were (permanent) Seekers after Truth. They sought diligently, persistently, carefully, cautiously, profoundly, with perfect honesty and nicely adjusted judgment--until they believed that without doubt or question they had found the Truth. That was the end of the search. The man spent the rest of his life hunting up shingles wherewith to protect his Truth from the weather. If he was seeking after political Truth he found it in one or another of the hundred political gospels which govern men in the earth; if he was seeking after the Only True Religion he found it in one or another of the three thousand that are on the market. In any case, when he found the Truth he sought no further; but from that day forth, with his soldering-iron in one hand and his bludgeon in the other he tinkered its leaks and reasoned with objectors."