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I am aware that I could show that something does not exist with the knowledge of an implication.

If B is true if A exists, then A does not exist if B is false

But if we have no implications, is it possible then?

Example: Can we prove that there is no Loch Ness Monster?

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    "There exists a universe-fungus that inhabits all regions of space with glowing strands no more than a meter apart." I'm pretty sure you can figure out how to show this doesn't exist. – Rex Kerr Nov 15 '13 at 17:43
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Actually, You Can Prove A Negative Sometimes

In general, sure, it can be difficult to disprove the existence of something on a universal level, because theoretically you would need total, infallible awareness of the entire universe to prove it with complete certainty. The problem here lies in that the scope of the concern and requirement of proof is essentially unbounded (extends to the entire continuum), and we lack the ability to search the entire universe.

However, if you are simply referring to existence in a local continuum (trying to prove the existence of something within a defined and searchable area and which itself is capable of being observed by humans), then the problem becomes possible to solve. Simply searching the problem space and finding the no existence of X would prove X does not exist (deductively).

Proof by Deduction vs Induction

The problem you are touching upon is whether you can deductively arrive at the conclusion that something does not exist, and you can in a searchable problem space. Only when the problem space becomes unsearchable in some way do we have to rely on inference (induction) rather than deduction; for example, if the search area is too big to possibly search, or our methods of observation [i.e., eyes, radar, etc.] lack 100% reliability [i.e., they could be used and but still potentially not see whatever you are looking for, maybe because the human blinked his eye and missed the sea monster, or the machine cannot see the monster when it is ghost form, whatever. The point is, if there is any possibility that the observation method could fail, however unlikely or odd, you move from deduction to induction, with the certainty of your conclusion directly related to the reliability of your method(s) of observation, the size of your search space, and your strategy for conducting that search (presuming your eyes/radar cannot see the entire space at once you will need to devise a plan to systematically investigate the entire area — and some strategies are of course more effective than others.).].

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    +1, removed my answer, as I think it was too redundant with yours and not nearly as complete. – user4634 Nov 16 '13 at 1:21

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