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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?
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).
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.).].
Look it all fall down to data. If data i.e. observation tell you something you should take it.
Non-existence can be proved only if you have complete information of the problem space.
To prove that dices don't have a seventh number go look at some dices. Look at as many as you want. Yes, you cannot be sure that all dices ever made have less than seven numbers but you can be sure that the dices you saw all have only six numbers.
Go see where they are made. Once you are sure they are made in machines and your data about previous machines tell you that machines make all its products sufficiently identical for our purpose here you can be more sure about dices you haven't seen.
P implies Q. Not Q therefore not P. This, the first equation of propositional logic is also based on data and nothing but data. If all our data tells us that all of the dozens of time you were sick your mama didnt send you school but you are in school today and your mama sent you here then you must not be sick today.
The base for this convenience is, as every one year old knows, things dont change on their own. All children play this hiding game to understand reality. Hide something somewhere they are sure nobody goes to. Then stand on guard around it as much as they can to be sure nobody went there before going examining the thing. Is it still there? Sure it is. Repeat the experiment a few hundred times. Sufficient for the child to conclude that things stay as they are unless some one change them (change their position, alter their condition etc). This conclusion never fail anybody.
One, classical, way of proving any proposition (claims of existence/nonexistence included) is to demonstrate that the proposition's negation leads to a contradiction (reductio ad absurdum).
So if I want to prove God doesn't exist, I first assume God exists and show that that leads to a contradiction (p & ~p). Ergo, I'm justified to then claim God doesn't exist.
P. S. Neutrons, I'm told, were the last of particles to be discovered (they're electrically neutral i.e. they're inert compared to electrons and protons; if all we could detect were charges, neutrons would exist but we would never know that they do).
