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A major concern in epistemology is just what we can know about existential claims, that is, claims about the existence (or lack of) something.

Suppose for example that I assert the following:

The cake is a lie. No physical cake exists.

This is a negative existential claim, meaning that I am making a claim that some thing (in this case, cake) does not exist physically. Whether the "concept" of a cake exists or not in some metaphysical way is irrelevant; I am only concerned with the existence of real, delicious cake. While this is a simple claim, a huge question in epistemology is whether I can prove this or not.

There are a number of ways I could go about trying to prove my statement. I could start looking around my room, under tables, in drawers, anywhere I could imagine in search of cake. I could recruit a team of volunteers with the promise of cake and set them off in search of said pastry to see if there is any to be found. I could make some sort of cake radar and plant gigantic antennas everywhere to investigate the outer reaches of space for the existence of cakes (see also SETI).

However, through any of this, does contemporary philosophy hold that I can ever prove that the cake is a lie?

As an aside, what about a restricted claim? What if I instead assert that:

No physical cake exists in this room.

Is there a consensus on whether the positional restriction changes the provability of the claim? What does the major literature have to say about this?

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    If anyone is wondering about my obsession with cake, it's to do with our (recently revised) community ad on the gaming.SE site. It will link to this question to provide an intriguing introduction for gamers. – commando Sep 16 '12 at 1:25
  • Another way to prove the non-existence of X is to assume X exists, then obtain a contradiction. See, for example, "Russell's Paradox." – Dan Christensen Feb 1 '15 at 4:41
  • I think your question is far broader than the answers---maybe I'm missing something. Your question is essential: assume I affirm there is a big planet between the Earth and the Moon, which is undetectable. You will probably not believe me, but you won't be able to prove that I'm wrong, because I made by hypothesis, the planet is undetectable! And then I could call you a believer, because you believe it does not exist, and could qualify you of believer as many times as I want just by inventing one more infalsifiable existence. I don't think proving non-existence is as easy as proving existence. – anderstood Feb 1 '15 at 6:32
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How you approach this question is going to vary significantly, depending on which particular school of epistemology you are interested in.

For most schools, the fact that it is a negative existential claim is not a particularly thorny part; we'd have the same difficulty trying to prove that there is a cake in this room.

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A dinosaur may be a more relevant example! You claim that "no dinosaur exist at the time" then you want to prove it, if you know what dinosaur is, the shape and size and behavior (for example if it has any ability to hide itself, get invisible or etc.) and else, then you could recruit a team to search for it, you can asymptotically get closer to the answer which may be yes or no. If you limit your question to a small place like in my home, then you have the chance to find your answer not only sooner but also with a larger certainty.

However, if you want to have a rigorous proof, maybe you should think of the object if there is any contradiction for it to exist in this world or not; that is, anything possible in this world is just possible, it may exist or not at the time but even if it doesn't exist now it could have been existed while ago or may come to existence while after. Dinosaurs' existence is not contradictory, so it may exist at the time and it may exist not at the time, you should search for them to find out the answer, but there are examples of what you can easily answer about Of course they don't exist!

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One can prove a nonexistential claim just as easily as proving an existential claim.

In both cases, you must define the proof system you are using, and many of them admit such claims.

The most accessible version is proof by contradiction in First Order Logic (FOL). In such a proof, you first prove that the negation of your claim creates a contradiction, then apply the rule of excluded middle to show that your statement (of non-existance) must be true.

My favorite non-existence proof is Godel's Incompleteness Theorems, which show that a vast swath of proofs people really want to create can never possibly be created. It bounds the limits of FOL very nicely.

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"There is no largest integer", or "largest prime" have purely logical proofs of their non-existence. These sidestep issues of justification that are required for more prosaic evidenced based inferences to non-existence.

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Showing a position is true or probably true is impossible. The idea that it is possible and desirable to prove ideas true or probably true (justificationism) is wrong. In reality, you can't prove any position or show it is probable. Any argument requires premises and rules of inference and it doesn't prove (or make probable) those premises or rules of inference. If you're going to say they're self evident then you are acting in a dogmatic manner that will prevent you from spotting some mistakes. If you don't say they are self evident then you would have to prove those premises and rules of inference by another argument that would bring up a similar problem with respect to its premises and rules of inference.

In reality all knowledge is created by conjecture and criticism. You notice a problem with your current ideas, propose solutions, criticise the solutions until only one is left and then find a new problem.

If you have an explanation that claims that some particular object is in a particular room, and you have an idea about how to test that claim, then you can go ahead and test it. If you test the claim and it fails the test, then either the claim was false or there was something wrong with the test. You explanation of how the test works will have implications for issues other than the outcome of that test and so can be tested by testing those independent implications.

See "Realism and the Aim of Science" by Karl Popper, Chapter I, "Logic of Scientific Discovery" by Popper, Chapters 1,2,4,5. "The Beginning of Infinity" by David Deutsch, Chapters 1,2 and 12. "The Fabric of Reality" by David Deutsch, Chapters 1,3,7.

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I think you can make a claim if you can break down what a cake constitutes and then prove that the constituents could not exist together.

Does A exist?

Let's say then that A is a composite of b and c.

If we can prove that for example that b and c cannot exist together, we could then conclude that A does not exist.

Of course now, the question then expands and one could ask:

Can b and c exist together?

But this is an easier claim - as it is 'positive'. We simple need to take b and put it with c - if they essentially disappear into a vacuum, we know that A does not exist.

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