Can one prove a negative existential claim?

This is just a simple matter of logics.

Proving an affirmation requires just one claim (e.g. cogito ergo sum). On the contrary, proving a negation implies verifying that all the universe of possibilities can't be (one must look for it in all rooms in all the buildings...).

There are assessments about negative possibilities of existence, which essentially reduce to prove a positive proposition. See evidence of absence [1] (e.g. can you see my belly? I ate the cake, so it does not exist anymore) and proof of impossibility [2] (e.g. the cake made of the flour I bought does not exist because the flour package is still here).

[1] https://en.wikipedia.org/wiki/Evidence_of_absence

[2] https://en.wikipedia.org/wiki/Proof_of_impossibility

The claim "you cannot prove a negative" is self-referentially (internally) inconsistent

If it were possible to prove that we cannot prove a negative then the existence of the proof would defeat the claim. Notice that the claim "you cannot prove a negative" is a negative. If it were true that negatives cannot be proven, then neither can the maxim "you cannot prove a negative".

Ex's of self-referentially inconsistent statements.:

1. There is no truth.
2. No one can know anything for certain.

etc.,

The Law of Non-Contradiction

It violates a foundational, rudimentary law of classical logic to assert both a proposition and its negation. A proposition X cannot both be true and false (at the same time, in the same sense, simultaneously).

The law of non-contradiction cannot be proven to be true because it must be assumed in order to write any such proof, including a proof of its own validity. That means that any attempt to prove it true will invoke it which would make the proof guilty of assuming its own conclusion and thus begging the question (circular reasoning).

• Null Hypothesis: H0: = "You cannot prove a negative"
• Alternative Hypothesis: HA: = "You can prove a negative"

Is H0 self-referentially contradictory?

The negation of the maxim H0 is not "All negatives can be proven", but rather that "A negative can be proven." Therefore the existence of one or more unprovable negatives does not render HA untenable. One single counter-example utterly destroys the claim that "No negatives can be proven", but no single example establishes it.

The maxim H0 "You cannot prove a negative" is defeated by one example of a provable negative, such as these:

1. There is no elephant in my kitchen.
2. The sun does not revolve around the earth.
3. There are no dragons.
4. The number e cannot be written as any fraction (because it is an irrational number)
5. One cannot divide by zero.

There are however negatives that one cannot prove! However, these examples do not establish the thesis H0 ("You cannot prove a negative"), which is a universal negative conclusion.

Note: "There are some negatives which cannot be proven" is a negative that can be proven; that does not justify the conclusion that "no negatives can be proven"

• Ex 1. "There are no intelligent extra-terrestrial beings (aliens)"
• Ex 2. "There are no other universes in the cosmos."

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.

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.

"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.

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.

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.

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!