News from the Vault: Proving the Negative, or, Why That Invisible Pink Unicorn Isn't

The TL;DR brief summary:
Propositions assert some argument is true, such as 2 + 2 ≠ 4. Such an argument must be both valid and sound. The validity of it, a question of knowledge, must first be evaluated. The soundness of it, a question of the actual existence of the subject, is evaluated secondarily. The first can be determined by inverting the statement, and if the statement is self-refuting, the statement must be true, and if not, it cannot be sound, and may be invalid as well. For a claim to be sound, it must be valid and its premises true.

The inversion here is 2 + 2 = 4, which is not self-refuting and thus disconfirms the statement as unsound ( it's premises are false ) and invalid as 2 + 2 = 4 by definition.
I've had several people tell me recently that you can't prove a negative. The most common metaphor to make this point "clear" is the assertion of some colorful, invisible, fantastic creation. "Disprove that", they smugly demand, apparently oblivious to the ease with which this is accomplished. I'll prove what the answer is shortly, but first I'll try to show why we can know it's a proof.
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First principles first: claiming something exists is a proposition. You won't be surprised to learn that the definition of "proposition" has a complex and contentious past which remains largely unresolved, however for our purposes here it isn't necessary to delve into the metaphysics of the word, as its common meaning suffices - a proposition is to set a claim or position before the claimant's audience.  In philosophical terms, it assert a truth claim which purports to describe reality and our knowledge of it.
Distinguishing these characteristics is a foundational principle of logical discourse. In philosophical terms, what is real is ontologic, while what is known is epistemic.
The demonstration or proof of such a claim can be either empirical, which is knowledge of a thing gained by direct sense experience of it, or rational, which is knowledge gained through logical analysis of a set of premises - pure reason in other words. Here is an important distinction, because empirical knowledge is derived from ontology, whereas rational knowledge is derived from epistemology. So, if a proposition is that some real object has certain characteristics, such a claim can be tested by observation of the object itself, via one's senses.  This is what makes empirical data objective: it can be observed and measured by anyone and will result in largely the same descriptions.
A proposition that claims an idea is true cannot be subjected to a test of the senses, however. We have only logic by which to determine the claim's truth value, which requires a two step process: first establish logical validity - that it's internally coherent, and that it is sound - that the conclusions are warranted by the premises.  As such, a rationalist claim may assert an ontological truth without sense experience by showing the claim's logical necessity and universality - what Kant called a priori knowledge.
Such knowledge is true by definition, and one way we can know that is by inverting the claim.  If the inversion is self contradictory, then the claim must be true. For instance, the assertion that 2 + 3 = 5 can be inverted by the statement 2 + 3 ≠ 5. Since the mathematical symbols for numbers are definitions, not object, we have certainty about their contents, and thus can know - also by definition - that the latter statement refutes itself. Therefore 2 + 2 must equal 4. It is the necessary conclusion, and is always true, in any circumstance.
Interestingly, this allows the reverse: any claim which is asserted to be necessary and universal can be disproved in the same way. All that needs be done is to provide a valid alternative. We can reverse the above example to do this, where the claim is 2 + 2 ≠ 4 and the inverse is 2 + 2 = 4, which is not self-refuting, and thus the original formulation is not only not necessary and universal, but is in this case false.
Now to our final component of evaluating propositions: a proposition can only be true, or false. It is a bivalent condition, not an open ended question.  Only one of those two conditions can result from the question, and anything else is essentially saying that the claim of "(2 + 2 = 4) = cabbage". It is an "answer" ( more specifically and accurately, a response ) which does not follow the question.
To return to those invisible pink unicorns, they do not exist.  Anywhere. This is a universal and necessary truth, as the following makes clear:
It is the case that there exists an invisible pink unicorn, where invisible means light passes through the object completely, and pink means all frequencies of color are absorbed except pink, which is reflected back to the view.
In other words, it takes the form It is true that [ 1 - transparent to photons ] + [ 2 - nottransparent to photons ] + [ 3 - unicorn ] exists. This statement refutes itself, as 1 and 2 cannot both be true simultaneously. Thus, the statement's inverse is necessarily and universally true. The inversion here is by replacing the "It is true that" with "it is not true that".
Here the logical form of the argument is invalid, and its ontological truth never gets evaluated.
There are means to prove the negative regarding ontological claims, as well. If the claim demands the subject exist under certain conditions, then recreating those conditions should produce the subject, and if not, then the claim is false as unsound, for example.
So now to questions:
1) Do you agree that we can be certain that 2 + 2 ≠ 5?
2a) If so, do you agree that claims of a god's existence can be known to be false?
2b) If not, do you assert that any proposition can be true?
Thanks for your time, guys & gals.

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