Timeline for Can one prove a negative existential claim?
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| Jan 26, 2022 at 0:00 | comment | added | Cort Ammon | Given a model of our world ℳ, I can easily write ℳ⊨¬∃"A pink flying hippo" and it is trivial to show that, in that model, there are no flying pink hippos. The model explicitly entails it. Implying it with a proof system, ℳ⊢¬∃"A pink flying hippo" can be much trickier. In fact, we often rely on closed world assumptions about our world in order to admit such a sentence (where we imply that a pink flying hippo only exists if we can prove it exists). Looking into closed world assumptions is fun trip down the fine lines that philosophy walks when it comes to logic. | |
| Jan 25, 2022 at 23:50 | comment | added | Cort Ammon | @GillBates That is true. Even if you look into the mathematics of logic, we see a distinction between a model which "entails" a sentence is true and a proof system that implies it. They are separate things in logic. Ideally one would find a model and a proof system such that the proof system is sound and complete. In that case, entailment and proof would be one and the same. However, as we found in the early 20th century, even First Order Logic over arithmetic falls short. It must be unsound or incomplete. Myself, I find that most people who care about such details also like arithmetic =) | |
| Jan 25, 2022 at 23:47 | comment | added | Gill Bates | I see that you are drifting into some kind of relativism. I'm starting to believe that trying to approach this problem from a purely logical standpoint is not fruitful. Logic deals with relationships between already established facts and statements. But it doesn't deal with the discovery of facts and knowledge, ontology. | |
| Jan 25, 2022 at 21:34 | comment | added | Cort Ammon | @GillBates Convincing would probably call for the proof systems of religions. It has astonishing capabilities in that regard. As for practically useful, we may have to beg to differ on that one. I find the ability to consider that pink flying hippos could exist to be extremely practical. How else do we step beyond our own assumptions? I've got an answer on WorldBuilding challenging the idea that physics need changes to permit magic. It's in the same vein. | |
| Jan 25, 2022 at 20:57 | comment | added | Gill Bates | The proof system that would allow to produce the most convincing and practically useful proof. | |
| Jan 25, 2022 at 15:29 | comment | added | Cort Ammon | @GillBates What proof system am I using? If I'm using a proof system where the existence of the referent of any noun phrases starting with the letter "p" is false, the proof is rather simple. On the other hand, if I am using the proof systems Russel referred to in his famous teapot scenario, it would be very difficult. Or perhaps the domain is restricted to that of a small box that does not fit a hippo, much less a flying one. | |
| Jan 25, 2022 at 4:20 | comment | added | Gill Bates | Can you demonstrate in practice how to prove that pink flying hippos do not exist? | |
| Jan 31, 2015 at 4:38 | history | answered | Cort Ammon | CC BY-SA 3.0 |