The links I provided were meant as references, not infallible sources. — TheMadFool
We can only prove what is true. So it is always easier to prove what is true, since there is no proof of a falsehood. That applies whether it's ExP or ~ExP. — TonesInDeepFreeze
I'm not faulting the article. I'm pointing out that the article says explicitly the exact opposite of how you described it. — TonesInDeepFreeze
We can only prove what is true. So it is always easier to prove what is true, since there is no proof of a falsehood. That applies whether it's ExP or ~ExP. — TonesInDeepFreeze
My answer would be, "it depends".I'm approaching the issue with an open mind without any preconceptions or prejudices. My aim was to discover for myself why the burden of proof has to be borne by those making a positive claim and not the one making a negative claim. — TheMadFool
I would hope that if X does not exist, it should be difficult to prove X does exist; otherwise, our proof method would be in question.By way of a possible reason, I found out that, insofar as existential claims are the issue, proving the positive is much, much easier than proving the negative. — TheMadFool
My answer would be, "it depends". — InPitzotl
Do you see the problem of proving a negative vis-à-vis god? To prove that god doesn't exist, one would have to have explored the entire universe - currently impossible - and even beyond - impossible. — TheMadFool
It depends on how reasonable the claim is.On what exactly? — TheMadFool
The question is supposed to be about burden of proof.Which is easier or conversely which is harder? — TheMadFool
It would appear to me that these are the same task. You start looking at dogs. You stop when either: (a) you have found a black dog, or (b) you searched all of the dogs on the planet. The task is no more made easier by asserting there's a black dog than it is made more difficult by asserting there isn't one.To assert [PA?], all I need is a single specimen of an A that is also a B (a black dog).
To assert UN, I need to find and examine each and every dog on the planet and check if they're black/not. — TheMadFool
I wouldn't think this would have to be said, but I'm making the assumption that ExP and ~ExP cannot both be true.Also, if discovery of proof proceeds by one-by-one examination of things, then yes, if ExP is true, then the sequence of proving by one-by-one examination for ExP is finite, while, if ~ExP is true, then the sequence of proving by one-by-one examination for ~ExP is indeterminate. And that holds with the example of "There is a black dog" vs. "There is not a black dog". They are not the same task. — TonesInDeepFreeze
If they cannot both be true, then I'm not sure you're telling me anything interesting or meaningful when you say they are not the same task. There's a task that may or may not halt at (a), and may or may not halt at (b). About all you are telling me is that if we count the possible tasks as two tasks, we get two. But you seem to acknowledge that the task cannot both halt at (a) and halt at (b). So, sure, if we count what doesn't happen as a different thing, we get two, but why is that interesting?They cannot both be true. — TonesInDeepFreeze
Sorry, you're just repeating yourself.If ExP is true, then that requires a task, call it TaskE.
If ~ExP is true, then that requires a task, call it TaskN.
TaskE and TaskN are different. — TonesInDeepFreeze
Sure, so say I write a program P to methodically check for counterexamples to the Goldbach conjecture (methodical in the sense that if there's a counterexample to be found it will check that counterexample in a finite amount of time). I'll grant that knowing whether P will halt or not is interesting. I'll grant that knowing if the GC is true or not is interesting. And I'll grant that the former is equivalent to the latter.And there's an analogy to it in mathematics [I'm simplifying somewhat]:
Let P be a computable property of natural numbers. — TonesInDeepFreeze
It depends on how reasonable the claim is. — InPitzotl
The question is supposed to be about burden of proof. — InPitzotl
(b) you searched all of the dogs on the planet. — InPitzotl
"'the task P if the Goldbach conjecture is true' is a different task than 'the task P if the Goldbach conjecture is false'"? — InPitzotl
There are no two integers p, q such that (p/q)^2=2. — InPitzotl
1. All As are Bs
2. No As are Bs
3. Some As are Bs
4. Some As are not Bs
I'm told that every proposition can be rephrased as one of the above. — TheMadFool
If ExP is true, then that requires a task, call it TaskE.
If ~ExP is true, then that requires a task, call it TaskN.
TaskE and TaskN are different.
— TonesInDeepFreeze
Sorry, you're just repeating yourself. — InPitzotl
You were told wrong — TonesInDeepFreeze
I don't have sufficient evidence to claim fairies don't exist. Do you? What is it?
— Down The Rabbit Hole
— Pinprick
It isn’t needed. — Pinprick
Do you have any evidence that they do exist? — Pinprick
I just meant it wasn’t a factor for determining burden of proof — Pinprick
It is to move from agnosticism. — Down The Rabbit Hole
There are no two integers p, q such that (p/q)^2=2. — InPitzotl
Indeed, you're right! There are occasions in which if a reductio ad absurdum is feasible, it's easier to prove a negative statement than a positive one. Unfortunately (if we want to know that is) or fortunately (if there are things we shouldn't know), a reductio ad absurdum isn't always possible. Do you agree then that in such cases it's easier to prove a positive existential claim than a negative claim that asserts no such thing as posited by the positive existential claim exists? I should've caught on earlier when you mentioned the horse running inside your fridge! :lol: Thanks. Will get back to you if I think of anything. — TheMadFool
I think you're focused too much on proof by contradiction.Proof by contradiction/indirect proof works well for both positive and negative claims. It doesn't favor one or the other. — TheMadFool
Not really, because your argument is making a false comparison. You're kind of committing the epistemic equivalent of a base rate fallacy.Coming to direct proofs, firstly, my argument that positive claims are easier to prove than negative ones, especially existential ones, stands. — TheMadFool
I think you're focused too much on proof by contradiction. — InPitzotl
We can also prove things like "there are no even numbers greater than 2 that are prime"; such is also an easy proof, but it does not require proof by contradiction per se... — InPitzotl
If you wish to measure the difficulty of proving something, you need to account for all methods of proof, not just proof by testing each case. — InPitzotl
If I'm trying to show there are black dogs, but it turns out there aren't, I still have to test every dog before I find out my mistake. If I'm trying to show there aren't any black dogs, but it turns out there are, I still stop early once I find the black dog. — InPitzotl
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