The links I provided were meant as references, not infallible sources. — TheMadFool

I'm not faulting the article. I'm pointing out that the article says explicitly the exact opposite of how you described it.

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

Be careful there, if T = god exists, then if A = god doesn't exist, ~T = A. "Not T" [negative statement] can be rephrased as "A" [positive statement] and likewise, "Not A" is "T". So, the question that pops into my head is which is the positive statement and which the negative. The simple answer is the positive statement is the one that after taking into account all the negation operations performed on it doesn't leave a residual, dangling negation affixed to it.

In the example above, A is ~T and T has no negation in it and so A has a residual negation left hanging and so A is a negative statement. T is however the statement, ~A and A has a negation, the two negations cancel each other out and we're left with the positive statement T.

1. A = ~T [dangling negation, negative statement]
2. T = ~A = ~~T = T [no residual negation, positive statement]

So, despite how negations can muddy the waters, we can still find out which statements are positive and which are negative as shown above. The rest follows as outlined in the other posts vide supra.

I'm not faulting the article. I'm pointing out that the article says explicitly the exact opposite of how you described it. — TonesInDeepFreeze

I don't know. Quite possible as I'm a bad reader. Will check!

I'm aware that with double negation we can turn any positive into a negation.

But that doesn't bear on the point I made:

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

Sorry, I was trying to work on the lacunae in my understanding. I figured it out to some extent thanks to you. :up: ~p isn't falsehood and I suppose your point rests on that being the case. It isn't.

Depends on the context, doesn't it? Whether or not it's indefinite.

Via John Watkins, where the domain of inquiry is indefinite:

(∀) empirical universal statements are falsifiable but not verifiable
(∃) existential statements are verifiable but not falsifiable

If you make a ∀ statement, then falsification is applicable. If you make an ∃ statement, then verification is applicable.

Claim (example): all swans are while
Burden (general): sufficient/relevant evidence is tentative/provisional/proportional falsifiable justification (unless the contrary is impossible)

Claim (example): there are evil-doers that cast magic spells on others
Claim (example): the Biblical Yahweh is real and intervenes
Burden (general): verify (unless the contrary is impossible)

I guess that also reiterates where the onus probandi is placed. Theists have to provide verification (when they wish to convince others), and when they fail (and have kept failing for centuries on end), others, including nonresistant nonbelievers, are equally justified in disregarding their extraordinary existential claims.

If the domain is local, like 180 Proof's elephant example, then it's a different matter.

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

My answer would be, "it depends".

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

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.

My answer would be, "it depends". — InPitzotl

On what exactly?

PA= Particular affirmative (positive existential claim): Some As are Bs e.g. Some dogs are black

UN = The negation of P is the universal negation (negative existential claim): No As are Bs e.g. no dogs are black

To assert PN, 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.

Which is easier or conversely which is harder?

This is a know problem in science - the difficulty with universal claims such as UN and UA (All As are Bs) lies at the heart of verificationism and falsfiability. UN can't be verified but it can be falsified, just like UA. PA, above, and PN (Some As are not Bs) are verifiable but difficult to falsify which, now that I think of it, proves my point if only with regard to PA.

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

I agree. This is why (and especially with regard to theological debates) I think it is more reasonable to maintain an agnostic position, at least until I can extract more information from my interlocutor. That is the key really. I wouldn't even bother to enter into the argumentation phase of the debate until my interlocutor has provided sufficient information about their position in order for me to derive a contradiction or reveal an absurdity entailed by the view.

I think that there are two important phases to a general debate: 1) clarity seeking; and 2) argumentation. The former is often overlooked and heavily underutilized (in my opinion). I think that before we delve into the structural validity of the arguments or the soundness of the arguments premises, that we should define all the terms of the debate proposition. If that proposition is anything like, "At least one God exists," then I would just let my interlocutor defeat their own position by requesting a definition of the term 'God' and relentlessly requesting further clarification until they flesh out a description that I can defeat.

It is no simple task to prove the negation to the proposition, "At least one God exists," but if you are able to flesh out what it is exactly that they are affirming, it can get much easier. For example, if your interlocutor defines 'God' as "A being who is omniscient, omnipotent and omnibenevolent," then you can derive a contradiction based on those terms. If God is all knowing, then God knows of the 'evils' in the world. And, if God is a perfectly moral being, then God is incapable of acting immoral. Lastly, if God is all powerful, the God has the power to rid the world of evil. Therefore, God cannot be all three of these things because God either is unaware of the evil of the world, indifferent to it, or is incapable of doing anything about it. That means that one of those terms entails a logical contradiction. Negation affirmed. Well done.

You do still have the burden of proof, but you shouldn't take and defend a position wherein you have not already satisfied this burden. Sometimes saying, "I don't know," is the most honest position to hold.

On what exactly? — TheMadFool

It depends on how reasonable the claim is.

Which is easier or conversely which is harder? — TheMadFool

The question is supposed to be about burden of proof.

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

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.

These principles have been offered, where the scope is not determined:

AxP is falsifiable but not verifiable
ExP is verifiable but not falsifiable

I think that is reasonable, if we take 'falsifiable' and 'verifiable' in a sense of 'definitively'. But if we admit degrees of falsification and degrees of verification, then perhaps we would adjust the above principles proportionately. But for the moment I'll take the notions in the sense of 'definitively'.

Also, reiterating what has already been mentioned:

~AxP is equivalent with Ex~P, so it is verifiable but falsifiable.
~ExP is equivalent with Ax~P, so it is falsifiable but not verifiable.

The relevant comparison is between proving ExP when it is true vs. proving ~ExP when it is true. (For a falsehood, not only is it difficult to prove, but it is impossible to prove.)

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.

So it has been claimed that this difference entails that the first burden is on ExP. It seems there might be something to that, but it is not self-evident and it requires support.

But we also want to consider cases where the scope is determinate and a context in which verification and falsification are not definitive but refer to degrees of verification and degrees of falsification. In either of those two frameworks, we can easily see that sometimes proving ExP when it is true is not "easier" than proving ~ExP when it is true.

/

Regarding whether there exits an omnipotent, omnipresent, omnibenevolent being, I'm not saying anything new here, but for me, the question requires specifying what would constitute empirical proof. If it's not an empirical matter, and unless the existence statement is shown to be a logical truth, then it seems it's a metaphysical or theological concern for which the notion of proof in the same sense of proving "there exists a black dog" doesn't even apply.

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

I wouldn't think this would have to be said, but I'm making the assumption that ExP and ~ExP cannot both be true.

They cannot both be true.

They cannot both be true. — 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?

Let me rephrase. They are the same task. But if ExP is true, then the task is sure to end, while if ~ExP is true, then its end is indeterminate.

This holds for this framework we're talking about - empirical search, one-by-one in an indeterminately large domain.

And there's an analogy to it in mathematics [I'm simplifying somewhat]:

Let P be a computable property of natural numbers.

If ExP is true, then we are ensured that in finite time we will find an x such that we prove Px is true, thus proving ExP.

But even if ~ExP is true, then we are not ensured that we will ever prove ~ExP (it might be the case that at all points in time, indefinitely, we don't know whether it's provable).

Let's look at Turing machine framework (I think I have this right):

Suppose P is a computable property of natural numbers. (Analogously, for purpose of this discussion, we suppose "this is a dog and it's black" is a definite enough statement that we can definitively declare when we find a black dog.)

Ask the machine for 'yes' or 'no' to "Is ExP true"?

If ExP is true, then the machine will answer 'yes'.

If ExP is false, then the machine might not halt.

If ~ExP is true (i.e. ExP is false), then the machine might not half.

If ~ExP is false (i.e. ExP is true), then the machine will answer 'yes'.

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.

And there's an analogy to it in mathematics [I'm simplifying somewhat]:
Let P be a computable property of natural numbers. — 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.

But what is so interesting in saying "'the task P if the Goldbach conjecture is true' is a different task than 'the task P if the Goldbach conjecture is false'"? And what meaningful thing is conveyed when you say that, as opposed to, say, just saying it's the same task, and we just don't know if it will halt or not?

It depends on how reasonable the claim is. — InPitzotl

Kindly restrict your comments to an Aristotelian format, provided below for you:

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. Might I remind you that the problem of the burden of proof/can't prove a negative problem are problems inherent in the nature of these statements. So, if you feel that I've got the wrong end of the stick somehow, you'll need to do it against the backdrop of these four statements.

The question is supposed to be about burden of proof. — InPitzotl

Of course, of course. I only offered a possible reason not the actual reason whatever that is but the reason I provided - difficulty in terms of practical considerations - is valid. If two people were in an argument, isn't it prudent to let the one who has the easier proof to go first? Why waste time? Time is money they say.

(b) you searched all of the dogs on the planet. — InPitzotl

Thank you for mentioning this "problem". It's a pseudo-problem though because think of what you've accomplished when "you searched all of the dogs on the planet" and found no black dogs? Well, you've proved "no dogs are black" (the negative claim corresponding to the positive existential claim, "some dogs are black") and that was tough, right?

"'the task P if the Goldbach conjecture is true' is a different task than 'the task P if the Goldbach conjecture is false'"? — InPitzotl

Let's go back the general question about ExP.

I'm not couching this as "The task for proving ExP when ExP is true is different from the task for proving ExP when ExP is false."

What I am saying is this: Proving ExP is "easy" only if ExP is true.

I wouldn't say ExP is easy. Because then someone may say, "It's not easy if it's false, because its's impossible, which is the ultimate not easy."

So I include the antecedent "If ExP is true".

And I'm not saying that is interesting. It's just necessary to be correct.

And to make meaningful comparison between proving ExP and ~ExP, we need to consider each when it is true.

2. No As are Bs — TheMadFool

There are no two integers p, q such that (p/q)^2=2.

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.

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

You were told wrong.

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

It's fair for you to have quoted me that way, since I did post it. But, just for the record, around the same time, I edited my post to not include that, as it's wrong, and I misspoke earlier when I said they are different.

You were told wrong — TonesInDeepFreeze

Give me an example that proves what I said is wrong.

For all x, y, z, if x=y and y=z, then x=z.

It's famous that monadic languages lack the expressiveness of dyadic languages, and that monadic logic is weaker than predicate logic with dyadic predicates.

So I responded to your challenge. Howzabout you respond to mine from previous posts?:

https://thephilosophyforum.com/discussion/comment/533894

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

It is to move from agnosticism.

Do you have any evidence that they do exist? — Pinprick

The first clip in this video looks pretty real: https://www.youtube.com/watch?v=ZswREtWpJrg

:wink:

I just meant it wasn’t a factor for determining burden of proof — Pinprick

I know what you meant. You are right - just because something is harder to prove (for example proving a negative) doesn't let the claimant off the hook.

It is to move from agnosticism. — Down The Rabbit Hole

Why? To me, you need a reason to believe something. If there is no reason, then disbelief is warranted. That is to say that the truth of the belief in question can be rejected, or denied.

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

First of all, thank you for that mathematical example of proof of a negative claim being easier than proving a positive claim. It was an eye-opener for me.

A coupla things that I want your opinion on:

1. Proof by contradiction/indirect proof works well for both positive and negative claims. It doesn't favor one or the other. If so, one really can't say that negative claims are, on the whole, easier to prove than positive ones.

2. Coming to direct proofs, firstly, my argument that positive claims are easier to prove than negative ones, especially existential ones, stands. Secondly, since positive claims precede their negation (~p can be only after p) and since to assert a proposition one needs proof, it follows that positive claims need to be proven first.

Proof by contradiction/indirect proof works well for both positive and negative claims. It doesn't favor one or the other. — TheMadFool

I think you're focused too much on proof by contradiction.

Essentially, I gather you're imagining a "proof by testing each case" kind of method. By your difficulty metric, the difficulty of a proof is proportional to the number of cases you have to test by that method. The weakness of this approach is simply that it only applies when you're using "proof by testing each case". The irrationality of square root of two can be demonstrated using proof by contradiction, but that just so happens to be one other proof method besides "proof by testing each case". 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... it can be proven by simply examining the definitions of even and prime, and working things out by theory.

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.

Coming to direct proofs, firstly, my argument that positive claims are easier to prove than negative ones, especially existential ones, stands. — TheMadFool

Not really, because your argument is making a false comparison. You're kind of committing the epistemic equivalent of a base rate fallacy.

Using the proof by testing each case method, you're comparing the work to prove ExP if ExP were true to the work to prove ~ExP if ~ExP were true; by doing so, you're ignoring what happens when you try to prove ExP if ~ExP were true and when you try to prove ~ExP if ExP were true. If you just take that into account, you would quickly realize that the claim is not what drives the difficulty you're talking about; but rather, the state of affairs is what drives it. 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.

I think you're focused too much on proof by contradiction. — InPitzotl

Quite naturally, no? Firstly, it's the method used in your example and secondly, the only method which makes proving a negative easier than proving the positive.

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

Possibly, but which - direct/indirect proof - is easier? I bet the latter (indirect proof) would turn out to be far, far easier. I have my own reasons for believing that.

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

In regard to difficulty in re existential claims that pertain to the physical, it goes without saying they're much easier to prove than their negations but, as your example shows, positive existential claims that are amenable deduction are sometimes harder to demonstrate than their negations.

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

Indeed, you're absolutely right but you need to understand or look at what it is exactly that you have proved here?

Suppose I wanted to prove S = some dogs are black. I begin looking for black dogs and either I find one or I don't. If I do find one, I've proven S and I stop, I don't have to check the rest of the dog population unless of course I'm really unlukcy and the dog which is black is the last dog I check. If I don't find any black dogs, I would have necessarily had to have gone through all the dogs and that proves ~S = No dogs are black. In other words, it's harder to prove S than ~S.

Imagine now I want to prove ~S = No dogs are black. As you already know, I have to see every single dog in this case. If I find a black dog, yes, I stop, but what does that prove? S! of course. In this case too proving S is easier than ~S.

There really is no point in debating this. Insofar as categorical statements are the issue, proving the positive, particular affirmative (Some A are B) is definitely easier than proving the negative, universal negation (No A are B). Experts agree on that and I defer to their expertise. Note that the caveat is only for direct proofs and also the claims have to be empirical.

Thank you for engaging with me. It's likely that I'm mistaken about all this but would appreciate your views on them nonetheless.