Proof by contradiction/indirect proof — TheMadFool

Be careful with the terms 'proof by contradiction' and 'indirect proof'.

This is the form of proof by contradiction (indirect proof):

Assume ~P
Derive contradiction
Infer P

This is not a from of proof by contradiction (indirect proof):

Assume P
Derive contradiction
Infer ~P

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. — TheMadFool

No, it does not follow. I've given you explanations for why it does not follow. You skip responding to the key points in the explanations.

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". — InPitzotl

Right, case-by-case in an indeterminate domain.

The irrationality of [the square root of] two can be demonstrated using proof by contradiction — InPitzotl

The ordinary proof that the square root of 2 is irrational is not a proof by contradiction. Assuming P, deriving a contradiction, then inferring ~P is not a proof by contradiction. It might seem that deriving a contradiction on the way to the conclusion is proof by contradiction, but 'proof by contradiction' refers to something more specific: Assuming ~P, deriving a contradiction, then inferring P.

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

But that is not an instance of demonstrating that there are no black dogs. It is better described as the process of discovery whether there are black dogs. That's different from demonstrating that there are no black dogs.

3. The house = the universe
4. The bear = god — TheMadFool

Some theists believe the bear = house. What say you then? I’m not disagreeing nor agreeing merely curious. For some it’s an argument where one says no we have two things a bear and a house - the bear either being internal to or external to the house. And others saying it’s just the bear and others saying it’s just the house.

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

[it's] the only method which makes proving a negative easier than proving the positive. — TheMadFool

As mentioned, 'proof by contradiction' is not the right term. And such cases can more comprehensively be described as 'deductively proving'. We may prove ~ExP deductively, either by definition (proving that there are no married bachelors), or from axioms or principles (proving that there is no rational number whose square is 2), or from facts taken as premises (proving that there is not a horse in the refrigerator from the premise that horses are not smaller than the space inside the refrigerator).

For deductive proofs of ~ExP, usually the method is to assume ExP, then derive a contradiction, then conclude ~ExP. This usually deploys modus tollens in this form: ((ExP -> Q) & ~Q) -> ~ExP, which is permitted by direct proof.

So I would restate your claim as: Other than deduction, there are no methods that make it easier to prove ~ExP than to prove ExP. (I will leave it as tacit that in such comparisons that we are concerned with relative difficulty only in context of which is true. If ExP is false, then necessarily it's easier to prove ~ExP, and if ~ExP is false, then necessarily it's easier to prove ExP.)

But "[it's] the only method which makes proving a negative easier than proving the positive" is not self-evidently true. It requires an argument. It is an ~ExP claim ("there does not exist a method that is neither deduction nor case-by-case in an indeterminate domain"), so notice that - contrary to your claim that ExP is necessarily claimed before ~ExP - this is an example where the ExP claim was not made first.

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. — TheMadFool

The deductions can be about physical facts. From premises about physical facts we may deductively reach conclusions.

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. — TheMadFool

(1) I will regard that in the context "except for deductions), (2) Your claim depends on whether there are methods other than deduction and case-by-case in an indefinite domain.

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 — TheMadFool

Unless you tell us the arguments of these experts, it's mere appeal to authority. So who are these experts and where can I read their arguments?

The ordinary proof that the square root of 2 is irrational is not a proof by contradiction. — TonesInDeepFreeze

https://en.wikipedia.org/wiki/Proof_by_contradiction#Irrationality_of_the_square_root_of_2

Definitions:

x is rational iff x equals a ratio of integers

x is irrational iff ~ x is rational

Theorems:

x is irrational iff ~ x equals a ratio of integers

if ~ x is irrational then x is rational (not intuitionistically acceptable)


(1) It's indirect if you put it this way:

Prove sqrt(2) is irrational

Suppose ~ sqrt(2) is irrational

Derive contradiction

So sqrt(2) is irrational (not inuitionistically acceptable)


(2) But that's not necessary, and it can be done without indirect proof (intuitionistically acceptable) this way:

Prove sqrt(2) is irrational

Prove ~ sqrt(2) is rational

Suppose sqrt(2) is rational

Derive contradiction.

So ~sqrt(2) is rational

So sqrt(2) is irrational

https://en.wikipedia.org/wiki/Proof_by_contradiction#Irrationality_of_the_square_root_of_2 — InPitzotl

Here are some more problems with that article:

(1) The sqrt(2) proof does not make clear its indirect form. Indirect in clear form would be to assume "~ sqrt(2) is irrational". But the proof assumes "sqrt(2) is rational".

Basically the same objection is mentioned by a participant in the Talk section for the article.

(2) Intuitionistic invalidity is mentioned only in passing as a clause in a sentence about excluded middle. The very important distinction, vis-a-vis intuitionism, between the two forms of deriving a contradiction is not mentioned. (That is, the article does not mention that "Suppose P, derive contradiction, infer ~P" is intuitionistically valid.)

And the article says "some intuitionist mathematicians do not accept [excluded middle]". Why only say 'some', why not say 'all'? (Or maybe the author knows of intuitionists who accept excluded middle?)

(3) The article mentions Cantor's diagonal argument as proof by contradiction. I don't recall whether Cantor himself used indirect proof (I tend to think he didn't), but even if he did, it should be noted that his argument does not require indirect proof and it is intuitionistically valid.

(4) There are a few good citations in the list of references, but some of them mention proof by contradiction only tangentially. And the rest of the list is lousy, including quite informal lecture bullet points and things like that.

/

What is the importance in math forums for keeping the distinction between the two forms clear? It is that often cranks disparage proofs by such as Cantor on the basis that they're "indirect" while these cranks have heard somewhere that indirect proofs are suspect (indeed, they are worse than suspect for intuitionists). But the proofs are actually not indirect, or if they are, they can be rearranged so that they are not indirect.

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. — TheMadFool

You want to prove S. So you're going to "set about trying to prove it" by commencing a task P. Essentially, P is a search algorithm; you're searching for a black dog.

So let's say there are n dogs. Here's a table:
$\begin{matrix} \text{row} & \text{claim} & \text{will prove} & \text{# dogs} & \text{# black} & \text{min} & \text{max} \\ \text{1} & \exists x : isblack(x) & \nexists x : isblack(x) & \text{n} & \text{0} & \text{n} & \text{n} \\ \text{2} & \nexists x : isblack(x) & \nexists x : isblack(x) & \text{n} & \text{0} & \text{n} & \text{n} \\ \text{3} & \exists x : isblack(x) & \exists x : isblack(x) & \text{n} & \text{1} & \text{1} & \text{n} \\ \text{4} & \nexists x : isblack(x) & \exists x : isblack(x) & \text{n} & \text{1} & \text{1} & \text{n} \\ \text{5} & \exists x : isblack(x) & \exists x : isblack(x) & \text{n} & \frac{n}{2} & \text{1} & \frac{n}{2}+1 \\ \text{6} & \nexists x : isblack(x) & \exists x : isblack(x) & \text{n} & \frac{n}{2} & \text{1} & \frac{n}{2}+1 \\ \text{7} & \exists x : isblack(x) & \exists x : isblack(x) & \text{n} & n-1 & \text{1} & 2 \\ \text{8} & \nexists x : isblack(x) & \exists x : isblack(x) & \text{n} & n-1 & \text{1} & 2 \end{matrix}$

Briefly, "row" is just a label; "claim" is what you are claiming and/or want to prove; "will prove" is what you'll wind up proving; "# dogs" is the number of dogs (set to n for all rows); "# black" is the number of those dogs that are black; "min" is the minimal number of dogs you check before you're done with P; and "max" is the maximum number of dogs you check before you're done with P.

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. — TheMadFool

In the table above, the min and max columns are metrics of difficulty. What drives both min and max to be n on rows 1 and 2 is the fact that # black is 0, not the fact that you're claiming S (row 1) or ~S (row 2). In fact, each pair of rows {1, 2}, {3, 4}, {5, 6}, and {7, 8} show the same min and max metric.

In other words, it's harder to prove S than ~S. — TheMadFool

But the claim has nothing to do with the difficulty (e.g., row 1 is exactly as difficult as row 2). The difficulty (how many things you need to search) depends on the state of affairs (in this view, how many black dogs there are). You don't know that state of affairs until you finish the task P, and once you do that, you no longer need burden of proof... it's already been met.

That chart seems to capture discovery not proof. For example, the min in row 4 is 1 only because we discover that there is a black dog and give up trying to prove that there is not one. But that is not the task. The task is to prove there is not a black dog.

Suppose someone says to you:

"I have two stacks of photographs. Each stack has 10000 pictures of dogs. In one stack there's at least one picture of a black dog. In the other stack there is no picture of a black dog. I am going to randomly give you one of the stacks. Now you have a choice:

You can choose to prove there is a picture of a black dog by going through the pictures until you find a picture of a black dog and then you may stop, and I pay you $500. But if you don't find a picture of a black dog to prove that there is one, then I pay you nothing.

or

You can choose to prove there is no picture of a black dog by going through all the pictures and not finding a picture of a black dog, and I pay you $500. But if you do find a picture of a black dog, then you may stop, but I pay you nothing."



The chance of being paid is even between the two choices. But, clearly, one should choose the best chance at having the shortest labor time - by choosing to prove there is a picture of a black dog. Because, if there is a picture of a black dog, then you get to quit when you find it. But if you choose to prove there is no picture of a black dog, and you don't find one, then you don't get to quit until you've gone through all the pictures.

In other words: If you choose "there is a picture of a black dog" then you can both win and have a chance of quitting early, even as early as looking at the first picture. But if you choose "there is no picture of a black dog" then you can't win unless you look through all the pictures.

If someone told me they chose "prove there is no picture of a black dog", I'd say "You must really like going through pictures of dogs, or you don't value your time, or you're stupid."

/

And I realize that this works no matter whether the domain is determinate or indeterminate (it's just that it's even worse for ~Ex Bx when the domain is indeterminate).

Possible outcomes:

ExBx

Black dog found before end. $500 and get to go home early.
Black dog found at end. $500.
Black dog not found. Wasted time trying.

~ExBx

Black dog found before end. Wasted time trying, but get to go home early.
Black dog found at end. Wasted time trying.
Black dog not found. $500.

If black dog found before end, advantage goes to ExBx.
If black dog found at end, advantage goes to ExBx.
If black dog not found, advantage goes to ~ExBx.

So advantage overall goes to ExBx, because it might be that ExBx has a chance to win $500 and go home early, while ~ExBx has only a chance to win $500 after going the full distance.

/

Put another way,

Suppose the chance of ExBx is the same as the chance of ~ExBx (i.e. we don't have reason to believe in advance that one is more likely than the other).

The expectation of the work to prove ExBx is less than the expectation of the work to prove ~ExBx, since proving ExBx might finish earlier than the end of the search, while proving ~ExBx will surely not finish earlier than the end of the search.

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. — Pinprick

To actively claim something does not exist, you have a burden of proof, and just because it's harder to meet your burden of proof, doesn't make it disappear. One should remain agnostic until they have sufficient evidence either way.

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.
— Pinprick

To actively claim something does not exist, you have a burden of proof — Down The Rabbit Hole

There is a difference between "S is false" and "I disbelieve S."

"S is false" in many contexts raises expectation of demonstration that S is false.

But "I disbelieve S" is merely an assertion that I am either withholding or rejecting belief that S is true. If one has not been shown adequate demonstration that S is true, then it may be reasonable to withhold or reject belief that S is true.

If someone says "There are kangaroos living at the North Pole", then I may say "I don't believe there are kangaroos living at the North Pole" without obligation of proving that there are no kangaroos living at the North Pole.

Yes, disbelief is consistent with agnosticism, and believing something is false is not. The former doesn't require evidence, but the latter does.

"There exists a fish with blue fins and a green body."

I don't assume that is true and I don't assume that it is false.

"There exists a striped kangaroo."

I assume that is false.

"There exists a fish with blue fins and a green body."

I don't assume that is true and I don't assume that it is false.

"There exists a striped kangaroo."

I assume that is false. — TonesInDeepFreeze

Based on your knowledge of kangaroos? This would be using evidence to reach a conclusion.

This would be using evidence to reach a conclusion. — Down The Rabbit Hole

You mentioned "sufficient evidence". I'm wondering whether we would deem my knowledge of kangaroos to be sufficient to have a reasonable belief that there do not exist striped kangaroos. If not then you would place a burden of proof on me if I say, "Striped kangaroos do not exist". If you did, then you might be right; I don't know.

The more poignant case is something like "There exists a billion ton being with intelligence, knowledge, physical strength, and moral purity a billion times greater than any human being."

If I say, "I don't believe that", then we agree that is reasonable.

But if I say, "That is false", then I have a burden to prove it is false?

The amount of evidence we should have to believe something, is a tough one. It's something I think about a lot.

Yes I was going to say, you gave me some low hanging fruit with the striped kangaroo argument :smile: I wouldn't be surprised at all if there were some striped kangaroos.

We would still have a burden of proof in the active claim that the billion ton being is false. As silly as it sounds, it may be that we are unable to obtain enough evidence to meet our burden of proof. I just don't think we should lower our burden of proof based upon the difficulty of obtaining evidence.

I wouldn't be surprised at all if there were some striped kangaroos. — Down The Rabbit Hole

I think you mean you would be surprised.

I just don't think we should lower our burden of proof based upon the difficulty of obtaining evidence. — Down The Rabbit Hole

That seems reasonable.

On the other hand, if an outlandish or "out of thin air" existence claim is asserted, it doesn't seem reasonable that the denier would have as great a burden to prove false as the assertor has to prove true.

/

Or consider recent history. A lot of Republicans, including the president, claim there was widespread voting fraud, but they have not produced convincing evidence. People who recognize the legitimacy of the election don't just say that the claim has no evidence but moreover that the claim is to be regarded as false.

I wouldn't be surprised at all if there were some striped kangaroos.
— Down The Rabbit Hole

I think you mean you would be surprised. — TonesInDeepFreeze

No, latest figures show almost 45 million kangaroos in Australia alone - I wouldn't be surprised if there were a few mutant striped ones. I definitely wouldn't feel confident saying there are no striped kangaroos, but I think you realise this wasn't the best example you could have given.

On the other hand, if an outlandish or "out of thin air" existence claim is asserted, it doesn't seem reasonable that the denier would have as great a burden to prove false as the assertor has to prove true. — TonesInDeepFreeze

Yes it feels deeply counter-intuitive. But for what reason other than the difficulty of obtaining evidence would the denier have a lesser burden of proof?

https://thephilosophyforum.com/discussion/comment/535447

I'm having second thoughts about this and I might need to retract that particular argument.

The table of outcomes is:

ExBx is true, and you can prove it and possibly go home early.
ExBx is false, and you can't prove it, and you won't go home early.

~ExBx is true, and you can prove it but you won't go home early.
~ExBx is false, and you can't prove it but possibly you can go home early.

Now it occurs to me that actually it is not clear how even to summarize those into one single advantage for ExBx side.

The only direct comparison is between the sides when it is true for that side. And that goes back to the point I made earlier.

if ExBx is true, then it is easier for the ExBx side than it is for the ~ExBx side when ~ExBx is true.

Yet, I still can't shake the intuitive feeling that, given that it is equally likely whether ExBx is true or ~ExBx is true, choosing to prove ExBx would be a better choice.

Since attempting to let this die didn't work:

That chart seems to capture discovery not proof. — TonesInDeepFreeze

What the chart indicates is what the chart was intended to indicate. It sounds like you're spinning tales about what it indicates. I'm not sure those tales are meaningful.

Suppose someone says to you: — TonesInDeepFreeze

I'm not sure what that entire scenario is about.

"I have two stacks of photographs. — TonesInDeepFreeze

You're thinking about this wrong. Let's just as a device call every place that a dog could be a "dog house". So if we want to find out if there's a black dog, we need to search all of the dog houses. Here, a dog house is analogous to a photo. Likewise, all of the dog houses is our analog to a stack of photos. In other words, there is only one stack of photos.

I think you want to imagine the stacks of photos as possibilities. But I find it incredibly difficult to relate to what you think you're doing when you pick a stack of photos. We don't get to pick what the contents of the dog houses are; all we get to do is search them.

You can choose to prove ...and I pay you $500 ... — TonesInDeepFreeze

Your hypothetical reward system is all messed up. Guessing when you don't know should be worthless. Finding out should be valued. You have that exactly backwards... your reward system rewards only guessing and lucking up.

But, clearly, one should choose the best chance at having the shortest labor time - by choosing to prove there is a picture of a black dog. — TonesInDeepFreeze

You've yet to actually argue against the critique... given it's the same search being done on the same dog houses, it's the same amount of effort regardless of what you pick. Imagining rigged rewards for guessing when you don't know and lucking up doesn't change the fact that it's just those dog houses with those dogs in it that we search in, and that doesn't change no matter what we wish up to be true before we do the search.

In fact, why do you even need to pick one to do the search in the first place? Why not simply do the search?

Doghouses don't hurt, but they're not necessary.

The question was "Which is easier to prove: ExBx or ~ExBx ?"

The only way that question makes sense is to compare ExBx when it is true vs. ~ExBx when it is true, because if ExBx is false then there's no proof of it and if ~ExBx is false then there is no proof of it.

If ExBx is true, then it is possible that it might be proven quickly. But if ~ExBx is true, then it can only be proven by showing all possible cases.

The question was "Which is easier to prove: ExBx or ~ExBx ?" — TonesInDeepFreeze

No, that's not the question. The question is whether it's easier to prove a negative claim or a positive claim.

Here's how TMF phrased it in the OP:

Suppose a theist claims that god exists, and you being an atheist claims the contrary, god doesn't exist. If now you're asked to prove god doesn't exist, that would be proving a negative. — TheMadFool

Joe claims there's a God. George claims there's no God. The former is a positive claim. The latter is a negative claim. Which of those two things is easier to prove?

Now I state the God thing here because TMF did, to tie it to the topic, but we can be a bit more neutral with something like... Joe claims the Goldbach conjecture is true. George claims the Goldbach conjecture is false. Which of those two things is easier to prove?

The only way that question makes sense is to compare ExBx when it is true vs. ~ExBx when it is true, because if ExBx is false then there's no proof of it and if ~ExBx is false then there is no proof of it. — TonesInDeepFreeze

1. It is easier to prove that the Four Color Theorem is false than it is to prove that the Four Color Theorem is true.
2. It is easier to prove that the Four Color Theorem is false if it is false than it is to prove that the Four Color Theorem is true if it is true.

The former is the real topic... that's the thing you claim you can't make sense out of. The latter, as I understand it, is the thing you're claiming is the only thing that makes sense.

So if you want to claim this makes sense, explain this to me. I know the FCT is true, and I know the proof of it was incredibly difficult. But when you talk about this thing called the proof of FCT being false if it is false, what sensible thing are you comparing the FCT's proof of being true to exactly?

Now if you want to compare proving a false thing false to proving a true thing true, that makes sense. But you're not telling me that. You're trying to tell me that you can compare the proof of a true thing being false to the proof of it being true, or the proof of a false thing being true to the proof of it being false, or maybe that simply not knowing whether you're comparing the proof of a true thing being false to the proof of it being true or you're comparing the proof of a false thing being true to the proof of it being false makes sense out of it somehow. And, no, it doesn't.

The question was "Which is easier to prove: ExBx or ~ExBx ?"
— TonesInDeepFreeze
No, that's not the question. The question is whether it's easier to prove a negative claim or a positive claim. — InPitzotl

An existential vs its negation.

I used 'black dog' only because it came into the discussion as an example.

Goldbach conjecture — InPitzotl

The juncture in the discussion I have recently been addressing is not of deductive determination, but rather empirical determination, on a case by case basis, in a finite domain.

You're trying to tell me that you can compare the proof of a true thing being false to the proof of it being true — InPitzotl

No, I am not.

or maybe that simply not knowing whether you're comparing the proof of a true thing being false to the proof of it being true or you're comparing the proof of a false thing being true to the proof of it being false makes sense out of it somehow. — InPitzotl

That's not what I've said.

An existential vs its negation. — TonesInDeepFreeze

Sure. But generally speaking we agree that one of them is true, and one of them is false. And with the metric/method under consideration, we don't know which is which until either we find the black dog, or we searched all of the dog houses among the single set of dog houses.

I used 'black dog' only because it came into the discussion as an example. — TonesInDeepFreeze

How nice of you, but "black dog" only came into the discussion as an example because the discussion started to be about black dog as an example.

The point in the discussion I have recently been addressing is not questions of deductive determination, but rather empirical determination in a finite domain. — TonesInDeepFreeze

Okay, so let's talk about dogs then. What exactly is your problem with my table, as it applies to the metric we were discussing in regards to empirical determination in a finite domain?

one of them is true, and one of them is false. And with the metric/method under consideration, we don't know which is which until either we find the black dog, or we searched all of the dog houses among the single set of dog houses. — InPitzotl

Yes. But that doesn't vitiate anything I've said.

How nice of you — InPitzotl

I don't see a basis for your sarcasm.

but "black dog" only came into the discussion as an example because the discussion started to be about black dog as an example. — InPitzotl

The thread didn't start with "black dog" and went for a while without it. Anyway, I don't know what you're driving at. You said that the question was not as I couched it, so I merely replied that the question indeed was which is easier to prove regarding a "black dog".

What exactly is your problem with my table — InPitzotl

I don't claim to understand what you intend to say with your chart. I can only say that the best I can glean from it is that it shows the difficulty in discovering whether there exists a black dog.

Meanwhile, again, what strikes me are these facts:

If there exists a black dog, then proving there exists a black dog might end early.

If there does not exist a black dog, then proving there does not exist a black dog will not end early.

Two more facts though that I admit that I don't know how to weigh in:

If there does not exist a black dog, then there is no proof that there exists a black dog, and trying to prove that there exists a black dog will not end early.

If there does exist a black dog, then there is no proof that there does not exist a black dog, but trying to prove that there does not exist a black dog might end early.

I don't see a basis for your sarcasm. — TonesInDeepFreeze

The basis is that you volunteered that you only talked about it because it was mentioned.

The thread didn't start with "black dog" and went for a while without it. — TonesInDeepFreeze

Whereas that's true, it was TMF that started both the thread and the black dog discussion.

You said that the question was not as I couched it, so I merely replied that the question indeed used the example of "black dog". — TonesInDeepFreeze

I'm saying something much more specific. The question in this thread is about the burden of proof as it applies to negative claims versus positive claims. The notion being suggested is that positive claims have a burden of proof because such claims are easier to prove. That is TMF's idea, and I think it's too generic to be correct. My suggestion as to where the burden lies is more: "it depends". In other words, a claim merely being negative or positive does not tell you which of the two claimants has a burden or what it is. In terms of TMF's easy theory, it doesn't even change the task, or how difficult it is to go about it (see below).

I don't claim to understand what you intend to say with your chart. — TonesInDeepFreeze

You replied to it. You said this:

That chart seems to capture discovery not proof. For example, the min in row 4 is 1 only because we discover that there is a black dog and give up trying to prove that there is not one. But that is not the task. The task is to prove there is not a black dog. — TonesInDeepFreeze

But let's take that as an example. Your task is to prove there is not a black dog. That is rows 2, 4, 6, 8.

If there exists a black dog, then proving there exists a black dog might end early. — TonesInDeepFreeze

That condition holds in rows 3, 5, 7.

If there does not exist a black dog, then proving there does not exist a black dog will not end early. — TonesInDeepFreeze

That's row 2.

If there does not exist a black dog, then there is no proof that there exists a black dog, and trying to prove that there exists a black dog will not end early. — TonesInDeepFreeze

That's row 1.

If there does exist a black dog, then there is no proof that there does not exist a black dog, but trying to prove that there does not exist a black dog might end early. — TonesInDeepFreeze

That condition holds in rows 4, 6, 8.

This is part of what the table is doing. If we can map what you mean to say to the rows, we can be precise. The other part of what the table is doing is showing you (at least in a hypothetical sketch) what all of the rows look like in terms of what you're meaning to say so you can see if you're missing something.

But we can proceed then to the next point. If we do apples to apples comparisons between what you're calling trying to prove ExBx and trying to prove ~ExBx, then we should rightfully start with states of affairs. If it turns out there are 0 black dogs, we're comparing 1 with 2. If it turns out there's 1 black dog, we're comparing 3 with 4, and so on. In all such cases, how difficult it is to either confirm or disconfirm your claim, whichever the case may be, is completely independent of whether your claim is the negative one or the positive one. It depends, instead, entirely on what the state of affairs is. You do maximal work on rows 1 and 2. Of the rows shown you're on average doing minimal work on rows 7 and 8.

And again might I emphasize that it's not necessary when doing P to be "trying to prove ExBx" nor to be "trying to prove ~ExBx"; you can do P without "trying to" confirm either theory... one might call this "trying to figure out whether ExPx is true or ~ExPx is true". We might phrase doing such a thing as making neither a negative claim nor a positive one, yet taking on the burden regardless.

I don't see a basis for your sarcasm.
— TonesInDeepFreeze
The basis is that you volunteered that you only talked about it because it was mentioned. — InPitzotl

I really don't get you. I didn't claim that I was "nice" to do that. Only that you said that the question was not "Which is easier to prove: ExBx or ~ExBx ?", so I replied that the existential was the question and I only referred to black dogs in particular because that was being discussed

a claim merely being negative or positive does not tell you which of the two claimants has a burden or what it is. — InPitzotl

I agree.

I don't claim to understand what you intend to say with your chart.
— TonesInDeepFreeze
You replied to it. You said this:
That chart seems to capture discovery not proof. For example, the min in row 4 is 1 only because we discover that there is a black dog and give up trying to prove that there is not one. But that is not the task. The task is to prove there is not a black dog. — InPitzotl

Yes, I said that is what it seems to me. I don't claim to understand nor to represent what you mean by it. Merely, that is what it seemed to me.

So that wasn't so difficult. — InPitzotl

For odd n, row n+1 - min max - is the same as row n. They are the same because, as far as I can tell, they don't capture the difference in the challenge of proof.

The situation is not:

"Team A, discover whether there is a black dog; and Team B, discover whether there is a black dog."

Rather the situation is:

"Team A, you win if you prove there is a black dog; and Team B, you win if you prove there is not a black dog. "

And in that situation, Team A might win early, but Team B cannot win early. Team A might prove their claim early, but Team B cannot prove their claim early. As far as I can tell, your chart doesn't capture that.

Only that you said that the question was not "Which is easier to prove: ExBx or ~ExBx ?", so I replied that the existential was the question and I only referred to black dogs in particular because that was being discussed — TonesInDeepFreeze

Here's the discussion leading up the black dogs:

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". — InPitzotl

On what exactly?

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

...and so on.

For odd n, row n+1 - min max - is the same as row n. They are the same because, as far as I can tell, they don't capture the difference in the challenge of proof. — TonesInDeepFreeze

But the reason they don't capture a difference in challenge is because the state of affairs is the same. You have the same number of total dogs and the same number of black dogs.

Team A, you win if you prove there is a black dog; and Team B, you win if you prove there is not a black dog. — TonesInDeepFreeze

Okay, you've made a claim that this is the situation. Back it up.

Tell me what "Team A wins" has to do with negative versus positive claims in relation to burden of proof.

Also, what about Team C, who just wants to figure things out without making claims? The guys who just mix the two chemicals and watches rather than pathetically trying to tell the chemicals what to do before they mix them? Are they just losers in this picture?

Here's the discussion leading up the black dogs — InPitzotl

So? it doesn't vitiate anything I said nor show a basis for your sarcasm.

the state of affairs is the same — InPitzotl

The facts are the same. But the question is not what the facts are, but what is the difficulty in proving the facts. What is the difficulty in proving ExBx when ExBx is true vs. the difficulty in proving ~ExBx when ~ExBx is true?

Team A, you win if you prove there is a black dog; and Team B, you win if you prove there is not a black dog.
— TonesInDeepFreeze
Okay, you've made a claim that this is the situation. Back it up. — InPitzotl

You're serious? It's a characterization of the problem if the context were a debate. If you don't like "team" and "win" then:

Person A sustains his claim when he proves there is a black dog. Person B sustains his claim when he proves there is not a black dog.

Tell me what "Team A wins" has to do with negative versus positive claims in relation to burden of proof. — InPitzotl

I might be corrected on this, but I don't recall making a claim about "burden of proof" in sense of a rhetorical obligation (as "burden of proof" is usually meant). Rather, I pointed out that Positive is easier in the particular sense that if Positive were correct then it might be proved earlier than Negative could be proved if Negative were correct.

what about Team C, who just wants to figure things out without making claims? — InPitzotl

They're discovering the facts, not claiming what the facts are, as opposed to the Positive claimer and Negative claimer who both are claiming what the facts are.

Your Team C seems to be a red herring.