But the question is not what the facts are, but what is the difficulty in proving the facts. — TonesInDeepFreeze

According to you, I cannot prove my claim if my claim is false. That implies that being able to prove the claim true in the first place requires my claim to be a fact. This is why you have to dance between two contradictory facts:

What is the difficulty in proving ExBx when ExBx is true vs. the difficulty in proving ~ExBx when ~ExBx is true?

...and the fact that this comparison requires dancing between two contradictory facts is just one of the things that makes this meaningless. There's also the fact that there's no meaningful way to measure "ExBx when ExBx is true" despite our having a metric, because that underspecifies what you're talking about.

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: — TonesInDeepFreeze

Wrong direction. I think burden of proof for claims applies in a wide variety of areas having nothing to do with winning debates. Furthermore, debates of the type you're describing seem to be relatively rare. The OP of this very thread had an example where a person's partner is trying to convince the person that there is a bear in their house... that's a claim with a burden, but there's no debate going on here... just the search for a bear. And that's not a win. The problem here is not that I dislike the word team, or the word win. It is that I think your view that this thread is about "winning debates" is cartoonish.

I might be corrected on this, but I don't recall making a claim about "burden of proof" in sense of a rhetorical obligation — TonesInDeepFreeze

But you said:

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. " — TonesInDeepFreeze

"Burden of proof" is literally in the title of this thread.

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. — TonesInDeepFreeze

They're invoking P and arriving at either a proof of ExBx or a proof of ~ExBx depending on what the state of affairs are. And by our metric they expend the same exact effort Team A or Team B would in proving it. So your red herring accusation doesn't hold up in terms of the difficulty of proving a negative claim or proving a positive claim.

According to you, I cannot prove my claim if my claim is false. That implies that being able to prove the claim true in the first place requires my claim to be a fact. — InPitzotl

Correct.

What is the difficulty in proving ExBx when ExBx is true vs. the difficulty in proving ~ExBx when ~ExBx is true?
The comparison is meaningless. Convince me otherwise. — InPitzotl

What you're asking requires that I repeat myself.

To prove ExBx, the prover might end early. To prove ~ExBx, the prover cannot end early.

I think burden of proof for claims applies in a wide variety of areas having nothing to do with winning debates. — InPitzotl

Agree. So what? I didn't say it has to be a debate. So, since you bridled at a debate, I also offered it in about as neutral terms as I can:

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. — TonesInDeepFreeze

And I didn't even opine about "burden of proof"; I only commented on comparative difficulty.

I might be corrected on this, but I don't recall making a claim about "burden of proof" in sense of a rhetorical obligation
— TonesInDeepFreeze
But you said:
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. "
— TonesInDeepFreeze — InPitzotl

So what? I didn't say anything there about who has "burden of proof".

Have you been thinking all along that I've been making some kind of argument about who has, or should have, the burden of proof or a greater burden of proof? I have not made such an argument. I didn't claim that the difference of difficulty implies or does not imply anything about who should have a burden of proof.

"Burden of proof" is literally in the title of this thread. — InPitzotl

So what? That burden of proof is the main subject of the thread doesn't entail that I can't also comment on individual points that have arisen. The point I have lately been commenting on has been the difference in difficulty between proving ExBx and proving ~ExBx. That difference was offered by a poster as reason to assign burden; but I have not gone on to claim one way or the other that that difference should be a reason for assigning burden. I only commented on the difference itself.

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.
— TonesInDeepFreeze
They're invoking P and arriving at either a proof of ExBx or a proof of ~ExBx depending on what the state of affairs are. And by our metric they expend the same exact effort Team A or Team B would in proving it. — InPitzotl

Team C is dedicated to discovery of whether ExBx or ~ExBx and in that discovery arises a proof of one or the other. That is not the same as Team A declaring ExBx and then whether they can prove it or Team B declaring ~ExBx and then whether they can prove it.

Whether Team C ends early depends on whether ExBx is true or ~ExBx is true.

Team A might prove its claim and end early only if ExBx is true.
.
Team B cannot both prove its claim and end early.

Those are three different situations.

If the discussion here is only about a Team C that is out to discover which is the case but not at the outset to make a claim one way or the other, then that it is a very different discussion from the one that had been presented here, which is that of opposing views being claimed, not just discovery. Of course, you're welcome to prefer to delve into the implications of a Team C situation, but it's not the situation I have been addressing.

What is the difficulty in proving ExBx when ExBx is true vs. the difficulty in proving ~ExBx when ~ExBx is true?
The comparison is meaningless. Convince me otherwise. — InPitzotl

What you're asking requires that I repeat myself. — TonesInDeepFreeze

Repeating the comparison doesn't get you any closer to convincing me that it's a meaningful comparison. Suppose I have a function f(x). I can say f(0) might be 1. I can say f(0) might be 2. I can say 1<2; that's comparing 1 to 2. But I propose that saying "f(0) if f(0) is 1 is less than f(0) if f(0) is 2" is gibberish.

Repeating the gibberish does nothing to advance the notion that it's meaningful. Repeating it is just superfluous. IOW, no, repeating yourself is neither required nor helpful.

So what? I didn't say anything there about who has "burden of proof". — TonesInDeepFreeze

That's entirely correct. You didn't say anything there about who has "burden of proof". And:

So what? That burden of proof is the main subject of the thread doesn't entail that I can't also comment on individual points that have arisen. — TonesInDeepFreeze

...that is also correct. That the burden of proof is the main subject of the thread doesn't entail that you can't also comment on individual points that have arisen.

The point I have lately been commenting on has been the difference in difficulty between proving ExBx and proving ~ExBx. — TonesInDeepFreeze

..but that is incorrect, or at least it's not the whole story. In this post:

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. " — TonesInDeepFreeze

...you're explicitly telling me what something you call "the situation" first is not, and second rather is. What is meant by declaring "the situation" to be that second thing and not that first thing you don't state, but there's some implication that you really, really want me to care about that second thing and to not care about that first thing.

When I ask you to connect "the situation" to the topic at hand, I'm not by doing so claiming you made that connection... rather, I'm prompting you to justify why I should be interested in this thing you're calling "the situation". If you want me to be interested in Team-A-winning, you need to sell it to me. Telling me it has nothing to do with that conversation is not a great sales pitch for my caring about it.

Whether Team C [could end] early depends on whether ExBx is true or ~ExBx is true.
Team A might prove its claim and end early only if ExBx is true.
Team B cannot both prove its claim and end early. — TonesInDeepFreeze

Sure, but there are symmetric descriptions of each of these things for Team A, Team B, and Team C in all of those scenarios. ~ExBx is identical to saying |{x:Bx}|=0. |{x:Bx}|=1 implies everyone might end early. |{x:Bx}|=2 and everyone will end early.

We can describe all sorts of scenarios involving Team {A|B|C} {confirming|disconfirming} {their|the} claim that {ExBx|~ExBx} under the condition |{x:Bx}|=k, 0<=k<=n.

At the heart of all of this shuffling of these variables, there's is the intentional carrying out of process J (see below), which was already described (it ends early when you see a black dog; the min/max number of steps is a function of the state of affairs).

If the discussion here is only about a Team C that is out to discover which is the case but not at the outset to make a claim one way or the other, then that it is a very different discussion from the one that had been presented here, which is that of opposing views being claimed, not just discovery. — TonesInDeepFreeze

To me, "discovery" versus "proof" is just a case of special labeling by you. The raw core of what is going on in terms of the cost of the thing and the thing being done that has that cost is that some entity undergoes some process J, which will end at some point when a black dog is discovered in a dog house or all dog houses have been searched, the former of which we get to label as the condition ExBx and the latter as the condition ~ExBx.

FYI, I'm changing the notation for the process from P (for process) to J (for justification).

If the discussion here is only about a Team C that is out to discover which is the case but not at the outset to make a claim one way or the other, then that it is a very different discussion from the one that had been presented here, which is that of opposing views being claimed, not just discovery. — TonesInDeepFreeze

I'm not interested in who is making claims, because it doesn't seem to affect how many steps J goes through, or what we are "J'd" in believing by the fact that J ended early or not whatever the case may be. ExBx is a positive claim. ~ExBx is a negative claim. I don't need claimants to give these those labels.

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 — InPitzotl

Excelente! :up:

Here's my own version of difficulty in re proof for the two statements, "some dogs are black" and "no dogs are black" [the former is a positive existential claim and the latter is the corresponding negative claim]

Difficulty: how many dogs we have to check. If a proof requires that we check all dogs, it's more difficult than a proof that doesn't require us to check only a few dogs

A. Proof of "some dogs are black": We begin searching for black dogs. There are three possibilities:
1. A black dog is found in the middle of the search. We didn't have to search all the dogs
2. There's only one black dog and that dog is the last dog we check. We searched all the dogs
3. There are no black dogs. We searched all the dogs

B. Proof of "no dogs are black":
1. We have to search all dogs (to make sure there are no black dogs)

Clearly, proving "no dogs are black" is more difficult, as defined above, than proving some dogs are black. See A1 and B1 vide supra.

"f(0) if f(0) is 1 is less than f(0) if f(0) is 2" is gibberish. — InPitzotl

And it's not a meaningful comparison to what I said.

That's entirely correct. You didn't say anything there about who has "burden of proof". — InPitzotl

So we'll disregard your comment about it, after I've pointed out it was not apropos.

That the burden of proof is the main subject of the thread doesn't entail that you can't also comment on individual points that have arisen. — InPitzotl

So we'll disregard your comment about it, after I've pointed out it was not apropos.

Also, my point stands that your sarcasm about my saying why I referenced a black dog was without basis.

What is meant by declaring "the situation" to be that second thing and not that first thing you don't state — InPitzotl

but there's some implication that you really, really want me to care about that second thing and to not care about that first thing. — InPitzotl

No, there is not. Please stop reading past what I actually posted to jump to your own incorrect conclusions about it. I have no interest in what you care about. I'm merely pointing out that there is a difference between (1) claims of opposing views about facts and (2) mere discovery about facts. You really don't see that?

If you want me to be interested in Team-A-winning — InPitzotl

"situation", "team", "winning", et. al are merely tropes to illustrate. I am not claiming that the discussion here is confined to talking about winning debates or being hired to prove the existence of a picture in a stack or any other particular illustration. I even made this clear when I said (twice) that we can reduce to more neutral terms:

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. — TonesInDeepFreeze

Whether Team C [could end] early depends on whether ExBx is true or ~ExBx is true.
Team A might prove its claim and end early only if ExBx is true.
Team B cannot both prove its claim and end early.
— TonesInDeepFreeze
Sure, but there are symmetric descriptions of each of these things for Team A, Team B, and Team C in all of those scenarios. — InPitzotl

The differences are:

(1) If ExBx is true, then Team A will prove its claim and might do so early.
(2) If ExBx is false, then Team A will not prove its claim and it won't fail early.

(3) If ~ExBx is true, then Team B will prove its claim but it won't do so early.
(4) If ~ExBs is false, then Team B will not prove its claim but it might fail early.

(5) If ExBx is true, then Team C will discover that it is true and might do so early.
(6) If ExBx is false, then Team C will discover that it is false but it won't do so early.

(1) compared with (3) gives difficulty more to Team A

The question that most interests me (and as it seems to me to be the question that most interests certain other people) is: What is the comparatively difficulty in proving ExBx and ~ExBx? That is the comparison between (1) and (3). And there's a difference.

But, as I mentioned, in a previous post, I admit that if we also consider (2) and (4) then it is not obvious how to weigh for comparison between Team and Team B. But the need to weigh for that pertained only to certain illustration I gave and which I later said I may need to retract it. And, again, for the question of proving a claim, (2) and (4) are not relevant in the same way that (1) and (3) are. If the proposition is false for a team, then they simply can't prove it so, of course, their difficulty to PROVE is maximal.

We can consider two possible worlds: World A in which ExBx is true and World B in which ~ExBx is true.

With World A , Team A will prove its claim and might do so early.

With World B, Team B will prove its claim but won't do so early.

You may take the subject of this discussion to be whatever you like, but where the sense is taken to be "how difficult is it to prove?", then it seems to me that it is more difficult for Team A.

Clearly, proving "no dogs are black" is more difficult, as defined above, than proving some dogs are black. See A1 and B1 vide supra. — TheMadFool

Analogous to A3, B is missing the case where you discover a dog earlier.

Despite the fact that you're trying to prove there are no black dogs (~ExBx), you don't initially know whether ExBx or ~ExBx. So say you start, you check the first dog, and it's not black. Everything is the same; you still don't know whether ExBx or ~ExBx. So say you check the second dog, and it is in fact black. But at that point everything changes. Suddenly, you know ~ExBx is false, and you know ExBx is true. You failed to prove your claim, but that piece of knowledge means your proof halted (in failure, but halted nonetheless).

You failed to prove your claim — InPitzotl

:chin:

We have two claims to consider: E = some dogs are black and N = no dogs are black. What I'm saying is it's easier to prove E than N for the simple reason that N requires a complete search of ALL dogs while E doesn't necessarily require that.

What I'm saying is it's easier to prove E than N for the simple reason that N requires a complete search of ALL dogs while E doesn't necessarily require that. — TheMadFool

Sure, but it's just as easy to disconfirm N as it is to prove E. Not only is it just as easy, but in our toy scenario it's literally the same thing. And it's just as easy to prove N as it is to disconfirm E. There will be some state of affairs, whatever it is... that might be E, or it might be N. If it is E, then P for both E and N are exactly as difficult as each other. If it is N, the P for both E and N are exactly as difficult as each other.

Given a state of affairs, the difference in effort has nothing to do with whether you're claiming E or N. The only thing that relies on whether you're claiming E or N is whether in the end you are confirming or disconfirming your claim. Does that make sense?

Sure, but it's just as easy to disconfirm N as it is to prove E. — InPitzotl

Yes but I'm not talking about disproving N which is equivalent to proving E. I'm interested in knowing whether it's easier to prove N or easier to prove E.

Yes but I'm not talking about disproving N which is equivalent to proving E. I'm interested in knowing whether it's easier to prove N or easier to prove E. — TheMadFool

But you can only prove N if N, and you can only prove E if E. Since N and E cannot both be true, the comparison between the proof of N and the proof of E is illegitimate.

We can side step this by just considering distinct claims. E=there exists a black dog, N=there does not exist a purple frog. Even here though, it still depends... how many dogs are there versus frogs? So we should fix that as well... for apples to apples comparisons, there are exactly as many of each. Finally, we'll suppose the set of affairs matches up... but we may as well overspecify while we're at it... there are no purple frogs, but there are at least 2 black dogs.

So now we have something that's meaningfully comparable; and indeed proving E is easier than proving N.

But you can only prove N if N, and you can only prove E if E. Since N and E cannot both be true as states of affairs, the comparison between the proof of N and the proof of E is illegitimate. — InPitzotl

You're correct that "some dogs are black" and "no dogs are black" are contradictory (both can't be true and both can't be false or, in other words, one has to be true and the other false). The whole point of this thread is to compare such pairs of statements (a positive statement and its negation, the corresponding negative statement) in re which is easier to demonstrate as a truth.

The whole point of this thread is to compare such pairs of statements (a positive statement and its negation, the corresponding negative statement) in re which is easier to demonstrate as a truth. — TheMadFool

But given we're talking about empirical claims, I think you get into trouble when you entertain comparing something real to something hypothetical. Would it be easier for me to prove the Goldbach conjecture is true, or to prove the Goldbach conjecture is false? The real answer is that I can only prove at most one of those things, and the other one, given I can't prove it, leaves me nothing to compare that proof to. Would it be easier for me to prove there is intelligent extra-terrestrial life in our galaxy, or to prove there isn't intelligent extra-terrestrial life in our galaxy? Again, the real answer is that I can only prove at most one of those two things (presuming it's well defined enough to be crisp).

he real answer is that I can only prove at most one of those two things — InPitzotl

Indeed but proving one disproves the other (contradictory).

Indeed but proving one disproves the other (contradictory). — TheMadFool

Exactly. That's why even though it takes n steps to prove there are no black dogs, if you find one on step 2 you can stop.

We can also phrase this in terms of philosophical knowledge. Maybe you believe the claim you're trying to prove. But before you prove it you don't have any real knowledge of it; you have an unjustified belief (UB). If you happen to be right, that's just an unjustified true belief (UTB). If you're wrong, it's a UFB.

The search ends as soon as you get justification, either for or against the thing you set out to prove. The person trying to prove there were no black dogs was all pepped up, fully prepared to look at all n dogs. Before step 1, this person had a UFB that ~ExBx. At step 1, same thing... it's a UFB that ~ExBx. But by step 2, the person attains a JTB that ExBx, which means ~ExBx is false.

Exactly. That's why even though it takes n steps to prove there are no black dogs, if you find one on step 2 you can stop. — InPitzotl

But then you haven't proven "there are no black dogs". You've proven "some dogs are black." :chin:

But then you haven't proven "there are no black dogs". You've proven "some dogs are black." — TheMadFool

Yes. But in proving "some dogs are black", you have proved your initial claim futile! Searching that third dog won't do you any good.

I think it's more than healthy to accept that you can be wrong... that's basically (ironically) the only proper way to be right. You discover that your wrong claim is wrong as fast as possible, then move on. Luckily for this guy, he was able to discard his false belief on step 2.

So we'll[you'll] disregard your comment about it, after I've pointed out it was not apropos. — TonesInDeepFreeze

FTFY.

And it's not a meaningful comparison to what I said. — TonesInDeepFreeze

I'll take that as a position statement, since you didn't bother convincing me of anything. That leaves my position that your comparison is meaningless untouched.

I have no interest in what you care about. — TonesInDeepFreeze

That is clearly false, because you keep replying to me and "merely stating" things directly to me.

I even made this clear when I said (twice) that we can reduce to more neutral terms — TonesInDeepFreeze

The neutrality of the terms has nothing to do with my lack of interest in what you're telling me.

Let's try this dimension. You and I both agree that the min and max steps J will take before halting depends on the number of dogs in that table. I argue, and actually show, that the min and max steps J will take before halting does not depend on what you set out to claim before you initiate J. Now you are comparing this:

(1) compared with (3) gives difficulty more to Team A — TonesInDeepFreeze

...where 1 and 3 respectively are:

(1) If ExBx is true, then Team A will prove its claim and might do so early. ... (3) If ~ExBx is true, then Team B will prove its claim but it won't do so early. — TonesInDeepFreeze

Your (1) as phrased is closest to 3 in my table. Your (3) is a great match to 2 in my table.

Row 3 in my table has a min/max difficulty of (1,n), because there is 1 black dog. Row 2 in my table has a min/max difficulty of (n,n), because there are 0 black dogs. Row 1 has the same min/max difficulty as Row 2, because despite the claim column not matching the will prove column, the # dogs column which is the real dependent variable is the same. Row 4 has the same exact difficulty as row 3. because despite the claim column not matching the will prove column, the # dogs column which is the real dependent variable is the same. Ignoring Rows 1 and 4 does not make the min/max difficulty dependent on the claim.

Furthermore, I don't think you even disagree with this. The only reason for me to state it is to make it crystal clear that this is what we agree on. Good!

And, again, for the question of proving a claim, (2) and (4) are not relevant in the same way that (1) and (3) are. — TonesInDeepFreeze

Consider this. We have Joe who always makes a negative claim, and George who always makes a positive one. It cost one dollar to do one J step. For apples to apples comparisons, George and Joe are going to attempt to prove every theorem that goes their way, and they will always check all of the metaphorical dog houses in the same order. Let's say there are several thousands of such claims. Then by the last claim, George and Joe paid the same amount of money trying to prove their negative and positive claims. Sure, if we ignore all of those times George paid $n to find out he was wrong and Joe paid $k<n to find out he was wrong, George might pay less total money than Joe. But that is not a real argument that positive claims are cheaper than negative ones.

And, again, for the question of proving a claim, (2) and (4) are not relevant in the same way that (1) and (3) are. — TonesInDeepFreeze

If (1) can happen to George (4) ipso facto can happen to Joe. If (3) can happen to Joe (2) ipso facto can happen to George. The symmetry here guarantees equal grounding for costs paid.

We can consider two possible worlds: World A in which ExBx is true and World B in which ~ExBx is true. — TonesInDeepFreeze

Sure, we can do that. But only one possible world is our actual one. But there's nothing stopping us from partitioning the actual world. There exists no black dogs... in Saskatchewan. There exists a black dog... in Uzbekistan. But how would this help, say, getting George to pay less money than Joe?

You may take the subject of this discussion to be whatever you like, but where the sense is taken to be "how difficult is it to prove?", then it seems to me that it is more difficult for Team A. — TonesInDeepFreeze

Only if you partition George and Joe's piles by what you want to call "that is a proof" do you have a chance that George pays less than Joe. But that requires you to cherry pick, and I literally mean requires. Without cherry picking there is no cost benefit.

There exists no black dogs... in Saskatchewan. There exists a black dog... in Uzbekistan. — InPitzotl

I don't want to have to spell or copy/paste those long place names every time in discussion.

Let ExUx stand for "there exists a black dog in Land U" and ~ExSx stand for "there does not exist a black dog in Land S. And let both be true. Let n = the number of cases, and the number of cases be the same for both.

Team U will prove its claim possibly in only 1 step. Team S will prove its claim only in n steps.

cherry pick — InPitzotl

I am comparing cherries (succeeding to prove) to cherries, not a blend of cherries and raspberries (failing to prove) to a different blend of cherries and raspberries. The blends are different because:

(2) If ExUx is false, then Team U will not prove its claim and it won't fail early.[/quote]

and

(4) If ~ExSx is false, then Team S will not prove its claim but it might fail early.

are different.

It is not clear how to compare the blend of (1) and (2) with the blend of (3) and (4).

I take the context to be comparing cherries to cherries. Otherwise it would depart from the ordinary question or disagreement people have about this subject. If we take away the immediate comparison of proof of one of two contradictory claims, then we take out the very thrust of talking about the subject.

I don't unquestioningly rely on Wikipedia for information or explanation, but this article does at least reasonably capture the context we often find:

"The difference with a positive claim is that it takes only a single example to demonstrate such a positive assertion ("there is a chair in this house" is proven by pointing to a single chair), while it is typically harder to demonstrate a negative assertion ("there is no chair in this house" requires a thorough search of the house, including any potential hidden crawl spaces)." https://en.wikipedia.org/wiki/Burden_of_proof_(philosophy)#Proving_a_negative

That is a difference in demonstration, not merely the sameness in discovery.

/

So we'll[you'll] disregard your comment about it, after I've pointed out it was not apropos.
— TonesInDeepFreeze
FTFY. — InPitzotl

I suspect that doesn't say what you meant it to say. If I am quoted as saying 'you' then 'you' would refer to you not me.

Anyway, 'we' was the editorial we. And, yes, your comments about "burden' should be disregarded.

And, you've not recognized that your other sarcasm was gratuitous. I mention that as it would help to know that I'm talking with someone who has the ability to recognize a rhetorical mistake.

I have no interest in what you care about.
— TonesInDeepFreeze
That is clearly false, because you keep replying to me and "merely stating" things directly to me. — InPitzotl

I reply to you, while also for whomever is reading, to express my thoughts, and hopefully to communicate. That doesn't entail that I'm interested in whether you care about any particular matter in the discussion.

The neutrality of the terms has nothing to do with my lack of interest in what you're telling me. — InPitzotl

The neutrality was to emphasize that my reasoning does not depend on particular tropes I used for illustration.

But I appreciate your candor in telling me that you're not interested in what I have to say.

Team U will prove its claim possibly in only 1 step. Team S will prove its claim only in n steps. — TonesInDeepFreeze

I've no problem with that; but to be more precise, we don't know U will prove its claim in 1 step. But we do know U will prove its claim in less than n steps.

I take the context to be comparing cherries to cherries. — TonesInDeepFreeze

You're mixing metaphors. Cherry picking is a type of selection bias where a person selects data that appears to confirm a conclusion (the metaphorical "cherry picking") while ignoring data that disconfirms it. "Cherries to cherries" sounds more like apples to apples (and its twin idiom "apples to oranges") which refers to comparing comparable things (in the case of apples to apples) or incomparable things (in the case of apples to oranges).

It is not clear how to compare the blend of (1) and (2) with the blend of (3) and (4). — TonesInDeepFreeze

I gave you that exact model. I'll back fill it with justification. If we're using a metric that taking 4 steps is half as difficult as taking 8 steps, then the thing we're measuring is how many steps we take. Hence, George and Joe both pay one dollar every time they take one step. So if George pays 5 dollars, it means he took 5 steps. If Joe pays 6 dollars, then George took less steps than Joe did.

Now in (1), George is taking steps to prove ExBx. In (2), George is also taking steps to prove ExBx. So your blend of (1) and (2) is basically George taking each step in process J. Likewise, in (3) Joe is taking steps to prove ~ExBx. And in (4) Joe is also taking steps to prove ~ExBx. So your blend of (3) and (4) is Joe taking each step in process J. Both George and Joe compare things in the same way because that allows us to meaningfully compare George to Joe (that's the "apples to apples" part).

So there's claim 1 out of the 10,000 claims that go by. It can be anything, but let's say there are 50 dogs, none of them black. That's George case 2 and it is Joe case 3. George pays 50 dollars. Joe pays 50 dollars. George didn't prove his theory, but Joe did, but, both George and Joe paid 50 dollars. So they paid the same thing.

Now claim 2 goes by... there are 1000 dogs, 900 of them black, and it so happens the dog is found on the 3rd try. This is a George case 1 and it is a Joe case 4. George pays 3 dollars. Joe pays 3 dollars. This time, George did prove his theory, but Joe didn't, but both George and Joe paid 3 dollars. The total paid so far is 53 for George, and 53 for Joe. And so on.

But, we note, George did not in fact prove his claim 1, and Joe did not in fact prove his claim 2. So let's only count George's second payment, and Joe's first. Now, the total we get so far is that George paid 3 dollars proving his theory. And Joe? He paid 50. Aha! So George in proving his theories is paying less than Joe in proving his theories. Right?

Wrong. This is cherry picking, i.e., selecting among the data the points that seem to confirm your theory while ignoring the points that seem to disconfirm it. Specifically our conclusion requires us to have ignored the 50 bucks George did indeed pay and the 3 bucks Joe did indeed pay. That we're counting it because "these are different things" and "George didn't prove anything in his first claim" is simply rationalizing the selection bias. Paying attention to only the cases where George and Joe managed to prove what they set out to prove is the selection bias.

But I appreciate your candor in telling me that you're not interested in what I have to say. — TonesInDeepFreeze

There's just the single point I'm uninterested in, without you telling me why I should be. If I were generally uninterested in what you have to say, I wouldn't be talking with you.

Team U will prove its claim possibly in only 1 step. Team S will prove its claim only in n steps.
— TonesInDeepFreeze
I've no problem with that; but to be more precise, we don't know U will prove its claim in 1 step. — InPitzotl

No, I said "possibly".

You're mixing metaphors. — InPitzotl

No, I'm not. I'm moving to a different metaphor.

"Cherries to cherries" sounds more like apples to apples — InPitzotl

Exactly.

George [...] Joe — InPitzotl

You unnecessarily change the names and symbols for the examples. I accepted your Land U and Land S and mentioned Team U and Team S. I'll stick with that, so that I don't have to keep reconfiguring the notation:

(1) If ExUx is true, then Team U will prove its claim and might do so early.
(2) If ExUx is false, then Team U will not prove its claim and it won't fail early.

(3) If ~ExSx is true, then Team S will prove its claim but it won't do so early.
(4) If ~ExSx is false, then Team S will not prove its claim but it might fail early.

(1) Is clearly easier than (3).

(4) is clearly easier than (2).

I said that I don't know how to evaluate both (1) and (2) against both (3) and (4). What I mean is, how to evaluate while preserving the sense that it's a matter of proving not just discovering.

Of course, if we just reduce everything to both teams going step by step through their respective domains, then we may wipe out any difference. But that does not capture the essence of the question of how difficult it is to prove, not just how difficult it is to discover. I repeat myself because your "cherry picking" is not a valid objection to the fact that proving something is the case is different from discovering whether something is the case.

Again, if proposition P is false, and I ask a person, "How difficult is a proof of proposition P?" then he may say that is a nonsensical question, because there is no proof of P.

You don't have to share my framework in this matter, as you prefer a framework that wipes out the distinction. But my is the framework that interests me, and I think it is the framework that usually interests other people when this subject comes up - otherwise people wouldn't correctly emphasize that indeed it is more difficult to prove the negation (refined to consideration of case-by-case examination in a finite domain, which refinement I will continue to leave tacit).

To succeed in proving P (that is, to prove P) is different from succeeding to discover whether P is true or false (that is, to discover whether P is true or false). The question that interests me is "What is the difficulty to prove ExUx compared with the difficulty of proving ~ExSx?" and not "How difficult is discovering whether ExUx is true?"

An analysis that doesn't account for that distinction seems to me to be inadequate.

Again, the comparison that interests me is this:

Suppose ExUx is true and ~ExSx is true, and both have the same number of cases to check. What is the difficulty in proving ExUx compared with the difficulty of proving ~ExSx.

/

There's just the single point I'm uninterested in — InPitzotl

Then I correct my remark to say, "I appreciate your candor in saying you are not interested in that point, which happens to be one of the main points in my remarks".

Meanwhile I note that you still won't recognize that your point about "burden" was irrelevant and that your earlier sarcasm was gratuitous.

I think I've lost interest.

What does it mean to be asked to prove a negative? — TheMadFool

Why not first disambiguate between formal and less formal construals of key terms such as ‘negative claim’ and ‘proof’? There are obviously many proofs which substantiate negative claims (e.g., Euclid’s theorem proves that there is no largest prime number). However, this is the case regarding a formal construal of ‘proof’ — as in a more mathematical or logical sense. A formal interpretation for ‘proof’ would be something like ‘the derivation of some conclusion from mathematical or logical axioms given certain rules of logical inference’ — for example, ‘if P then Q,’ ‘P,’ ‘therefore, Q’ or by showing that Modus Ponens is itself a logical truth derivable from the axioms of logic. Whereas, in a less formal sense, an interpretation for ‘proof’ could be something like ‘a demonstration that need not follow from mathematical or logical truths and thus needn’t be logically guaranteed in this way” — so we can show things inductively, for example, or abductively, or just offer some support considered sufficient for the demonstration or acceptance of some conclusion. These ‘proofs’ are legitimate in a folk logic sense whereby the standards are set by a community to govern their own public discourse through conventional methods. However, at least as far as I tell, these informal uses by no means supersede the stronger formal sense.

Similarly, the informal use of the term ‘negative claim’ is but a colloquialism of a stronger, formal sense which can be interpreted as ‘any proposition where the main operator is a negation sign (¬),’ — so the proposition ‘not P’ (‘¬P’) is a ‘negative,’ for instance. On the other hand, it can otherwise be more charitably interpreted as ‘a proper subset of propositions where the main operator is a negation sign (¬), and the subset where that negation sign is followed by an existential quantifier’ — as exemplified by the proposition ‘there does not exist P’ (‘¬ ∃x P(x)’). Forgive my novice attempts at notation.

What about burden of proof? The received wisdom is that the person making a positive claim is the one who must produce the proof. This squares with what I've said. It's harder to prove a negative existential claim than a positive one; thus, if only because its easier, the burden of proof falls on those making positive existential claims. — TheMadFool

This no wisdom, rather it is pseudo-logic from a folk-logic understanding for how the burden of proof falls upon all existential claims — which are all claims affirming the existence (or lack thereof) of something. Principles such as Hitchens’s razor or the Sagan standard forward tenable objections to positive existential claims (e.g., ‘that ‘X’ exists.’) via dismissal on the grounds that their supporting evidence fails to provide sufficient warrant for to hold their position (e.g., ‘there is no evidence to support the existence of ‘X’.’). Here it is more appropriate to withhold any judgments committing us to either positive or negative existential claims, whereas general negative existential claims may form untenable objections to general positive existential claims since their negations cannot be substantiated, and thus fail to meet the burden of proof. This is because with informal arguments whether or not a piece of evidence meets the burden of proof required to substantiate a claim is determined by whatever standards are found acceptable by the community in which the public discourse is taking place. As I said earlier, there are formal arguments (e.g., logical syllogisms, mathematical theorems, etc.) which require mathematical or strictly logical proofs, and such casual domains of public discourse whereby the standard for evidence to meet the burden of proof is typically determined in the context of community standards and conventions are inferior models.

Evidence of absence is evidence of any kind that suggests something is missing or does not exist, whereas the absence of evidence is simply failing to find any evidence to support that something actually exists. As Sagan put it in his book, The Demon-Haunted World:, the expression “absence of evidence is not evidence of absence” is a critique of the “impatience with ambiguity” exhibited by appeals to ignorance. The appeal to ignorance is the claim that whatever has not been proven false must be true, and vice versa. A better analogy is Russell's teapot is an analogy, of which he specifically applied in religious contexts, insofar as it illustrates the philosophical burden of proof lies upon a person making empirically unfalsifiable claims, rather than shifting the burden of disproof to others. It goes to show that an agnostic position, which is in many cases more tenable and intellectually honest, suffices in regards to forwarding theistic; general, positive existential claims. In the words of Bertrand Russell:

I ought to call myself an agnostic; but, for all practical purposes, I am an atheist. I do not think the existence of the Christian God any more probable than the existence of the Gods of Olympus or Valhalla. To take another illustration: nobody can prove that there is not between the Earth and Mars a china teapot revolving in an elliptical orbit, but nobody thinks this sufficiently likely to be taken into account in practice. I think the Christian God just as unlikely — Bertrand Russell

Why not first disambiguate between formal and less formal construals of key terms such as ‘negative claim’ and ‘proof’? There are obviously many proofs which substantiate negative claims — Cartesian trigger-puppets

Good call but that's I was hoping other, more knowledgable, folks would do. What is, after all, a negative claim and what do we mean by proof?

To me, an affirmative claim is one that says how the world is and a negative claim is one about what the world is not. For example, "the sky is blue" states what the sky is and "the sky is not blue" what the sky is not.

A proof is a logical argument (inductive/deductive) that establishes the truth of a claim, positive/negative.

Let's look at some postive and negative claims and, what will be the cornerstone of my argument, their logical translations:

Gx = x is God

1. God exists: $(\exists x)(Gx)$


2. God does not exist: $(\forall x)(\neg Gx)$

God exists is an affirmative statement and is translated in logic with the existential quantifier ($\exists$) i.e. we only need one thing that is a god to prove it.

God does not exist, in logic, requires the universal quantifier ($\forall$) and to prove this statement we need to show how each and everything in the universe is not God.

It's easier to prove God exists than God does not exist or, negatively expressed, it's next to impossible to prove God does not exist. Hence, we can't prove a negative.

Gx = x is God

1. God exists: (∃x)(Gx)

2. God does not exist: (∀x)(¬Gx)

God exists is an affirmative statement and is translated in logic with the existential quantifier (∃
∃) i.e. we only need one thing that is a god to prove it.

God does not exist, in logic, requires the universal quantifier (∀∀) and to prove this statement we need to show how each and everything in the universe is not God.

It's easier to prove God exists than God does not exist or, negatively expressed, it's next to impossible to prove God does not exist. Hence, we can't prove a negative. — TheMadFool

Why would you need a universal quantifier for the negating proposition of an existential claim? Actually, let me just approach this less formally. Tell me where I going wrong in the following scenario:

1)
• Interlocutor 1: “God exists”

• Me: “What do you mean by ‘God’? Could you provide a definition?”

(Argumentation is pointless until we define terms. If “God” is defined as the God of Einstein an Spinoza, we may agree.)

2)
• Interlocutor 1: “The Christian God of course!”

• Me: “The Christian God is far too vague. There is significant diversity regarding the way Christian denominations define ‘God’.”

(I will dwell in the clarifying stage of the conversation until the term ‘God’ is clearly defined.)

3)
• Interlocutor 3: “God is the supreme being and creator of everything that exists. His existence is eternal and necessary. He is omnipotent, omnipresent, omniscient, and omnibenevolent. His involvement and love for us is both imminent (of this world) and transcendent (beyond this world).”

• Me: “The divine properties your predicating upon the being of God in your definition have elements which make them either non-absolute, or mutually exclusive. If God is all powerful, then He is either not all good, and morality is arbitrarily dictated on his whim, or He is not all powerful. If whatever God does is good simply because He did it, and if He is not bound by any limitations and is free to do whatever, then God has no moral system, no moral obligations, and no moral standards. On the other hand, if whatever God does is good because he has to, then he may be all good but he cannot be all powerful because morality makes him limited.”

“A being cannot possess the divine properties of all goodness, all powerful, and all knowing if there exists evil and suffering. There is evil. If God is unaware that there is evil, then he cannot be all knowing. If He is aware but cannot prevent it, then he cannot be all powerful. If he is aware and able to prevent evil but doesn’t, then he cannot be all good.”

(In most cases no arguments are needed. Terms are meaningless until they are clearly defined, so unless God is defined as “the universe” or “that which is beyond our abilities to conceive” or some other such metaphysical axiom, as the clarification brings more and more definition and less vagueness, eventually one of two things will happen. One, you reveal a contradiction, a physical impossibility, a mathematical impossibility, or fail to substantiate an empirical claim… easy — or, two, a clear definition is provided with no logical, physical, mathematical, empirical, etc. problems that you see… so you agree.)

Just like with the existential claim “there exists a microscopic teapot between the orbits of Mars and Jupiter” — and moreover, with an incoherent existential claim “there exists a duooc46hee57orch#bbdu56zzfzz+?54”. There is no need for a refutation, rebuttal, contradiction or even criticism tbh… because the term is noncognitive and unintelligible. Could it possibly mean something? Sure, but the onus of clarifying as well as the subsequent burden of proof, all notwithstanding is not on us. You very rarely need to ‘prove a negative’ but In the cases where you have to, make them do the work.

I was assuming the philosophical analysis you describe above was behind us, territory already covered. Once we get past all that, when and if we do, the issue boils down to what I alluded to in my post - existential vs. universal quantification.

I'm grateful though for the reminder on how philosophy is actually done. It helps novices like me to stay on track. :up:

I am the novice. I know much of what I say must surely be riddled with flaws, and I am incapable (as are you and everyone else) of being perfectly accurate. You have taken the time to show me some of those flaws before and for that I respect you. Whether you did so for reasons other than helping me or not, It doesn’t matter because I too am (if you indeed are) motivated by selfish reasons.

existential vs. universal quantification. — TheMadFool

Let me see if I’m understanding you… If I make such statements as ‘There is a universe,’ or ‘The universe exists,’ what I’m doing is making a positive claim (describing the way things are) by predicating a property to an object (the object being the ‘universe’ and the property being predicated upon it being ‘existence’), which requires the logical notation of an existential quantifier. But, it seems your telling me otherwise with regards to these statements negations? ‘There is not a universe,’ or ‘The universe does not exists,’ would just be the same with the exception of the negation sign. Universe is pretty much the best comparison for what most predicate upon God (besides transcendence possibly but we can’t yet generalize a metaphysical difference beyond logical possibilities)

I need a keyboard for notation. So I’ll not attempt a sloppy one.

Where Ux = x is a universe,

C = A universe exists = $(\exists x)(Ux)$

~C = No universes exist = $(\forall x)(\neg Ux)$

Where the universe = c,

The universe exists = $(\exists x)(Ux \wedge x = c)$

The universe does not exist = $(\forall x) \neg(Ux \wedge x = c)$

:flower: