Proving A Negative/Burden Of Proof
I've heard it being said more times than I care to count that anyone who demands proving a negative is being silly. This idea seems connected somehow to the notion of burden of proof.
Since these ideas are encountered most often in matters religious, I'll talk about them in that context. The results of these maxims or rules of thumb may vary in different situations.
That out of the way...
What does it mean to be asked to prove a negative?
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.
An analogy is in order...
Suppose you and your partner live together in a beautiful home somewhere. Your partner claims there's a bear in the house. If that claim is true, your partner should be able to show you the bear - fae would take you to the location where the bear is, point to it, and probably yell at you, "there! bear!". Imagine now yourself denying that there's a bear in the house. How would you prove to your partner that, "there isn't a bear in the house"? Well, you would have to take your partner to every single room in your house and show that there's no bear in any one of them. Quite,clearly, your task is more difficult compared to your partner's - you had to take your partner to all the rooms in your house while your partner only had to lead you to the room with the bear.
Make the following substitutions:
1. Your partner = theist
2. You = atheist
3. The house = the universe
4. The bear = god
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.
Before I go any further be alert to the fact that proving a negative is an issue in the setting of an existential claim - to demonstrate something doesn't exist is nigh impossible compared to the doing the opposite, proving that thing exists. After all, if you assert something exists, you would have proven it to yourself, and that means you know where that thing is.
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.
Comments...
Since these ideas are encountered most often in matters religious, I'll talk about them in that context. The results of these maxims or rules of thumb may vary in different situations.
That out of the way...
What does it mean to be asked to prove a negative?
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.
An analogy is in order...
Suppose you and your partner live together in a beautiful home somewhere. Your partner claims there's a bear in the house. If that claim is true, your partner should be able to show you the bear - fae would take you to the location where the bear is, point to it, and probably yell at you, "there! bear!". Imagine now yourself denying that there's a bear in the house. How would you prove to your partner that, "there isn't a bear in the house"? Well, you would have to take your partner to every single room in your house and show that there's no bear in any one of them. Quite,clearly, your task is more difficult compared to your partner's - you had to take your partner to all the rooms in your house while your partner only had to lead you to the room with the bear.
Make the following substitutions:
1. Your partner = theist
2. You = atheist
3. The house = the universe
4. The bear = god
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.
Before I go any further be alert to the fact that proving a negative is an issue in the setting of an existential claim - to demonstrate something doesn't exist is nigh impossible compared to the doing the opposite, proving that thing exists. After all, if you assert something exists, you would have proven it to yourself, and that means you know where that thing is.
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.
Comments...
Comments (168)
Your argument boils down to, absence of evidence is evidence of absence which, fortunately or not, is not as good as you seem to think it is. For instance, back in the heydays of exploration, as Europe likes to call it, there was absolutely no evidence that germs caused diseases and yet here we are in the, medically speaking, antibiotic age. Many more similar stories are available at the click of a mouse.
To sum up, absence of evidence is evidence of absence ain't really the appropriate response to theistic claims of god's existence.
By the way, you're barking up the wrong tree. What I really want to do is investigate the rationale behind,
1. Demanding that someone prove a negative is asking that person to do the impossible and thus the expression, "you can't prove a negative".
Since you seem more concerned about the example (theism vs atheism) than what it's supposed to illustrate (you're not supposed to ask someone to prove a negative), you should rest easy in that this response is used often and to good effect by atheists.
2. The burden of proof falls on the one making a positive claim which in the case of religious debates means the theist has to furnish the proof and not the atheist.
I have to admit I was trying to make an argument but non sequitur/strawman??? Please kindly expand and elaborate, if you don't mind.
What's your take on,
1. You can't prove a negative
2. Burden of proof
?
This is a genuine inquiry, attempted in good faith.
Proof you either haven't read my first post thoroughly or can't understand what you've read.
Your entire argument hinges on this: absence of evidence is evidence of absence. I gave an appropriate response to that.
True that I might not have understood what you wrote but set that aside for the moment and, if it isn't too much trouble, answer the questions I posed to you in the previous post. Dumb it down for me will ya? Thanks.
:rofl: Says you! :rofl:
That didn't come out right! :)
I meant to say it takes brains to dumb things down!
180's a good guy, but understandably, he tends to get a bit emotional/sensitive to defending his atheism.
Of course, we know EOG topics by nature are quite emotional undertakings anyway. Ironically enough, human feeling/sentience/phenomena itself, was found by Atheist Simon Blackburn, to be quite an abstract metaphysical undertaking... .
But getting back to proving a negative, is it true that the absence of evidence isn’t the evidence of absence?
Quoting TheMadFool
The only concern with that analogy that I can see would be objectivity v subjectivity. Meaning, what kind of truth are we relying on here to prove a negative? While it is true that if an Atheist makes a positive statement that 'God does not exist", he is put in a precarious untenable position of proving or defending same, since you would still have to grapple with which methods of logic to apply in making that case.
For instance, in the real world (so to speak), if you wanted to argue from inference, Modus Tollens, phenomenology, the religious experience, ineffable truth's, real subjective truth's, so on and so forth, that is one method. But, if we are discussing ontological argumentations based on the a priori, then that's another method, which of course is generally less persuasive in either direction, positive or negative (with the exception of analogizing to QM, uncertainty principle, bivalence, non-contradiction, etc. where an existential theist-Christian existentialist like myself- could poke holes in absolute kinds of thinking/reasoning).
But then another irony rears it head there. And that's because the a priori also encompasses mathematical abstract structures as found in the natural world. And as such, in ontology that includes abstract features of consciousness, like human sentience and the like. So now we're back to metaphysics and to 180's highly emotionally charged defensive reactions :joke:
Lot's of ironies in life.
Nice OP TMF!
A digression: If God is the universe, then everywhere you look . . . .
But yeah, anyone can prove A = A; but how to prove A does not = -A. It's not my burden because I never made the claim.
"One cannot prove a negative" and "absence of evidence is not evidence of absence" are not always true. An example would be using sonar on a pond and finding no fish - you have proven a negative (there are no fish in the pond), and absence of evidence of the fish is evidence the fish are absent.
However, I don't see how we can rule out the existence of god/s. Even if we have one universe and it is finite, and we can search it fast enough to ensure it hasn't moved to a part we have already searched, the god/s could be in another dimension we can't access.
I think Occam's Razor might make god/s less likely, but it is not enough to shift me from agnosticism, to an active belief that there is no god.
@180 Proof
For what it’s worth, I think you’re both correct. 180’s example assumes that if a god exists, then miracles (and presumably other such phenomena) must also exist. Kind of like how if mules exist, then horses and donkeys must exist, only you’re reasoning backwards rather than forwards. Maybe that’s not the best example, but that’s all I can come up with at the moment. Anyway, if god’s existence necessarily entails whatever phenomena, and that phenomenon is lacking, then it’s logical to conclude god does not exist.
I think with TMF, there’s no assumption of any entailed consequences of god’s existence. Perhaps something like the Deistic conception of god that doesn’t interact with the physical world. In this case, there are no necessarily entailed consequences of god’s existence, therefore there’s nothing to point to to imply god doesn’t exist. The lack of the existence of miracles in this case doesn’t imply god doesn’t exist.
Quoting TheMadFool
I think the burden of proof falls on the person making the positive claim because it asserts something beyond the default position, which is skepticism. I don’t think the difficulty of providing proof is a factor at all.
Quoting Down The Rabbit Hole
Are you also agnostic on the existence of fairies? I think you’re using the wrong razor.
We prove negations often.
You mentions bears. I'll mention termites. If you call an inspector to your house, and he reports "No termites", then you may say, "What's your basis? What's your proof?" And you shouldn't have to pay him if he just says, "Well, I can't be expected to prove a negative, now can I?" No, he may show you photos of the areas and surfaces or whatever. Or he may give as evidence his attestation that he examined the areas.
So there are instances where the burden of proof does go to person who claims a negation.
Good one! The default position is skepticism but with the caveat that that's not the best concept to describe the epistemic state in question. I prefer to call it a tabula rasa kinda condition - a blank slate on which neither the proposition "god exists" nor "god doesn't exist" has been written. Skepticism proper is a state of doubt regarding propositions, their truth value to be precise and hence requires for there to be at least one proposition that can be true/false/undecided/undecidable.
That out of the way, your post, although not explicitly mentioning the point, made a lightbulb go off in my head. A negative statement can't be discussed/analyzed prior to a positive statement that's subject to a similar treatment. Before negation can be performed and a negative statement obtained, there must be a preexisting positive statement that can be negated. Ergo, positive statements precede negative statements and since every statement must be proved, it follows that the burden of proof rests squarely on the shoulders of one making a positive statement.
Quoting Pinprick
I don't have sufficient evidence to claim fairies don't exist. Do you? What is it?
Remember what you said here:
Quoting Pinprick
That's a non sequitur. Yes, to have a negation there is first a statement to be negated. But that doesn't entail anything about burden of proof.
I gave this some thought and here's what I found out.
We start off with a proposition (p) & the negation of that proposition (~p) and discover that p v ~p is the ground epistemic state.
From here, our choices are either to prove p or to prove ~p. We can do either of them as there are no obvious reasons to prefer one over the other.
However, the catch is, appears to be, that the default epistemic stance is ~p i.e. in all cases you will be asked to prove p. Hence the rules of thumb, you can't prove a negative & the burden of proof falls on the one making the positive claim. If ~p were not assumed, we would be at liberty to take either arm of the disjunction, p v ~p and the aforementioned maxims wouldn't exist or wouldn't be part of the narrative of critical thinking like it is.
Why, you may wonder?! Is there a good reason? Is it advantageous in other ways like being easier and thus an energy/time-saving strategy? If not all that, is it an intuition and can we make sense of it?
Let's look at the choices we have:
1. Maybe Is or Maybe Is not (p v ~p)
2. Is (p Positive)
3. Is not (~p Negative)
Our journey begins at 1 (above) but proof of Is (p) has precedence over proof of Is not (~p) [can't prove a negative, the burden of proof falls on the one making the positive claim]
As far as I can tell, to the extent that it makes sense to me, the epistemic state 1. p v ~p is worrying because it represents a state of uncertainty perhaps best described as we could be wrong (~p could be the case). Since our fear is getting it wrong (~p) , why not assume that we are (~p)? It's like being uncertain whether there's a burglar in the house; the best course of action is to assume there is one.
1. Negation is an operation i.e. it needs for there to be something which can then be negated e.g. to get to ~p, we need a p first.
2. From 1, p comes first.
3. No flat assertions are permissible i.e. I can't state p unless I have proof.
4. From 2 and 3, since p comes first, at the very least time-wise, proof of p necessarily precedes the proof of its negation ~p.
Not true.
Quoting TheMadFool
So you assert.
Quoting TheMadFool
That's a good example against your argument. We don't assume there is a burglar in the house, since we don't want to be constantly running to the front door to escape or constantly taking whatever defensive measures one would take against a burglar.
Wrong. Just to merely state a sentence does not require proving the sentence.
You are correct that in a formation sequence, P precedes ~P. But that does not entail that in a proof sequence P must precede ~P.
Why would you state a sentence p?
To assert p is true.
If so, you need to justify p.
Different reasons:
To assert it.
To mention that someone else asserted it.
To wonder about it.
To mention it as a topic for discussion.
To mention it as a possible topic for discussion.
To stipulate a proposition to be the subject of a formal debate.
To mention that you will use it as the antecedent for a conditional.
To enter it as the first line of proof of its negation.
Etc.
And if it is to assert it, one can assert it without proving it. People do it all the time. It's not even always reasonable to expect proof:
If I say "There is a traffic jam to avoid on that street" but not supply proof, then one may respond "Thank you for that information, I'll avoid that street" and thus grant the usefulness of my unproved assertion.
/
And I repeat what I just posted, but you skipped:
Quoting TonesInDeepFreeze
You're oversimplifying this. Ignore the negative part and focus on burden. Compare the following claims:
All of these are negative claims, but they are clearly not equivalent.
Quoting TheMadFool
That's a bad analogy. You're trying to prime the pump by using an agreed upon extant entity (bear) in an unlikely place (house), but that's precisely what makes the analogy bad. Bears are demonstrably extant entities that would fit in a house... they are more like squirrels in a fridge.
But the above negative list isn't complete. It's possible we might both disagree on the existence of gremlins, but we might agree on the test for the gremlin... open the fridge and look. If you see a gremlin, there's one in the fridge. If you don't, there isn't one.
I'm not sure god in the fridge fits this criteria... I suspect you believe god is omnipresent, but god being in a fridge looks exactly like what an atheist would expect a fridge without a god in it to look like. If that were the case, and you were expecting the atheist to show you there's no god in the fridge, what exactly is it you expect the atheist to show you? I would argue this is incredibly different than the bear in the house scenario, where not only do we all believe bears exist, but we agree what it would look like to see a bear in a room.
Possibly but the Wikipedia page on burden of proof/can't prove a negative uses the same analogy. I'm quite content with that.
Quoting InPitzotl
Quoting InPitzotl
I didn't mention any equivalences between negative statements and how am I oversimplifying the matter?
Quoting TonesInDeepFreeze
Not so I'm afraid. Whenever you declare p, you are in fact asserting p is true and that can't be done without evidence. If I had said ~p then I would need to prove that too but p was here first and so...prove p. Also, asking someone to prove ~p doesn't help your case at all. If that someone is unable to prove ~p then that doesn't mean p is true (argumentum ad ignorantiam).
I didn't say 'declare P' in the sense of 'declare P to be true'.
I mean 'state P' in the sense of writing it or saying it. Not necessarily to state that it is true. I gave you examples.
A burden of proof of P does not follow from the mere fact that syntactically ~P can't be formed without first forming P.
You just skip recognizing decisive examples and arguments against you.
Suppose I want to prove proposition P and I use argument T, in the context of T, P is true or so the claim is. Then the following is true:
1. IF P is true with respect to argument T THEN argument T is sound
2. Argument T is unsound
Ergo,
3. P is false with respect argument T [1, 2 modus tollens]
In other words, successfully refuting argument T is tantamount to proving P is false but, note, only with respect to argument T. People don't have to prove ~P, refuting the argument that attempts to prove P is the same as proving ~P but with the caveat that ~P only with respect to the argument that attempts to prove that P.
Suppose there are two people (two propositions, p, ~p) in a line, and both are required to pay a fee (both need proof), shouldn't the first in the line pay the fee first (prove p first) and only then the second person (prove ~p second)?
(1) I don't think so, not necessarily. There could be better, more relevant factors used
(2) It is not even an operational analogy for the matter at hand anyway.
(3) Also, do you get to move ahead in the line by proving things? If so, the people claiming provable positives would always get to move ahead. (I guess if you don't give a proof then you're kicked out of the line altogether and don't get inside the fancy sexy nightclub reserved for good provers.)
Q1. Does p come first or does ~p come first?
A1. p of course.
Q2. Is proof required for a proposition?
A2. Yes, all propositions require proof.
Q3. So p requires proof and ~p requires proof?
A3. Yes.
Q4. Which should be proved first p or ~p?
A4. That which comes first of course.
Q5. Why?
A5. p was asserted first.
Q6. So what if p was asserted first?
A6. No assertion can be made without proof.
Q7. And?
A7. p was asserted before ~p. Necessarily that proof of p must be produced before proof of ~p
You explicitly did exactly that:
Quoting TheMadFool
...that's the same quote in the post you replied to.
My bad. Yes, I did but since you raised an objection, it kinda,threw me off. What's the problem with "negative statements" being equivalent? That you assert
this self-contradictory statement Quoting InPitzotl
played a big role in my confusion.
Note you say "all of these are negative claims." That's an equivalence if ever there was one!
It is indeed! They are equivalent in that they are all negative claims. But I don't think they have equivalent levels of burden. I don't see the self-contradiction. 4 and 7 are equivalent modulo 3, but you'd better believe I'd prefer 7 red velvet cup cakes to 4!
On what basis do you claim that is?
I don't understand the question (quite frankly, I have problems even parsing it). Are you asking why I think claims of the non-existence of something are negative claims, or are you asking why I think not all negative claims have equivalent burdens?
Why?
I don't need to check the fridge... there isn't enough room in my fridge for a horse to run in it.
So, you mean to say the positive statement, "a horse is in the fridge" is harder to prove than the negative statement, "a horse is not in the fridge"? The latter seems to follow in an immediate sense from the fact that a horse can't fit in a fridge. Thumbelina (2001 - 2018)
No; I mean that some negative claims, like "there is no horse running in my fridge", can be reasonably held with less burden than other negative claims, like "there is no mold growing on butter in my fridge".
Quoting TheMadFool
The negative claim is about a horse running, not fitting, in my fridge. But we can bypass this. It is hypothetically possible that there is a horse none of us know about, which is so small, that it can indeed actually run in my fridge. But for such a horse to actually be running in my fridge, hypothetical isn't good enough... it must be actual. That's still possible. But lest you forget, I'm not talking about what's possible or what burden any particular claim has... I'm talking about the equivalence of burden between negative claims. It would be quite surprising, for good reason, to find a horse running in my fridge. It would not be nearly as surprising to find molded butter.
I think this is limited to existential statements, but yeah that’s another way to look at it.
There are certain things that we usually believe in even if we can’t see them because we have a compelling explanation for why we wouldn’t be able to see them. For example, most scientifically educated people believed in the existence of black holes even before we actually managed to see or detect them. It seems that the reason why most educated people rightfully believed in black holes before they could see them is because there was a compelling explanation why we wouldn’t be able to see those things right away. Namely, that you would need to have special technology to detect black holes.
By contrast, I tend to think that if there really was a God that loved us and wanted to have a relationship with us that we should be able to easily see him because there doesn’t seem to be a compelling explanation for why someone that loves us wouldn’t just immediately reveal themselves to us. Personally, I think it’s more likely than not that such a loving entity wouldn’t make themselves hidden. Most Christians probably disagree. Therefore, I tend to think that this is pretty good evidence against the existence of the Judeo-Christian God. So, I think this sort of dispute is more about whether or not we should expect to see something like a God if such entities existed.
It isn’t needed. Do you have any evidence that they do exist? If not, then the reasonable thing to do is not believe they exist.
Quoting Down The Rabbit Hole
I just meant it wasn’t a factor for determining burden of proof, but yes it is difficult (I would actually claim it’s impossible) to prove something doesn’t exist. There are exceptions, of course when the scope is narrow (I.e. there is no money in my pocket at this time).
What page is that? The page I found says we can prove a negative.
Quoting TheMadFool
We don't know that P was asserted before ~P.
I assert the following statement:
It is not the case that there exists a rainbow colored kangaroo doing yoga in the White House Oval Office now.
That statement is ~P where P is:
There exists a rainbow colored kangaroo doing yoga in the White House Oval Office now..
And P was not asserted before ~P.
The best you could correctly say is that, with the language formation rules, we cannot formulate ~P without first formulating P. But it's a naked non sequitur to claim that the syntactical formation rules entail rules for discourse. Not not only is it not the case that P must be asserted first, but it is also not the case that the fact ~P cannot be syntactically formed without first forming P entails that P must be proven first.
(2) Certain negations have positive equivalents.
"It is not the case that the death penalty should be continued."
is equivalent to
"The death penalty should be abolished."
And
"The death penalty should be continued."
is equivalent to
"It is not the case that the death penalty should be abolished"
So, in such an example, there wouldn't even be a way using by your rule to claim which should be proven first.
(3) You skipped my counterexample to your claim that assertion requires proof:
Quoting TonesInDeepFreeze
~P cannot be understood without first understanding P. But that does not entail that P must first be proved. Your "Ergo" is a non sequitur.
Negation is an operation. It needs a proposition i.e. before I negate p and get ~p, the proposition p has to be there. Right? Just think of it, "not cat" makes no sense if "cat" doesn't exist as an idea. I rest my case.
The links I provided were meant as references, not infallible sources.
I myself have said over and over and over that you can't form ~P without first forming P.
But, and I've said this over and over and over, that does not entail that you must first prove P.
Your "I rest my case" is empty.
And you skipped my other points, yet again.
Suppose we're having a debate. The topic is whether god exists or not. The first proposition that kicks off the debate is without doubt the proposition, "god exists" for the reason that the negation of a proposition comes only after the proposition has been stated in positive form. In other words, the debate begins with "god exists". We know that every proposition must be supported/demonstrated with an argument. Ergo, necessarily that "god exists" must be proven before "god doesn't exist."
I haven't debated all that much in life but last I checked, the team for the motion makes the first move, followed by the team against the motion. This is a big clue in and of itself.
Sorry if you feel differently. Everyone is entitled to his/her own opinion but read the post preceding this one. It should be crystal clear why p needs to be proved before ~p.
I rest my case.
It's crystal clear that your method is to just keep insisting you're right without addressing the arguments.
It might seem awkward for the subject of that debate to be couched in the negative, but it is not logically necessary that it be couched in the positive. It would not defy logic to start with the proposition "It is not the case that there is an omniscient, omnipotent, all-good being".
The team that starts is called the 'Affirmative' but I don't know that it is the case that the proposition itself in an academic debate must be couched in the positive.
And the conventions for academic debates don't govern controversy and discourse universally.
Also, I gave you an example where the negative has a positive equivalent and vice versa. You ignored that.
No, I didn't miss your point. I dismissed it. This was quite explicit in the last post... you explicitly asked if I meant that the positive claim was harder to prove than the negative claim. And I explicitly said no, that I meant that some negative claims can be reasonably held with less burden than other negative claims.
Quoting TheMadFool
But TMF, it's not that I deny how easy this is to prove, it's that I deny its relevance to burden of proof. If Joe says there's no horse running in his fridge, I would accept that claim without checking. If Frank says there's no molded butter in his fridge, I would not accept that claim without checking. Therefore I place different burdens on different negative claims. How easy it is to check the claim is irrelevant; in fact, it's easier to show there's no horse running in a fridge than it is to show there's no molded butter.
Quoting TheMadFool
And who would that be?
Quoting InPitzotl
I don't base my burden of proof here on what's possible; rather, it's based on what's reasonable:
Quoting InPitzotl
Put it this way. Allow me to describe a game. We take 1,000,000 fridges (all nice and plugged in and operational, like mine is). Every time we find a fridge with molded butter in it, you pay me 20 bucks, but only on one condition. If we ever find a single of these 1,000,000 fridges with a horse running in it, you pay me nothing; instead, I pay you 5,000 bucks. I feel safe playing this game.
You have raised objections to my argument re why positive claims have priority over negative claims with regard to which must be first tackled in the sense proved. I responded adequately in my humble opinion to those objections. Just so you know, you actually haven't argued your stand on the issue. Thanks for the engaging discussion, I enjoyed it a lot. I hope it was the same for you. :wink: :smile:
(1)
Quoting TonesInDeepFreeze
You evaded that that is a counterexample to your claim that P must be asserted before ~P is asserted. Instead you just intoned again your non sequitur that syntactical formation entails order of proof.
(2)
Quoting TonesInDeepFreeze
You evaded that the items in the list (except the first one) are examples answering your challenge that to state a proposition is to assert it.
You skipped completely the example of an assertion that has value without having been proven.
(3)
Quoting TonesInDeepFreeze
You skipped that entirely.
(4) Quoting TonesInDeepFreeze
You skipped that completely.
My point has been that your arguments are specious. That doesn't not require "taking a stand" on anything other than what I have said.
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.
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. This qualifies difficulty level as a good reason why those making positive existential claim should bear the burden of proof. I'm not claiming that this is the reason but it definitely is a reason. If given a choice between easy and hard, common sense would have you choose the easy (way out).
Fine, have it your way then. G'day.
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.
Then let's compare a true positive with a true negative.
ExP
and
~ExQ
it is not true that in all cases, ExP is easier to prove than ~ExQ. It would depend on P and Q.
I'm not faulting the article. I'm pointing out that the article says explicitly the exact opposite of how you described it.
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 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:
Quoting TonesInDeepFreeze
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.
My answer would be, "it depends".
Quoting 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.
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.
Quoting 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.
It depends on how reasonable the claim is.
Quoting TheMadFool
The question is supposed to be about burden of proof.
Quoting 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.
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.
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?
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).
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'.
Sorry, you're just repeating yourself.
Quoting 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?
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.
Quoting 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.
Quoting 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?
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.
There are no two integers p, q such that (p/q)^2=2.
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.
You were told wrong.
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.
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
Quoting Pinprick
It is to move from agnosticism.
Quoting Pinprick
The first clip in this video looks pretty real: https://www.youtube.com/watch?v=ZswREtWpJrg
:wink:
Quoting 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.
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.
Quoting 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.
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.
Quoting 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.
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.
Quoting 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.
Quoting 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.
Quoting 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.
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
Quoting 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.
Right, case-by-case in an indeterminate domain.
Quoting 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.
Quoting 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.
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.
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.
Quoting TheMadFool
The deductions can be about physical facts. From premises about physical facts we may deductively reach conclusions.
Quoting 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.
Quoting 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?
https://en.wikipedia.org/wiki/Proof_by_contradiction#Irrationality_of_the_square_root_of_2
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
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.
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:
[math]\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}[/math]
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.
Quoting 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.
Quoting 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).
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.
Quoting 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.
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.
Quoting TonesInDeepFreeze
Based on your knowledge of kangaroos? This would be using evidence to reach a conclusion.
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 think you mean you would be surprised.
Quoting 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.
Quoting 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.
Quoting 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?
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.
Quoting 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.
Quoting TonesInDeepFreeze
I'm not sure what that entire scenario is about.
Quoting 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.
Quoting 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.
Quoting 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.
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:
Quoting 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?
Quoting 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.
An existential vs its negation.
I used 'black dog' only because it came into the discussion as an example.
Quoting 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.
Quoting InPitzotl
No, I am not.
Quoting InPitzotl
That's not what I've said.
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.
Quoting 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.
Quoting 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?
Yes. But that doesn't vitiate anything I've said.
Quoting InPitzotl
I don't see a basis for your sarcasm.
Quoting 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".
Quoting 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.
The basis is that you volunteered that you only talked about it because it was mentioned.
Quoting TonesInDeepFreeze
Whereas that's true, it was TMF that started both the thread and the black dog discussion.
Quoting 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).
Quoting TonesInDeepFreeze
You replied to it. You said this:
Quoting 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.
Quoting TonesInDeepFreeze
That condition holds in rows 3, 5, 7.
Quoting TonesInDeepFreeze
That's row 2.
Quoting TonesInDeepFreeze
That's row 1.
Quoting 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 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
Quoting InPitzotl
I agree.
Quoting 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.
Quoting 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.
Here's the discussion leading up the black dogs:
Quoting TheMadFool
Quoting InPitzotl
Quoting TheMadFool
...and so on.
Quoting 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.
Quoting 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?
So? it doesn't vitiate anything I said nor show a basis for your sarcasm.
Quoting 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?
Quoting 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.
Quoting 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.
Quoting 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.
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:
...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.
Quoting 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.
Quoting TonesInDeepFreeze
But you said:
Quoting TonesInDeepFreeze
"Burden of proof" is literally in the title of this thread.
Quoting 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.
Correct.
Quoting 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.
Quoting 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:
Quoting TonesInDeepFreeze
And I didn't even opine about "burden of proof"; I only commented on comparative difficulty.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting TonesInDeepFreeze
That's entirely correct. You didn't say anything there about who has "burden of proof". And:
Quoting 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.
Quoting TonesInDeepFreeze
..but that is incorrect, or at least it's not the whole story. In this post:
Quoting 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.
Quoting 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).
Quoting 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).
Quoting 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.
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.
And it's not a meaningful comparison to what I said.
Quoting InPitzotl
So we'll disregard your comment about it, after I've pointed out it was not apropos.
Quoting 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.
Quoting InPitzotl
Quoting 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?
Quoting 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:
Quoting TonesInDeepFreeze
Quoting 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.
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).
: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.
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?
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.
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.
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.
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).
Indeed but proving one disproves the other (contradictory).
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.
But then you haven't proven "there are no black dogs". You've proven "some dogs are black." :chin:
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.
FTFY.
Quoting 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.
Quoting TonesInDeepFreeze
That is clearly false, because you keep replying to me and "merely stating" things directly to me.
Quoting 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:
Quoting TonesInDeepFreeze
...where 1 and 3 respectively are:
Quoting 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!
Quoting 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
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.
Quoting 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?
Quoting 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.
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.
Quoting 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.
/
Quoting 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.
Quoting 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.
Quoting 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.
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.
Quoting 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).
Quoting 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.
Quoting 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.
No, I said "possibly".
Quoting InPitzotl
No, I'm not. I'm moving to a different metaphor.
Quoting InPitzotl
Exactly.
Quoting 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.
/
Quoting 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.
Quoting 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.
Quoting 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:
Quoting Bertrand Russell
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: [math](\exists x)(Gx)[/math]
2. God does not exist: [math](\forall x)(\neg Gx)[/math]
God exists is an affirmative statement and is translated in logic with the existential quantifier ([math]\exists[/math]) i.e. we only need one thing that is a god to prove it.
God does not exist, in logic, requires the universal quantifier ([math]\forall[/math]) 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.
Quoting 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'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.
Quoting 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 = [math](\exists x)(Ux)[/math]
~C = No universes exist = [math](\forall x)(\neg Ux)[/math]
The universe exists = [math](\exists x)(Ux \wedge x = c)[/math]
The universe does not exist = [math](\forall x) \neg(Ux \wedge x = c)[/math]
1. God doesn't exist
and
2. God exists but we haven't come across the evidence that He does.
Compare the above to
3. God exists [ I have proof that God exists]
We can't actually prove God doesn't exist (a negative) for we can't tell if it's because of my ignorance (of the proof).
For positive claims, there's no such complication.
A "God" without definite, sine qua non, predicates renders this statement Not Even Wrong. Nonetheless, I disagree with what I think you're saying, which is that negative proofs are not possible ...
Quoting 180 Proof
So which predicated g/G are we even talking about?
Use the relevant predicates to make it right? :chin: It matters not to the point which is we can't prove a negative. It's just an example of a positive statement.
Quoting 180 Proof
Woops. Looks like I made a boo-boo.
Anyway, let's look at some other example to avoid getting bogged down in the atheism-theism debate.
1. X is a poison.
2. X is not a poison.
There must be something special about negation (not). It just doesn't make sense to say that positive and negative statements are equally easy to prove, as if the negation (not) makes no difference at all.
The word god is so bloody vague. Do you think that in neophyte philosophical discussions such as these it should be clearly spelt out what kind of theism or deism is being referred to? We have no properties to explore here or any kind of connection from this notion of god to any existing branch of theism. It seems rather empty.
Pardon the oversight. I hadn't slept well the day before. I see your point how predicates should inform us about search parameters and if the search turns up empty, we can (via modus tollens à la falsifiability claim nonexistence of entity assigned the predicates). I hope I got that right.
Quoting tim wood
That's right! I can't find something if I don't know what I'm looking for. Did I catch your drift?
Quoting tim wood
Yes, from an Aristotelian logic perspective negation is different in terms of its scope (the correct concept would be distribution). See below for more.
Quoting tim wood
I've never completely understood positive, negation and complements.
1. X is a poison (positive)
2. X is not a poison (negative)
3. X is a non-poison (positive using a complement viz. non-poison)
Venn diagrams, as you suggested, reveal more.
All I can say for now is negation should have some kinda impact on provability (possibility, ease, and so on).
:up:
True -- maybe there really is an elephant sitting on your chest. Maybe one day evidence will emerge that shows this to be the case. If this is what you end up concluding, then something has gone terribly wrong. Try identifying where the problem occurs.
Part of it, in my view, is that logic, theory, propositions, abstraction, generalizations, etc., can only take us so far. I see a useful distinction in theory and practice, and it can extend to this example.
In theory, absence of evidence doesn't "prove" something isn't there -- whether God, or the elephant, or the spaghetti monster -- there's just no practical reason to believe it's there, and no reason to believe any evidence will ever show up that will demonstrate that it's there.
There's also the issue of why a claim like this is even being made, which is a more interesting point. 180 made the claim about an elephant. He conjured it up out of thin air to prove a point. Should we waste any time whatsoever wondering about whether or not it's true? Likewise, should we devote any more time about the claims of Semitic peoples that have been handed down to us over millennia? Also: we generally agree about what an elephant is, yet we have almost no idea about God. The word is empty and almost completely meaningless. It persists in its use, however, and many people find it important -- so that fact alone perhaps makes it worth spending time on, but for psychological reasons.
It actually is a pretty good argument. I'm not sure where the saying "absence of evidence isn't evidence of absence" came from, but its simply incorrect: not only is absence of evidence evidence of absence, that absence of evidence is evidence of absence is a provable theorem of probability theory. And how strong of evidence it is, depends on the likelihood or the expectation of the presence of a particular sort of evidence, if the proposition in question were true.
Now, people often confuse/conflate "evidence" and "proof" and so in at least some cases I think that what people mean when they say that absence of evidence isn't evidence of absence is that evidence of absence isn't proof of absence. And that is true enough. But it is evidence, the only question is just how strong or compelling it is.
And so showing that the evidence we would expect if Christianity (or anything else) were true (special creation, a moral world order, efficacy of prayer, miracles, etc) is absent, is a strong argument against the truth of Christianity.
[quote=Wikipedia]Russell's teapot is an analogy, formulated by the philosopher Bertrand Russell (1872–1970), to illustrate that the philosophic burden of proof lies upon a person making empirically unfalsifiable claims, rather than shifting the burden of disproof to others.[/quote]
If we're asked to prove a negative, the problem is we will have to sometimes try and prove that which is essentially unfalsifiable.
There could be someone in D or there could be no one in D. You're uncertain.
If you assume there's someone in D, you'll pull out your gun.
If you assume there's no one in D, you'll keep your gun in its holster.
If you don't assume either way, how would you act?
1. the burden of proof rests on the positive statement
2. the burden of proof rests on the claim
Imagine a few people sitting at a table in a restaurant. There is a large window such that the diners can see that it is raining outside. A woman enters the restaurant with wet hair. For some reason, one of the persons at the table says, “Her hair got wet in the rain.”
Someone else at the table says, “That’s not true.”
I think everyone would feel that the burden of proof rests on the second speaker, though the first speaker has made a positive claim.
I am afraid that in practice we feel that the burden of proof rests with the statement that is farthest from common sense. Since common sense is demonstrably defective, this criterion is sloppy, sheeplike, and depressing. Can we come up with a good workable criterion for the burden of proof?
Exactly. The common understanding (in cases when there is one) typically has at least experiential if not evidentiary data to back it up. Proposing an alternative requires "proof" to counter the common understanding.
This is basically correct. The burden of proof is on the claim that is contentious or contrary to the prevailing consensus, and this could also be expressed in terms of common sense.
Quoting Jedothek
No, it's not, and there's an important point at play here. In philosophy today people like to follow Descartes and think that everything ought to be crystal clear and perfectly certain. They think
This is a completely wrongheaded way to think about precision. Not everything is or should be apodictic, and the burden of proof is one of those things. The burden of proof is itself little more than a loose convention with respect to debate and dialogue. It cannot be ascertained in an apodictic way; it is not susceptible to a high degree of certainty; it is not a very important concept in the first place; and it is itself just as sloppy as notions such as consensus and "common sense." Simpler: if the burden of proof were not a sloppy concept, then it would require a non-sloppy alternative; but the burden of proof is a sloppy concept.
Quoting Aristotle, Nicomachean Ethics, I.iii