Let's put this common refrain to rest~ something that I hear all the time in arguments concerning religion.  It goes something like this:

" Remember, you can't prove a negative!"

Really people?

This has become much more common, especially in the arguments amongst Atheists and agnostics concerning certainty, and it really puzzles me how people can be skeptical and free-thinkers, yet take to an idea so easily and not question it.  I will elaborate on this a little more once I have the time, but let me start all of those "can't prove a negative" types off with a question~ " I am not sitting at my desk."  Thats a negative claim.  Are you telling me that there is no way to confirm or disprove that?

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"That which can be asserted without evidence, can be dismissed without evidence." 

— Christopher Hitchens

I never heard of that quote. I will be using it quite frequently. Thanks for bringing it to my attention.

When the claim you can't prove a negative is made, an important qualification is left out. It should be: "You can't prove a universal negative". By a universal negative, I mean something that does not exist. The implication is that if something truly does not exist, and you go looking for it, you will find nothing. But it could be that you just haven't looked in the right places - there are so many places to look - the furthest reaches of outer space.


The trick which theism uses is to make 'God' have properties which lead that supposed entity to be beyond our reach so to speak. You don't see 'God' because 'God' is not usually visible. You don't hear 'God' because 'God' is not usually audible. Of course the claim is that a select few have seen and heard 'God', but for the rest of us, we can't have someone else's experience - not unless, (the theist will say), 'God' wills it.


The only way to prove that 'God' does not exist, is to take some well defined property of 'God', one which must be detectable in ordinary human experience, and show that it is false. For example, 'God' answers prayers; or 'God' does miracles. If we can show that prayers are never answered, and miracles never done, then that shows that a god for whom prayer answering and miracle working are properties, such a god does not exist.


I'm sure theists have ways of explaining all that away too. All I can say is that I have never detected anything which is incontrovertible evidence that there is a god in existence, so there might as well not be. And I guess that theists would blame that on me.

There is slight misinterpretation of what a negative implies in the scientific sense going on here.  "The room is not occupied," is not a negative.  Being occupied or not occupied has nothing to do with the state of being a room.  Therefore, being unoccupied says nothing negative about the room.  Let's take a positive statement:  Spiders have 8 legs.  This statement can be proven true by finding one spider with 8 legs.  Having 8 legs is definitely part of the state of being a spider.  In order to have a negative, we must state something that frames a negative state about being a spider, such as, "No spiders have six legs."  No matter how many eight legged spiders you find, you will never prove this statement.  Even if you found every spider in the universe, you could not prove that you had done so since the possibility of one more spider you have not found cannot be ruled out.  You cannot prove a negative.  Of course, a single six legged spider disproves our theory, but we weren't talking about disproving stuff.

Rich made a good clarification that really gets to the root of this topic, but I would like to take it a step further and demonstrate that you it is possible to prove a negative if you address certain underlying conditions properly. First, here is how I would restate the central question:


1. Is it possible to exclude a possibility without an exhaustive search?

2. If not, is it possible to determine if a search has been exhaustive?


The traditional approach is to implicitly answer 1. as No, an exhaustive search is necessary, and then focus on 2. as the primary question. From this point, there are numerous different ways to attempt an answer, but they all boil down to one issue:


Limitation: Given an incomplete set, the contents of the set are not sufficient to determine the scope of the set. 


Or put another way, if you don't know everything, you can't determine how far away you are from knowing everything. Since the potential range of information is infinite, and we can only acquire a finite amount of information, we can never determine if we've been exhaustive when attempting to prove a negative. If we accept that 1. is a No.


I would argue the contrary-- that 1. can be answered Yes, and that accepting an implicit No answer artificially produces the above paradox. The key switch in this line of thinking is to consider the known information as a cohesive range of data rather than an aggregate of unrelated data points. I would present this directly as: 


Constraint: For any set containing dependent associations, each new element must satisfy every such association. 


So to work off of Rich's example, any attributes which must be satisfied to be a spider will necessarily be satisfied by any newly-discovered spider species. That's what makes positive statements easier to (dis)prove, since they match directly against known attributes. The mistake that is often made with negative statements is assuming that these attributes are a random set, rather than a mutually-compatible set. 


Essentially, a spider doesn't just "happen" to have a certain number of legs and the word "spider" isn't a circular definition, both are driven by empirical demonstrations (a real spider) showing that possessing 8 legs is physically compatible with all the other known attributes of spiders. 


So a much better answer to 1. would be Yes, an exhaustive search can be avoided if a possible attribute would be incompatible with already-known attributes. This changes the focus of 2. from an infinite range of possible data points to a finite number of possible combinations. This can still lead to a massive number of possibilities, but it will always be a finite amount.


Back to the spiders again for a simple demonstration of this approach. Let's take the negative statement, "No species of spider can grow only one leg". This is slightly different than Rich's wording because it contains 2 attributes within the statement (to avoid haggling over the meaning of "spider"). These are:


Attribute A: Possesses only one leg.

Attribute B: Survives well-enough to reproduce.


Without exhaustively checking every single species of spider, this statement can be proven by demonstrating that the locomotive function of a spider is compromised by possessing only one leg. This leads to a solid proof:


Answer: No known spider species can survive with only one leg, therefore Attribute A is incompatible with Attribute B. This conclusively excludes any possibility of a species possessing both attributes which also qualifies as a spider (satisfies the control attributes). 


I would consider this to be a fairly simple, efficient way to prove negative statements. It also has the secondary benefit of short-circuiting the ambiguity of most "unknown to science" arguments. You still can't make the generalization "There are no possible gods," but you can say "There are no possible gods which are inifinitely large, yet existing" because the attributes of 'existing' can be logically incompatible with infinite size. This requires both sides of the argument to select at least one solid, foundational attribute to define any entities they reference-- and anyone who can't provide one isn't equipped to have an argument on any grounds.


That's my bit. It's definitely a non-traditional approach, but I don't see any logical flaws in proving a negative statement by this method, and I personally find it extremely useful for a lot of the tougher topics. How does this sit with the rest of you? Do you see any ways it could be refined or expanded to tackle other similar issues?

I contend that you cannot eliminate the possibility with any search unless the search finds the thing searched for. Just because you did a search, no matter how exhaustive, it doesn't mean that you would not have found it if you searched a little more. The possibility is not removed entirely by any kind of search. It may be diminished, but never removed. Therefore, you cannot prove an negative.
@ John:  Fail.  Please see Rich's comment above.

If you state that there are no elephants anywhere, and you didn't have any evidence of elephants at all, you could say you cannot prove a negative. Do you see what I mean?


If you ask me what I know about whether there is a God, I would have to say I can be 99.9999% sure. If you ask me if I believe there is a God, I would say I 100% believe there is no God. 

If you are honest with yourself, and you ask yourself whether you know, according to science you can never really know with 100% certainty because you require evidence and you cannot prove a negative.

But if you ask yourself whether you believe there is a God or not, you can reach 100% because believing requires no evidence.

About that... I wonder if you should instead be asking about the personal identity and behaviors that sustain the concept of a god-- because it's not the concept itself that matters, it's only the places where it interfaces with themselves that people actually consider. The rest is often just mental goo to support the behaviors they'd like to protect.

Well, Cane, that's exactly the point of my argument: the incompleteness limitation (from above) makes it impossible to determine if a search has been exhaustive, so you can't use an exhaustive search to prove a negative. 


You can, however, take a different approach and prove that a possibility is incompatible with the current set of attributes-- which means that combination cannot occur at all, completely regardless of how much searching you've done. That's how you bypass the incompleteness limitation and prove a negative statement.


The only real constraint in this approach is that you must have a plurality of attributes to consider, so you can't ask a single-property question like "Can there be red?" The question has to express at least one control attribute to compare against, so a viable question would be "Can the human eye perceive red?" The former question cannot be answered, but this one can demonstrated as true (by a search) or false (by incompatibility). 


Or to take John D's example, "My office is smaller than an elephant; I can therefore conclusively exclude the possibility of an elephant being in my office." No search is necessary if you can show that an incompatibility between attributes makes the possibility self-contradictory (since the statement implied a full-size, adult elephant).


The main way to bypass exclusions is to find "near-matches" that avoid the conflicting attribute by suggesting alternates like a pygmy elephant, baby elephant, time-travelling elephant with an invisibility cloak, etc-- but the exclusionary statement still stands and you will not find a full-size, adult elephant in the current office.

Hello Drake,

To whittle all that down you come to this statement: Knowledge requires evidence, believing requires no evidence. 

You can believe that something cannot exist, and I am not saying it is not a valid belief, but you can never prove it with the scientific method. You can convince someone to believe in "proof" of a negative, but all you are really saying is due to the rules of logic, it cannot exist. There is no evidence either way to substantiate the proof of a negative. See what I mean. It is language vs. evidence.

The first bit was way in the wrong direction, but the second part was just about right. Here's the miniature version of my argument:


Logical deduction can be applied to positive proof to exclude new possibilities by pre-emption. Essentially, if you have an "A or B" situation and you know A is true, then you have also proven "not B".


Besides that, I am somewhat leery of how you're using 'belief' and 'evidence' because it comes across as somewhat dogmatic. Is it the scientific method which determines the truth or falsity of data, or is it the rules of logic which judge the data collected by the scientific method? If is the former, then logical deduction probably can't qualify as proof (or knowledge) in your philosphy.

If A or B, A is true, then not B requires that you prove A. How do you prove A?

If you use your prior "knowledge"(what I call belief), you can convince yourself of the truth of A, or in other words, you can believe that A is true.

If you get evidence of A, that can strengthen your belief that A is true, but then what is knowledge? I guess I do not know what most people define knowledge as. Is it memories of something being true? I don't know.

I do know that without evidence, we cannot really determine if something is true or not. To get the identity of A, I need evidence of A, even if that evidence is only light bouncing off of A and reaching my eyes causing me to form a belief that A exists.


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