Clutch

If I’ve already posted something to this effect, my apologies. It’s bugged me for a while; I can’t recall if I’ve said anything about it, though. I keep seeing people write that “you can’t prove a negative”. This is wrong.

First things first. The notion of proof is not univocal; equivocating on it is one of the typical problems here. So let’s clearly distinguish logico-mathematical proof from empirical proof. The former is absolute (at least, relative to the system of axioms), while the latter is by definition defeasible. Some people don’t like applying the notion of proof to the empirical case, but this is standard usage; the police proved that Ted Bundy murdered people – though of course that proof is hostage to further possible discoveries.

The point, then, is that as long as you stick to one conception of proof or the other, there is no principled reason to claim that one cannot prove a non-existence claim.

Consider logico-mathematic proof. Here, in fact, contrary to the initial claim, non-existence proofs abound. There exists no round square. There exists no spherical cube. There exists no even prime greater than 2. In this sense, then, one can indeed prove negatives. Lots of them, at that.

So what about the empirical case? Suppose I say, “There is no elephant in this room”. Can I prove that negative? By empirical standards, yes. I can look all around, including behind the Chesterfield cushions, and I can fail to find an elephant.

“But,” comes the reply, “what if your eyes are deceiving you? What if you’re hallucinating? What if a cunning paint job has put the elephant in living room camo? What about exceedingly tiny elephants?”

Well, yes. What about all those things? These are (among other things) reasons why empirical standards of proof are in principle defeasible. But notice that this has exactly nothing specifically to do with proving a negative. For consider the positive case, in which I say, “There is an elephant in this room”, and go on to empirically establish this by looking around and finding an African elephant balancing on the TV. Notice that the same or cognate questions apply. What if your eyes are deceiving you? What if you’re hallucinating? What if a cunning paint job makes it look like there’s an elephant there?

The possibility of being mistaken is just another aspect of the fact that empirical standards of proof do not amount to logico-mathematical certainty. But this point applies equally to positive and negative existence claims. Within the standards for empirical proof, one can indeed prove negatives with the same confidence that one can prove positives; defeating conditions for evidence in the negative case are defeating conditions in the positive case as well.

While the above remarks show that you can indeed prove negatives in both the logical and empirical cases, this might be thought to apply only to empirical cases (like within a room) where the search area is both finite and practically surveyable. The remaining issue is whether there isn’t a special class of negatives that cannot be proven, because this would require generalizing over an indefinite unsurveyed domain. So, for example, “There exist no hippogriffs” might be evidentially unhappy in a way that “There is no elephant in this room” is not, since the former quantifies over all of spacetime, while the latter quantifies over this finitely and practically surveyable room.

Here, at least, the positive and negative claims do appear to be on an unequal footing. “There exists a planet of hippogriffs” is open to a sudden dramatic degree of proof, up to empirical standards, simply by the discovery of a planet of hippogriffs. “There exists no planet of hippogriffs”, however, is not. Does this mean that one cannot prove at least this kind of negative?

This is tricky stuff, and I’m not entirely sure what to think, but my strong suspicion is that even here it does not mean this. Notice that if the statement is a well-formed empirical conjecture, the notion of a hippogriff must be well-defined: eg, having the size, hair, and dietary properties of horses, while having exactly one pair of wings, and so forth. And we can learn an awful lot about what environment such an organism would require. Then we can apply highly confirmed theories to the task of predicting how many such planetary environments might exist, the probability of hippogriffs evolving on any one of them… and so forth, again. That is, many kinds of evidence can lead us to raise or lower the probability that our negative existence claim is correct. The confidence that such evidence can engender in us will then be a matter of our confidence that the information at our disposal is a truly representative sample of the universe. But scepticism about whether this is true is simply the familiar scepticism of the Problem of Induction: how do we know that our observed cases are reliable guides to the unobserved cases?

The Problem of Induction is not a reason to doubt that observed cases are reliable guides to unobserved cases, though; at least, I’ve never heard anyone claim this. Rather, the nature of the problem is explanatory. Given this reliability, what the heck explains it? As long as we do in fact think it reliable to take observed instances as representative of unobserved instances, and to the extent that we do, then we should see no problem in seeing indefinite negative empirical generalizations as open to degrees of proof. Which is, in the empirical case, all you ever get anyhow.

So here there is a disparity between positive and negative claims, but not a disparity that would bear out anything so strong as the claim that you cannot prove a negative.

Feather

Other than that your "round square" and "spherical cube" are really not proofs so much as definitions, I concur.

To prove a negative, all one needs is to show that the statement "p is not true," where p is a positive assertion, is true for all cases where p is sensible.

Of course, if p is sensible for an infinity of cases, then the statement "p is not true" is not provable.

And, of course, by "sensible" I mean rational, coherent, etc. (I.e. it wouldn't make much sense to try and prove the statement "pink giraffes exist is not true" by demonstrating that purple alligators exist).

Clutch

Well, in the logico-mathematical case, proofs are entirely a matter of what follows "by definition", given axioms. If someone says that there exists a natural number with exactly two immediate successors, I can disprove this existence claim by showing it to be inconsistent with the Peano-Dedekind axioms that characterize the natural numbers. In the simplest cases, hence those chosen for examples, this seems so straightforward as not to merit the term "proof". But Wiles and co'.s proof of Fermat follows "by definition" to no lesser extent.

tronvillain

Actually, it's fairly easy to prove a negative. Here is a proof for the negative "There is not an elephant in this room.":

If there is an elephant in this room, then I will see it. I do not see an elephant in this room; therefore, there is not an elephant in this room.

The problem is that any uncertainty about the premise "If there is an elephant in this room, then I will see it" destroys the proof, and it becomes a matter of probability. I do not have proof that there is not an elephant in this room, I have overwhelming evidence that there is not an elephant in this room - I simply do not use the two interchangeably.

tronvillain

Oh, and the problem of induction is that "If p then q, q therefore p" is not logically correct. The only apparent solution is to admit that induction is a matter of probability rather than proof.

Doubting Didymus

The problem is that 'you cannot prove a negative' is not, nor was it ever, a rule of logic.

The rule is supposed to be: 'you can not prove an unrestricted negative'.

This is the difference between your claims 'there is no elephant in this room' and the claim 'there are no hippogriffs'. One is restricted, the other is not. We can search the room and prove 'no elephants here', but to prove that there are no hippogriffs we would need to search an infinite number of locations. While it is possible in theory to search all locations, in reality it can not be done.

So, we can not prove the unrestricted negative that 'there are no hippogriffs' and we can not prove the claim that 'there is no god'. They are both statements that must be DISproved, or restricted.

Edited to note: Your application of probability to the existence of hippogriffs relys on being able to restrict the negative. A truly unrestricted negative form of this statement would be: 'no magic hippogriffs exist'. You can no longer talk about such a being being improbable, as it overcomes any restrictions by magic.

[ September 03, 2002: Message edited by: Doubting Didymus ]</p>

Clutch

tron, thanks. I thought I pretty much said that, but maybe not. However, I'm not sure I agree that the PoI just amounts to the fact that affirming the consequent is a fallacy. Inductive arguments are deductively invalid, but in the most general terms the invalidity is just some failure of relevance or other. For inductive inferences to the best explanation, affirming the consequent may be the shape of the invalidity. For inductive inferences about causal correlation, it's probably the less formal post hoc ergo propter hoc, which implicates observational correlation in a way that the material conditional does not. And for simple inductive generalizations to unobserved cases, it is just a non-sequitur. So I incline to stick with the standard diagnosis of the PoI: Inductive arguments are deductively invalid (by some failure of relevance or other) without the Principle of the Uniformity of Nature, and there is no non-circular way of justifying that principle.

didymus, I also thought I covered that ground -- except that I didn't tie unrestrictedness to infinity. I said "indefinite", which leaves open whether the search space is infinite or simply of unknown size.

In any case, the infinitude of the search space does not impinge on my final point, which was that we can have empirical evidence for what the universe is like, modulo the problem of induction. That the search space is infinite does not entail that it contains all manner of everything; it may well be uniform. Sampling regions of it can indeed provide rational grounds for forming beliefs about the whole thing, including those about non-existence.

Keith Russell

Greetings:

One can easily prove a negative in a closed system.

I can easily prove that I didn't take calculus last semester, or that this mall does not contain a Baskin-Robbins.

Keith.

Doubting Didymus

If you do this, you are restricting the negative. You do so because your subject, the hippogriff, is defined as having physical constraint. The true unrestricted negative form, as I said, should be: Magic hippogriffs do not exist. There are no restrictions on this statement. It cannot be proven.

A very good example of a completely unrestricted negative statement is 'undetectable things do not exist'.

Edited to add: Does anyone recognise this statement? Who is everyones favourite undetectable thing?

[ September 04, 2002: Message edited by: Doubting Didymus ]</p>

Cosimo

Dark matter?