# Not being able to prove a negative.

Ok, this is probably a very stupid question, I admit, and when somebody gives me the answer, I’ll probably slap my forhead and say Doh! :eek:
Anyway, why can’t you prove a negative? For example, if a religious person tells an athiest to prove that God doesn’t exist, the athiest will shoot back with “You can’t prove a negative.”
Let me try
1 + 1 does not equal 3
Allright, let’s see, there’s one 1, there’s the second 1…that’s two.
Haven’t I proven that 1 + 1 doesn’t equal 3 ?
(Yes, this question is serious, laugh at me if you must, but at least answer this question. Thank you.)

Yes, there are SOME negatives you can prove, such as “Joel is not standing in front of me right now.” But in the type of situation that this phrase is used for, you can’t.

In other words, proof of anything requires you to be able to touch, see, taste, etc. the item or other clues as to its existence. But just because you don’t touch, see, etc., an item doesn’t mean it doesn’t exist, it just means that you have no proof of it existing. So it may or may not exist.

Is that above paragraph confusing enough?

It depends. It is not always impossible to prove a negative. For example, the Christian god can be shown to be logical contradictory if orthodox Christian doctrines are used, but many Christians then just would weasel out by saying logic does not apply to his god.

There are a lot of factors involved. First of all, to have any sort of proofs to work there must be a common ground (e.g. whether the Christian god is contrained by logic or not), clear definitions, and a context to work from. Secondly, it depends on whether you can construct a deductive or just an inductive proof.

Thanks for both your answers. Just to clarify, this post wasn’t about whether or not God exists, that was just an example.
Anyway, I’m guessing then, that it’s more accurate to say, that you can’t prove a negative about the unknown, right ?

Nope. You’ve only proven that 1+1 equals 2. You’ve said nothing about 3.

You can’t say anything about the unknown

Certain negatives can be proved by providing a counter example. When this is not possible, it is generally accurate to state that “you can’t prove a negative.” We can show that the statement “no even primes exist” is false by producing the number 2. We can’t disprove the statement “God exists” by producing an anti-God or what have you.

friedo, good point, but I didn’t write down everything I wanted to. The part:
Allright, let’s see, there’s one 1, there’s the second 1…that’s two.
Haven’t I proven that 1 + 1 doesn’t equal 3 ?

Should have said
Allright, let’s see, there’s one 1, there’s the second 1, but hmmm, I don’t see a third one…so that’s only two.
Haven’t I proven that 1 + 1 doesn’t equal 3 ?

My fault for not doing a better job of checking my message before posting.

When people say you can’t prove a negative they generally don’t mean it in the sense of the OP. What’s generally meant by the phrase is that it is very difficult to prove the non-existence of a non-existent object. People who want to believe in the object will claim that the lack of observable evidence of its existence doesn’t prove that the evidence doesn’t exist somewhere beyond immediate observation.

1+1=3 for very large values of 1.

That may sound a little silly, but consider the following:
x=1.35 and y=1.35. In the real world we always round to some number because after that the precision gets a little silly, so let’s say we are rounding to integers. In this case, x=1 and y=1, but x+y=3. (1.35+1.35=2.7, which if you round to integers you get 1+1=3).

So, 1+1 can equal 3.

Actually, you’ve proven it only given the assumption that addition will give the same result every time. Additionally, you are assuming that 2 is not equal to 3 (don’t laugh! The assumption that 2=2 is called the reflexive postulate!). In normal math, this is implicit, but you can invent any kind of math you like. The math police won’t arrest you. If there’s a math theorist out there, perhaps he could tell us the name of the “same result” assumption, but I’m afraid my terminology runs out fairly quickly!

Just what constitutes “proof” is a big issue in math - and in general philosophy and logic, I assume. I think - and again, I must note my terminology is rusty - that your proposition is that 1+1 = 3. You claim to have disproved this by showing 1 + 1 = 2, and so your technique is called disproof by counterexample.

http://www.csc.tntech.edu/~srini/DM/chapters/examples/ex2.1.4.html (Note that an upside-down A means “For all”).

In terms of the original question “Prove there is no God”, then this technique won’t work. You might analyze the problem as follows: Either there is a God, or there is no God and no other conditions are possible (this is “the law of the excluded middle”). Thus, the proposition “There is no God” can be proved either directly (a direct proof that there is no God) or indirectly (by showing that an assumption that there is God leads to logical contradictions). In practice, hard to do either way since one can make whatever assumptions one chooses - and look how many assumptions are built into your proof that 1 + 1 is not 3!

So if you can’t find your car keys in the morning, you shouldn’t necessarily assume they don’t exist…but you might!

A negative might be impossible to prove by evidence. However negatives can be proved by logic using reductio ad absurdum which is a fancy way of saying that you make an assumption that such and such is true and then show that this assumption leads to a contradiction. The contradiction shows that there is something wrong with the assumption, namely such and such must not be true.

The proof that there is no ratio of two whole numbers that equals the square root of two is just such a proof and isn’t very difficult. In fact it is so easy that even I can do it and I’m no mathematician.

A negative can be proven if there is a mutually exclusive condition that can be proven false.

For example, “There is no highest prime number.” This is a negative statement that proposes that there are is an infinite number of primes of ever increasing value. This being a negative proposition, it cannot be proven directly. However, there is a mutually exclusive condition, that being “There is a highest prime number.” We now have two mutually exclusive conditions, one of which must be true and the other false. If we can prove either of these to be true or false, we have a proof for both of them.

The first statement cannot be proven true as it involves identifying all the members of an infinite set. If one could identify the one highest prime number, this would prove the statement false, but this also is not possible, as it would involve proving that all higher numbers are not prime, another negative proposition.

Take the second statement, “There is a highest prime number.” This cannot be proven true, for reasons stated above, but it can be proven false. Without going into detail, if we assume that it is true, and make logical deductions based on that assumption, we eventually come upon a contradiction that cannot be resolved. Thus the statement is proven false, and because we have two mutually exclusive conditions, one of which must be true, we have proven the negative.

Alibi defenses in criminal trials are based on this idea. Assume that Bob is accused of robbing a liquor store at 8:30 p. m. Saturday. Bob cannot directly prove that he didn’t rob the store (he doesn’t have to, but that’s a different discussion). He can prove it indirectly by proving a mutually exclusive condition to be true: Bob was in a bar across town at the time of the murder, where 30 people saw him drunkenly attempt to hit on a woman way out of his league and get thrown out by a bouncer. If Bob sufficiently proves he was elsewhere, he has proven a mutually exclusive condition to be true, thus proving the original accusation to be false.

The problem with applying this to certain philosophical debates is finding the mutually exclusive condition and proving it. There is no provable condition which necessarily excludes the existence of a creator. Suppose I could offer concrete, uncontrovertable proof that the universe is 12-15 billion years old, started with a big bang, that the stars and planets are remnants of that early chaos, that life on earth evolved over billions of years, and that man and apes evolved from common ancestors. Had I uncontrovertable proof of all this, it would not disprove the existence of God, because the Big Bang, evolution, old universe, etc., and the existence of God are not mutually exclusive.

There are two kinds of athiests. Dogmatic, or “hard” athiests, actively beleive that there is no God, and some can be somewhat evangelical in this belief. Skeptical, or “soft” athiests, do not believe there is a God because there is insufficient evidence to come to that conclusion. The athiest in your OP sounds like a hard athiest. A soft athiest would more likely respond with something like, “I don’t have to prove anything. The burden of proof is on you to prove that there is a God.”

Others have dealt above with proof of negatives by logic. It is also possible to prove some negatives, in a practical sense. If you say to me “prove there is no elephant in my office at this time” there will be no real difficulty. I need only search for a large closely defined object in a small closed system. We can look the room and prove there is nothing of that size in my room at the moment with very little effort, to our mutual satisfaction.

However, the real issues in most metaphysical arguments of the type mentioned in the OP are twofold.

Firstly, we live in an open ended universe.

Secondly it is possible to imagine things that have very few if any readily identifiable characteristics (e.g. to take an extreme example we can all imagine the existence of an invisible, non-corporeal, soundless, odourless, tasteless god).

Against that background it is easy to see that the class of objects that can be proven not to exist by searching the whole of space and time is narrow indeed.

In a practical sense, what that leads to is this: if you assert that something exists or has happened but I do not agree, I can search forever and never find that thing, but you will always just assert that I have not looked hard enough.

John picks his nose and it is recorded on video. We can prove that John has picked his nose.

Fred claims he has never picked his nose. He can’t prove it because we don’t have video of every moment of Fred’s life.

This example illustrates the relative ease of proving a positive compared to the virtual impossibility of proving a negative.

This is a great example. Go ahead. Prove it.

Maybe Joel has turned invisible. Maybe you’re blind. Maybe you see him but have neurologic damage and think he’s a hat. Maybe the person who is in front of you has just change their name to Joel. Maybe your own disbelief is keeping you from seeing Joel (the Paranormal defense). Maybe Joel is short and you’re looking over his head. Maybe Joel’s in a blackhole in front of you, and information about him can’t get out. Maybe you’re not really where you think you are, but in a lab hooked up to a virtual reality life, and Joel is standing right in front of you (Vanilla Sky style). Maybe you have a good proof, but I don’t buy into your rigid Western logic and refuse to believe it has validity. Maybe Joel is in front of you, on the other side of the wall.

…and so on and so forth, limited only by the creativity of possible “outs”. You would never be able to definitively prove all of the above. Each time you tried, there would be more “outs” in your attempt to prove that the first “out” was incorrect.

Ireal world situations, and even some in math and logic, you can’t disprove a negative. There are always things that could have happened to make your experiment non-perfect, always things that weren’t controlled for, always assumptions you made that might not be valid.

There are situations where can assume that we have a pair of mutually exclusive statements, but in fact they are merely opposite extremes; disproving one of the statements does not automatically validate the other.

A nitpick. Proof is really only proof within a formal system. In the first order predicate calculus with equality, every statement can be shown to be true or false. That’s the Completeness Theorem, first proved by Goedel IIRC, and not to be confused with his Incompleteness Theorem, which refers to mathematics. Mathematics is based on logic, but adds a new predicate (set or class membership) whose properties are established by axioms. There are unprovables in math, but they are not called “negatives.”

This,

is mostly accurate. He cannot prove Joel is standing before him unless “Joel” is an object in some formal system. Empirical science is not such a system, and it properly deals with falsification, not proofs.

A philosophy major friend of mine once told me “You can’t prove a negative!” is (paraphrasing) the tactic of a person who wants to weasel out of an argument without actually changing the subject. Yes, I’m aware this is an anecdote, not a cite.

But doesn’t that imply it’s just as difficult to prove a positive? Proving a positive has the same issues of control and assumptions as with a negative.

The concept of “not proving a negative” doesn’t really derive from the difficulty in proof itself, more from the difficulty in proof of non-existence.