You can't prove a negative... right? Wrong.

[Will]

It’s often said you can’t prove a negative. This is an interesting piece of folk philosophy. The truth is that you can prove a negative. The law of non-contradiction â€" something can both be and not be â€" is provable and is a basic law of logic. In fact, if you couldn’t prove a negative then the statement you can’t prove a negative would be false because it is a negative statement. One can always use the law of double negation to turn any positive statement into a negative. Take a statement you agree with, say 1 + 1 = 2, then by the law of double negation we have it is not the case that 1 + 1 does not equal 2. Obviously this is true and it’s a negative statement. So, one can prove a negative statement. What about proving that something doesn’t exist? This is often what people mean when saying you can’t prove a negative. Here it’s important to determine what one means by proof. If one means that you cannot prove deductively the existence some entity, then they are saying that given some set of premises you cannot prove deductively that something doesn’t exist. This too is false, for example:

Argument 1.

1. If the Balrog exists we see Balrog spores.
2. We do not see Balrog spores.
3. Therefore the Balrog does not exist.

This argument is valid, it relies on the inference rule modus tollens, the conclusion is a consequence of the premises. If one denies the truth of the premises one denies the truth of conclusion. One thing to note is that the form of the argument, the valid deductive part is content less, that is we could insert anything into the argument form and it would still be valid, we could set it out thus:

Argument 2.

1. If P then Q
2. NOT Q
3. Therefore NOT P

In our first deductive argument P was the statement the Balrog exists and Q the statement we see Balrog spores. We could substitute for P the statement Bill likes coffee and Bill goes to the café for Q in the above argument to obtain:

Argument 3.

1. If Bill likes coffee then Bill goes to the café.
2. Bill doesn’t go to the café.
3. Therefore Bill doesn’t like café.

Now, you might say, but Bill doesn’t go to the café but he likes Coffee, so the conclusion doesn’t follow, or that the Balrog does exist, we just haven’t been looking in the right place to find any Balrog spores. Deductive proofs only get you what you put into them. If the premises are true, then a valid argument guarantees a true conclusion. You can always disagree with the premises of a valid deductive argument and thus deny the conclusion. In some cases it might be reasonable to do so, especially concerning Balrogs. So, do we then have to prove the premises to be true? Well, in some cases we can, but somewhere along the line we have to accept some premises as being true to avoid an infinite regress of premise proving.

We can often get premises that people accept as true from observation. For example, the universe exists; there are people in the living room, and so on. If we accept that the premises are true, and feed these into a valid deductive argument then we can prove the truth or falsity of an argument. This is what was done above proving that Balrogs don’t exist. However, we need observations to support the premises. Having no evidence for Balrogs and a fortiori Balrog spores means that few people are going to accept that the argument proves anything except that I’ve read the Lord of the Rings once too often. So, deductive arguments alone cannot prove existence or non-existence, being empty formalisms. We need evidence and observations about the universe. What is there? Now we are relying on induction. We have up to this point in time never seen any trace of a Balrog. It doesn’t follow from this that we will never see any trace of a Balrog. Induction is indispensable and though not deductively valid, has up to this point in time worked. The previous sentence was itself based on induction. We can never deductively show that the laws of the universe will continue in a uniform manner tomorrow because we cannot observe tomorrow but instead rely on the past. We cannot demonstrate that induction is deductively valid, but we have to rely on induction in any case and to deny that the sun will rise tomorrow doesn’t seem reasonable.

Deductive proofs for the existence of God attempt to do something similar to proving a negative, they attempt to prove that something, God, does exist from premises that are supposedly reasonable. The arguments as presented are valid. If the proof succeeds, then as a matter of logical consequence it must necessarily be true, at least given the premises and that system of logic. If we turn the argument around we see that if the argument isn’t necessarily true then the proof hasn’t succeeded, this is known as the contra positive. We see from the above that proving the existence of something requires evidence. Empty propositions only show the validity of an argument, not that it succeeds.

Hume in his dialogs on Natural religion had this to say:

…there is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by arguments a priori. Nothing is demonstrable, unless the contrary is a contradiction. Nothing, that is directly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no Being whose contradiction is demonstrable.

What Hume is getting out here is that some fact of this universe could have been otherwise. It would not be a contradiction to suppose that I did not exist. My existence, a matter of fact, is only known from observation. Like induction, it is not deductively guaranteed, but thankfully, is a matter of fact. Therefore if we can conceive of something not existing, without contradiction, it does not necessarily exist and must be shown by observation. If an argument for the existence of God succeeded, then God’s existence would be necessary. The contra positive is that if God’s existence is not necessary then God does not exist. Taking these premises we have:

Argument 4.

1. Nothing is demonstrable, unless the contrary is a contradiction.

2. Whatever we conceive as existent, we can also conceive as non existent.

3. Deductive arguments for the existence of God attempt to demonstrate the existence of God.

4. There is no contradiction in denying that God exists. This follows from 2.

5. Therefore, deductive arguments for the existence of God fail as demonstrations because if they were sound, they would necessarily follow and could not be denied without contradiction.

6. God’s existence is not necessary, this follows from 2, we can conceive God as not existing and 5, deductive arguments for the existence of God can be denied without contradiction.

7. Ontological arguments argue that if God exists, then God necessarily exists.

8. If God doesn’t necessarily exist, then God doesn’t exist. Contra positive of 7.

9. God doesn’t exist. This follows from 6 and 8.

The first premise seems reasonable. A contradiction is when we have the same statement true and not true at the same time. I.e. I exist but I don’t exist is a contradiction. If any attempted demonstration leads to the conclusion that I exist and at the same time that I don’t exist, it fails.

The second premise states the obvious fact that we can conceive of unicorns existing and we can conceive of them not existing. The same goes for dinosaurs, people and even God. In fact, even when I conceive of God existing it has no bearing on the matter, no more than my conceiving that there is no God. Existence is not a predicate as Kant said. Just conceiving something necessarily existing is no more existence than conceiving it not existing.

The third premise states what deductive arguments set out to do. Demonstrate or deduce the existence of God using formal logic.

The fourth proposition follows from the second premises. We can conceive of God existing and of God not existing with equal ease.

The fifth proposition follows from the premises 1, 3 and 4. That is there is no contradiction in denying the existence of God, which means that demonstrations (deductive arguments) that try to prove the existence of God are false.


The sixth proposition states that if God’s existence were necessary, we could not conceive of God’s non existence. Also, if God’s existence were necessary, then the conclusion any valid demonstration of God’s existence would lead to a contradiction.

The seventh statement just describes what ontological arguments attempt to do. They attempt to show that God has some property of necessary existence, and therefore if God exists, God must necessarily exist.


The eighth statement is the contra positive of the seventh. If God doesn’t necessarily exist, then God doesn’t exist at all.

The ninth statement follows from the sixth and eighth statements. If the premises are acceptable, the argument valid, then the conclusion necessarily follows.

http://philosophicalneuron.blogspot.com/2009/05/silly-argument-against-existence-of.html
... emphasis mine.

This is what I've been looking for. This is the airtight ontological argument for God not existing. It's not fallacious as all of the ontological arguments from apologists.

Thoughts?

[VanReal]

To quote Jack McCoy "Prove to me that you haven't read that book."

[samuel]

I think premise 7 of argument 4 fails since Ontological arguments don't prove that a god exists.
check out this breakdown of ontological arguments: http://www.youtube.com/watch?v=YLQ0DnUTyX8

[Whitney]

I think premise 7 of argument 4 fails since Ontological arguments don't prove that a god exists.
check out this breakdown of ontological arguments: http://www.youtube.com/watch?v=YLQ0DnUTyX8

Hey Samuel, I'm glad you decided to join our discussion.  Feel free to post an introduction when you have the time.

[AlP]

I agree with samuel on premise 7.

2. Whatever we conceive as existent, we can also conceive as non existent.

Also, who is "we"? If this argument hinges on there being nobody who is unable to conceive of the non-existence of a thing that they can conceive of as existent, then this might be a flaw. God would be a example for some people. The universe would be one for me. I cannot conceive of the non-existence of the universe but I can easily conceive of its existence. It's strange, I can conceive of any number of universes greater or equal to one. But I have to be in one. So zero doesn't work for me. My thinking might be flawed but my point is that if "we" includes everyone then it only takes one person who is mistaken to potentially undermine the argument.

[PipeBox]

I agree with samuel on premise 7.

2. Whatever we conceive as existent, we can also conceive as non existent.

Also, who is "we"? If this argument hinges on there being nobody who is unable to conceive of the non-existence of a thing that they can conceive of as existent, then this might be a flaw. God would be a example for some people. The universe would be one for me. I cannot conceive of the non-existence of the universe but I can easily conceive of its existence. It's strange, I can conceive of any number of universes greater or equal to one. But I have to be in one. So zero doesn't work for me. My thinking might be flawed but my point is that if "we" includes everyone then it only takes one person who is mistaken to potentially undermine the argument.

"We" is anyone who accepts the premise, where their acceptance is free to hinge on their understanding/knowledge/observations.  I'd say it's unfortunate, but that's how it works.

As to the ontological arguments that prove that if a god exists, its existence must be necessary, I suppose that is tautologically true, if it's a creator god.  For anything else, I suppose I'm not well read enough to make a clear call on premise 7 of argument 4.  Still, if we accept the premise, then we reach the conclusion.

[Sophus]

Hmmmm..... prove to me that you can't defy gravity. Prove to me that my name isn't Bob. Prove to me that Joe didn't murder Sally.

But what if I say: Prove to me the theory of gravity. Prove to me that my name is Sophus. Prove to me that Joe is innocent.

For example, that last question - in a court case the evidence should point one way or another. "Where was Joe on the night of the crime?" Discovering his location will prove one idea and disprove another. You see, I don't think we're trying to prove positives or negatives. I think we're just looking for proof. Given the observables we should be able to come to a conclusion. That conclusion can be made to appear as either a positive or a negative. Certain assertions or claims that are beyond the observable may be unprovable but however still provable or disprovable in theory. Am I making sense?

We need to establish some other things first:

 1. Can all truths be proven? Or can, at least, all truths be proven in theory?
 2. What qualifies as proof?

And please critique me on this! I haven't even convinced myself of this idea yet.

Btw, doesn't this belong in the Philosophy section?