My earlier post on this topic has generated some questions, so, I am going to explain this matter again, and this time I will explain it like my readers are kindergarteners.
"Proving a negative" means giving compelling reasons to accept a negative proposition. Negative propositions are the ones that include negations in them, such as the words not and no. In categorical logic we distinguish four proposition types, A, E, I and O.
A propositions are universal and affirmative: All S is P.
E propositions are universal and negative: No S is P.
I propositions are particular and affirmative: Some S is P.
O propositions are particular and negative: Some S is not P.
One way to define what we mean by "proving a negative" is based on these proposition types. We could say that proving a negative means providing compelling reasons to accept a proposition of either the O or E variety.
It is often rather easy to think up ways to provide compelling reasons for accepting O type propositions. Here are some examples.
Some men are not happy.
Some birds are not brown.
Some clubs are not participating.
Some people are not old.
Compelling reasons to accept one of these propositions merely require pointing out examples that support the assertion, such as a creature that is definitely a bird and which is also any color except brown, or at least one man who is unhappy, etc.
Finding compelling reasons to accept a statement of the E variety is another matter. These statements deny that there are any shared members between two classes. For example:
No fish are birds.
No man is an island.
Nobody is happy.
Nothing is wrong.
To deny one of these statements is much easier than proving one. The denials of E propositions are always propositions of the I variety. One would assert that there is at least one fish that is also a bird, that there is a man who is also an island, that there is some person who is happy, and finally that there is at least one thing that is wrong.
But giving good reasons to accept an E proposition is another matter. Am I supposed to pick up every fish and examine it to make sure it is not a bird? Or should I check every bird to make sure it is not a fish? Do I have to check every island to make sure it is not a man?
The trouble here has to do with universality, not with negation. The proposition that all bachelors are unmarried cannot be demonstrated by examining all bachelors, because that is impractical. On the other hand, the meaning of the word bachelor is simply an unmarried male. So, it would follow that when the term is used with that meaning, all members of the set of bachelors are in fact unmarried. So, we can know about the truth of universal propositions, at least sometimes, by knowing about the meaning of the words they employ.
That would be at least one way to give good reasons for accepting an E proposition. If you know enough about the meaning of words like fish and bird, you can demonstrate convincingly that no fish is a bird just by examining their definitions.
Another way is by differentiating qualities. For example, you say "All fish have gills, but no birds have gills, therefore no fish is a bird". If gills are in fact crucial to counting as a fish, then not having gills would prove that something is not a fish. Similarly, if having human DNA is crucial to being human, then you can prove that no robot is human by arguing that all humans have human DNA, but no robots have human DNA, therefore, no robots are human.
The argument form here is either EAE or AEE. That is, from two universal propositions, one of which is an A and the other an E proposition, you can validly derive a universal negative proposition, that is, an E proposition.
But if we can validly derive E propositions, then we must be able to prove negatives. I have defined "proving a negative" as giving compelling reasons to accept a negative proposition. Valid deductions are compelling reasons, in fact, they are among the most compelling kinds of reasons known to human minds.
Because we can in fact give compelling reasons to accept some E propositions, it is false to say that "you cannot prove a negative". That irritating and ignorant sentence should be rejected by all thinking beings. Of course you can prove a negative. You can prove that no robot is human. You can prove that no bird is a fish. You can also disprove gods by arguing that all living things are mortal, but no gods are mortal, therefore no gods are living things.
So, one way to prove a negative is by using AEE or EAE deductions.
Another way to prove a negative is called modus tollens. The insight behind this type of argument is that false consequences falsify, or that all propositions with false consequences are false. The scheme of the reasoning is this:
If P, then Q
Not Q
Therefore, not P
In this scheme, variables stand for whole propositions. So the argument asserts that if the the first proposition (P) is true, the second one (Q) would be. But the second premise asserts that proposition Q is not true, and the argument concludes that therefore neither is proposition P.
To explain this let us start small and discuss what we mean when we say that propositions have consequences. By a proposition's consequences we mean that if a proposition is true, certain other propositions have to be true as well. For example, if the proposition that Peter is in Berlin is true, then the proposition that Peter is in Paris cannot be true, and the proposition that Peter is not in Paris must be true. In addition, the proposition that Peter is not in London must be true, along with the proposition that Peter is not in New York, etc. These are real consequences that follow from the truth of a single proposition. Because you know the names of hundreds of places, you can think up hundreds of sentences that must be true if the statement that Peter is in Berlin is true.
The truth of a proposition forces other propositions to be true or false. In particular, if a proposition is true, then its denial has to be false. For example, if Zebras are mammals is true, then "Zebras are not mammals" has to be false. And if the proposition that John Elway won two superbowls is true, then the proposition that "John Elway won zero superbowls" has to be false, while the statement that "it is not the case that Elway won zero superbowls" is true.
Now let's talk about false consequences. I've just said that one of the consequences of a proposition's being true is that assorted other propositions have to be true while others have to be false.
When we say that false consequences falsify we mean something specific: we mean that a consequence that should be true turns out to be false. When this happens, the proposition being tested is falsified. Let's look at an example.
Someone says John Elway won five superbowls. If this statement was true, records of superbowls would show it. But the records do not show it. They show that Elway played in five superbowls, but lost three of them. He won two. Again, if he had won five, records would show that he won five. More explicitly, if the proposition that he won five was true, then the proposition that the records show he won five would also be true. But it is not. Therefore, the consequence is false, and the initial proposition is falsified. Let's look at the scheme for all of this.
If Elway won five superbowls (P), then the records show him winning five (Q).
The records do not show him winning five. (not Q)
Therefore, Elway did not win five superbowls. (not P)
What we mean when we say that "false consequences falsify" is that when a consequence that should be true fails to be true, the proposition it is a consequence of has been falsified. When we say that "all statements with false consequences are false," we mean that when the truth of P necessitates the truth of Q, and Q fails to be true, P has been falsified. In other words, we mean modus tollens. A better way to put it is that "all propositions with false consequences are false" is a beautiful miniaturization of modus tollens.
It is thought that all instances of disproof can be represented as uses of modus tollens reasoning.
Now, if you are an atheist like me, you might want to use one or more of these forms of reasoning to throw hard balls across the theist's plate. This is not so much to change the theist's mind, as to give your audience permission to change theirs. Try proving a negative by using modus tollens:
If the bible was inspired by a god, it would be an extremely well written book.
It is not a well written book at all.
Therefore, the bible was not inspired by a god.
If there are any gods, there would be signs of them.
But there are no signs of them,
Therefore, there are no gods.
If heaven is real, there would be souls coming back to earth.
There are no souls coming back to earth.
Therefore, heaven is not real.
Or try using EAE or AEE reasoning to prove a negative.
All living things are mortal.
No gods are mortal.
Therefore, no gods are living things.
All gods are immortal.
No living things are immortal.
Therefore, no gods are living things.
All existing things are natural.
No gods are natural.
Therefore, no gods are existing things.
No existing things are unnatural.
All gods are unnatural.
Therefore, no gods are existing things.
