||The English used in this article may not be easy for everybody to understand. (September 2011)|
A theory is a group of linked ideas intended to explain something, especially an explanation based on basic ideas that do not depend on facts about the thing to be explained. For instance, to explain why gold is yellow, a good theory should not base itself on the fact that people all observe gold to be yellow.
A theory in science (in contrast to a theory in layman's terms) is "a logical, systematic set of principles or explanation that has been verified—has stood up against attempts to prove it false."  For example, Darwin's Theory of Evolution is a system of ideas that points to humans and apes having evolved from a common ancestor. This conclusion is based on evidence that supports it.
Logical consistency [change]
One basic thing needed by a theory is not to defeat itself. If someone argued that every action a human being takes in life is predetermined before that person is even born, and also argued that a human being will determine his or her future according to the acts that he or she freely wills, there is a big problem. If the first half is true then the person cannot "freely will" anything, so the second half cannot be true. If the second half is true then not every action a human being takes in life is predetermined, so the first half is not true. One or the other half of this theory must be false, but the person who made the argument give us no way to decide which part is right. All that other people have left is the idea that there is a conflict between the idea of everything being determined even before one is born, and the idea that people can freely will to do certain things. In short, to say,"It is not true that the sky is blue, and it is true that the sky is blue," is to give the listener an idea with one hand and then take it away with the other hand.
This idea is called the principle of non-contradiction, not saying the opposite of what you just said. If we consider any statement, A (perhaps A stands for "The Apple has a worm in it."), then A and the contradiction of A is always false. (In logical symbolism, the single statement "A ∧ ¬A" has a truth value of 0 or false.)
Practical consistency [change]
A theory that can produce a statement that is not true, a theory that can predict something that does not in fact happen, or that predicts that something will happen but it does not happen, is not a correct theory. This idea has a big hole to fall into. Scientists can look at experiment after experiment, and they may always find that what theory says is true. Years may pass, and then somebody looks at one more experiment. That experiment shows that the theory is false, and every time they do that experiment it shows that the first time was not some kind of accident. The philosopher Karl Popper gave the example of people in Europe before 1492 who wanted to give scientific descriptions of birds. One bird they worked on was the swan. Somebody proposed the idea, "All swans are white." All of the swans that were checked in their century were white. When Europeans first sailed to Australia, the first swan that they saw was black. Suddenly the old scientific description of swans had to be changed.
So it is never possible to prove conclusively that some theory is correct. A "black swan" may come with the next experiment. It is possible to prove some theories are incorrect. Science makes progress by using one theory until it fails, trying to understand why that theory failed, and then making a better theory. Some theories are very well confirmed, that means that they have been tried over and over again and have never yet failed. When a theory is well confirmed people trust its predictions.
It is possible to prove conclusively that some theory is incorrect. Proving a theory is incorrect makes it possible to find a better theory and thus to make progress.