What should we mean when we say we have evidence for something we believe? If I point to a fossil and say "That is evidence for evolution" and a creationist says "No, it is not evidence for evolution," is there some way for an impartial third partly to decide which of us is right?
I tend not to like courtroom analogies when debating scientific issues, but we might as well start with a look at how the law defines evidence. According to California state law, evidence is "testimony, writings, material objects, or other things presented to the senses that are offered to prove the existence or nonexistence of a fact" (http://law.justia.com/codes/california/2010/evid/100-260.html).
Scientists do not generally appeal to testimony or writings to prove anything. They deal with material objects and other things -- observations, broadly speaking.
But what do we mean by proving a fact, and how does evidence do that? It is often said that science never proves anything.
Proof is a flexible word. It does not mean quite the same thing to a scientist as it does to a mathematician or logician, and what it means in a courtroom is yet something else. What it means in a general way in all those contexts, though, is a logically based justification for some proposition that we are asked to believe.
In a criminal trial, the evidence is supposed to prove the defendant guilty of whatever charge was filed against him. It is supposed to justify a belief that he committed the crime of which he is accused, and to justify it well enough to overcome an initial presumption of his innocence.
How does that work, logically speaking?
To prove an allegation is to demonstrate that denying the allegation would entail a contradiction, that it is not logically consistent to accept the evidence while believing the allegation to be false.
Bringing in a bit of symbolism, we can explain it this way: Given a fact F, and a proposition P, we say that F is evidence for P if
Symbolically, this would be rendered: If F is evidence for P, then ~P => ~F. (For some comments on the rudiments of logic, see my essay on square circles.)
If the implication is a logical certainty, then we say that F is conclusive evidence for P. Otherwise, we qualify the claim accordingly: F is evidence that P is probably true (strong evidence), or merely could be true (weak evidence), or whatever.
F might be not just one fact but a set of facts, none of which individually is strong evidence but collectively imply a high probability that the proposition is the only reasonable explanation for their existence. Consider a murder trial in which it is established that the defendant owned the murder weapon, which was found at the scene with his fingerprints on it, and that he was at the scene when the murder was committed, and that he had a motive for killing the victim, and that the victim's blood was on clothing that the defendant was wearing when the murder occurred. Those facts are very hard to account for except by supposing the defendant killed the victim. It is very strong evidence of his guilt. Is his innocence a logical impossibility? No, it is not, but if we know nothing else, we think it unreasonable to believe he is innocent.
Mere speculation about facts that could exist is irrelevant if we don't know of their existence. The evidence that we are aware of may well lead us to a false belief. It's a chance we sometimes have to take as human beings. We cannot know everything that might have some logical relevance to our beliefs. We can only make what seems to be a reasonable effort to get all the facts we can, realizing that we can always change our minds later if we get more facts that don't happen to be available to us right now.
What is a reasonable effort? That depends on a lot of things, among them the apparent consequences of believing an error. We should try a lot harder to avoid a costly mistake than to avoid a minor inconvenience, and that means searching very hard for information of the sort that would contradict what we think we already know.
But at some point we have to get on with life, too. At some point we have to say to ourselves: If there were facts that would change my mind, I would probably have found them by now.
That does not mean assuming that those contrary facts don't exist. It does not mean suppposing that our current beliefs, based on all the facts we have found so far, cannot be wrong. It just means we will leave it to others, who are convinced that there are contrary facts yet to be found, to continue the search for them.
We may also note that the mere assertion that some fact F is evidence for a proposition P does not make it evidence.
Suppose I make some such assertion -- "F is evidence for P" -- and you wish to refute it. Your options include the following.
Nothing in science is proved with perfect certainty. No matter how much evidence there is for some assertion or what kind of evidence it is, it is never logically impossible for the assertion to be false notwithstanding the facts in evidence.
This is in contrast to mathematics, where, for a given set of axioms (analogous to evidence), a valid proof for any theorem makes it impossible for the theorem to be false.
The reason for this is that in the deductive logic of mathematics, a valid argument establishes that a denial of the conclusion absolutely contradicts at least one premise of the argument. If the conclusion is false, then at least one premise unavoidably must also be false. Under the rules of deductive logic, no reconciliation is possible.
The inductive logic of science involves only probabilities, not certainties. A strong inductive argument establishes only a high probability that, assuming its premises are true, its conclusion is true. That is to say, it is highly improbable that the conclusion is false if the premises are true. The stronger the argument, the higher the probability that if the conclusion were false, at least one of the premises would also have to be false.