"Proof" is one of those words that are abused nearly to the point of meaninglessness. I generally only use it in math-related contexts.
I prefer the word "evidence" over "proof". So, instead of saying "This test score proves you didn't do your homework", I'd rather say "This test score is evidence that you didn't do your homework". It's just more accurate to say it that way.
Does this not only push back the issue? What guarantees that I won't abuse the word "evidence" like people abuse "proof", as a buzzword to throw around when I need to make my position appear stronger? This suggests that I need a firm definition of the word "evidence".
So, then, this is my definition of "evidence": given two positions, some new information counts as evidence for the position that better anticipated or explains that information. That is to say, the information counts as evidence for the position that predicted (or could have predicted) it with higher probability, and as evidence against the position that predicted it with lower probability.
Some of you may note that this is simply the Bayes factor in Bayesian inference. If you're familiar with Bayesian probability, then you can simply take the Bayes factor as my definition of "evidence", consider me a Bayesian, and give the rest of this post only a cursory read.
For everyone else, let me provide some examples of evidence according to this definition.
Let's say that Alice fails a test. One position holds that she studied hard, and another position holds that she didn't study. Between these two positions, the "didn't study" position is better able to anticipate Alice failing. That is to say, she had a higher chance of failing the test if she didn't study than if she did. Therefore her failing counts as evidence that she didn't study (and as evidence against her having studied).
Let's say that Bob is accused of murder. His DNA is found at the site of the crime. This counts as evidence that Bob is guilty, because the position that he's guilty is better able to anticipate his DNA being found at the crime site. It is more likely that his DNA will be found at the crime site if he's guilty than if he's not. That is to say, if he's guilty, there's a relatively higher probably that his DNA is found at the crime site, and if he's innocent, there's a relatively lower probability that his DNA is found at the crime site.
Note that it's possible that a particular piece of evidence points in one direction, yet is not enough to lead to a firm conclusion. Let's say that Carol claims that she's psychic. She asks you to think of a number between 1 and 100, and then guesses it correctly. This counts as evidence that she is, in fact, psychic. It'd be fairly strong evidence too, at a Bayes factor of 100:1. However, even this evidence may not be enough to convince you that she's psychic, if you were very skeptical of that position before this experiment. This is a valid way to think - you can believe that a particular piece of evidence points one way, and yet choose to reject that position in the end because it was unlikely in the first place and the evidence wasn't enough. This is where familiarity with Bayesian probability is would be very helpful - you'd be able to put numerical values to all these statements. But for now, these examples are simply a way to get a qualitative feel for what I mean by evidence.
If you flip a coin once and it lands heads, it counts as very weak evidence that the coin is a trick coin that will always land heads, because that position is better able to anticipate a coin landing heads. But you should not yet be convinced by this evidence, since such trick coins are rare and therefore you'd need much stronger evidence than a single coin flip, which has a Bayes factor of 2:1. However, after 20 consecutive heads, the Bayes factor would have increased to 1,048,576:1, and you'd have a much stronger case for believing that this coin will always land heads.
I hope this is enough to give you a solid sense of what I mean by evidence. I will be using that word often, starting with the next posts of this series.
You may next want to read:
Science as evidence for Christianity against atheism (introduction) (Next post of this series)
The Gospel: the central message of Christianity
How physics fits within Christianity (part 1)
Another post, from the table of contents