Not too long ago, the distinction between proof and evidence was offered to me. Evidence being a series of arguments that, if sound, point towards the truth of Christianity. Evidence has an objective sense about it. Arguments that are sound do provide evidence of their conclusion. However, a lot of the time, the conclusion offered is not exclusive.
Proof is the more subjective cousin of evidence. Proof may consist of evidence, it may not. Proof is what convinces people of the truth of a claim. Many people are convinced of the truth of things without any evidence, while others have lots of evidence. Either way, the truth of that something has been proven to them.
When a person claims that an argument "doesn't prove anything," they are typically saying that that particular argument is not persuasive to them. Unfortunately, we tend to interpret that same statement as the person saying that there is no evidence for the conclusion. I discovered this mistake when I attempted to show the logical path to the conclusion. The person wasn't looking for evidence, they were looking for something to convince them specifically.
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Faithful Thinkers: Evidence For vs. Proof Of