A judge must have reasons
Unlike a jury, when a judge acts as the finder of fact, they must give reasons for their decisions on the facts. This includes what credit they give to any piece of evidence, including witness testimony.
"I found the witness to be credible when he testified that he was in Alabama on January 30th, and not credible when he testified that he was never there" is unlikely to be an adequate reason and might be overturned on appeal. The judge needs to go a step further and add "because ...". Providing that the reason given is a conclusion a reasonable person could make on the evidence, the decision will be appeal proof. That is, the judge has to state why they prefer the testimony that he was in Alabama over the testimony that he wasn't. For example:
- the statement that he was there on a specific day is more likely than the general statement that he had never been there,
- the circumstances of the testimony say that the general statement had been given in direct evidence, but the specific answer was in response to a question asked in cross-examination,
- that the specific statement was given to police in an interview at a time closer to the date than the general denial at trial
Now, if the judge genuinely cannot decide — for example, if they conclude that the witness is incredible — they fall back on who bears the onus of proof. Who needs to prove that the witness was or was not in Alabama?
Let's make this concrete: let's say it's vital to the plaintiff/prosecution's case that the witness had to be in Alabama on 30 January. If they can't prove that, they lose. If the only evidence they have of this is this witness's testimony, they lose because they haven't met their burden of proof. If it's impossible to say he was there, they haven't proved that he was.
Conversely, if they need to prove that he wasn't in Alabama on 30 January, they also lose because, equally, if it's impossible to say he wasn't there, they haven't proved that either.