First: see How is a judge to evaluate a witness's credibility?
Appealability of credibility findings is extremely limited
A fact-finder's determination of credibility is not an appealable issue unless it can be traced to:
- an error of law (e.g. if a judge found that a complainant is less worthy of belief because of his or her sexual history); or
- what is known as a "palpable and overriding error" in Canada or "clear error" in the United States (e.g. the judge based an adverse credibility finding on a
piece of evidence that just didn't exist — like the judge writing that the witness said X and Y, but the transcript shows that the witness never said Y).
See footnote 141 of Moffat v Edmonton (City) Police Service, 2021 ABCA 183 for a good summary of the standard in both the US and Canada.
But as long as there is no error of law and no palpable and overriding error (or clear error), credibility findings will not be overturned on appeal. A reviewing court will not simply replace the judge's credibility finding because they would have found differently.
Credibility findings are not all or nothing
Also, credibility findings do not necessitate accepting all or none of a witness's testimony. This is made explicit in typical jury instructions:
you may disregard so much of the testimony as you find was untruthful, and accept so much of it as you find to have been truthful and accurate
Less than full explanation is not the same as completely arbitrary
Last, given your title and introductory paragraph, it seems you may be conflating "completely arbitrary" and not "fully explaining." These are wildly different circumstances. The adequacy of a trial judge's reasons is assessed on a functional basis. The Supreme Court of Canada has said (and this approach is similar elsewhere):
The question is whether the reasons are sufficient to allow for meaningful appellate review and whether the parties’ “functional need to know” why the trial judge’s decision has been made has been met. The test is a functional one.
In determining the adequacy of reasons, the reasons should be considered in the context of the record before the court. Where the record discloses all that is required to be known to permit appellate review, less detailed reasons may be acceptable. This means that less detailed reasons may be required in cases with an extensive evidentiary record, such as the current appeal. On the other hand, reasons are particularly important when “a trial judge is called upon to address troublesome issues of unsettled law, or to resolve confused and contradictory evidence on a key issue”, as was the case in the decision below. In assessing the adequacy of reasons, it must be remembered that “[t]he appellate court is not given the power to intervene simply because it thinks the trial court did a poor job of expressing itself”.
Less than a full explanation does not turn a credibility determination into a completely arbitrary one (e.g. accepting testimony based on a flip of a coin).