There's a great John Stuart Mill quotation from On Liberty related to this. In the UK it used to be the case that you were barred from testifying in court if you declared yourself an atheist.

> Under pretence that atheists must be liars, it admits the testimony of all atheists who are willing to lie, and rejects only those who brave the obloquy of publicly confessing a detested creed rather than affirm a falsehood.

In law school we were taught that swearing on the Bible (and/or belief in God or some higher power) had been thought necessary to give competent testimony because that belief ensured truthfulness (or made it more likely) on the premise that, even if a falsehood were not detected and punished in court, the higher power would surely punish it later or in the afterlife. (In other words, fear-based testimony.)

And we were taught that the need to swear or affirm truthfulness nowadays simply evinces the requisite understanding of the seriousness of testimony in court / under oath — and, by extension, that the testimony must therefore be truthful in order to duly perform the judicial function of arriving at an accurate understanding of past events and realities.