# Is "it holds that" correct?

One can clearly say "He holds that A=B", meaning "he strongly believes that A=B". But can one say impersonally, in a scientific text, "It holds that A=B" or "There holds A=B" if one means "It is true that A=B"?

I think it's a phrase that can convey the notion of proving a theory, for example, "If it can be proved that Smith was in the building at 3pm, then it holds that he had time to steal the turnip and still be at his desk by 3:05"

Well, it holds that A = B is fine. It holds, in such a context, is impersonal, just as it seems that is an impersonal use.

But when it comes to Jane holds that A = B, the verb holds is being used in a different way. It is equivalent to believes or asserts or perhaps holds strongly that A = B.

But then there is this second use of hold, where the verb is used intransitively.

Attacked relentlessly by overwhelming forces, the line held or held firm.

This intransitive usage is common in both mathematics. So, for example, we hold that

If A = B and B = C, then A = C

If anyone doubted that claim, we can provide a reductio ad absurdum to show that if my claim is untrue, then A does not equal A, which is a self-contradiction. So

The proposition that I hold does, hold. But this use usually (perhaps only) applies to arguments (in the logical or mathematical sense). Ordinary truth claims like “it is raining” Re not said to hold, even though we can say “Once many people held that the earth was flat.”