What are the Roles of "Quin" and "Sit" in "fieri non potest quin sit"?

Question

In the question on Sherlockian logic, Batavulus, in his answer gave an alternative translation of the clause "it must be believed"/ "one must believe it", which is:

"fieri non potest quin sit credendum." =

"it cannot happen that it may not be believed."

The English was a quick off-the-cuff translation (by me); an appalling double-negative. It implies doubt, from subjunctive "sit" = "may"/ "might" (character Sherlock Holmes was not in any doubt) and the key seems to be the meaning of "quin".

Joonas (CHAT: 3/2/2021) suggested:

"there is no option but to be believed."

The beauty of this is that it not only captures the required meaning, but by adroit sleight-of-hand (the translator's art) it negates any need to define "quin" & "sit"--brilliant!

QUIN

The conjunction "quin" functions in a number of contexts; more than one may be applicable e.g. "(so) that..not" which would create another double-negative. Alternatively, "why not?". Although a question appears to be reinforcing the doubt, Allen & Greenough p.449 (2b): "...and quin with a Present Indicative may have the force of a command:-

"quin accipis" (Ter. Haut. 832) = "Here, take it" ("Why not take it?").

The translation would then be:

"It cannot happen it is to be believed."

This is the opposite of what is required and ignores "sit".

QUESTION

What are the roles of "quin" and "sit" in "fieri non potest quin sit credendum"?

Answer 1

Quin behaves very differently in independent and subordinate clauses. In the Terence quote quin accipis it is used independently and means roughly "why not". In the sentence in question it introduces a subordinate clause and does not mean "why not". A&G discuss the use of subordinate clauses introduced by quin and quominus at some length.

On that A&G page you will find this example sentence:

Fierī nūllō modō poterat quīn Cleomenī parcerētur.
It was out of the question that Cleomenes should not be spared.

This is the main example, and we can simplify it:

Fieri non potest quin moriamur.
It is not possible that we would not die.
≈ There is no option for us but to die.

This is a good construction to remember through a simple example like this. If we replace morimur ("we die") with credendum est ("it must be believed"), we find:

Fieri non potest quin credendum sit.
It is not possible that it would not have to be believed.
≈ There is no option (for it) but to be believed.
≈ It has to be believed.

(Quin can introduce a negative relative clause, being equivalent to qui/quae/quod non. But this is not the case here, as the subordinate clause you have is not relative. The relevant entry in A&G now is §558.)

Without negatives a similar thought could be expressed like so:

Fieri potest ut credendum sit.
It is possible that it would have to be believed.

In this sense quin ≈ ut non in this use. This replacement is not universally valid, though. The kind of quin you have contains a negation and can only be used when the dominant clause has a negation.