How can 'but' mean 'not'?

Question

_{Primary Source: Benjamin N. Cardozo; Nature of the Judicial Process (1921).
Secondary Source: Lief H. Carter, Thomas F. Burke; Reason in Law (9 ed 2016), p. vii (before the Table of Contents).}

 What is it that I do when I decide a case? To what sources of information do I appeal for guidance? In what proportions do I permit them to contribute to the result? In what proportions ought they to contribute? If a precedent is applicable, when do I refuse to follow it? If no precedent is applicable, how do I reach the rule that will make a precedent for the future? If I am seeking logical consistency, the symmetry of the legal structure, how far shall I seek it? At what point shall the quest be halted by some discrepant custom, by some consideration of the social wel- fare, by my own or the common standards of justice and morals? Into that strange compound which is brewed daily in the caldron of the courts, all these ingredients enter in varying proportions. I am not concerned to inquire whether judges ought to be allowed to brew such a compound at all. I take judge-made law as one of the existing realities of life. There, before us, is the brew. [1.] Not a judge on the bench but has had a hand in the making.

The context implies 1 to mean:

2. Not a judge on the bench has NOT had a hand in the making [of law].
   = 3. Every such judge has had such a hand.

But comparing 1 with 2 evokes this question: how does but mean NOT?
The above published in 1921, I recognise this syntax's possible obsolescence.

P.S.: The polyfunctionality of 'but' has troubled me: ELU, Linguistics.

Answer 1

Not a(n) X but Y is an obsolescent expression meaning every X Y.

The original meaning of "but" is "outside", and in most of its meanings it has some sort of negative force. But (!) only in this construction (I think) does it act as a negator on a clause.

It seems to be covered in the OED entry under meaning 12: "With general sense ‘that not’, Latin quin. After negative and questioning constructions."

12a is: "a. In a simple attributive clause belonging to a n. or pronoun in the main sentence: That..not.", example (1846):

Hardly a man passes by but he must add a wreath to it.

12b. is "With omission of the pronominal subject or object of the subord. clause, so that but acts as a negative relative: That..not, who..not. (Latin quin.)", and I think the example from Keats, 1820, is parallel with the one in the question:

Not a man but felt the terror in his hair.