Trying to find a grammar rule that explains heavy use of ‘but’ instead of ‘that doesn’t’ in older English

I have seen quite a few sentences that look like this in 18th and 19th century literature:

There was hardly a lawyer of repute but took up the question, and had an opinion as to Lizzie’s right to the necklace.

― Anthony Trollope, The Eustace Diamonds (1871)

And as for canes, hardly a man but has a half-dozen or more of all styles, colors, and weights.

― Mark Sibley Severance, Hammersmith: His Harvard Days (1878)

And after this matter it is said we come to the abstract idea of man, or, if you please, humanity, or human nature; wherein it is true there is included colour, because there is no man but has some colour, but then it can be neither white, nor black, nor any particular colour, because there is no one particular colour wherein all men partake.

― George Berkeley, The Principles of Human Knowledge (1710)

These buts are very akin to that doesn’t or doesn’t have. (Note that in the first sentence, Trollope means that reputable lawyers ᴅɪᴅ take up the question, as revealed by context, since he goes on to list a couple of solicitors and attorneys with opinions.)

Then there are a few easier explained that use but strictly instead of that:

I thank your Lordship for your invitation to Chilbolton, but I fear it is impracticable this year, It is not impossible but that next year I may have the honour of waiting on your Lordship at Sᵗ. Asaph, If I go to Ireland I certainly will go that way.

― Sir Joshua Reynolds, Letter LXXVII¹ (1784)

But using that rule on Trollope’s sentence would turn the sentence on its head completely.

I would like to learn more about the former usage, and what kind of rules explain it.

Solution 1:

A famous sentence having this rather archaic but is:

It never rains but it pours.

And I've always thought that this but means "unless":

It never rains unless it pours.

Although the former is sort of a set phrase and thus much more productive, we do have some attested examples of the latter in Google Books:

Considering that but and except are frequently treated as synonyms, and more importantly that except has an archaic meaning of "unless" (as in She never offered advice, except it were asked of her.), it's not surprising at all that the archaic but also has the meaning of 'unless'.

Admittedly, though, the syntactic behavior of this archaic but is not as clear-cut as I'd hope it to be, so you can't always replace the but with unless and make the sentence flawless in Present-day English. But you get the point.

Solution 2:

This is an example of what I call the nobbut construction. All of the examples cited involve negatives (hardly, not, no) which invoke the special "except" meaning of but in their scope. This is quite common outside of literature, though usually not to this length.

• No one but Harry would have suggested that.
• He wants nothing but the best.

Compare these with ungrammatical non-negatives:

• *Someone but Harry would have suggested that
• *He wants something but the best

Since nobbut constructions are used to single out something, they're available for specialized uses like clefts.

• Nothing remains but to thank you for your attention.

To repeat: this has nothing to do with that, or relative clauses; it has to do with negative polarity.