Proof

Proof is related to prove. Curiously, the split didn't happen in English, but all the way back in Latin (from probare "to prove" came proba "a proof"). Both words then independently made their way into English.

Devoicing

Both English words originally ended in a v sound. Prove was originally prēven, and proof was preove. Over the centuries, the v in preove got devoiced into an f.

English is no stranger to the devoicing of a final consonant. See believe - belief, relieve - relief, advise - advice, bathe - bath, house /houz/ - house /hous/, use /yooz/ - use /yoos/. Unlike German though, English does not systematically devoice the final consonant, nor is there a rule to produce a noun from a verb by devoicing. In modern English, these are simply related but distinct words to be memorized, though it is often the noun that gets a voiceless ending while the verb has a voiced ending.

Testing

Prove originally meant "to test, to try, to examine", not necessarily "to provide evidence that something is correct". The Latin word that gave us prove also gave us probe, making them doublets of each other (words that got imported twice). Spanish speakers will notice that probar could mean both.

The sense of "testing" is preserved in the phrase proving ground, where weapons are experimented with or tested; proofing (or proving) dough, where yeast is tested to see if it's still alive; as well as proofreading, which means reading a proof (a test copy). The word later extended to mean "to confirm it holds up to a standard", giving us the modern sense of proof (as in a mathematical proof), as well as the suffix -proof (as in waterproof, "confirmed impervious against water").

The proof is in the pudding

Plenty of articles (1, 2, 3) have been written about this one. Here's the TLDR:

• The "correct" idiom is the proof of the pudding is in the eating, and dictionaries agree (Webster, Wiktionary).
• The word proof here means "to test", as in, eating the pudding is ultimately the only way to test whether the pudding is good.
• Pudding here is not a dessert, but blood sausages, and a badly made one can make you sick.

A version of the "correct" form appeared in writing in Camden's Remaines (p. 319 in-book), published in the early 1600s. The earliest written record of the shortened version is generally said to be an issue of The Farmer's Magazine (p. 294 in-book), published in 1867.

The shortened version does make at least one earlier appearance that we know of, in Henry Dircks's novel Joseph Anstey (p. 335 in-book), published in 1863. In it, a character had just been told of an improbable family relation (so-and-so's wife turned out to be so-and-so's long-lost sister), and he exclaims:

Well, this is the most astonishing affair of all the romances in life that ever came to my personal knowledge! [...] after dinner we will talk over all this strange mystery. [...] this surely cannot be a dream, if it is, it's the most like reality of anything I ever met with. The proof is in the pudding—or the turkey if you please, so I will even ring for it.

In other words, the author does indeed use the phrase "the proof is in the pudding", but what he means by it is, let's have dinner, eat some pudding and some turkey too, so that I can confirm this is reality and not a dream. The word "proof" here doesn't appear to mean "to test" but "to provide evidence".

Is this usage pre-established or made up by the author? Did this influence the evolution of the idiom "the proof is in the pudding"? It's hard to say.

The exception that proves the rule

A literal reading of this idiom doesn't quite make sense. An exception literally proves the rule doesn't hold; that's what an exception is. But again, idioms don't have to make sense, and Fowler's A Modern Dictionary of English Usage, published in 1922, already noted five different ways people used this phrase (p. 176 in-book), including the speaker joking or emphasizing how rare an exception is.

An explanation that sometimes comes up is that the word prove here merely means "to test": an exception tests the rigor of the rule, and finds it lacking. This is almost certainly a modern-day invention that does not track with the history of the idiom.

The more widely accepted meaning is this: having an exception shows that there's a rule in effect where not excepted. For example, "no parking 9am - 5pm" implies parking is allowed before 9am and after 5pm; "Taco Tuesday" implies we don't have a tradition of having tacos on other days. Here, the word prove has the sense "to provide evidence", not "to test".

This is a real legal principle, though it often goes by different names. The US supreme court seems to favor expressio unius est exclusio alterius (New Latin for expression of one is the exclusion of another), for example in this opinion from 2023.

The phrase exception that proves the rule first entered the English language in non-Latin form in the 1600s; we have written records of it from a play (p. 237 in-book) and a pamphlet (p. 14 in-book). The origin of this phrase is often attributed to Cicero's Pro Balbo, though he never used the word prove (or the Latin version of it):

Quod si exceptio facit ne liceat, ubi <non sit exceptum, ibi> necesse est licere.

(... if the exception makes it unlawful, where there's no exception, there it necessarily is lawful.)

The phrase evolved over the centuries, sometimes using the word prove (or the Latin equivalent), sometimes not, but always in the sense that "the exception implies a rule where not excepted", until the phrase escaped legal writing and people started using the shortened form in other ways.

Liquor strength

When you look at a bottle of liquor and it says 80 proof, the word proof here also dates back to the "testing" sense. In the US, the proof number is simply twice the ABV (alcohol by volume). This is the result of a simplification of an older British system.

The proof system dates back to 1500s England, when the authorities taxed strong liquor more than weak liquor. The word proof referred to a burn test they used to determine the strength of the liquor, sometimes burning it on its own, sometimes mixing it with gunpowder. If it ignited, it was strong enough to be taxed extra. Liquor that burned was called "above proof", and one that didn't was called "under proof".

Later, this process was made more precise, and liquor strength began to be quantified. Liquor that was just strong enough to burn became known as "100 proof", which was defined as a liquor that weighed 12/13 (twelve thirteenths) the weight of pure water of the same volume (this is called relative density, also known as "specific gravity"). Under this definition, 100% pure alcohol came out to about 175 proof.

When the US standardized its own proof system, they chose to base it on ABV rather than relative density. As part of the change, they also redefined 100% pure alcohol to be 200 proof. Why didn't they just make it such that 100% ABV = 100 proof? I haven't found any specific reasons, but I suppose it's a compromise between simplifying the measurement and maintaining some backward compatibility with the old system. People were used to the idea of "100 proof" being "strong but still safe to drink". Redefining it to mean 100% alcohol would have been confusing or even unsafe.

France, on the other hand, used the Gay-Lussac scale, named after the chemist who co-discovered, among other things, that water can be created using exactly two parts hydrogen, one part oxygen. The Gay-Lussac scale is simply ABV, and it's measured in degrees. Thus, 100 ºGL = 100% ABV = 200 proof (US).

Sidenote: ABV vs. V/V

If you mix alcohol with water, the resulting liquid will shrink a bit. Not a lot, but enough for there to be two different measurements of alcohol concentration. ABV is the more popular one, but V/V (also known as volume fraction) isn't rare.

A liquor that's 50% ABV (sometimes notated as alc/vol) is one that's equivalent to taking 50 volume units (mL, fluid ounces, etc) of pure alcohol, and mix in enough water such that the resulting liquid is 100 units.

A liquor that's 50% V/V (sometimes notated as vol/vol) is one that's equivalent to taking 50 units of pure alcohol, and 50 units of pure water, and mixing them together. The resulting liquid will take up less than 100 units of volume due to the shrinking. In other words, alcohol that's 50% V/V is slightly stronger than one that's 50% ABV.