"}so what’s the linguistic relationship between -proof as in “immune to” (“waterproof”, “fireproof”) and proof as in potency (alcoholic proof) and proof as in “authoritatively establishing as true”?
First, I\u2019d like to note that proof (along with it\u2019s related verb, prove) is one of those Latin-by-way-of-Old-French-and-Norman-Invasion words which possesses a more-direct-from-Latin doublet, in the form of \u201cprobe.\u201d
Per the Online Etymological Dictionary:
early 13c., preove “evidence to establish the fact of (something),” from Anglo-French preove, Old French prueve “proof, test, experience” (13c., Modern French preuve), from Late Latin proba “a proof,” a back-formation from Latin probare “to prove” (see prove). “The devocalization of v to f ensued upon the loss of final e; cf. the relation of v and f in believe, belief, relieve, relief, behove, behoof, etc. [OED].
Meaning "act of proving” is early 14c. Meaning “act of testing or making trial of anything” is from late 14c., from influence of prove. Meaning “standard of strength of distilled liquor” is from 1705. In photography from 1855. Typographical sense of “trial impression to test type” is from c. 1600. Numismatic sense of “coin struck to test a die” is from 1762; now mostly in reference to coins struck from highly polished dies, mainly for collectors.
Adjectival sense (proof against) is recorded from 1590s, from the noun in expressions such as proof of (mid-15c.), hence extended senses involving “tested power” in compounds such as fireproof (1630s), waterproof (1725), fool-proof (1902), etc. Shakespeare has shame-proof. Expression the proof is in the pudding (1915) is a curious perversion of earlier proof of the pudding is in the eating (1708), with proof in the sense “quality of proving good or turning out well” (17c.); perhaps an advertiser’s condensed form of the original.
But, from Wikipedia\u2019s\u00a0\u201cAlcohol proof\u201d:
The term proof dates back to 16th century England, when spirits were taxed at different rates depending on their alcohol content. Similar terminology and methodology spread to other nations as spirit distillation, and taxation, became common. In England spirits were originally tested with a basic “burn-or-no-burn” test, in which alcohol that would ignite was said to be “above proof” and alcohol which would not was said to be “under proof”.[1] The breakpoint between under and over proof was defined as 100 and was the basis for taxation. Because alcohol’s flammability is highly dependent on temperature, 100 proof defined this way could range anywhere from 40\u201390% ABV in normal air temperatures; at 22 \u00b0C (72 \u00b0F) 100 proof would be 60% ABV.[2]
Another early method for testing liquor’s alcohol content was the “gunpowder method.” Gunpowder was soaked in a spirit, if the gunpowder could still burn the spirit was rated above proof. This test relies on the fact that potassium nitrate (a chemical in gunpowder) is significantly more soluble in water than in alcohol.[3] While less influenced by temperature than the simpler burn-or-no-burn test, gunpowder tests also lacked true reproducibility. Factors like the grain size of gunpowder and the time it sat in the spirit impact the dissolution of potassium nitrate and therefore what would be defined as 100 proof. However, the gunpowder method is significantly less variable than the burn-or-no-burn method and 100 proof defined by it is traditionally defined as 57.15% ABV.