provable adj.

c. 1200, prēven, pruven, proven "to try by experience or by a test or standard; evaluate; demonstrate in practice," from Old French prover, pruver "show; convince; put to the test" (11c., Modern French prouver), from Latin probare "to make good; esteem, represent as good; make credible, show, demonstrate; test, inspect; judge by trial" (source also of Spanish probar, Italian probare, and English probe), from probus "worthy, good, upright, virtuous."

This is from PIE *pro-bhwo- "being in front," from *pro-, extended form of root *per- (1) "forward," hence "in front of," + root *bhu- "to be," source also of Latin fui "I have been," futurus "about to be;" Old English beon "to be;" see be.

From early 13c. as "render certain, put out of doubt," also "establish the validity or authenticity of a will, etc." By c. 1300 as "test and find worthy, virtuous, false, etc.," also "find out, discover, ascertain; prove by argument." By mid-14c. as "check the accuracy of." The meaning "be found to be (a hero, coward, etc.) by experience or trial" is by late 14c.

The word had many more senses and broader application in Middle English than Modern English: "to experience; to strive, endeavor; act, accomplish; thrive, succeed." Also in Middle English in a now-obsolete sense of "approve, sanction, praise" (c. 1300; compare approve). Related: Proved; proven; proving. Proving ground "place used for firing cannons for making ballistics tests and testing powder" is by 1837.

1560s, raliabill, "that may be relied on, fit to be depended on, trustworthy," originally Scottish; see rely + -able. Not common before 1850, and often execrated thereafter in Britain as an Americanism because it involves a use of -able different from its use in provable, etc., and not warranted in classical Latin. But it is defended (by OED, Century Dictionary, etc.) on grounds of the suffix's sense in available, laughable, livable, dependable; indispensable, etc. Related: Reliably; reliableness. As a noun, "a reliable person, beast, or thing," by 1890 (with old).

Reliable expresses what cannot be expressed by any other one word. Moreover, it conveys an idea of constant occurrence. He who would be exact has, indeed, no alternative, if he avoids it except a periphrasis. [Fitzedward Hall, "On English Adjectives in -Able," 1877] 

early 15c., from un- (1) "not" + provable (adj.).