The "messed-up meaning of inflammable" problem results because the "in" in inflammable comes from the Latin inflammare and not from the prefix in- meaning negation. (The il- in illogical, im- in immoral and the ir- in irreproducible are kin to the in-.) Flammable meanwhile comes from the Latin flammare to flame. It's the hodge podge nature of English that is at fault.

The long-standing English word for "capable of going up in flames" is inflammable. It comes from the same root as enflame and inflame, where the Latin in- means the English in, as in in flames.

In Latin there are two different prefixes in-, the one meaning "in", and the negative one (related to English un-). So the word inflammable looks (falsely) like it's the opposite of a word flammable.

But the word flammable is effectively a recent invention. Although flammability is recorded from the seventeenth century (in that great word-coiner Sir Thomas Browne, who however also used inflammability with the same meaning), it never caught on. Never took fire, so to speak. Webster 1913 briefly notes it and marks it [Obs.] (obsolete). The OED has quotes from 1813 and 1867, then the next one is 1959: it's from the BSI (British Standards Institute), suggesting the adoption of the word flammable for safety reasons, in place of inflammable.

Likewise, flammability has two sporadic quotations in the seventeenth century, but next appears in 1942, and in 1963 the BSI again call for the word flammability to be adopted in place of its dangerous synonym inflammability.

And what if something couldn't catch flame? What did we used to call it? Non-inflammable.

In*flam"ma*ble (?), a. [CF. F. inflammable.]

1.

Capable of being easily set fire; easily enkindled; combustible; as, inflammable oils or spirits.

2.

Excitable; irritable; irascible; easily provoked; as, an inflammable temper.

Inflammable air, the old chemical name for hydrogen.

© Webster 1913.