1640s, "something which is inconsistent;" 1650s as "quality of being inconsistent," from in- (1) "not, opposite of" + consistency. Related: Inconsistencies. Inconsistence (1630s) is marked "Now rare or Obs." in OED.
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."
In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
1590s, "firmness of matter," from Medieval Latin consistentia literally "a standing together," or directly from Latin consistentem (nominative consistens), present participle of consistere "to stand firm, take a standing position, stop, halt," from assimilated form of com "with, together" (see con-) + sistere "to place," causative of stare "to stand, be standing," from PIE root *sta- "to stand, make or be firm."
Meaning "state of being in agreement or harmony" (with something) is from 1650s; meaning "steady adherence to principles, patterns of action, etc." is from 1716. Meaning "harmonious connection, as of the parts of a system" is from 1787.
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Definitions of inconsistency from WordNet
the relation between propositions that cannot both be true at the same time;