Irreconcilable, a.
Etymology: [Pref. ir- not + reconcilable: cf. F. irréconciliable.]
Not reconcilable; implacable; incompatible; inconsistent; disagreeing; as, irreconcilable enemies, statements. -- Ir*rec"on*ci`la*ble*ness, n. -- Ir*rec"on*ci`la*bly, adv.
Shades of meaning
Fernald has no entry of its own for irreconcilable; he lists it under incongruous.
He lists it without comment; the entries above explain its neighbors.