Etymology
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unsound (adj.)

early 14c., of persons, "diseased, wounded," from un- (1) "not" + sound (adj.). Similar formation in Middle Low German unsund, Middle Dutch ongesont, German ungesund. Meaning "morally corrupt" is recorded from c. 1300; that of "not mentally healthy" is from 1540s. Sense of "not based on reasoning or fact" is attested from 1590s. Related: Unsoundly; unsoundness.

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Definitions of unsound

unsound (adj.)
not in good condition; damaged or decayed;
an unsound foundation
unsound (adj.)
not sound financially;
unsound banking practices
unsound (adj.)
containing or based on a fallacy;
an unsound argument
Synonyms: fallacious
unsound (adj.)
suffering from severe mental illness;
of unsound mind
Synonyms: mentally ill / unstable
unsound (adj.)
physically unsound or diseased;
an unsound limb
unsound teeth
Synonyms: bad / unfit
unsound (adj.)
of e.g. advice;
From wordnet.princeton.edu