1. in a manner that is not authorized by law or rules; unlawfully
2. in a manner that is not logically valid or properly reasoned

Examples

• The company illegitimately obtained the patent by bribing officials.
• He was accused of illegitimately accessing confidential files without authorization.
• The argument illegitimately assumes that correlation proves causation.
• The regime illegitimately seized power in a military coup.
• She claimed the evidence was illegitimately obtained and should be excluded from trial.
• The conclusion was illegitimately drawn from incomplete data.