Illogical (adjective)

1. Not logical; lacking in a coherent system of reasoning.
2. Not in agreement with the principles of logic.

Origin:

Late 16th century: from french illogique or late latin illogicus, from greek illogikos, from a- "not" + logikos "pertaining to reason".

Examples:

1. His argument was illogical and didn't make any sense.
2. The conclusion was illogical and not supported by the evidence.
3. The thought process was illogical and led to a flawed decision.
4. The reasoning was illogical and not based on facts.
5. The decision was illogical and not in the best interest of the company.