Correct spelling, explanation: the word illogical and other words from this word family come from the Greek language. The origin of illogical is logos, which was a Greek word meaning reason. Nevertheless, illogical entered English not from Greek, but from Old French, which happened in the 14th century. Illogical combines the -il prefix and the adjective logical.

Definition of illogical:
1. adjective – when something is unreasonable, often directed by emotions and not by thorough thinking;
Staying up late when you know you have to get up early in the morning seems illogical to me.

Phrases with illogical:
Some commonly used phrases with illogical are: sound illogical, entirely illogical, rather illogical;
The theory that he presented at the meeting sounded so illogical that everybody started leaving the room.
It’s entirely illogical not to take your A-Levels only because you are afraid of failing.
Skipping classes just before your final exams is rather illogical, don’t you think?

Incorrect spelling, explanation: when an adjective starts with an -l letter, a common prefix added to it when creating the opposite is -il. However, many users of English are wrong to assume that when we add this prefix, we can omit one -l letter. This happens in the case of illogical, which is often misspelled as ilogical. The -l letter is always doubled in case of such words so the only correct spelling is illogical.