Illation is a noun that means an inferred conclusion.
The Latin word illatio (ee LAH cho) means ‘to bring in or carry.’ When we arrive at an illation, we are bringing in a conclusion based on what we’ve inferred.
When Gene hinted that Thursday might be a good day to cancel work and allow all employees to stay home and have cake while opening presents, the illation I gathered was that Thursday was his birthday. Gene never was subtle when it came to dropping hints.