Can you solve it? Logicians in a line

A head-scratching hat puzzle

UPDATE: Read the solutions here

Today's puzzle retrospectively commemorates UNESCO's World Logic Day, which took place last week. (The date, January 14, is both the day Kurt Godel died and the day Alfred Tarski was born, a calendrical coincidence that links the pre-eminent logicians of the twentieth century.)

It is a logic puzzle and, as is typical for the genre, concerns a group of clever people wearing colourful headwear.