Ibn Sina's syllogistic: a logic between modal and many-valued
Although the proof theory of Aristotle's categorical syllogisms has been studied to exhaustion for two thousand years or so, there has been strangely little significant work on the mathematical structures underlying them. (A rare exception is Haskell Curry in 1936.) Ibn Sina's syllogisms require us to go to a higher level of generality, which is helpful for understanding the mathematical scenery. Ibn Sina's generalisation of Aristotle is natural and could in theory be applied to any logic, though I don't know where else it might be useful.