What are our intellectual debts to the Middle Ages?

A fair bit of philosophy and logic (and theology, which they were bound up with) was done in the West, and was built on subsequently. The De dicto and de re distinction is Thomas Aquinas’ handiwork, for example.

European nationhood is mostly a Romantic era creation, but its raw materials came out of the Middle Ages. As others have alluded to, modern Western literature and art in some aspects was built on mediaeval foundations (though a lot of it was also reinvented in the Renaissance, based on classical foundations).

The Middle Ages kept a critical mass of the Classics around, although it is fair to say that they did not make as much use of the literature as they did of the philosophy (and history, at least in the East).