What could "transcendental rationalism" be?

Disclaimer: The words “transcendental” and “rationalism” have been used in a variety of ways both within and without the Western philosophical tradition. I will be limiting myself to the parts of the Western tradition that gave rise to the most common view that goes by the name “transcendental rationalism.”

To understand the term, we’re going to need a little bit of history. First, one of the major philosophical debates of the early modern period (basically, the 16th – 19th centuries, with special emphasis on the 17th century) was that of empiricism vs. rationalism. This started out as a debate over the foundations of knowledge: is the primary source of knowledge our senses or our reason?

This debate evolved over the years, particularly as the sciences developed and the empiricists were more and more able to explain things purely in terms of sensory experience. By the time we get to Kant, empiricism had become the view that all knowledge comes from experience, whereas rationalism had become the view that not all knowledge comes from experience.

Kant started out as a rationalist, but then had a bit of a crisis when he started reading Hume that he thought affected both rationalism and empiricism: regardless of whether experience is primary or secondary, it is always mediated/filtered by our sensory organs and cognitive faculties—which means there will always be a gap between the world as we know it and the world as it is.

This wasn’t an entirely new idea, but Kant had a new approach: transcendental reasoning (aka transcendental argumentation). The basic idea here is to start with a fact about our experience and then work out the necessary conditions for that experience (i.e., what must be true in order for that given fact about our experience to be the case). In simpler terms: if you know you have experience X, and if you determine that experience X is only possible under condition Y, then you know that condition Y obtains.

The transcendental rationalist runs with this and claims that there is knowledge that we can gain purely through transcendental reasoning, and that at least some of this knowledge is not ultimately grounded in experience. In other words, it’s a form of rationalism that builds itself on the notion of transcendental reasoning.