> How can something be both logically sound and violently at odds with reality?

Firstly, logical soundness has little to do with reality; here's a logically sound argument:

All mammals nurse their Young

Ducks are mammals

Therefore ducks nurse their young

Obviously the statement "Ducks are mammals" is false, garbage-in, garbage-out, but the logic itself is sound.

Now there was an entire field of philosophy called "Rationalism" that had a chief tenet that innate ideas and deductive reasoning was superior to sensory information. Rene Descartes formulated "I think therefore I am" because any empirical evidence of the existence of an item is suspect, but the logical statement that "if a thought occurs, then some being must be doing the thinking" is irrefutable.

Be aware that there are many epistemologies besides empiricism.