In this article I investigate a neglected form of radical skepticism that questions whether any of our logical, mathematical and other seemingly self-evident beliefs count as knowledge. ‘A priori skepticism,’ as I will call it, challenges our ability to know any of the following sorts of propositions: (1.1) The sum of two and three is five. (1.2) Whatever is square is rectangular. (1.3) Whatever is red is colored. (1.4) No surface can be uniformly red and uniformly blue at the same time. (1.5) If ‘if p then q’ is true and ‘p’ is true, then ‘q’ is true. (1.6) No statement can be both true and false at the same time and in the same respect. (1.7) If A is taller than B, and B is taller than C, then A is taller than C. (1.8) Everything is identical to itself. (1.9) If the conclusion of an inductive argument is contingent, it is possible for the premises of that argument to be true and its conclusion to be false. (1.10) George W. Bush could have been a plumber. (1.11) George W. Bush could not have been a prime number. (1.12) ‘2 + 3 = 5’ is necessarily true.