Kantian epistemology was challenged only many years later, not by philosophers but by logicians like George Boole, Gottlob Frege, Giuseppe Peano, C.S. Peirce and Bertrand Russell and Alfred North Whitehead.
In the 17th C. already Gottfried Leibniz philosopher at Univ. of Leipzig had proposed that if numbers were assigned to concepts then formal rules for manipulating numbers could effectively manipulate the corresponding concepts.
In 1854 the English mathematician George Boole developed this idea in his book The Laws of Thought. The Boolean system enabled operations (like, AND, OR, NOT) to be performed on propositions.
In 1879 Gottlob Frege expanded the system to deal with predications: thus predicate calculus developed by end of 19th C. Frege tried to show that all arithmetic could be reduced to logic. 2
Critical idea: Information can be represented in symbolic notation. Formal logical operations applied to these symbols can simulate reasoning.
Building on Frege's insights, Whitehead and Russell (Principia Mathematica 1910-1913) sought to derive all of mathematics from the basic laws of logic (as reformulated in the century since Kant's time). The motivation for this program was explicitly anti-Kantian, for Russell and Whitehead wanted to discredit the Kantian a priori notion that mathematical knowledge was dependent on experience. (This work led to closer ties between empirical science, logic and mathematics. Russell believed that most traditional philosophical questions could be expressed in logical terms and could either be solved in those terms or shown to be insoluble.)
Sensory data + Tools of logic give Account of external world
Ludwig Wittgenstein, Austrian philosopher in Tractatus logico-philosophicus, 1921, tried to demonstrate a logical structure implicit in language. In his later work he saw language as the source of problematic issues; took a pessimistic view of psychology.
See Gardner MNS for - Logical empiricism, Carnap program, unraveling of LE - Ryle, Wittgenstein, Austin, Rorty, Putnam, Fodor.
Logical empiricism inspired Behaviourism (to be discussed in tutorials of philosophy and psychology)