 Although opponents of classical foundationalism are not always eager to admit it, we suspect that the primary dissatisfaction with classical foundationalism lies with the difficulty the view has avoiding radical skepticism. On infallible belief, infallible justification, or direct acquaintance theories of foundational justification, there is precious little included in the foundations of knowledge. Most classical foundationalists reject the idea that one can have noninferentially justified beliefs about the past, but the present disappears into the past in the blink of an eye. How can one even hope to get back the vast body of knowledge one pre-philosophically supposes one has, if one’s epistemic base is so impoverished? If the second clause of the Principle of Inferential Justification were accepted, the problem is even more serious. One might be able to convince oneself that one can know noninferentially the principles of deductive reasoning, but deduction will not take one usefully beyond the foundations of knowledge and justified belief. As Mill (1906: 126) argued, there is a very real sense in which one doesn’t advance one’s knowledge significantly employing a form of reasoning that takes one only to conclusions that were implicitly contained in the conjunction of one’s premises. To advance beyond foundations we will inevitably need to employ non-deductive reasoning and according to PIJ that will ultimately require us to have noninferential knowledge of propositions describing probability connections between evidence and conclusions that are not logically implied by the evidence. It is not absurd on the face of it to suppose that one can have noninferential a priori knowledge of probabilistic connections, but it is perhaps an understatement to say that the view is not popular (see Russell 1948 for an excellent discussion of this issue).