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Chapter 7 presents an account of the nature of a priori knowledge specifically, in three steps: a discussion of the nature of apriority; a consideration of the a priori–a posteriori distinction and its eleven major varieties; and an examination of the nature of transcendental idealism as the metaphysical foundation of an adequate theory of a priori knowledge. The majority of the chapter is devoted to critically clearing a place in logical space for a contemporary Kantian theory of rational intuitions, Kantian Intuitionism, as the core of the book’s account of a priori knowledge. Furthermore, in the context of working out solutions to the three versions of The Benacerraf Dilemma, this chapter discusses the nature and epistemic status of intuitions, with special reference to rational intuitions, and offers a corresponding critique of the contemporary critique of intuitions by Experimental Philosophy.

We respect certain epistemic norms — including (roughly) that one should tentatively believe things to have the colors they seem to have, should conform to modus ponens and the principle of non-contradiction, and should reason in accord with induction. But what could make these (or certain alternative norms, perhaps) the right ones? What could explain their correctness? This chapter scrutinizes the most commonly offered answers to this question. Special attention is devoted to the ‘semantogenetic’ strategy whereby the correctness of such norms is grounded in the meaning-engendering features of words. (For example, it's sometimes said that the axioms of arithmetic should be accepted because they implicitly define the primitive arithmetical terms). But amongst the competing strategies briefly examined here are proposals that invoke rational intuition, those that are militantly internalistic, those that prioritize considerations of reliability, and those that aim to ground the facts of epistemic rationality ‘constructively’ in normative commitments implicit in our linguistic activity. It is argued that none of these approaches is remotely adequate. And it is suggested, in conclusion, both that the correctness of the above-mentioned norms cannot be explained, and that this result should be neither surprising nor worrying.

Chapter 8 presents solutions to all three versions of The Benacerraf Dilemma—namely, The Original Benacerraf Dilemma, The Extended Benacerraf Dilemma, and The Generalized Benacerraf Dilemma—and develops a new structuralist account of the nature of mathematics and logic (Kantian Structuralism) alongside a new rational-intuition-based account of the nature of mathematical and logical a priori knowledge (Kantian Intuitionism), comparing and contrasting the two differing accounts with Charles Parsons’s theory. The chapter then moves on to work out a positive or anti-skeptical, innatist, rational-intuition-based solution to The Extended Benacerraf Dilemma. Following this, it then goes on to generalize that solution to The Generalized Benacerraf Dilemma. These solutions and accounts, in turn, presuppose and indirectly vindicate a Kant-inspired rational-intuition-based approach to meta-philosophy.

This chapter argues for a competence-based account of rational intuitions, whether of the basic human variety, or of the cultural sort. It draws on insights of Moore and Wittgenstein in its attempt to broaden the scope of rational intuitions and of a priori knowledge

Is there a true form of Dogmatism consistent with Mentalism and Foundationalism? Michael Huemer has developed an unqualified form of Dogmatism that is consistent with Mentalism and Foundationalism, but unqualified Dogmatism is false. Michael Bergmann has developed a qualified form of Dogmatism that is consistent with Foundationalism but inconsistent with Mentalism. I organize my search for a true form of Dogmatism consistent with Foundationalism and Mentalism as follows. After a brief examination of Dogmatism in both its unqualified and qualified forms, I show that two major arguments for unqualified Dogmatism are unsuccessful and that the view is open to compelling counterexamples. With qualified Dogmatism as the remaining option, I explore how best to develop it in the context of Mentalism and Foundationalism, focusing on one approach, the Knowledge How Proposal, in particular.