 Those who hold that intuitions can justify because they are based on the understanding think, for instance, that our understanding the concept knowledge is the source of our intuition that a correct lucky guess is not knowledge. Hawthorne’s criticisms of basing a priori justification on the understanding (as Boghossian forthcomingb proposes) is that there seem to be instances of a priori justification that require skill in “sophisticated proof techniques that are quite obviously not preconditions of anything in the domain” (Hawthorne 2013: 213). For instance, sophisticated proof techniques were required to prove Fermat’s Last Theorem even though it is easy to understand what the theorem says, namely, that the equation \(x^n + y^n = z^n\) has no solutions when n is a positive integer greater than two. A possible response to this objection is to expand the notion of enabling experiences to include those needed to acquire intellectual skills which are needed to employ intellectual seemings in reasoning (see, secs.  4.1  above and  4.5  below) and are different from perceptual skills. But possessing those techniques without also understanding the connections between the premises would not yield justification. It is possible to hold that the important difference between a priori and a posteriori justification is that once the concepts and the relevant techniques are acquired, nothing more is needed for a priori justification (that is the sense in which a priori justification is independent of experience), but further experience is needed for a posteriori justification. Further, Hawthorne’s criticism seems not to affect whether intellectual intuitions that are evoked by thought experiments (as in standard Gettier cases) can provide evidence since no special techniques are required there.