First, the assumption that knowledge requires certainty comes at a heavy cost, as it rules out so much of what we commonly take ourselves to know. Second, as many contemporary rationalists accept, intuition is not always a source of certain knowledge. The possibility of a deceiver gives us a reason to doubt our intuitions as well as our empirical beliefs. For all we know, a deceiver might cause us to intuit false propositions, just as one might cause us to have perceptions of nonexistent objects. Descartes’s classic way of meeting this challenge in the Meditations is to argue that we can know with certainty that no such deceiver interferes with our intuitions and deductions. They are infallible, as God guarantees their truth. The problem, known as the Cartesian Circle, is that Descartes’s account of how we gain this knowledge begs the question, by attempting to deduce the conclusion that all our intuitions are true from intuited premises. Moreover, his account does not touch a remaining problem that he himself notes (1628, Rule VII, p. 7): Deductions of any appreciable length rely on our fallible memory.
A more plausible argument for the Intuition/Deduction thesis again assumes that we know some particular, external world truths, and then appeals to the nature of what we know, rather than to the nature of knowledge itself, to argue that our knowledge must result from intuition and deduction. Leibniz (1704) tells us the following.
The senses, although they are necessary for all our actual knowledge, are not sufficient to give us the whole of it, since the senses never give anything but instances, that is to say particular or individual truths. Now all the instances which confirm a general truth, however numerous they may be, are not sufficient to establish the universal necessity of this same truth, for it does not follow that what happened before will happen in the same way again. … From which it appears that necessary truths, such as we find in pure mathematics, and particularly in arithmetic and geometry, must have principles whose proof does not depend on instances, nor consequently on the testimony of the senses, although without the senses it would never have occurred to us to think of them… (1704, Preface, pp. 150–151)
Leibniz goes on to describe our mathematical knowledge as “innate,” and his argument may be directed to support the Innate Knowledge thesis rather than the Intuition/Deduction thesis. For our purposes here, we can relate it to the latter, however: We have substantive knowledge about the external world in mathematics, and what we know in that area, we know to be necessarily true. Experience cannot warrant beliefs about what is necessarily the case. Hence, experience cannot be the source of our knowledge. The best explanation of our knowledge is that we gain it by intuition and deduction. Leibniz mentions logic, metaphysics and morals as other areas in which our knowledge similarly outstrips what experience can provide. Judgments in logic and metaphysics involve forms of necessity beyond what experience can support. Judgments in morals involve a form of obligation or value that lies beyond experience, which only informs us about what is the case rather than about what ought to be.
The strength of this argument varies with its examples of purported knowledge. Insofar as we focus on controversial claims in metaphysics, e.g., that God exists, that our mind is a distinct substance from our body, the initial premise that we know the claims is less than compelling. Taken with regard to other areas, however, the argument clearly has legs. We know a great deal of mathematics, and what we know, we know to be necessarily true. None of our experiences warrants a belief in such necessity, and we do not seem to base our knowledge on any experiences. The warrant that provides us with knowledge arises from an intellectual grasp of the propositions which is clearly part of our learning. Similarly, we seem to have such moral knowledge as that, all other things being equal, it is wrong to break a promise and that pleasure is intrinsically good. No empirical lesson about how things are can warrant such knowledge of how they ought to be.