According to metaphysical realism, there are truths that we can understand but cannot even in principle come to know, whereas according to anti-realism, there are no such truths. The goal of this paper is to point out some of the problems that beset attempts to justify such metaphysical theses about our cognitive limits. It is argued that we should be agnostic about each of them. We are not in a position to know anything about what does and what doesn't lie beyond the limits of our knowledge. On the one hand, argument for anti-realism are based on controversial conceptual analyses of truth or they proceed from a controversial theory about the nature of understanding. On the other hand, arguments for metaphysical realism depend on the controversial principle of bivalence.
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