 That is because in intuitionistic logic the quantifier exchange rule, \(\neg \forall x\P[x] \vdash \exists x \neg\P[x],\) is not unrestrictedly valid. Importantly the expression of non-omniscience at line \(12, \neg \forall p(p \rightarrow Kp)\), is only classically, and not intuitionistically, inconsistent with line \(10, \neg \exists p(p \wedge \neg Kp)\). So the intuitionist anti-realist can consistently give expression to the truism that we are non-omniscient (with line 12) while accepting the intuitionistic consequence derived at line 10. In effect, the anti-realist admits both that no truths are unknown and that not all truths are known. The satisfiability of this claim on intuitionistic grounds is demonstrated by Williamson (1988, 1992). 