It was recognized already by the authors of Principia Mathematica that using the law of excluded middle , or its equivalent, proof by contradiction, to prove the law of excluded middle involves a vicious circle . In view of this, it is astonishing that the critics of Brouwer's rejection of the law of excluded middle claimed that his rejection leads to a third truth value , which is inconsistent, and that Church had to correct his fellow logicians by restating that their argument involves a vicious circle. The first section of this chapter is concerned with the intuitionistic interpretation of proofs by contradiction, i.e., of apagogical proofs. This analysis will put the intuitionistic rejection of the law of excluded middle in perspective. The second section treats of some philosophical and metaphysical aspects of the law of excluded middle. The third section consists of a critique of formalism and set-theoretical Platonism as approaches to the foundations of mathematics.