In October, 1944 G. E. Moore gave a talk to the Moral Science Club in Cambridge that contained a sentence that has become known as “Moore's paradox”. In the ensuing sixty plus years, Moore's paradox has generated an extensive literature. Wittgenstein, who at times made caustic remarks about Moore's intelligence, immediately wrote to Moore, urging him to publish his “discovery”. In his letter, Wittgenstein explained why he thought that Moore's finding was so important.
You have said something about the logic of assertion. Vis: It makes sense to say “Let's suppose: p is the case and I don't believe that p is the case”, whereas it makes no sense to assert “I-p is the case and I don't believe that p is the case.” This assertion has to be ruled out and is ruled out by “common sense”, just as a contradiction is. And this just shows that logic isn't as simple as logicians think it is. In particular: that contradiction isn't the unique thing people think it is. It isn't the only logically inadmissible form and it is, under certain circumstances, admissible. And to show that seems to me the chief merit of your paper.(Cited in Monk 1990: 545)
As Ray Monk has pointed out, this was not how Moore himself saw it. He thought that, as the paradox did not issue in a formal contradiction, it was an absurdity for psychological, rather than for logical, reasons – an interpretation that Wittgenstein vigorously rejected.