Wittgenstein: Meaning and Mind (An Analytical Commentary on the Philosophical Investigations, Vol. 3, Part 2: Exegesis 243-427)

This is an excellent volume, containing detailed exegesis of the crucial part of the Philosophical Investigations, in which the private language argument is discussed. There is no other commentary which is as lucid and helpful as this one, on the most famous and important philosophical argument of the 20th century.