Kripke's exposition of Wittgenstein's private language argument from Philosophical Investigations is lucid, if not rigid. He sees the argument as best stated when Wittgenstein concludes in a remark, "Hence, it is not possible to obey a rule 'privately': otherwise, thinking one was obeying a rule would be the same as obeying it." To Kripke, this poses "the most radical and original skeptical problem that philosophy has seen to date." It ranges far beyond the mental/sense-related expressions for which people generally reference the private language argument, and casts doubt over whether any necessary meanings for our past behaviors can hold by facts at all.
At this meetup, I'd like to see if we can appreciate the private language problem as Kripke does, as the weight he gives to it may not seem immediately apparent to everyone. We will live-read much of the section titled, "The Wittgensteinian Paradox", as we are more interested in formulating the problem than in whether Kripke's "skeptical solution" (provided in the other section) is an acceptable one.
I have linked the book. I recommend reading through the short introduction and longer first section if you have time.