A very good question. +1
It is said ( according to Wiki ), Wittegenstein read Brothers Karamazov 50 times when he engaged in WW1.
His view says,
Compare Dostoyevsky's view here to that of Protagoras: If the individual man, if the individual human being, is the measure of all things, that is to say if there is no true or false as we normally use those words (for 'truth' and 'falsity' are objective concepts, independent of the individual subject), then, in Dmitri's words "Everything is permissible" -- because the individual is also the only measure of good and evil (which therefore like truth and falsity do not exist).... However, for isn't there a "however"? "But you know that's only words." -- "All my life I wanted it not to be "only words"."
But is it "only words"? For haven't we a fundamental sense of fairness that Dmitri's words do violence to ["Forgive us our wrong-doing as we forgive those who do wrong to us"]; doesn't his contention do violence to our human nature -- for no one with this moral sense says "I can do it" = "It is good"? No one (except the ones Hume called "monsters") says, if there is no punishment for wrong-doing, then I shall be cruel and selfish ... again, as if the good were not really good and so worth doing for itself, but that one had to be threatened with punishment in order to do what one recognizes to be good. There is a fundamental flaw in Dostoyevsky's account; I would say, he is being perverse (false to his own feelings) when he says "All things are permissible", as if he himself would ever have permitted himself all things! "If all laws were to disappear tomorrow, all public order, the philosopher would go on living as he had always done." That is the difference between Greek philosophy and Dostoyevsky's religious (if, that is, religion must be about reward and punishment) account.
I personally think he ( Wittgenstein ) read too far.
Personally I think Dmitri of Brothers Karamazov is just representing "Russian folks ( especially poor people then )" in 19th century's Russia (n empire )) when Russian Empire met **Western thoughts, particularly socialism, in another words, I personally think Dostoevsky portrayed Dmitry as an ordinary folks back then while Ivan the Intelligentia.
For example, this phrase by Wittgenstein
But is it "only words"? For haven't we a fundamental sense of fairness that Dmitri's words do violence to "Forgive us our wrong-doing as we forgive those who do wrong to us"
I think he developed ( perhaps ) his idea beyond the original "image" by Dostoevsky, in my opinion.