In a game of chess, we may decide who is white by putting the white king in one fist, the black king in the other, and then having our opponent pick a hand. This does not seem an essential feature of the role of the king in chess, even if this process of selection were written into the rules. Similarly, it seems inessential that two different uses of "is" (as identity in "twice two is four" and as copula in "the rose is red") should both employ the same word.
The rest of Part I deals with questions of psychology and the mind in a less directed way than many of the themes in the earlier sections of the Investigations. Wittgenstein jumps around more than in the earlier sections, and it is difficult to define any particular thread that runs throughout. This commentary will focus primarily on his discussion of the belief that fire burns; the following section will deal with some of the more general themes that run through the later sections of Part I.
It makes perfect sense to talk about my "belief" that fire will burn, but when we view it as a belief, we are tempted to ask all sorts of questions that we might ask about other beliefs. If you say, "I believe the Red Wings will win the Stanley Cup," I can ask you why you believe this, and you can give many reasons, ranging from the team's recent trade acquisitions to an admission that you simply want them to win. Questions of reason, justification, and explanation seem to be a part of the grammar of the word, "believe." But if, by analogy, we then ask the same questions about my belief that fire will burn me, the same is not true. I don't reason this belief through, or justify it to myself. At the same time, the belief is not unreasoned or unjustified, but that in this case, questions of reason and justification have no place.
In On Certainty, Wittgenstein uses the metaphor of a door and its hinge to explain the relationship between propositions such as, "I believe the Red Wings will win the Stanley Cup," and propositions such as, "I believe that fire will burn me." Questions of doubt, justification, and so on, can only rotate about the fixed "hinge" of propositions such as "I believe that fire will burn me," "my hand will not go right through the table when I press my finger against it," or "the world existed five minutes ago." Doubt, justification, and so on can only function because there is a fixed ground upon which there is no need for doubt, and justification. If I am to have a substantial debate with somebody over the Red Wings' prospects in the playoffs, we must be able to agree on what counts as a reason, what might convince one of us of the other's view. Fruitful discussion relies on the fact that justification comes to an end: there are certain things that we simply agree upon. If I say I doubt that fire will burn me, what possible reasons could you give me to convince me otherwise? You could stick my hand in a flame and say, "see!" but I could reply that although I was burnt in this case, I do not believe that it will necessarily happen again next time. We cannot doubt the assertion that Napoleon won the battle of Austerlitz by asking, "how do you know the world did not spontaneously come into being five minutes ago?" If we did, we would sacrifice our ability to communicate.
Wittgenstein is not making a sharp distinction between two kinds of propositions. We can extend the metaphor of the door to point out that the bit of the door near the hinge moves very little, while the outer part of the door moves a great deal. Different propositions are open to varying levels of doubt, where propositions such as "the world existed five minutes ago" stand as a limiting case.
To return to the initial question, of what is wrong with saying, "I believe that fire will burn me": it is that this is not a belief in the sense that we normally take the word. There is nothing wrong with stating it as a belief, so long as we recognize that we are not then authorized to ask the same sorts of questions we might ask of the belief that the Red Wings will win the Stanley Cup. Wittgenstein does not try to answer the question of how my belief that fire will burn me is justified, or what sort of shape that belief takes in my mind. Rather, he leads us to recognize that we are heading down the wrong path in asking these questions. His investigations may feel unsatisfying, because he takes away without giving. He advises us not to follow these lines of questioning, but then leaves us where we started before we began questioning. He takes us back to where we started; but he leaves us with a deeper understanding of where exactly that is.