 New difficulties still arise if one accepts that propositions form a type (as they are the only objects of which it can be meaningfully asserted that they are true or false). First of all, there are obviously at least as many propositions as objects (just consider the map associating with \(x\) the proposition expressed by \((x = x)\); p. 527). On the other hand, if it is possible to form types of propositions, there are more types of propositions than propositions, by Cantor’s argument. Then we can inject types of propositions into propositions by means of the notion of logical product. Let \(m\) be a class of propositions and let \(\Pi m\) be the proposition “every proposition of \(m\) is true” (regarded as a possibly infinitary conjunction); then, if \(m\) and \(n\) are different classes, the propositions \(\Pi m\) and \(\Pi n\) are different, i.e., the map associating to \(m\) its product \(\Pi m\) is injective. Therefore, if we consider the class