1.1. In two previous papers a consistent theory of real numbers has been outlined by the author, using a system K′. This latter system is an extension of a system K, which is “basic” in the sense that every finitary (recursively enumerable) subclass of its well-formed expressions is in a certain sense represented in it. The system L described below is a further extension of K. The system K′ lacks two important features possessed by L: a symbol for a special kind of implication (or “conditionality”) and a symbol for the modal concept “necessity.” The presence of the implication symbol, and the additional assumptions that go with it, make available in L various kinds of restricted universal quantification not available in K′, for example, universal quantification restricted to the real numbers of the author's theory of real numbers.

1.2. If ‘~[a & ~a]’ is a theorem of L, then the proposition expressed by ‘a’ may be said to L-satisfy the principle of excluded middle. I t is always the case that ‘a’ L-satisfies the principle of excluded middle (or rather that the proposition expressed by ‘a’ does so) if and only if ‘a’ or ‘~a’ is a theorem of L. An example of a proposition that does not L-satisfy the principle of excluded middle is that expressed by ‘’, namely the proposition that asserts that the class of classes that are not members of themselves is a member of itself.