On the interpretation of intuitionistic number theory

1. Let P be some property of natural numbers. Consider the existential statement, "There exists a number n having the property P." To explain the meaning which this has for a constructivist or intuitionist, it has been described as a partial judgement, or incomplete communication of a more specific statement which says that a certain given number n, or the number n obtainable by a certain given method, has the property P.2 The meaning of the existential statement thus resides in a reference to certain information, which it implies could be stated in detail, though the trouble is not taken to do so. Perhaps the detail is suppressed in order to convey a general view of some fact. The information to which reference is made should be thought of as possibly comprising other items besides the value of n or method for obtaining it, namely such items as may be necessary to complete the communication that that n has the property P. Consider next the generality statement, "All numbers n have the property P." The accompanying explanation which has been given for this is that it is a hypothetical assertion about whatever particular n might be given. We now propose, without excluding this motif, likewise to regard the generality statement as an incomplete communication of a more specific statement, namely of one which gives an effective general method for obtaining, to any particular value of n, the information implicit in the assertion that that n has the property P. As a third example, consider the implication, "A implies B." This we now propose to interpret intuitionistically as an incomplete communication of