Principia mathematica, vol. 1 (of 3)

In the above proposition (*50·33), the hypothesis is equivalent to the hypothesis that more than one object exists of the type in question. This can be proved for all except the lowest type. For the lowest type, we can only prove the existence of at least one object: this is proved in *24·52. For the next type, we can prove the existence of at least two objects, namely and ; these are distinct, by*24·1. For a class of order , we can prove the existence of objects. But for the class of individuals we cannot prove, from our primitive propositions, that there is more than one object in the universe, and therefore we cannot prove . We might, of course, have included among our primitive propositions the assumption that more than one individual exists, or some assumption from which this would follow, such as But very few of the propositions which we might wish to prove depend upon this assumption, and we have therefore excluded it. It should be observed that most philosophers, being monists, deny this assumption.