 The moral to be drawn here is that, even if Basic Law V were consistent, it is not exactly clear how its right side analytically implies the existence of extensions. In the end, we may need some other way of justifying our knowledge of principles like Basic Law V, that imply the existence of abstract objects – the justification discussed so far seems to contain a gap. Even if we follow Frege in conceiving of extensions as ‘logical objects’, the question remains: how can the claims that such objects exist be true on logical or analytic grounds alone? We might agree that there must be logical objects of some sort if logic is to have a subject matter, but if Frege is to achieve his goal of showing that our knowledge of arithmetic is free of intuition, then at some point he has to address the question of how we can know that numbers exist. We’ll return to this issue in the final subsection.