Sentence:  Suppose we have any axiom of the form: (2) For every Tweedledee there is a <WSD>Tweedledum</WSD> R-related to it. Axioms of the form (2) cut two ways. They can be construed as putting a limit on the Tweedledees: only those exist for which there is an R-related <WSD>Tweedledum</WSD>. On the other hand, they can be seen as postulating the existence of a rich collection of <WSD>Tweedledums</WSD>. There are so many of them that there is at least one R-related to every single Tweedledee. 
Meaning of the word marked with <WSD> tags:  One of a pair (the other of the pair being "Tweedle-dee") of nominally different (often: but practically identical) things.