The quantification and identity principles

When I was a student I was taught that there were intimate connections between identity, quantification, and number.

First quantification and number. I was taught that for there to be something – anything at all – is for there to be at least one such thing. For there to be an F, or for there to be Fs, or for something to be F, is for there to be at least one thing or entity that is F. (I don’t mean anything special by ‘thing’. Everything is a thing – ‘thing’ is for me just a completely general count noun.) We might put this by saying that for any kind to be instantiated is for there to be at least one thing that instantiates it. But the claim is not meant to require the existence of kinds or other universals. Nor is it meant to allow the possibility of nonkinds that could be instantiated without being instantiated by at least one thing. (Being red might be a nonkind.) So it might be better to appeal to a schema:

The claim is that every possible instance of this schema is true. Call this the quantification principle.