This is a formalisation of one aspect of emergence: that a set of properties of individual objects (I0,..In) do not express properties of aggregate wholes (W0..

This is a formalisation of one aspect of emergence: that a set of properties of individual objects (I0,..In) do not express properties of aggregate wholes (W0..Wn). But (1) this fails to formalise this notion, since we're not talking about two languages 'plucked from thin air' -- they are langauges whose domains are I-terms and W-terms (a metaphysical, not a logical, constraint). And (2) it's not a useful formalisation of this really either, since the discussion is why, not that (this is taken as a given). Suppose that weak emergence is true, then the failure of the I-language to express the W-language is an illusion -- rather there's just some very large number of terms involving I that W reduces to. Suppose strong emergence is true, then no amount of I-terms will express a W-term. Which of these is the case cannot be settled as a matter of logic, so the construction of two languages (I-lang) and (W-lang) begs the question. If you say emergence is simply W-inexpressible in I, then you're begging the strong view. (Incidentally, I take the strong view).