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).<p>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 <i>metaphysical</i>, not a logical, constraint).<p>And (2) it's not a useful formalisation of this really either, since the discussion is <i>why</i>, not <i>that</i> (this is taken as a given).<p>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.<p>Suppose strong emergence is true, then no amount of I-terms will express a W-term.<p>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.<p>(Incidentally, I take the strong view).
Emergence is simply repeating patterns observed in aggregates.<p>There is nothing that "emerges" in the physical universe, instead things emerge in an observer's model of the world.
What is an example of emergence that is not self-similarity, and given the diminishing entropy of self-similar structures and processes, how is emergence not just a mean reversion in the entropy of a system? We could say that information is what emerges from a system as the result of the mean reversion of its entropy over time.