I had some heated arguments with “emergentists”, who tend to define strong emergence as something like:
Strong emergence (definition 1): a property of the system that can not be derived from the properties of its parts and their effects on each other when so combined.
Of course, using this definition, strong emergence would obviously exist.
But in my view this definition of strong emergence makes not much sense philosophically. We introduced a technical term and now apply this term to certain properties of systems, but in reality we are just referring to our epistemological state (i.e. what we can know with the current science and computational methods).
With such a definition there would not be a “discrete” difference between weak and strong emergence.
It would be much more interesting and useful to insert the words “in principle” into the definition:
Strong emergence (definition 2): a property of the system that can in principle not be derived from the properties of its parts and their effects on each other when so combined.
In this case, the difference between weak and strong emergence would be clear-cut and “discrete”.
Do cases of such strong emergence exist? Well, that's a controversial point.
I would disagree with you, for example, about consciousness. There we have some serious hints that definition 2 really applies, because arguably a contradiction is involved in trying to explain something inherently subjective (conscious experience) by something inherently objective (the physical brain).
But this is a matter of debate, because one person's proof by contradiction is another person's argument from ignorance.
Still it is surely the case that the usual analogies don't work for consciousness. So if physicist Steven Weinberg says something like
I see nothing about the human mind any more than about the weather that stands out as beyond the hope of understanding as a consequence of impersonal laws acting over billions of years.
he misses the point because he just asserts, without any explanation, that the problem of reducing subjective consciousness to what's going on in the objectively observable brain (the “hard problem of consciousness” as philosopher of mind David Chalmers called it) is not at all different than reducing the complex behavior of weather to the fundamental physical laws and interactions between gas molecules – which are both objective phenomena.
