In other words, a strongly emergent phenomenon must not be reducible to the constituent parts. The emergent phenomenon must exhibit properties that are not reducible to the interaction of the constituent parts. Would you agree with this definition?
Well its very difficult to have a cristal clear idea of what every author means :-) , this is a very difficult topic, here are some lines reflecting my own views over this subject. Minimally defined a process (or a property) is emergent if it does not exist at the immediately lower hierarchical level (that of its components) and cannot be 'guessed' easily 'apriori', though in principle maybe it is reducible theoretically to this lower hierarchical level (including the laws 'governing' it). That is being emergent, in the minimal interpretation, is also fully compatible with theoretical reductionism, at least in principle, at a lower organizational level.
Strong emergence of a phenomenon, in my acception, means the impossibility, even in principle, to make the complete logical reduction to the laws governing an inferior hierarchical level + the characteristics of all the constitutive parts of that level + some [usually presupposed unique] conditions at limit + some 'bridging principles' applied to smaller components. Even a 1:1 simulation on a computer is not helpful, the only way to realize that the strongly emergent property exist being its direct observation in the real world (or perceive it subjectively-in the case of some subjective experiences of consciousness).
Related with this problem John Searle [in the case of consciousness] makes the difference between 2 types of reductionism: causal and ontological. Thus when discussing subjective experiences and consciousness in general he wrotes somewhere: "...though
causally reducible [my note: in other words consciousness supervenes on the physical, the neural network of the brain] it is
ontologically irreducible ,...a complete description of the third person objective features of the brain would not be a description of [my note: ALL] of its first person subjective features...[for ex. a vague, diffuse, sensation of discomfort, envy, hate etc]".
In my own terms I think his
'causal reduction' is caught well by the term
'supervenience on the physical', the neural network of the brain more exactly, and its
'ontological irreducibility' by the term
'strong emergence' as I've defined it above (in Searle's view even a one to one computer simulation in terms of dynamics of the lower organizational level cannot raise by itself some subjective experiences, thus we cannot say that consciousness is a weakly emergent phenomenon).
Further Searle's says that consciousness does not have causal powers beyond those of its neurobiological base indeed but he argues additionally, enough persuasively, that we are still fully entitled to talk of 'downward causation', thus epiphenomenalism and, more generally, the problem of how could some non-physical properties supervenient on the physical produce downward causation are avoided.
I think Paul Davies says roughly the same when he argues that life might be a strongly emergent phenomenon: life is caused entirely by the lower hierarchical levels, life and the laws of complexity governing it (yet to be found) do not involve vitalism or supernatural, being however strongly emergent; some of the properties of living creatures being not reducible logically, even in principle, to the 'lower' laws of chemistry and physics (though these complexity laws could be found on a purely phenomenal base, we can know them by direct observations at the higher organizational levels); this being the way nature works, stop, a new basic principle of nature. There is nothing magical here and certainly this type of strong emergence is fully compatible with evolution through natural selection, Behe's definition of 'irreducible complexity' used by his version of Intelligent Design program being very far away.
The attempts of some scientists to claim that [strong] reductionism is granted, at least in its 'in principle' form, that everything can be reduced in principle to the lowest laws of the universe is far from having sufficient reasons though of course it deserves provisionally the status of first choice research program. As a digression here, but relevant to a great extent to the discussion reductionism vs emergentism, after reading Motl's reviews of some books at amazon.com, one of them concerned with the possibility of strong emergence, his thoughts regarding the pilot-wave interpretation of QM, his intransigency towards loop quantum gravity and yes especially his huge misconceptions regarding the philosophy of science (see for example his comments to Kuhn's book 'The structure of scientific revolutions'- I myself do not agree with the incommensurability thesis but underdetermination and theory ladenness are enough to make me be very cautious to the emphatic claim that science will ever be cumulative and that we know for sure that science lead us to the Truth, in absolute) I recommend to all people I know to read the books he rated very low, labeled as being 'a waste of time and money' :-). There is a 'law' among scientists the so called 'Law of Ortega' stating that the adavances of science have been made by a huge number of average scientists making very small steps ahead, the real breakthroughs being made by real 'giants'. Well the history of science show us that these 'giants' were also enough good philosophers, capable to work well outside the 'orthodox paths' of their times, personally I do not think that the next huge 'breakthrough' will be made by too rigid scientists, on the contrary those 'working' usually [also] well far from the 'normal science' of their time are much better placed, indeed there is no sufficient reason to even think now that the science of tomorrow will ever be cumulative, only some minimal epistemological loss being always possible (as Motl takes for granted)...
Now I think it would be interesting to go even further with the problem of reductionism / emergentism of conscious experiences, the most promising example, from what we know now, of a possible strongly emergent phenomenon (as I've defined it above). In what follows I will write some characteristics of consciousness (some of them incompatible) admitting minimally that the mental is not something totally different from the physical (or that 'everythnig that is' were purely mental) and overlooking the different variants of [still tenable] interactionist dualism. Further I'd try to identify various existing positions regarding the nature of consciousness by indicating the properties with which they agree.
0. Consciousness has properties distinct from those of the [far enough] sub-components (it does not exist at the neuronal level, even at the level of some higher agglomerations of neurons).
1. Consciousness has only physical properties.
2. Some properties of consciousness are non physical.
3. Consciousness has strongly emergent properties (irreducible logically, some subjective experiences for ex.).
4. Consciousness has at most 'weak' emergent properties.
5. Consciousness is supervenient on the physical (in particular I mean here supervenient on the neural network of the brain).
6. Consciousness is not supervenient on the neural network.
7. Consciousness is incapable of downward causation (epiphenomenal).
8. Consciousness is capable of downward causation but it has no causal powers in addition to the causal powers of the underlying neurobiological base.
9. Consciousness is capable of downaward causation having genuine causal powers which cannot be explained in terms of neurobiological base.
From what I understand we have (the numbers in parantheses are in doubt):
a. The british empiricists of the early 20-th century accept ---> 0 + 1 + 3 + 5 + 9
b. Chalmers ---> 0 + 2 + 3 + 6 + (9)
c. identity theorists ---> 0 + 1 + 4 + 5 + 8
d. non reductive physicalists (holding that some properties of consciousness are non physical but nonwithstanding this their downward causal powers are explanable, at least in principle, though today no such explanation exists, in terms of neurobiological base) ---> 0 + 2 + 4 + 5 + 8
e. John Searle ---> 0 + 1 + 3 + 5 + 8
f. property dualists (minimal) ---> 0 + 2
g. functionalists (minimal) ---> 0 + 5