The thrust of Humphreys’ paper is that the philosophical concept of 'supervenience', which is often brought up when there is talk of reductionism vs anti-reductionism, is not sufficient, and that emergence is a much better bet for the anti-reductionistically inclined.
The Stanford Encyclopedia of Philosophy defines supervenience thus: “A set of properties A supervenes upon another set B just in case no two things can differ with respect to A-properties without also differing with respect to their B-properties. In slogan form, ‘there cannot be an A-difference without a B-difference.’”
A typical everyday example of supervenience is the relation between the amount of money in my pockets (A-property) and the specific make up of bills and coins I carry (B-property). While I am going to have the same amount of money (say, $20) regardless of the specific combination of coins and bills (say, no coins, 1 $10 bill and 2 $5 bills; or 4 25c coins, 9 $1 bills and 1 $10 bill), it is obvious that the total cannot possibly change unless I change the specific makeup of the coins+bills set (the opposite is not true, as we have just seen: we can change the composition of coins+bills without necessarily changing the total).
Again according to the SEP, “Supervenience is a central notion in analytic philosophy. It has been invoked in almost every corner of the field. For example, it has been claimed that aesthetic, moral, and mental properties supervene upon physical properties. It has also been claimed that modal truths supervene on non-modal ones, and that general truths supervene on particular truths. Further, supervenience has been used to distinguish various kinds of internalism and externalism, and to test claims of reducibility and conceptual analysis.”
Let’s say, for instance, that you think that mental properties supervene on the physical properties of the brain. What that means is that the same mental outcome (say, thought X, or feeling Y) could — in principle — be multiply instantiated, i.e., obtained by way of different brain states. This undermines a simplistic reductionism that would want to proclaim a one-to-one correspondence between physical and mental, but it still means that any change in the latter requires a change in the former, which is perfectly compatible with a physicalist interpretation of mental phenomena.
Humphreys claims that while accounts deploying supervenience often do so with an anti-reductionist aim, supervenience itself is no big foe of reductionism, for two reasons: (i) “If A supervenes upon B, then A is nothing but B’ talk”; and (ii) “if A supervenes upon B, then because A’s existence is necessitated by B’s existence, all that we need in terms of ontology is B.” I think that’s just about right, which explains why I’ve always felt that supervenience is an interesting philosophical concept, but has little to do with the debate about reductionism.
Well, what is supervenience good for, you might say? Humphreys gives the example of aesthetic judgment: “If aesthetic merit supervenes upon just spatial arrangements of color on a surface, and you attribute beauty to the Mona Lisa, you cannot withhold that [same] aesthetic judgement from a perfect forgery of the Leonardo painting.” Supervenience, then, becomes a way to assess consistency in the attribution of concepts, but has nothing interesting to say about ontological relationships, which is where the meat of the reductionism / anti-reductionism debate lies.
So for Humphreys one needs emergence, not just supervenience, to move away from reductionism. Fine, but we are still left with the need for a reasonable articulation of what emergent properties are. The author proposes a list of characteristics of emergence, though not all of them are necessary to identify a given phenomenon as emergent:
1) Novelty: “a previously uninstantiated property comes to have an instance.”
2) Qualitative difference: “emergent properties ... are qualitatively different from the properties from which they emerge.”
3) Absence at lower levels: “an emergent property is one that could not be possessed
at a lower level — it is logically or nomologically  impossible for this to occur.”
4) Law difference: “different laws apply to emergent features than to the features from which they emerge.”
5) Interactivity: “emergent properties ... result from an essential interaction between their constituent properties.”
6) Holism: “emergent properties are holistic in the sense of being properties of the entire system
rather than local properties of its constituents.”
Having thus set the stage, Humphreys goes on to consider some candidate examples of emergent properties. Interestingly, his first is none other than quantum entanglement, which provides the physical basis for higher level phenomena like superconductivity and superfluidity. According to Humphreys, quantum entanglement itself satisfies the 5th and 6th criteria (interactivity and holism), while when the phenomenon is considered as an explanation for, say, superconductivity, it minimally satisfies also criteria 1, 2 and 4 (novelty, qualitative difference and law difference).
The article then moves to a discussion of the general point that emergent properties can only manifest themselves in macroscopic systems, because they “enjoy properties that are qualitatively different from those of atoms and molecules, despite the fact that they are composed of the same basic constituents ... [properties] such as phase transitions, dissipative processes, and even biological growth, that do not occur in the atomic world.”
This is important because Humphreys then derives from his analysis a conclusion that is very much like the one Batterman arrived at in his paper, though beginning from a completely different starting point: “emergent properties cannot be possessed by individuals at the lower level because they occur only with [practically] infinite collections of constituents. Some of the most important cases of macroscopic phenomena are phase transitions, such as the transition from liquid to solid.” Hence the theoretical relevance of the mathematical singularities that describe phase transitions, which we have encountered in the first essay of this series.
The point is worth rewording more clearly: the reason mathematical singularities such as infinities pop up in description of emergent phenomena is because emergent phenomena occur when the number of components of a system is very large, effectively approaching infinity. Which in turns explains why only complex systems (of certain types) display emergent phenomena. Neat, no?
I know, I know, you are itching for less theory and more examples. Humphreys obliges, discussing the case of spontaneous ferromagnetism occurring below the Curie temperature. To wit:
“If one takes a ferromagnet whose Hamiltonian is spherically symmetric, then below the Curie temperature the system is magnetized in a particular direction, even though because of the spherically symmetric Hamiltonian, its energy is independent of that specific direction. This divergence between the symmetry exhibited by the overall system and the symmetry exhibited by the laws governing its evolution is an example of spontaneous symmetry breaking. We have here a case where there is a distinctively different law covering the N > infinity system than covers its individual constituents. This is exactly the kind of difference of laws across levels of analysis that we noted earlier as one criterion of a genuinely emergent phenomenon.”
To recap, supervenience — despite the crucial role it plays in many philosophical discussions — is not in fact a way to describe non reducible phenomena, for which task one really needs the more robust concept of emergence, with convincing examples to accompany it. This concept can be articulated in terms of Humphreys’ six criteria, and turns out to approximate Batterman’s approach based on the mathematics of phase transitions.
 For something to be nomologically impossible means that if instantiated it would violate a law of nature.