Properties of a complex physical system are emergent just in case they are neither (i) properties had by any parts of the system taken in isolation nor (ii) resultant of a mere summation of properties of parts of the system

Sloganistically, a system exhibits emergent properties when those properties are more than the sum of its parts' properties. Emergence is a notion cherished by those philosophers of mind who are mental realists, and physicalists who nonetheless reject the reducibility of the mental to the physical. Mental emergentists, then, posit that mental properties emerge from certain complex sets of physical properties, for instance, physical properties of human brains. Thus they hold that mental events are not identical to any brain events but instead emerge from them.
Ernest Nagel (1961) and Brain McLaughlin (1992) cite Mill's 'Of the Composition of Causes' chapter of System of Logic (1843) as the locus classicus on the notion of emergence. For Mill, the key to the distinction between emergent and non-emergent properties centers on a distinction regarding two different ways in which conjoint causes can produce an effect: Non-emergent properties are effects that are mere summations of the effects of each of the causal conjuncts, whereas emergent properties are effects that are not sums of the effects of each causal conjunct. These respective notions might be best conveyed by the following examples.
A paradigmatic example of an effect best construed as non-emergent is the way that multiple force vectors sum to propel a body in a given direction. Mill writes:
If a body is propelled in two directions by two forces, one tending to drive it to the north and the other to the east, it is caused to move in a given time exactly as far in both directions as the two forces would separately have carried it; and is left precisely where it would have arrived if it had been acted upon first by one of the two forces, and afterwards by the other. This law of nature is called, in dynamics, the principle of the Composition of Forces: and in imitation of that well-chosen expression, I shall give the name of the Composition of Causes to the principle which is exemplified in all cases in which the joint effect of several causes is identical with the sum of their separate effects. (1843, p. 428)
A key determinant of whether a behavior is emergent on this view is whether removing any of the causal conjuncts prevents the remaining conjuncts from contributing their effects to the remaining system. If not, then the behavior of the system in question is non-emergent. If so, then it is emergent. Mill offers as examples of emergent effects chemical reactions. Consider the following chemical process: CH4 + 2O2 --> CO2 + 2H2O (Methane + oxygen produces carbon dioxide + water). For Mill, the products of such chemical reactions are not, in any sense, the sum of the effects of each reactant (McLaughlin, 1992, p.60).
While the mechanics underlying chemical reactions are understood well enough today to render Mill's point dubious, we can see why the above chemical reaction would impress Mill and his contemporaries as significantly different in kind from the Composition of Forces for moving bodies. In the case of the chemical reaction, the resulting compounds exhibit properties significantly different from those of the reactants. For instance, methane is violently combustible, whereas carbon dioxide and water are not. This contrasts sharply against the case of a north-westerly moving object being propelled by two forces--one towards the north, the other towards the west--insofar as the subsequent motion is so obviously decomposable into the effects of the conjoint causes. A very live possibility to consider in connection with these examples is that an enhanced understanding of the processes that underlie some observed property of a system may show that system not to be an example of emergence. That is, an increase of knowledge about the way certain effects are obtained may reveal that certain effects are decomposable into the effects contributed by subcomponents of that system. Mill's chemical examples fail as properly emergent for just this reason. With the development of quantum mechanical explanation, we have been able to see how chemical reactions are composed of additive properties of individual electrons (McLaughlin, 1992, p.89).
Pete Mandik

Response to Mandik:
I disagree with the definition of emergent properties. This definition would automatically make the shape of a composite body an 'emergent' property. Consider a brick in the shape of a cube. The parts of this object are, let's say, molecules. Now (1) None of the molecules is cubical in shape, and (2) "cubical" is not the 'sum' of the shapes of the molecules, nor the sum of anything else. (In fact, only quantities can be sums. Since 'cubical' isn't a quantity, it can't be the sum of anything.)

Thus, the existence of emergent properties would be trivial.

Instead, we should take Broad's definition from _The Mind and its Place in Nature_. Roughly: A property, P, of a composite object, O, is emergent if it is not metaphysically necessary that an aggregate composed of parts having exactly the (intrinsic) properties that the parts of O in fact have and arranged in the way that the parts of O are in fact arranged, should have P. (Broad says, if you were given all the intrinsic properties of the parts plus their arrangement, you could not predict the properties of the whole. I think this is what he means.)
Michael Huemer

 Response to Huemer

While I agree that any good definition of "emergence" should exclude the shapes of bricks from counting as emergent properties, I disagree that my definition fails to do so. Superimpose a coordinate system (such as Descartes') on a brick, and it becomes a simple exercise to see that the particular way that the brick occupies space is a sum of the ways its parts occupy space. Given the techniques of analytic geometry, particular shapes can be converted into particular quantities and summed all day long.
As regards the definition attributed to Broad, the following problem arises. According to that definition, my mental properties would count as emergent if and only if it is metaphysically possible that my microphysical doppleganger lacks (qualitatively identical) mental properties. Thus, Broad's definition of "emergence" is inconsistent with many formulations of psychophysical supervenience, which, I think, would strike many contemporary emergentists as an unhappy result.
Pete Mandik