He is the area editor for philosophy of science at the Internet Encyclopedia of Philosophy

With Harald Atmanspacher, he is the editor of Between Chance and Choice: Interdisciplinary Perspectives on Determinism, papers from an international workshop in June, 2001, sponsored by the Institut für Grenzgebiete der Psychologie und Psychohygiene of Freiburg.

Also with Atmanspacher, Bishop developed a theory of "contextual emergence," in which the description of properties of a lower level of emergence offers necessary but not sufficient conditions to rigorously describe the properties of the next higher level. The sufficient conditions are provided by "contextual" and contingent conditions.

Bishop and Atmanspacher distinguish between levels of description and levels of reality, i.e. between epistemological and ontological frameworks for reduction and emergence. They say:

Broadly speaking, descriptive terms are subjects of epistemological discourse while elements of reality are subjects of ontological discourse. Both types of discourse are used in reductionist and emergentist approaches. The concept of reference establishes a connection between descriptive terms and described elements of reality (leaving aside difficult questions about reference itself).

If one wants to have the option of ontic elements at each level of description rather than only at one or a few fundamental levels, a straightforward and strictly reductive scheme for interlevel relations becomes impossible and must be relaxed. The way in which ontic and epistemic descriptions are related to each other motivates contextual emergence as a viable alternative.

In order to clearly distinguish between different concepts of reduction and emergence, it is desirable to have a transparent classification scheme, so that the basic characteristics of these concepts can be discussed coherently. A useful approach toward such a classification is based on the role which contingent contexts play in reduction and emergence. More precisely, the way in which necessary and sufficient conditions are assumed in the relation between different levels of description can be used to distinguish four classes of relations:

("Contextual Emergence in the Description of Properties," Foundations of Physics, 36, no. 12 (2006): 1753-1777.)

Bishop and Atmanspacher situate contextual emergence in a matrix of related pictures of reductionism and emergence. He says the way in which necessary and sufficient conditions are assumed in the relation between different levels of description can be used to distinguish four classes of relations:

(1) The description of properties at a particular level of description offers both necessary and sufficient conditions to rigorously derive the description of properties at a higher level. This is the strictest possible form of reduction. It was most popular under the influence of positivist thinking in the mid-20th century.

(2) The description of properties at a particular level of description offers necessary but not sufficient conditions to derive the description of features at a higher level. This version is called contextual emergence, because contingent contextual conditions are required in addition to the lower-level description for a rigorous derivation of higher-level properties.

(3) The description of properties at a particular level of description offers sufficient but not necessary conditions to derive the description of features at a higher level. This version includes the idea that a lower-level description offers multiple realizations of a particular property at a higher level, which is characteristic of supervenience.

(4) The description of properties at a particular level of description offers neither necessary nor sufficient conditions to derive the description of properties at a higher level. This represents a form of radical emergence insofar as there are no relevant conditions connecting the two levels whatsoever.

("Contextual Emergence in the Description of Properties," Foundations of Physics, 36, no. 12 (2006): 1753-1777.)