![]()
Chapter 1
Conversations on Contextuality
Ehtibar N. Dzhafarov*
Purdue University, USA
Janne V. Kujalaâ
University of Jyväskylä, Finland
Dramatis personĂŚ:
EXPOSITOR, trying to present and clarify the main points of a certain view of contextuality.
INTERLOCUTOR, skeptical but constructive.
AUTHORS, supportive of Expositor but sympathetic to Interlocutor (remain off-stage except for occasionally inserting footnotes).
Conversation 1
EXPOSITOR: My dear Interlocutor, as we have agreed, we will discuss a certain approach to probabilistic contextuality. Its authors call it, perhaps not too descriptively, the Contextuality-by-Default theory (CbD).1 I think I should begin by giving you an informal overview of the main ideas.
INTERLOCUTOR: My dear Expositor, I always find an informal presentation of ideas a dubious exercise. If I do not understand the presentation clearly (which happens often), it is never clear to me whether this is because it was dumbed down so much as to become deficient of information, or because the ideas themselves are deficient. Nevertheless I should let you proceed.
EXP: Let me try. Objects (or things, or properties â choose what you like) are measured under varying conditions, called contexts. The measurements are generally random variables, and their identity is defined by what is measured (object) and under what conditions it is measured (context). As a result, the same object measured in different contexts is represented by different random variables: it is meaningless to ask why they are different (hence the designation âcontextuality-by-defaultâ). Moreover, measurements made in different contexts, whether of the same object or of different objects, do not have a joint distribution. One cannot, e.g., speak of their correlation or of the probability with which they have the same value. All measurements made within one and the same context, however, are jointly distributed. The overall picture we have therefore is one of stochastically unrelated to each other islands of jointly distributed measurements (âbunches of random variablesâ). Is this sufficiently clear?
INT: I am not sure. How does one define âobjectsâ and âcontextsâ?
EXP: Primitives of a theory cannot be explained conceptually except in their relations to other primitives of the theory, and their operational meaning may be outside the theory. The âobjectsâ and âcontextsâ are such primitives: formally, they are no more than labels defining the identity of a measurement (so that each measurement is defined by two labels, one for âwhatâ and another for âin what contextâ).
INT: Perhaps we could clarify this with examples.
EXP: Here is an example. Suppose we pose two Yes/No questions to randomly chosen people and record their responses. The questions can be asked verbally or presented in writing. Intuitively, a question asked is the âobjectâ being measured, the presentation mode (verbal or written) is context, and the response to a given question by a randomly chosen person is measurement, a random variable labeled by the question asked and by its presentation mode.
INT: So we have four random variables, if I understood you correctly: response to question A presented verba...
