Abstract
One of the central goals of science is to find consistent and rational representations of observational data: a map of the world, if you will. How we do this depends on the specific tools, which are often mathematical. When dealing with real-world situations, where the data (or “territory”) has a random component, the mathematical tools most commonly used are those grounded in probability theory, defined in a precise way by the Russian mathematician Andrei Kolmogorov. In this paper we explore how experimental data (the “territory”) can be represented (or “mapped”) consistently in terms of probability theory, and present examples of situations, both in the physical and social sciences, where such representations are impossible. This suggests that some “territories” cannot be “mapped” in a way that is consistent with classical logic and probability theory.
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Other values for certain and impossible events may be given, e.g. \(p(\varOmega )=100\) or \(p(\emptyset )=-1\), but the choice of 1 and 0 for them makes the overall expressions for the calculus of probabilities simpler (Jaynes 2003).
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To be precise, he used an algebra over \(\varOmega \) which did not necessarily include all the subsets. As importantly, later on it was discovered that for certain sets we need to require the algebra to be countable, i.e. a \(\sigma \)-algebra, but such nuances are not relevant for our discussions.
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i.e. if I have the belief that tomorrow it will rain with probability 0.5 and that a movie I want to watch has probability 0.3 of being interesting, and so on, I can rationally decide whether I will go to the beach or to the movies. Of course, probabilities only offer a measure of belief, and not whether I would extract more pleasure from one activity or another. To model this, economists use what is known as utility, a part of rational choice theory (Anand et al. 2009).
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Some authors, most notably Dzhafarov and Kujala (2017), label this type of contextuality as “direct influences,” to highlight the idea that the experimental context directly influences the variables in question. They reserve the label “contextuality” only for what we call here “hidden contextuality.”.
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The term “explicit” and “hidden” contextuality was, as far as the authors know, proposed by Kurzyński (2017).
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Another way is to use upper probabilities (Suppes and Zanotti 1991; de Barros and Suppes 2010). For example, Holik, Saenz, and Plastino showed that if we relaxed the requirement for a Boolean algebra and instead allowed for orthomodular lattices, one would get upper probabilities instead of standard probabiilty theory (Holik et al. 2014).
- 8.
Of course, here we mean negative numbers as counting of actual objects. As is well known, other useful interpretations of negative numbers such as placement on the number line were introduced that make sense.
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Acacio de Barros, J., Oas, G. (2018). Mapping Quantum Reality: What to Do When the Territory Does Not Make Sense?. In: Wuppuluri, S., Doria, F. (eds) The Map and the Territory. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-72478-2_17
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