Contextuality Analysis of Impossible Figures.

Víctor H Cervantes,Ehtibar N Dzhafarov

doi:10.3390/e22090981

Abstract

This paper has two purposes. One is to demonstrate contextuality analysis of systems of epistemic random variables. The other is to evaluate the performance of a new, hierarchical version of the measure of (non)contextuality introduced in earlier publications. As objects of analysis we use impossible figures of the kind created by the Penroses and Escher. We make no assumptions as to how an impossible figure is perceived, taking it instead as a fixed physical object allowing one of several deterministic descriptions. Systems of epistemic random variables are obtained by probabilistically mixing these deterministic systems. This probabilistic mixture reflects our uncertainty or lack of knowledge rather than random variability in the frequentist sense.

Highlights

Our main purpose is to illustrate the use of epistemic random variables using objects that are naturally described in a deterministic way, but not uniquely
In Appendix B to this paper we report several measures of contextuality computed for our systems of epistemic random variables, but in the main text we focus on one measure only, introduced here for the first time, a hierarchical version of thecontextuality measure CNT2 -NCNT2 described in References [3,4]
The alternative impossible triangle is contextual under both representations, with CNT22 = 0.5 under Option 1, and CNT22 = 1.5 under Option 2, both values being lower than the corresponding ones for the Penrose triangle

Summary

Introduction

Our main purpose is to illustrate the use of epistemic random variables using objects that are naturally described in a deterministic way, but not uniquely. (More generally, any two Rcq , Rcq0 ∈ R with c 6= c0 are stochastically unrelated That is, they do not possess a joint distribution.) The set Rc is called the bunch corresponding to context c and the set Rq is referred to as the connection for content q. The vector of maximal probabilities with which two random variables in a connection could both equal 1 (if they possessed a joint distribution). 0’s) assigned to all random variables Rcq in the system, and M is an incidence (Boolean) matrix [3,5]: in the row of M corresponding to a given element of p∗ , say, Pr( Rcq1 = .

Hierarchically Measuring Contextuality

The Penrose Triangle

Other Impossible Figures

Escher’s “Ascending and Descending”

Conclusions