Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies
Contextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to inconsistently connected systems (those with disturbance) by defining it...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-03-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/25/4/581 |
| _version_ | 1827745188793024512 |
|---|---|
| author | Ehtibar N. Dzhafarov Janne V. Kujala |
| author_facet | Ehtibar N. Dzhafarov Janne V. Kujala |
| author_sort | Ehtibar N. Dzhafarov |
| collection | DOAJ |
| description | Contextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to inconsistently connected systems (those with disturbance) by defining it in terms of the systems’ couplings subject to certain constraints. Such extensions are sometimes met with skepticism. We pose the question of whether it is possible to develop a set of substantive requirements (i.e., those addressing a notion itself rather than its presentation form) such that (1) for any consistently connected system, these requirements are satisfied, but (2) they are violated for some inconsistently connected systems. We show that no such set of requirements is possible, not only for CbD but for all possible CbD-like extensions of contextuality. This follows from the fact that any extended contextuality theory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> is contextually equivalent to a theory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="sans-serif">T</mi><mo>′</mo></msup></semantics></math></inline-formula> in which all systems are consistently connected. The contextual equivalence means the following: there is a bijective correspondence between the systems in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="sans-serif">T</mi><mo>′</mo></msup></semantics></math></inline-formula> such that the corresponding systems in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="sans-serif">T</mi><mo>′</mo></msup></semantics></math></inline-formula> are, in a well-defined sense, mere reformulations of each other, and they are contextual or noncontextual together. |
| first_indexed | 2024-03-11T05:03:34Z |
| format | Article |
| id | doaj.art-e019228a61b44cd19e2be4574f269abf |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-11T05:03:34Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-e019228a61b44cd19e2be4574f269abf2023-11-17T19:08:06ZengMDPI AGEntropy1099-43002023-03-0125458110.3390/e25040581Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other SatisfiesEhtibar N. Dzhafarov0Janne V. Kujala1Department of Psychological Sciences, Purdue University, West Lafayette, IN 47907, USADepartment of Mathematics and Statistics, University of Turku, FI-20014 Turun yliopisto, FinlandContextuality was originally defined only for consistently connected systems of random variables (those without disturbance/signaling). Contextuality-by-Default theory (CbD) offers an extension of the notion of contextuality to inconsistently connected systems (those with disturbance) by defining it in terms of the systems’ couplings subject to certain constraints. Such extensions are sometimes met with skepticism. We pose the question of whether it is possible to develop a set of substantive requirements (i.e., those addressing a notion itself rather than its presentation form) such that (1) for any consistently connected system, these requirements are satisfied, but (2) they are violated for some inconsistently connected systems. We show that no such set of requirements is possible, not only for CbD but for all possible CbD-like extensions of contextuality. This follows from the fact that any extended contextuality theory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> is contextually equivalent to a theory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="sans-serif">T</mi><mo>′</mo></msup></semantics></math></inline-formula> in which all systems are consistently connected. The contextual equivalence means the following: there is a bijective correspondence between the systems in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="sans-serif">T</mi><mo>′</mo></msup></semantics></math></inline-formula> such that the corresponding systems in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">T</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="sans-serif">T</mi><mo>′</mo></msup></semantics></math></inline-formula> are, in a well-defined sense, mere reformulations of each other, and they are contextual or noncontextual together.https://www.mdpi.com/1099-4300/25/4/581contextual equivalencecontextualityconsistent connectednessconsistificationconnectionsdisturbance |
| spellingShingle | Ehtibar N. Dzhafarov Janne V. Kujala Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies Entropy contextual equivalence contextuality consistent connectedness consistification connections disturbance |
| title | Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies |
| title_full | Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies |
| title_fullStr | Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies |
| title_full_unstemmed | Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies |
| title_short | Contextuality with Disturbance and without: Neither Can Violate Substantive Requirements the Other Satisfies |
| title_sort | contextuality with disturbance and without neither can violate substantive requirements the other satisfies |
| topic | contextual equivalence contextuality consistent connectedness consistification connections disturbance |
| url | https://www.mdpi.com/1099-4300/25/4/581 |
| work_keys_str_mv | AT ehtibarndzhafarov contextualitywithdisturbanceandwithoutneithercanviolatesubstantiverequirementstheothersatisfies AT jannevkujala contextualitywithdisturbanceandwithoutneithercanviolatesubstantiverequirementstheothersatisfies |