### Abstract

One of us previously proposed a model of internal measurement that is not only an interaction between internal elements but also involves the notion of an indefinite observer measuring the interaction. In this model, all interactions encompass the notion of detection. The latter can be expressed as a process generating a contradiction. Hereby a fixed point becomes invalid after inducing the contradiction. Instead of embedding this process as an instance along the time axis, we here represent the logical jump via a contradiction (fixed point). Specifically a new operation is defined which replaces the operation by which a fixed point is induced, and we show that both a set as a fixed point and this particular operation on a set can give rise to an orthomodular lattice that is a model of quantum logic. The new procedure is a mixture of inducing a fixed point and addressing a fixed point. It can be compared to mixing up indicating an element with indicating a set consisting of elements, that is, to Russel's paradox. Finally, we point out that orthomodularity can ubiquitously be found in the evolutionary complex systems because it is generally relevant to resolving a paradox or a logical jump.

Original language | English |
---|---|

Pages (from-to) | 261-274 |

Number of pages | 14 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 126 |

Issue number | 3-4 |

Publication status | Published - 1999 Feb 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- Complex systems
- Fixed point
- Orthomodular lattice

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*126*(3-4), 261-274.

**Orthomodular lattice obtained from addressing a fixed point.** / Gunji, Yukio; Ito, Gen.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 126, no. 3-4, pp. 261-274.

}

TY - JOUR

T1 - Orthomodular lattice obtained from addressing a fixed point

AU - Gunji, Yukio

AU - Ito, Gen

PY - 1999/2/15

Y1 - 1999/2/15

N2 - One of us previously proposed a model of internal measurement that is not only an interaction between internal elements but also involves the notion of an indefinite observer measuring the interaction. In this model, all interactions encompass the notion of detection. The latter can be expressed as a process generating a contradiction. Hereby a fixed point becomes invalid after inducing the contradiction. Instead of embedding this process as an instance along the time axis, we here represent the logical jump via a contradiction (fixed point). Specifically a new operation is defined which replaces the operation by which a fixed point is induced, and we show that both a set as a fixed point and this particular operation on a set can give rise to an orthomodular lattice that is a model of quantum logic. The new procedure is a mixture of inducing a fixed point and addressing a fixed point. It can be compared to mixing up indicating an element with indicating a set consisting of elements, that is, to Russel's paradox. Finally, we point out that orthomodularity can ubiquitously be found in the evolutionary complex systems because it is generally relevant to resolving a paradox or a logical jump.

AB - One of us previously proposed a model of internal measurement that is not only an interaction between internal elements but also involves the notion of an indefinite observer measuring the interaction. In this model, all interactions encompass the notion of detection. The latter can be expressed as a process generating a contradiction. Hereby a fixed point becomes invalid after inducing the contradiction. Instead of embedding this process as an instance along the time axis, we here represent the logical jump via a contradiction (fixed point). Specifically a new operation is defined which replaces the operation by which a fixed point is induced, and we show that both a set as a fixed point and this particular operation on a set can give rise to an orthomodular lattice that is a model of quantum logic. The new procedure is a mixture of inducing a fixed point and addressing a fixed point. It can be compared to mixing up indicating an element with indicating a set consisting of elements, that is, to Russel's paradox. Finally, we point out that orthomodularity can ubiquitously be found in the evolutionary complex systems because it is generally relevant to resolving a paradox or a logical jump.

KW - Complex systems

KW - Fixed point

KW - Orthomodular lattice

UR - http://www.scopus.com/inward/record.url?scp=0040032691&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040032691&partnerID=8YFLogxK

M3 - Article

VL - 126

SP - 261

EP - 274

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -