<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the “classical core” of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.</p>
</abstract>

We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the “classical core” of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Łukasiewicz Logic and the Divisible Extension of Probability Theory"}

We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability...

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Published Online: Jan 01, 2022

Page range: 119 - 128

Received: Nov 20, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0008

Keywords
divisible extension of probability theory, Boolean logic, random variable, observable, statistical map, stochastic channel, Łukasiewicz logic, full Łukasiewicz tribe, conservative observable

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Łukasiewicz Logic and the Divisible Extension of Probability Theory

Published Online: Jan 01, 2022

Page range: 119 - 128

Received: Nov 20, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0008

Keywordsdivisible extension of probability theory, Boolean logic, random variable, observable, statistical map, stochastic channel, Łukasiewicz logic, full Łukasiewicz tribe, conservative observable

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
divisible extension of probability theory, Boolean logic, random variable, observable, statistical map, stochastic channel, Łukasiewicz logic, full Łukasiewicz tribe, conservative observable