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## Uncertainty: two probabilities for the three states of neutrosophy

##### Authors Names :   Philippe Schweizer   1 *

1  Affiliation :  Independent researcher, Switzerland

Email :  flippe2@gmail.com

Doi   :   https://doi.org/10.54216/IJNS.020104

Abstract :

Uncertainty is inherent to the real world: everything is only probable, precision like in measurements is finite, noise is everywhere... Also, science is based on a modeling of reality that can only be approximate. Therefore we postulate that uncertainty should be considered in our models, and for making this more easy we propose a simple operational conceptualization of uncertainty. Starting from the simple model of associating a probability p to a statement supposed to be true our proposed modeling bridges the gap towards the most complex representation proposed by neutrosophy as a triplet of probabilities. The neutrosophic representation consists in using a triplet of probabilities (t,i,f)  instead of just a single probability. In this triplet, t represents the probability of the statement to be true, and f it's the probability to be false. The specific point of neutrosophy it that the probability i represents  the probability of the statement to be uncertain, imprecise, or neutral among other significations according to the application. Our proposed representation uses only 2 probabilities instead of 3, and it can be easily translated into the neutrosophic representation. By being simpler we renounce to some power of representing the uncertain but we encourage the modeling of uncertainty (instead of ignoring it) by making this simpler. Briefly said, the prepare the path towards using neutrosophy. Our proposed representation of uncertainty consist, for a statement, not only to add its probability to be true p, but also a second probability pp to model the confidence we have in the first probability p. This second parameter pp represents the plausibility of p, therefore the opposite of its uncertainty. This is the confidence given to the value of p, in short pp is the probability of p (hence the name pp), This is simple to understand, and that allows calculations of combined events using classical probability such as based on the concepts of mean and variance. The stringent advantage of our modeling by the couple (p,pp) is that experts can be easily interrogated to provide their expertise by asking them simply the chance they give to an event a occur (this is p) and the confidence they have in that prediction (which is pp). We give also a formula to transform from our model to the neutrosophic representation. Finally, a short discussion on the entropy as a measure of uncertainty is done.

Keywords :

Uncertainty. neutrosophy , probability , representation of uncertainty , entropy

References :

[1] Smarandache Florentin (1995) New Trends in Neutrosophic Theory and Applications, Pons asbi, Brussels.

[2] Smarandache Florentin (2017) Plithogeny, Plithogenic Set, Logic, Probability, and Statistics, Pons asbi, Brussels.

[3] Zadeh L. A. (1965) Fuzzy sets. Information Control, 8(1965), 338–353.

[4] Atanassov Krassimir T. (2017) Intuitionistic Fuzzy Logics, Springer, Berlin.

[5] Kalman, R. E. (1960) "A New Approach to Linear Filtering and Prediction Problems". Journal of Basic Engineering. 82: 35–45.

[6] Smarandache, F. (1998) Neutrosophy/Neutrosophic probability, set, and logic. American Research Press: Santa Fe, NM, USA.

[7] Patrascu V.  (2013) Neutrosophic information in the framework of multivalued representation, The 21th Conference on Applied and Industrial Mathematics, CAIM 2013, Bucharest, Romania, September 19-22, 2013, arXiv: 1412.4802.

[8] Patrascu V. (2016) Refined Neutrosophic Information Based on Truth, Falsity, Ignorance, Contradiction and Hesitation, Neutrosophic Sets and Systems, Volume 11, pp. 57-66.