Dominik prefers a nomenclature that emphasizes the connections of consonant structures with fuzzy numbers and possibility distributions because these topics were the fertile ground from which he came to a new calculus for consonant structures. He and Ander therefore call the structures fuzzy numbers or possibility distributions.
Scott argues that this choice is a decidedly poor idea from a branding perspective because probabilists and quantitative scientists and engineers are prone to surprisingly vehement if rather unthinking criticism of things fuzzy or possibilistic. As a matter of rhetorical effectiveness, it makes sense to avoid the baggage of these criticisms.
A perhaps stronger academic argument to distinguish the new objects from fuzzy numbers and possibility distributions is that consonant structures constitute a new kind of mathematical object with novel and important implications vis-à-vis the false confidence theorem (Balch et al. 2019). They also have their own distinct arithmetic. The calculus for consonant structures is not that defined by fuzzy arithmetic described by Kaufmann and Gupta (1985), and it is not that defined by possibility theory as described by Dubois and Prade (2012). Consonant structures also differ from fuzzy and possibilistic objects in various important other ways. They fully realize the confidence curves of Birnbaum (1961) and filter credal sets to represent and exploit modal information to improve calculations. They facilitate the incorporation of consonant confidence structures (Balch 2020) into practical calculations. All of these connections intertwine consonant structures with purely probabilistic conceptions, so that, as Dominik himself admits (Hose 2022), these consonant structures are really all about probability. Yes, they have similarities to fuzzy numbers, and they are arguably special cases of possibility distributions, but we are using them for problems quite distantly related to those that motivated Zadeh. Although Joslyn (1995) noted that consonant structures constitute consistent possibility distributions, they also constitute confidence curves, a concept predating possibility in the modern sense by several decades. It might be hard to convince a post-modern audience, but, in fact, the use of fuzzy numbers formed as Birnbaum confidence curves was actually rejected by Lotfi Zadeh himself (pers. comm. in re Ferson 1996). Lotfi spent a lot of effort in distinguishing fuzzy from probability as he did not want his ideas to be reduced to special cases of probability.
References
Balch, M.S. (2020). New two-sided confidence intervals for binomial inference derived using Walley's imprecise posterior likelihood as a test statistic. International Journal of Approximate Reasoning 123: 77-98. https://doi.org/10.1016/j.ijar.2020.05.005
Balch, M.S., R. Martin, and S. Ferson (2019). Satellite conjunction analysis and the false confidence theorem. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475: 20180565. https://doi.org/10.1098/rspa.2018.0565
Birnbaum, A. (1961). Confidence curves: an omnibus technique for estimation and testing statistical hypotheses. Journal of the American Statistical Association 56: 246-249.
Dubois, D., and H. Prade (2012). Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press.
Ferson, S. (1996). Reliable calculation of probabilities: accounting for small sample size and model uncertainty. Intelligent Systems: A Semiotic Perspective, National Institute of Standards and Technology, Gaithersburg, Maryland.
Hose, D. (2022). Possibilistic Reasoning with Imprecise Probabilities: Statistical Inference and Dynamic Filtering. Ph.D. thesis, Universität Stuttgart. https://dominikhose.github.io/dissertation/diss_dhose.pdf
Joslyn, C. (1995). In support of an independent possibility theory. Foundations and Applications of Possibility Theory, pages 152-164, G. de Cooman, D. Ruan, and E.E. Kerre, (eds.), World Scientific, Singapore.
Kaufmann, A., and M.M. Gupta (1985). Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York.