Lower and upper probability calculations using compatibility measures for solving Zadeh's challenge problems

In this paper, we present solutions to Zadeh's challenge problem on calculating linguistic probabilities. First, we argue that Zadeh's solution to this problem via the Generalized Extension Principle is very difficult to implement. Then, we use a syllogism based on the entailment principle to interpret the problem so that it can be solved by calculation of pessimistic (lower) and optimistic (upper) probabilities via Linguistic Weighted Averages. We use a pessimistic and an optimistic compatibility measure to calculate such probabilities. Then, we choose vocabularies for heights and linguistic probabilities that are involved in the problem statement. The vocabularies are modeled using interval type-2 fuzzy sets. We calculate optimistic (upper) and pessimistic (lower) probabilities, which naturally would be interval type-2 fuzzy sets. Finally, we map the pessimistic and optimistic probabilities into linguistic probabilities present in the vocabularies, so that the results can be comprehended by a human. We investigate viable alternatives for the pessimistic and optimistic compatibility measures, and also solve a similar problem with a different hypothesis.