The M-Notation and Ignorance vs Uncertainty

Abstract

The three fundamental formal differences between the BP chain set logic and the M-chain set logic are,

• 1. The BP chain set logic operates only with the three ‘pure probability values’ 0, m and 1. In contrast, the M chain set logic operates, in addition, with interval-valued probability values such as 0m, m1, and 0m1. These are used to represent ignorance in addition to uncertainty (see sect. 10.2). Ignorance is due to insufficient information supply concerning the probability distribution (in terms of the pure probability values 0, m and 1) over the universe of chains.

  Three ways in which interval-valued probabilities enter the M logic are,

  • (a) In connection with IF THEN information chain sets (see item 6 of sect. 9.2.2 and especially chapter 14).

  • (b) In connection with the (prolongation and) expansion of chain sets where they replace the use of Bayes postulate (see sect. 12.2).

  • (c) In connection with answers to questions. These may also be interval-valued due to insufficient information supply (see sect. 10.3.3).

• 2. The BP logic operates with only one type of conjunctive updating of probabilities, namely the procedure defined in sect. 3.7.1. We call the procedure of that section ‘updating of probabilities of type 2’. It corresponds closely to the AND connective of propositional calculus. The M logic operates, in addition, with ‘type 1’ updating of probabilities (see sections 11.2, 11.3).

• 3. Every information chain set with one or more interval-valued probabilities in its probability row can be decomposed into an information chain set with several pure probability rows (see sect. 9.2.2, item 6). Due to the insufficiency of the supplied information we do not know which of these is the correct one. Conjunction with additional information supply can decrease the number of possible pure probability rows (see fig. 9.1 and fig. 11.1).