Subjective Logic - Properties

Properties

In case the argument opinions are equivalent to binary logic TRUE or FALSE, the result of any subjective logic operator is always equal to that of the corresponding propositional/binary logic operator. Similarly, when the argument opinions are equivalent to traditional probabilities, the result of any subjective logic operator is always equal to that of the corresponding probability operator (when it exists).

In case the argument opinions contain degrees of uncertainty, the operators involving multiplication and division will produce derived opinions that always have correct expectation value but possibly with approximate variance when seen as Beta/Dirichlet probability distributions. All other operators produce opinions where the expectation value and the variance are always equal to the analytically correct values.

Different composite propositions that traditionally are equivalent in propositional logic do not necessarily have equal opinions. For example in general although the distributivity of conjunction over disjunction, expressed as, holds in binary propositional logic. This is no surprise as the corresponding probability operators are also non-distributive. However, multiplication is distributive over addition, as expressed by . De Morgan's laws are also satisfied as e.g. expressed by .

Subjective logic allows extremely efficient computation of mathematically complex models. This is possible by approximating the analytically correct functions whenever needed. While it is relatively simple to analytically multiply two Beta distributions in the form of a joint distribution, anything more complex than that quickly becomes intractable. When combining two Beta distributions with some operator/connective, the analytical result is not always a Beta distribution and can involve hypergeometric series. In such cases, subjective logic always approximates the result as an opinion that is equivalent to a Beta distribution.