Soft Computing Evaluation Logic

2.4Graded Conjunction/Disjunction and Logic Modeling of Simultaneity and Substitutability

All models are wrong but some are useful.

—G.E.P. Box

Mathematical models of simultaneity (conjunctive logic aggregators) and substitutability (disjunctive logic aggregators) are the point of departure in all studies of logic. In GL simultaneity and substitutability are modeled using the graded (or generalized) conjunction/disjunction (GCD). According to GL conjecture (see Section 2.1.8 and P₂₀ in Section 2.2.3) we need nine special cases of GCD. In this chapter we study mathematical properties and various implementations of GCD aggregators.

General aggregators of n variables are defined using Definition 2.1.0. This general definition includes functions that have no applicability in logic and we introduced a more restrictive Definition 2.1.1 to specify logic aggregators (LA) as functions of two or more variables that support nondecreasing monotonicity in all variables, boundary conditions for truth and falsity, and sensitivity to positive truth and incomplete truth. Aggregators based on that definition are suitable for GL applications. Logic aggregators can be either idempotent or non‐idempotent and GCD supports both idempotent and nonidempotent aggregators that are models of simultaneity and substitutability, as shown in Fig. 2.4.1. In GL applications idempotent models of simultaneity and substitutability are more frequently used than nonidempotent models (hyperconjunction and hyperdisjunction) ...