توسعه منطق فازی با موانع زیاد Extending fuzzy logics with many hedges

1. Introduction Previous works on adding hedges to logical systems of mathematical fuzzy logic (MFL) [9] include those by Hájek [18], Vychodil [37], and Esteva et al. [14]. In MFL, hedges are called truth-stressing or truth-depressing depending on whether they strengthen or weaken the meaning of the proposition. The intuitive interpretation of a truth-stressing (resp., truth-depressing) hedge on a chain of truth values is a subdiagonal (resp., superdiagonal) non-decreasing function preserving 0 and 1. Such functions are called hedge functions. Hájek [18] introduces an axiomatization of a truth-stressing hedge vt as an expansion of Basic Logic (BL) [17], and the resulting logic is called BLvt . Vychodil [37] extends BLvt to a logic BLvt,st with a truth-depressing hedge st dual to vt. The logics are shown to be algebraizable and enjoy completeness w.r.t. the classes of their chains, but are not proved to enjoy standard completeness in general. Moreover, Hájek and Vychodil’s axiomatizations do not cover a large class of hedge functions [14]. Hence, Esteva et al. [14] propose weaker axiomatizations over any core fuzzy logic for a truth-stressing hedge or/and a truthdepressing one, which do not impose any more constraints on hedge functions, and the axiomatizations are proved to enjoy standard completeness. This work proposes two axiomatizations over any propositional core fuzzy logic for multiple truth-stressing and truth-depressing hedges, one for non-dual hedges and the other for dual ones. The axiomatizations not only cover a large class of hedge functions but also have all completeness properties of the underlying core fuzzy logic w.r.t. the class of their chains and distinguished subclasses of their chains, including standard completeness. The axiomatizations are also extended to the first-order level. Moreover, we show how to build linguistic fuzzy logics based on the axiomatizations and hedge algebras [32,33] for representing and reasoning with linguistically-expressed human knowledge. The remainder of the paper is organized as follows. Section 2 gives an overview of MFL, previous axiomatizations for hedges, linguistic truth domains and operations on them. Section 3 presents an axiomatization for multiple non-dual hedges while Section 4 provides an axiomatization for multiple dual ones. Section 5 shows how to build linguistic fuzzy logics. Section 6 extends the axiomatizations to the first-order level. Section 7 discuses related work. Finally, Section 8 concludes the paper.