Canonical Extension of a Crisp Hilbert Logic

Abstract

As observed in Section 6 in Chapter 3, any crisp (abstract) logic can be extended to a fuzzy (abstract) logic in a canonical way. In this chapter we will consider an extension principle which fits the Hilbert systems well (see Gerla [1995]). Namely, given any crisp H-system S = (A, R) and a crisp rule r ∈ R, we define the fuzzy rule r* = (r′r″) by setting

$$r' = r{\kern 1pt} \;,r''\left( {{x_1},...{x_2}} \right) = {x_1} \wedge ...{x_n},$$

where ∧ denotes the minimum operation. We call r* the canonical extension of r.