A logical approach to fuzzy truth hedges

Abstract

The starting point of this paper are the works of Hájek and Vychodil on the axiomatization of truthstressing and-depressing hedges as expansions of Hájek's BL logic by new unary connectives. They showed that their logics are chain-complete, but standard completeness was only proved for the expansions over Gödel logic. We propose weaker axiomatizations over an arbitrary core fuzzy logic which have two main advantages: (i) they preserve the standard completeness properties of the original logic and (ii) any subdiagonal (resp. superdiagonal) non-decreasing function on [0, 1] preserving 0 and 1 is a sound interpretation of the truth-stresser (resp. depresser) connectives. Hence, these logics accommodate most of the truth hedge functions used in the literature about of fuzzy logic in a broader sense.

On minimal sets of graded attribute implications

“…Recent results on hedges and their treatment in fuzzy logics in the narrow sense can be found in [12].…”

“…In particular, we use complete residuated lattices [13] with linguistic hedges [12,19,29] for the job. In our setting, (2) can be seen as generalization of (1) if all the degrees a 1 , .…”

Monoidal functional dependencies

“…The approach has been later extended and more developed in [5] by considering formalizations of linguistic hedges [16,45] as additional parameters of semantics of the if-then rules.…”

Fuzzy inequational logic

“…ξ M (t, t) = 1 for any ξ M ∈ Mod( ). In order to see that (40) is satisfied, take t, t , t ∈ T F (X) and observe that (7), (20), and (35) together with the fact that a ⊗ i∈I b…”