Substructural Fuzzy-Relevance Logic

Abstract

This paper proposes a new topic in substructural logic for use in research joining the fields of relevance and fuzzy logics. For this, we consider old and new relevance principles. We first introduce fuzzy systems satisfying an old relevance principle, that is, Dunn's weak relevance principle. We present ways to obtain relevant companions of the weakening-free uninorm (based) systems introduced by Metcalfe and Montagna and fuzzy companions of the system R of relevant implication (without distributivity) and its neighbors. The algebraic structures corresponding to the systems are then defined, and completeness results are provided. We next consider fuzzy systems satisfying new relevance principles introduced by Yang. We show that the weakening-free uninorm (based) systems and some extensions and neighbors of R satisfy the new relevance principles.

Logics for Finite UL and IUL-Algebras Are Substructural Fuzzy Logics

“…In this paper, we prove that UL ω and IUL ω are standard complete by Wang's constructions in [5] and [6], which are some generalizations of the Jenei and Montagna-style approach for proving standard completeness for monoidal t-norm-based logic MTL [7] and the proof of the standard completeness for IMTL given by Esteva, Gispert, Godo, and Montagna in [8]. These constructions have been extended by Yang in [9][10][11][12].…”

The Weak Variable Sharing Property

“…Yang's proof can be found as theorem 2.ii in [14]. As it stands it is correct had it only been claimed to hold for RM 3 rather than for RD.…”

“…11 However, there are no RM 3 -interpretation I such that I(∼(p → p) → (q → q)) = −1, nor any I such that I ((r ∧ ∼r) → (∼(p → p) → (q → q))) = −1 since both these formulas are theorems of RM and so are both valid in the RM 3 -matrix. This, then, reopens the question whether logics like RD, as well as the other logics [14] calls "relevant fuzzy logics," do in fact satisfy (WVSP). Additionally, whether RUE satisfies (WVSP) is also an open question.…”

“…There are two interesting sublogics of RM which both fail to satisfy the variable sharing property, but for which the status of the weak version is unsettled, namely RUE and RD-R augmented by, respectively, the axiom A ∧ ∼A → B ∨ ∼B and (A → B) ∨ (B → A). 2 A proof to the effect that RD-"fuzzy R"-satisfies (WVSP) was put forth by Yang in [14]. That proof, however, is faulty.…”