Abstract

One of the main problems in t-norm fuzzy logic's meta-theorems is that despite the strong completeness of BL’s extensions such as Łukasiewicz (Ł), Gödel (G) and Product (Π) logics (i.e., Multi-valued, Gödel and Product standard algebras on [0,1] interval) in the propositional approach, in the first-order approach, given their standard chains and corresponding algebras, they aren't complete and strongly complete. One solution to this problem is that the first-order approaches of different fuzzy logics are complete and even strongly complete with respect to non-standard single chains. But despite the success of this method in proving the strong completeness of many fuzzy logics such as TM, NM, BL, SBL, Ł and their first-order extensions, G and its first-order extensions, Π and its first-order extensions, the n-contraction logics SBLn, and Every finite valued extension of BL (such as finite valued Łukasiewicz (Łn) and finite valued Gödel (Gn)), there are three open problems: (1) Are MTL, IMTL, PMTL, WNM and their first-order extensions, (strongly) complete w.r.t. a single chain? (2) Although SMTL and SBL are strongly chain complete, Are SMTL∀ and SBL∀ also strongly completeness w.r.t. a single chain? And, (3) does chain completeness entail strong chain completeness or not? Answering these open problems and proving the completeness or incompleteness of these logics is the main purpose of this study, which will be achieved by some algebraic and meta-logical strategies.