A Proof of the Standard Completeness for the Involutive Uninorm Logic

Abstract

In this paper, we solve a long-standing open problem in the field of fuzzy logics, that is, the standard completeness for the involutive uninorm logic IUL. In fact, we present a uniform method of density elimination for several semilinear substructural logics. Especially, the density elimination for IUL is proved. Then the standard completeness for IUL follows as a lemma by virtue of previous work by Metcalfe and Montagna.

Group Representation for Even and Odd Involutive Commutative Residuated Chains

“…Recently it is claimed to be proved in[52] by using proof-theoretic methods. However, the scientific community has doubts about the correctness of the proof, see the remark of the author himself in [52, second section in page 43].…”

On Neutrosophic Offuninorms

“…This union is manifold advantageous, the most evident one being that we have provided a new aggregator operator to these sets. As we mentioned in the introduction, there exists a wide variety of fuzzy uninorm applications, namely, Decision Making [9,14,15], DNA and RNA fusion [9], logic [17], Artificial Neural Networks [16], among others. Uninorm is more flexible than t-norm and t-conorm because it includes the compensatory property in some cases, which is more realistic for modeling human decision making, as was experimentally proved by Zimmermann in [21].…”

“…The definition of uninorm-based implicators is not new in literature, they can be seen in [41] (pp. 151-160) for fuzzy uninorms, in [17] it is extended for type 2 fuzzy sets, in [24] for L*-fuzzy set theory, and in [25] for neutrosophic uninorms. In the present paper, uninorm-based offimplicators are defined, however, we only counted on symbolic offimplication operators (see [31], p. 139).…”

Involutive Uninorm Logic with Fixed Point enjoys finite strong standard completeness

“…For any element (a, g) in X i−1 gr its second coordinate g is an element of G i−1 by (16), and since G i−1 is discretely ordered, the upper neighbor s of g exists in G i−1 , hence (a, s), which is the upper cover of (a, g) in X i−1 by (18), is in X i−1 gr , too. An analogous argument works for the lower cover of (a, g).…”

“…Its standard completeness has remained an open problem, which has stood against the attempts using the widely used embedding method of [13] or the density elimination technique of [15]. It is fair to mention, however, that there is already a claimed proof for the standard completeness of IUL in [18], but the presented proof-theoretic proof via density elimination is very hard to read and even harder to check, and as a consequence, as of today, the result is not generally recognized 1 .…”