Belnap's Four-Valued Logic and De Morgan Lattices

In my paper [3], recently published in this Journal, I study several aspects of the relations between Belnap's logic B and the class DM of De Morgan lattices. Some of these relations concern a Gentzen system for Belnap's logic, denoted by G B , which was introduced in 1988 by my colleague Ventura Verdú and me (see [5] and [6]); indeed, the study of this Gentzen system within the theory of full models developed in [4] is one of the main objects of my paper. In the final part of Section 4 I compare this Gentzen system with another, better known one denoted by G BL , specially in the framework of the theory of algebraizability of Gentzen systems developed in [9, 10]. The reader should take note that an algebraic study of a non-structural and multiple-conclusion version of the Gentzen system G BL , denoted by G B , has been published by Mr. Alexej Pynko in [8] (already quoted in [3]). There, in comments following Corollary 3.6 on page 449, G B is compared to G B (there denoted by G FV) and results parallel to those in Proposition 4.10 of [3] are obtained. Further, it is worth noticing that the results in Theorems 4.11 and 4.12 of [3] were first obtained by Pynko in his unpublished manuscript [7]. In it he develops a theory of algebraizability of sentential-like logical systems of the most general kind, which encompasses Willem Blok and Don Pigozzi's original theory [1], its extension to k-dimensional deductive systems [2], and also its extension to Gentzen systems by Jordi Rebagliato and Ventura Verdú [9, 10], on which [3] relies. In the final Section 4.5 of [7] the three above mentioned Gentzen systems are presented, together with the structural and multiple-conclusion versions G BC and G BC of G B and G BL , respectively. Then Theorem 4.88 of [7] states the same as Theorem 4.11 of [3], that is, the strong algebraizability of G B (denoted by G BC in [7]), with the variety DM of De Morgan lattices as its equivalent algebraic semantics. And Theorem 4.98 of [7] states the non-algebraizability of G BC , which is proved from the strong algebraizability of G BC ; since it is obvious that the same fact and proof hold for their single-conclusion fragments, one can consider that this result also contains the non-algebraizability of G BL …