Cut-Down Operations on Bilattices ∗ Thomas Macaulay Ferguson CUNY Graduate Center 365 Fifth Avenue New York, NY 10016 United States ABSTRACT In this paper, Melvin Fitting’s notion of a “cut-down” operation on a bilattice is considered. It is shown that the logic of cut-down operations on bilattices is equivalent to Harry Deutsch’s four-valued S fde , demonstrating that S fde is as intimately connected to the theory of bilattices as is the Dunn-Belnap logic E fde . The paper concludes by observing that the “not ¬” operation of Carlos Dam´ asio and Lu´ ıs Pereira serves as a cut-down in Fitting’s sense and that the logic of Dam´ asio-Pereira cut-down operations on the bilattice NINE is equivalent to Richard Angell’s AC. 1. INTRODUCTION: BILATTICES AND CUT-DOWNS This paper examines the notion of a cut-down operation on a bilattice, as defined by Melvin Fitting in [1], in which an epistemic interpretation of the operations of Kleene’s weak three-valued logic (equivalent to Bochvar’s “internal” logic Σ 0 ) is provided. Two types of cut-downs will be considered, and the logic of such operations will be described. The notion of a bilattice was introduced by Matthew Ginsberg in [2]. Definition 1. A bilattice B is a structure 〈B, ≤ k , ≤ t , ¬〉 where: B is a nonempty set ≤ k and ≤ t are partial orderings of B such that both 〈B, ≤ k 〉 and 〈B, ≤ t 〉 are complete lattices ¬ : B → B is a function such that – ¬¬a = a – If a ≤ t b then ¬b ≤ t ¬a – If a ≤ k b then ¬a ≤ k ¬b The orderings ≤ t and≤ k are often referred to as the “truth” and “information” orderings, respectively. Meet and join with respect to ≤ t are denoted by “∧” and “∨” while meet and join with respect to ≤ k are denoted by “⊗” and “⊕,” respectively. Moreover, the definition assumes that for all bilattices B and a,b ∈ B, meets and joins modulo both ≤ k and ≤ t exist and that there exist distinct tops and bottoms modulo each relation. Definition 2. With respect to two lattices L, R with orderings ≤ L and ≤ R , respectively, the Ginsberg-Fitting product L⊙R is a bilattice 〈L × R, ≤ k , ≤ t , ¬〉 where for 〈a 0 ,b 0 〉, 〈a 1 ,b 1 〉∈L×R: * This is the author’s version of a paper published in the Proceedings of the 45th International Symposium on Multiple-Valued Logic. Some corrections, editing, and typesetting differences exist between this and the final version. The final version of the paper was published as: Ferguson, T.M. Cut-Down Operations on Bilattices. In: Proceedings of the 45th International Symposium on Multiple-Valued Logic (ISMVL 2015). IEEE Computer Society Press, Los Alamitos, CA, 2015. 24–29. The final publication is available at the IEEE Computer Society Press via http://dx.doi.org/10.1109/ISMVL.2015.14 .