On axiomatizing Shramko-Wansing's logic
The trilattice \(SIXTEEN_3\) introduced by \textit{Y. Shramko} and \textit{H. Wansing} [J. Philos. Log. 34, No.~2, 121--153 (2005; Zbl 1094.03012)] is a generalization of the famous bilattice \(FOUR_2\). In \(SIXTEEN_3\) there are two partial orders defined on the underlying set \textbf{16} of generalized truth values: the truth order \(\leq _t\) and the falsity order \(\leq _f\). Each ordering gives rise to (i) a conjunction, a disjunction, and a negation connective (thereby providing a truth vocabulary and a falsity vocabulary) and (ii) a semantical consequence relation (truth entailment \(\models _t\) and falsity entailment \(\models _f\)). In [Shramko and Wansing, loc. cit.] it was shown that \(\models _t\) in the truth vocabulary and \(\models _f\) in the falsity vocabulary both coincide with first-degree entailment, alias Belnap and Dunn's useful four-valued logic. The problem of axiomatizing truth entailment and falsity entailment in the full vocabulary containing both the truth and the falsity connectives was left open in [Shramko and Wansing, loc. cit.]. Odintsov solves this problem by first representing \(SIXTEEN_3\) as a twist-structure over the two-element Boolean algebra. He then shows that \(\models _t\) and \(\models _f\) are intersections of certain pairs of consequence relations defined as first-degree axiom systems. In order to be able to apply Miura's Theorem on axiomatizing the intersection of two superintuitionistic logics axiomatized with modus ponens as the only rule of inference, Odintsov introduces truth implication and falsity implication interpreted as the residuum of truth (falsity) conjunction with respect to \(\leq _t\) (\(\leq _f\)). The main result consists of axiomatizations of truth and falsity entailment in the extended languages containing truth implication, falsity implication, or both.
scientific article; zbMATH DE number 5552258