Abstract
A deterministic weakening \(\mathsf {DW}\) of the Belnap–Dunn four-valued logic \(\mathsf {BD}\) is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic \(\mathsf {DW}\) is formalized as a sequent calculus. The completeness and decidability of \(\mathsf {DW}\) with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for \(\mathsf {DW}\) is described, and found to be a subvariety of Berman’s variety \(\mathcal {K}_{1,2}\). Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that \(\mathsf {DW}\) is the logic of a nine-element distributive lattice with a negation. Moreover, \(\mathsf {BD}\) is embedded into \(\mathsf {DW}\) by Glivenko’s double-negation translation.
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Presented by Jacek Malinowski; Received June 19, 2017.
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Ma, M., Lin, Y. A Deterministic Weakening of Belnap–Dunn Logic. Stud Logica 107, 283–312 (2019). https://doi.org/10.1007/s11225-018-9792-x
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DOI: https://doi.org/10.1007/s11225-018-9792-x