From Belnap-Dunn Four-Valued Logic to Six-Valued Logics of Evidence and Truth

The main aim of this paper is to introduce the logics of evidence and truth LET_K^+ and LET_F^+ together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics LET_K and LET_F^- with rules of propagation of classicality, which are inferences that express how the classicality operator ∘ is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with 2 more values that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for LET_K is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for LET_K^+ is then obtained by imposing restrictions on the semantics of LET_K . These restrictions correspond exactly to the rules of propagation of classicality that extend LET_K . The logic LET_F^+ is obtained as the implication-free fragment of LET_K^+ . We also show that the 6 values of LET_K^+ and LET_F^+ define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that LET_K^+ is Blok-Pigozzi algebraizable and that its implication-free fragment LET_F^+ coincides with the degree-preserving logic of the involutive Stone algebras.