  The main aim of this paper is to introduce the logics of evidence and truth
LETK+ and LETF+ together with a sound, complete, and decidable six-valued
deterministic semantics for them. These logics extend the logics LETK and LETF-
with rules of propagation of classicality, which are inferences that express
how the classicality operator o 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 LETK is obtained by means of Nmatrices based on
swap structures, and the six-valued semantics for LETK+ is then obtained by
imposing restrictions on the semantics of LETK. These restrictions correspond
exactly to the rules of propagation of classicality that extend LETK. The logic
LETF+ is obtained as the implication-free fragment of LETK+. We also show that
the 6 values of LETK+ and LETF+ 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 LETK+ is Blok-Pigozzi
algebraizable and that its implication-free fragment LETF+ coincides with the
degree-preserving logic of the involutive Stone algebras.
