Belnap–Dunn Modal Logic with Value Operators
2020
The language of Belnap–Dunn modal logic
$${\mathscr {L}}_0$$
expands the language of Belnap–Dunn four-valued logic (having constant symbols for the values 0 and 1) with the modal operator
$$\Box $$
. We introduce the polarity semantics for
$${\mathscr {L}}_0$$
and its two expansions
$${\mathscr {L}}_1$$
and
$${\mathscr {L}}_2$$
with value operators. The local finitary consequence relation
$$\models _4^k$$
in the language
$${\mathscr {L}}_k$$
with respect to the class of all frames is axiomatized by a sequent system
$$\mathsf {S}_k$$
where
$$k=0, 1, 2$$
. We prove by using translations between sequents and formulas that these languages under the polarity semantics have the same expressive power on the level of frames with the language
$${\mathscr {L}}_0$$
under the relational semantics for classical modal logic.
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