Provability multilattice logic

ABSTRACT In this paper, we introduce provability multilattice logic and multilattice arithmetic which extends first-order multilattice logic with equality by multilattice versions of Peano axioms. We show that has the provability interpretation with respect to and prove the arithmetic completeness theorem for it. We formulate in the form of a nested sequent calculus and show that cut is admissible in it. We introduce the notion of a provability multilattice and develop algebraic semantics for on its basis, by the method of Lindenbaum-Tarski algebras we prove the algebraic completeness theorem. We present Kripke semantics for and establish the Kripke completeness theorem via syntactical and semantic embeddings from into and vice versa. Last but not least, the decidability of is shown.