Disjunction and existence properties in modal arithmetic

Abstract We systematically study several versions of the disjunction and existence properties in modal arithmetic. First, we newly introduce three classes $\mathrm {B}$ , $\Delta (\mathrm {B})$ $\Sigma formulas arithmetic basic them. Then, prove implications between properties. In particular, among other things, that for any consistent recursively enumerable extension T $\mathbf {PA}(\mathbf {K})$ with $T \nvdash \Box \bot $ -disjunction property, -existence property are pairwise equivalent. Moreover, notion -soundness theories {K4})$ is equivalent to -soundness.