Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Disjunction and existence properties in modal arithmetic   We systematically study several versions of the disjunction and the existence
properties in modal arithmetic. First, we newly introduce three classes
$\mathrm{B}$, $\Delta(\mathrm{B})$, and $\Sigma(\mathrm{B})$ of formulas of
modal arithmetic, and study basic properties of them. Then, we prove several
implications between the properties. In particular, among other things, we
prove that for any consistent recursively enumerable extension $T$ of
$\mathbf{PA}(\mathbf{K})$ with $T \nvdash \Box \bot$, the
$\Sigma(\mathrm{B})$-disjunction property, the $\Sigma(\mathrm{B})$-existence
property, and the $\mathrm{B}$-existence property are pairwise equivalent. From
this result, we obtain a refinement of Friedman and Sheard's result. Moreover,
we introduce the notion of the $\Sigma(\mathrm{B})$-soundness of theories, and
prove that for any consistent recursively enumerable extension of
$\mathbf{PA}(\mathbf{K4})$, the modal disjunction property is equivalent to the
$\Sigma(\mathrm{B})$-soundness.

31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn