<p><InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textsf{BS4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">BS</mi> <mn mathvariant="sans-serif">4</mn> </mrow> </math></EquationSource> </InlineEquation> is a natural Belnapian conservative extension of Lewis’ modal system <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textsf{S4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">S</mi> <mn mathvariant="sans-serif">4</mn> </mrow> </math></EquationSource> </InlineEquation> via strong negation. In [<CitationRef CitationID="CR23">23</CitationRef>] it was proved that the translation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{T}_{\textbf{B}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>T</mtext> <mi mathvariant="bold">B</mi> </msub> </math></EquationSource> </InlineEquation> that naturally generalises the Gödel–Tarski translation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{T}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>T</mtext> </math></EquationSource> </InlineEquation> embeds faithfully Nelson’s logic <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textsf{N4}^{\bot }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">N</mi> <msup> <mn mathvariant="sans-serif">4</mn> <mi>⊥</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textsf{BS4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">BS</mi> <mn mathvariant="sans-serif">4</mn> </mrow> </math></EquationSource> </InlineEquation>. So it is natural to define a modal companion of a logic extending <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textsf{N4}^{\bot }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">N</mi> <msup> <mn mathvariant="sans-serif">4</mn> <mi>⊥</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> as an extension of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textsf{BS4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">BS</mi> <mn mathvariant="sans-serif">4</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper we construct a representation of an <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textsf{N4}^{\bot }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">N</mi> <msup> <mn mathvariant="sans-serif">4</mn> <mi>⊥</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>-lattice similar to the representation of a Heyting algebra as an open elements algebra for a suitable topoboolean algebra. Using this algebraic result we construct a wide class of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\textsf{N4}^{\bot }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">N</mi> <msup> <mn mathvariant="sans-serif">4</mn> <mi>⊥</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>-extensions, elements of which have modal companions. In particular, all <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\textsf{N3}^\bot $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">N</mi> <msup> <mn mathvariant="sans-serif">3</mn> <mi>⊥</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>-extensions have modal companions. Also we prove that there are a continuum of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\textsf{N4}^{\bot }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">N</mi> <msup> <mn mathvariant="sans-serif">4</mn> <mi>⊥</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>-extensions that have no modal companions.</p>

On Modal Companions of Logics with Strong Negation

期刊论文
发表日期：

2026年2月12日

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Dmitry M. Anishchenko

摘要

$\textsf{BS4}$ $BS 4$ is a natural Belnapian conservative extension of Lewis’ modal system $\textsf{S4}$ $S 4$ via strong negation. In [23] it was proved that the translation $\textrm{T}_{\textbf{B}}$ $T_{B}$ that naturally generalises the Gödel–Tarski translation $\textrm{T}$ $T$ embeds faithfully Nelson’s logic $\textsf{N4}^{\bot }$ $N 4^{⊥}$ into $\textsf{BS4}$ $BS 4$ . So it is natural to define a modal companion of a logic extending $\textsf{N4}^{\bot }$ $N 4^{⊥}$ as an extension of $\textsf{BS4}$ $BS 4$ . In this paper we construct a representation of an $\textsf{N4}^{\bot }$ $N 4^{⊥}$ -lattice similar to the representation of a Heyting algebra as an open elements algebra for a suitable topoboolean algebra. Using this algebraic result we construct a wide class of $\textsf{N4}^{\bot }$ $N 4^{⊥}$ -extensions, elements of which have modal companions. In particular, all $\textsf{N3}^\bot $ $N 3^{⊥}$ -extensions have modal companions. Also we prove that there are a continuum of $\textsf{N4}^{\bot }$ $N 4^{⊥}$ -extensions that have no modal companions.