In this paper, we introduce a variant of second-order propositional modal logic interpreted on general (or Henkin) frames, SOPMLH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}$$\end{document}, and present a decidable fragment of this logic, SOPMLdecH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}_{dec}$$\end{document}, that preserves important expressive capabilities of SOPMLH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}$$\end{document}. SOPMLdecH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}_{dec}$$\end{document} is defined as a modal loosely guarded fragment of SOPMLH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}$$\end{document}. We demonstrate the expressive power of SOPMLdecH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}_{dec}$$\end{document} using examples in which modal operators obtain (a) the epistemic interpretation, (b) the dynamic interpretation. SOPMLdecH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}_{dec}$$\end{document} partially satisfies the principle of non-Fregean logic: two different atomic propositions with the same truth value can have different contents. In SOPMLdecH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}_{dec}$$\end{document}, we also define relating connectives and show that the weak Boethius’ Thesis built using these connectives is a valid formula of SOPMLdecH\documentclass[12pt]{minimal}
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\begin{document}$$SOPML^{\mathcal {H}}_{dec}$$\end{document}.
