## Modal logic with names

G. Gargov and V. Goranko. Modal logic with names. *Journal of Philosophical Logic*, 22(6):607–636, 1993.

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#### Abstract:

We investigate an enrichment of the propositional modal language $\scr L$ with a `universal' modality $\blacksquare$ having semantics $x\vDash\blacksquare\phi$ iff $\forall y(y\vDash \phi)$, and a countable set of `names'---a special kind of propositional variables ranging over singleton sets of worlds. The obtained language $\scr L\sb c$ proves to have great expressive power. It is equivalent with respect to modal definability to another enrichment $\scr L\Boxed\not=$ of $\scr L$, where $\Boxed\not=$ is an additional modality with the semantics $x\vDash\Boxed\not=\phi$ iff $\forall y(y\not= x\to y\vDash\phi)$. Model-theoretic characterizations of modal definability in these languages are obtained. We also consider deductive systems in $\scr L\sb c$. Strong completeness of the normal $\scr L\sb c$-logics is proved with respect to models in which all worlds are named. Every $\scr L\sb c$-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties such as [in]completeness, filtration, the finite model property, etc. from $\scr L$ to $\scr L\sb c$ are discussed. Finally, further perspectives for names in a multimodal environment are briefly sketched.

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@ARTICLE{gargov93:_modal, author = {G. Gargov and V. Goranko}, title = {Modal logic with names}, journal = {Journal of Philosophical Logic}, year = {1993}, volume = {22}, pages = {607--636}, number = {6}, abstract = { We investigate an enrichment of the propositional modal language ${\scr L}$ with a `universal' modality $\blacksquare$ having semantics $x\vDash\blacksquare\phi$ iff $\forall y(y\vDash \phi)$, and a countable set of `names'---a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ${\scr L}\sb c$ proves to have great expressive power. It is equivalent with respect to modal definability to another enrichment ${\scr L}\Boxed{\not=}$ of ${\scr L}$, where $\Boxed{\not=}$ is an additional modality with the semantics $x\vDash\Boxed{\not=}\phi$ iff $\forall y(y\not= x\to y\vDash\phi)$. Model-theoretic characterizations of modal definability in these languages are obtained. We also consider deductive systems in ${\scr L}\sb c$. Strong completeness of the normal ${\scr L}\sb c$-logics is proved with respect to models in which all worlds are named. Every ${\scr L}\sb c$-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties such as [in]completeness, filtration, the finite model property, etc. from ${\scr L}$ to ${\scr L}\sb c$ are discussed. Finally, further perspectives for names in a multimodal environment are briefly sketched. } }

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