## Modal logic with names

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

#### 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.

@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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