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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.

BibTeX: (download)

@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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