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Hybrid Completeness

P. Blackburn and M. Tzakova. Hybrid Completeness. Logic Journal of the IGPL, 6(4):625–650, 1998.

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Abstract:

In this paper we discuss two hybrid languages, and provide them with complete axiomatizations. Both languages combine features of modal and classical logic. Like modal languages, they contain modal operators and have a Kripke semantics. Unlike modal languages, in these systems it is possible to `label' states by using binders to bind special state variables. This paper explores the consequences of hybridization for completeness. As we shall show, the challenge is to blend the modal idea of canonical models with the classical idea of witnessed maximal consistent sets. The languages discussed provide us with two extreme examples of the issues involved. In the one case we can combine these ideas relatively straightforwardly with the aid of analogs of the Barcan axioms coupled with a modal theory of labeling. In the other case, although we can still formulate a theory of labeling, the Barcan analogs are not valid. We show how to overcome this difficulty by using COV, an infinite collection of additional rules of proof which has been used in a number of investigations of extended modal logic.

BibTeX: (download)

@ARTICLE{blackburn98:_hybrid_compl,
  author = {P. Blackburn and M. Tzakova},
  title = {Hybrid Completeness},
  journal = {Logic Journal of the IGPL},
  year = {1998},
  volume = {6},
  pages = {625--650},
  number = {4},
  abstract = {
	In this paper we discuss two hybrid languages, and provide them with
	complete axiomatizations. Both languages combine features of modal
	and classical logic. Like modal languages, they contain modal operators
	and have a Kripke semantics. Unlike modal languages, in these systems
	it is possible to `label' states by using binders to bind special
	state variables.
	This paper explores the consequences of hybridization for completeness.
	As we shall show, the challenge is to blend the modal idea of canonical
	models with the classical idea of witnessed maximal consistent sets.
	The languages discussed provide us with two extreme examples of the
	issues involved. In the one case we can combine these ideas relatively
	straightforwardly with the aid of analogs of the Barcan axioms coupled
	with a modal theory of labeling. In the other case, although we can
	still formulate a theory of labeling, the Barcan analogs are not
	valid. We show how to overcome this difficulty by using COV, an infinite
	collection of additional rules of proof which has been used in a
	number of investigations of extended modal logic. }
}

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