Frontiers of Philosophy in China 5 (2):294-311 (2010)

Abstract
Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic logics can be transferred to modal logics.
Keywords model theory  first-order logic  modal logic  graded modalities
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DOI 10.1007/s11466-010-0017-2
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From a Logical Point of View.W. V. O. Quine - 1953 - Harvard University Press.
From a Logical Point of View.Richard M. Martin - 1955 - Philosophy and Phenomenological Research 15 (4):574-575.
Modal Logic.Patrick Blackburn, Maarten de Rijke & Yde Venema - 2001 - Studia Logica 76 (1):142-148.

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