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Modal LogicAn Introduction to its Syntax and Semantics$
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Nino B. Cocchiarella and Max A. Freund

Print publication date: 2008

Print ISBN-13: 9780195366587

Published to Oxford Scholarship Online: October 2011

DOI: 10.1093/acprof:oso/9780195366587.001.0001

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Semantics for S5

Semantics for S5

(p.71) Chapter 5 Semantics for S5
Modal Logic

Nino B. Cocchiarella

Max A. Freund

Oxford University Press

The chapter shows how a completeness theorem for S5 can be proved by means of a “cut down” or secondary semantics. It is significant that the modal logic S5 can be shown to be complete on both the sentential and quantified levels only by allowing the notion of “all possible worlds” of a logical space to have a secondary meaning in which the possible worlds of that space might be “cut down” to a proper subset of those worlds. The difference regarding the notion of “all possible worlds” are referred to as one between a primary and secondary semantics for modal logic. Possible worlds are not individuated in terms of their nonmodal facts, but depend on their modal facts as well in a secondary semantics.

Keywords:   completeness theorem, S5, cut down semantics, modal logic, secondary semantics, possible worlds

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