This chapter is constituted by two parts. The ¯rst part comprising Sections 1-5 was written by Torben Brauner and the second part comprising Sections 6-11 was written by Silvio Ghilardi. In the first part of the chapter, we give an introduction to first-order modal logic. We present a selection of logics involving constant domains, increasing domains, and varying domains, and moreover, we present a ¯rst-order intensional logic as well as a ¯rst-order version of hybrid logic. One criteria for selecting particular logics has been the availability of sound and complete proof procedures, typically axiom systems and/or tableau systems. We have compared the first-order modal logics under consideration to fragments of sorted first-order logic via appropriate versions of the standard translation. In the second part of the chapter, we review both positive and negative results con- cerning fragment decidability, Kripke completeness and axiomatizability. Modal hyperdoctrines are then introduced, as a unifying tool for analyzing the alternative semantics proposed in the literature. These alternative semantics range from specific semantics for non-classical logics (like metaframes), to interpretations in well-established mathematical frameworks (like topological spaces and toposes). Finally, the strict relationship between toplogical semantics and D. Lewis' counterpart semantics is closely investigated and an axiomatization is presented.

First Order Modal Logic / T. Brauner, S. Ghilardi (STUDIES IN LOGIC AND PRACTICAL REASONING). - In: Handbook of Modal Logic / [a cura di] Patrick Blackburn, Johan van Benthem, Frank Wolter. - Amsterdam : Elsevier, 2007. - ISBN 9780444516909. - pp. 549-620 [10.1016/S1570-2464(07)80012-7]

First Order Modal Logic

S. Ghilardi
2007

Abstract

This chapter is constituted by two parts. The ¯rst part comprising Sections 1-5 was written by Torben Brauner and the second part comprising Sections 6-11 was written by Silvio Ghilardi. In the first part of the chapter, we give an introduction to first-order modal logic. We present a selection of logics involving constant domains, increasing domains, and varying domains, and moreover, we present a ¯rst-order intensional logic as well as a ¯rst-order version of hybrid logic. One criteria for selecting particular logics has been the availability of sound and complete proof procedures, typically axiom systems and/or tableau systems. We have compared the first-order modal logics under consideration to fragments of sorted first-order logic via appropriate versions of the standard translation. In the second part of the chapter, we review both positive and negative results con- cerning fragment decidability, Kripke completeness and axiomatizability. Modal hyperdoctrines are then introduced, as a unifying tool for analyzing the alternative semantics proposed in the literature. These alternative semantics range from specific semantics for non-classical logics (like metaframes), to interpretations in well-established mathematical frameworks (like topological spaces and toposes). Finally, the strict relationship between toplogical semantics and D. Lewis' counterpart semantics is closely investigated and an axiomatization is presented.
Settore M-FIL/02 - Logica e Filosofia della Scienza
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/35594
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