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ALGEBRAIC SEMANTICS FOR RELATIVE TRUTH, AWARENESS, AND POSSIBILITY

Part of: Algebraic logic General logic

Published online by Cambridge University Press:  28 September 2023

EVAN PIERMONT Open the ORCID record for EVAN PIERMONT [Opens in a new window]
EVAN PIERMONT*
Affiliation:
DEPARTMENT OF ECONOMICS ROYAL HOLLOWAY UNIVERSITY OF LONDON EGHAM TW20 0EX, UK
*
E-mail: evan.piermont@rhul.ac.uk
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Abstract

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth is relative. In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics.

Keywords

relativized Boolean algebrasawareness framesawareness logics

MSC classification

Primary: 03B42: Logics of knowledge and belief (including belief change) 03G27: Abstract algebraic logic 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 17 , Issue 1 , March 2024 , pp. 154 - 177
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Board, O., & Chung, K.-S. (2007). Object-Based Unawareness. Minneapolis: Department of Economics, University of MinnesotaGoogle Scholar
Bueno, O., & de Souza, E. G. (1996). The concept of quasi-truth. Logique et Analyse, 39(153/154), 183–199.Google Scholar
Dekel, E., Lipman, B. L., & Rustichini, A. (1998). Standard state-space models preclude unawareness. Econometrica, 66(1), 159–173.CrossRefGoogle Scholar
Fagin, R., & Halpern, J. Y. (1988). Belief, awareness, and limited reasoning. Artificial Intelligence, 34(1), 39–76.CrossRefGoogle Scholar
Haimo, F. (1951). Some limits of Boolean algebras. Proceedings of the American Mathematical Society, 2(4), 566–576.CrossRefGoogle Scholar
Halpern, J. Y., & Piermont, E. 2019. Partial awareness. Proceedings of the AAAI Conference on Artificial Intelligence, 33, 2851–2858.CrossRefGoogle Scholar
Halpern, J. Y., & Rêgo, L. C. (2008). Interactive unawareness revisited. Games and Economic Behavior, 62(1), 232–262.CrossRefGoogle Scholar
Halpern, J. Y., & Rêgo, L. C. (2009). Reasoning about knowledge of unawareness. Games and Economic Behavior, 67(2), 503–525.CrossRefGoogle Scholar
Halpern, J. Y., & Rêgo, L. C. (2013). Reasoning about knowledge of unawareness revisited. Mathematical Social Sciences, 65(2), 73–84.CrossRefGoogle Scholar
Heifetz, A., Meier, M., & Schipper, B. C. (2006). Interactive unawareness. Journal of Economic Theory, 130(1), 78–94.CrossRefGoogle Scholar
Heifetz, A., Meier, M., & Schipper, B. C. (2008). A canonical model for interactive unawareness. Games and Economic Behavior, 62(1), 304–324.CrossRefGoogle Scholar
Kachi, D. (2002). Validity in simple partial logic. Annals of the Japan Association for Philosophy of Science, 10(4), 139–153.CrossRefGoogle Scholar
Lemmon, E. J. (1966). Algebraic semantics for modal logics ii. The Journal of Symbolic Logic, 31(2), 191–218.CrossRefGoogle Scholar
Li, J. (2009). Information structures with unawareness. Journal of Economic Theory, 144(3), 977–993.CrossRefGoogle Scholar
Lloyd Humberstone, I. (1981). From worlds to possibilities. Journal of Philosophical Logic, 10(3), 313–339.CrossRefGoogle Scholar
Modica, S., & Rustichini, A. (1994). Awareness and partitional information structures. Theory and Decision, 37(1), 107–124.CrossRefGoogle Scholar
Modica, S., & Rustichini, A. (1999). Unawareness and partitional information structures. Games and Economic Behavior, 27(2), 265–298.CrossRefGoogle Scholar
Monk, J. D. (2018). The mathematics of Boolean algebra. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Fall 2018 Edition). Metaphysics Research Lab, Stanford University.Google Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8(1), 219–241.CrossRefGoogle Scholar
Setlur, R. V. (1970). On the equivalence of strong and weak validity of rule schemes in the two-valued propositional calculus. Notre Dame Journal of Formal Logic, 11(2), 249–253.CrossRefGoogle Scholar
van Benthem, J., Bezhanishvili, N., & Holliday, W. H. (2016). A bimodal perspective on possibility semantics. Journal of Logic and Computation, 27(5), 1353–1389.Google Scholar