2026-09-19T06:31:30Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1475662023-01-20T03:56:16Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
McGee, Vann
Wu, Xinhe
Massachusetts Institute of Technology. Department of Linguistics and Philosophy
2023-01-19T19:59:02Z
2023-01-19T19:59:02Z
2022-09
2022-09-30T20:14:18.011Z
https://hdl.handle.net/1721.1/147566
https://orcid.org/0000-0003-4267-6212
Boolean-valued models generalize classical two-valued models by allowing arbitrary complete Boolean algebras as value ranges. The goal of my dissertation is to study Boolean-valued models and explore their philosophical and mathematical applications.
In Chapter 1, I build a robust theory of first-order Boolean-valued models that parallels the existing theory of two-valued models. I develop essential model-theoretic notions like "Boolean-valuation", "diagram", "elementary diagram", and prove a series of theorems on Boolean-valued models, including the (strengthened) Soundness and Completeness Theorem, the Löwenheim-Skolem Theorems, the Elementary Chain Theorem, and many more.
Chapter 2 gives an example of a philosophical application of Boolean-valued models. I apply Boolean-valued models to the language of mereology to model indeterminacy in the parthood relation. I argue that Boolean-valued semantics is the best degree-theoretic semantics for the language of mereology. In particular, it trumps the well-known alternative - fuzzy-valued semantics. I also show that, contrary to what many have argued, indeterminacy in parthood entails neither indeterminacy in existence nor indeterminacy in identity, though being compatible with both.
Chapter 3 (a collaboration with Bokai Yao) gives an example of a mathematical application of Boolean-valued models. Scott and Solovay famously used Boolean-valued models on set theory to obtain relative consistency results. In Chapter 3, I investigate two ways of extending the Scott-Solovay construction to set theory with urelements. I argue that the standard way of extending the construction faces a serious problem, and offer a new way that is free from the problem.
Ph.D.
Massachusetts Institute of Technology
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http://rightsstatements.org/page/InC-EDU/1.0/
Boolean-Valued Models and Their Applications
Thesis
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Doctoral
Doctor of Philosophy
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<Abstract>Boolean-valued models generalize classical two-valued models by allowing arbitrary complete Boolean algebras as value ranges. The goal of my dissertation is to study Boolean-valued models and explore their philosophical and mathematical applications.
In Chapter 1, I build a robust theory of first-order Boolean-valued models that parallels the existing theory of two-valued models. I develop essential model-theoretic notions like "Boolean-valuation", "diagram", "elementary diagram", and prove a series of theorems on Boolean-valued models, including the (strengthened) Soundness and Completeness Theorem, the Löwenheim-Skolem Theorems, the Elementary Chain Theorem, and many more.
Chapter 2 gives an example of a philosophical application of Boolean-valued models. I apply Boolean-valued models to the language of mereology to model indeterminacy in the parthood relation. I argue that Boolean-valued semantics is the best degree-theoretic semantics for the language of mereology. In particular, it trumps the well-known alternative - fuzzy-valued semantics. I also show that, contrary to what many have argued, indeterminacy in parthood entails neither indeterminacy in existence nor indeterminacy in identity, though being compatible with both.
Chapter 3 (a collaboration with Bokai Yao) gives an example of a mathematical application of Boolean-valued models. Scott and Solovay famously used Boolean-valued models on set theory to obtain relative consistency results. In Chapter 3, I investigate two ways of extending the Scott-Solovay construction to set theory with urelements. I argue that the standard way of extending the construction faces a serious problem, and offer a new way that is free from the problem.</Abstract>
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