Elohim, David (2017) Epistemic Modality and Hyperintensionality in Mathematics. [Preprint]
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This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topicsensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic twodimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper)intensions. Chapter 4 advances a topicsensitive twodimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μcalculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μcalculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μcalculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a groundtheoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the twodimensional conceivability argument against physicalism. The topicsensitive epistemic twodimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting.
Chapters 812 provide cases demonstrating how the twodimensional hyperintensions of hyperintensional, i.e. topicsensitive epistemic twodimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic ChurchTuring Thesis, the modal axioms governing the modal profile of Ωlogic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic ChurchTuring Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to firstorder abstraction principles in order to make firstorder abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the firstorder abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ωlogic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic twodimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory.
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