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David Elohim

University of St Andrews, Philosophy, Graduate Student

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*Please note that I changed my name, from Hasen Joseph Khudairi and Timothy Alison Bowen, to David Elohim, in April, 2024. Please cite my published book and papers under `Elohim, David'.

AOS:

Mathematical and Philosophical Logic [set theory; philosophical applications of (i) modal logic, especially modal algebra, coalgebra, and the modal $\mu$-calculus (ii) dynamic epistemic logic, (iii) hyperintensional semantics, and (iv) two-dimensional semantics]

Philosophy of Mathematics (modality and hyperintensionality in mathematics; set theory; category theory; homotopy type theory; mathematical practice)

Epistemology (epistemic logic and epistemic modal and hyperintensional semantics; modal epistemology; epistemology of mathematics; conceivability; the apriori)

Metaphysics (modal ontology; mathematical objects; consciousness; grounding; hyperintensionality)

Philosophy of Mind (intentional content; consciousness; the language of thought)

AOC:

Philosophical Linguistics;

Cognitive Science;

Ethics;

Feminist Philosophy

Education:

Ph.D. Student, Philosophy. Arché Philosophical Research Centre for Logic, Language, Metaphysics, and Epistemology, University of St Andrews. (2014-2017) [withdrew, owing to physical illness]

Research Groups:

History and Philosophy of Logic and Mathematics (2015-2017; Convener for the group in the Fall 2016 semester)

Arché Logic Group (2014-2017)

Models, Modality, and Meaning (2014-2015)

Metaphysics (2014-2017)

Visiting Ph.D. Student. Australian National University. (2017) [declined, owing to physical illness]

M.A., Philosophy. Columbia University. (2010-2012)

B.A. (Hons.), Philosophy. Johns Hopkins University. (2005-2008)
Phone: +1 7814923429

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University of Southern California

University of Bristol

Universitat de Barcelona

Carnegie Mellon University

Academy of Sciences of the Czech Republic

Silvio Valentini

Università degli Studi di Padova

VSB - Technical University of Ostrava

Bjørn Jespersen

Utrecht University

InterestsView All (9)

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Interviews by David Elohim

Autobiographical Interview

I wasn't interviewed by "What is it like to be a Philosopher?", but I thought that it would be wo... more

Interview with Timothy Williamson

HK: The continuum hypothesis (CH) claims that any infinite set of real numbers is in 1-1 correspo... more HK: The continuum hypothesis (CH) claims that any infinite set of real numbers is in 1-1 correspondence with either the natural numbers or the real numbers. From the work of Kurt Gödel and Paul Cohen, we know that CH can't be proved or refuted by current methods of mathematical proof. Thus, given that CH is either true or false, either CH or its negation is a mathematical truth which can't be proved by current methods. However, in your paper "Absolute Provability and Safe Knowledge of Axioms", you argue that no mathematical truth is absolutely unprovable, by the following line of thought. Suppose that A is a true interpreted mathematical formula which eludes present human techniques of provability. First, you argue that since A is a mathematical truth, it is metaphysically necessary. You then enjoin one to consider the following scenario: a metaphysical possibility in which there is a species which can and does prove A. You write: "In current epistemological terms, their knowledge of A meets the condition of safety: they could not easily have been wrong in a relevantly similar case. Here the relevantly similar cases include cases in which the creatures are presented with sentences that are similar to, but still discriminably different from, A, and express different and false propositions; by hypothesis, the creatures refuse to accept such other sentences, although they may also refuse to accept their negations."

Books by David Elohim

Forms of Luminosity: Epistemic Modality and Hyperintensionality in Mathematics

Forms of Luminosity: Epistemic Modality and Hyperintensionality in Mathematics, 2017

This book concerns the foundations of epistemic modality and hyperintensionality and their applic... more 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 and hyperintensional cognitivism and modal and hyperintensional expressivism. I also develop a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional 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 μ-calculus 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 μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus 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 ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting.

Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional 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 Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing 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 higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic 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 two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the 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 in 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.

Epistemic Modality and Hyperintensionality by David Elohim

$Research paper thumbnail of Fixed Points in the Epistemic Hyperintensional $\mu$-Calculus and the KK Principle$

Fixed Points in the Epistemic Hyperintensional $\mu$-Calculus and the KK Principle

This essay provides a novel account of iterated epistemic states. The essay argues that states of... more This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing iterated epistemic states on the model of fixed points in the modal $\mu$-calculus. Despite the epistemic indeterminacy witnessed by the invalidation of modal axiom 4 in the sorites paradox -- i.e. the KK principle: $\square$$\phi$ $\rightarrow$ $\square$$\square$$\phi$ -- a hyperintensional epistemic $\mu$-automaton permits fixed points to entrain a principled means by which to iterate epistemic states and account thereby for necessary conditions on self-knowledge. The hyperintensional epistemic $\mu$-calculus is applied to the iteration of the epistemic states of a single agent instead of the common knowledge of a group of agents, and is thus a novel contribution to the literature.

Cognitivism about Epistemic Modality and Hyperintensionality

This essay aims to vindicate the thesis that cognitive computational properties are abstract obje... more This essay aims to vindicate the thesis that cognitive computational properties are abstract objects implemented in physical systems. I avail of Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory, in order to specify an abstraction principle for two-dimensional (hyper-)intensions. The homotopic abstraction principle for two-dimensional (hyper-)intensions provides an epistemic conduit for our knowledge of (hyper-)intensions as abstract objects. Higher observational type theory might be one way to make first-order abstraction principles defined via inference rules, although not higher-order abstraction principles, computable. The truth of my first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable i.e. Turing computable and the Turing machine being physically implementable. Epistemic modality and hyperintensionality can thus be shown to be both a compelling and a materially adequate candidate for the fundamental structure of mental representational states, comprising a fragment of the language of thought.

A Modal Logic and Hyperintensional Semantics for Gödelian Intuition

This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the p... more This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the modal $\mu$-calculus. Via correspondence results between fixed point modal propositional logic and the bisimulation-invariant fragment of monadic second-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the notion of 'intuition-that'. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions and hyperintensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages. Hyperintensionality is countenanced via a topic-sensitive epistemic two-dimensional truthmaker semantics.

Modal and Hyperintensional Cognitivism and Modal and Hyperintensional Expressivism

This paper aims to provide a mathematically tractable background against which to model both moda... more This paper aims to provide a mathematically tractable background against which to model both modal and hyperintensional cognitivism and modal and hyperintensional expressivism. I argue that epistemic modal algebras, endowed with a hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics, comprise a materially adequate fragment of the language of thought. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are categorically dual. I examine five methods for modeling the dynamics of conceptual engineering for intensions and hyperintensions. I develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. I examine then the virtues unique to the modal and hyperintensional expressivist approaches here proffered in the setting of the foundations of mathematics, by contrast to competing approaches based upon both the inferentialist approach to concept-individuation and the codification of speech acts via intensional semantics.

Hyperintensional Conceivability, Grounding, and Consciousness

This paper provides a rebuttal to the argument in Elohim (2018) in `Synthese'. Elohim provides a ... more This paper provides a rebuttal to the argument in Elohim (2018) in `Synthese'. Elohim provides a novel hyperintensional, ground-theoretic regimentation of the proposals in the metaphysics of consciousness. He then argues that Chalmers' (2010) intensional two-dimensional conceivability argument against physicalism is unsound, in light of the hyperintensional metaphysics of consciousness. Thus, intensional conceivability cannot be a guide to hyperintensional metaphysics. This paper demonstrates that a multi-hyperintensional version of epistemic two-dimensional semantics can be countenanced, and is sufficient for conceivability to be a guide to metaphysics in the hyperintensional setting such that Chalmers' argument, hyperintensionally construed, is in fact sound.

Conceivability, Essence, and Haecceities

This essay aims to redress the contention that epistemic possibility cannot be a guide to the pri... more This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I introduce a novel epistemic two-dimensional truthmaker semantics. I argue that the interaction between the two-dimensional framework and the mereological parthood relation, which is super-rigid, enables epistemic possibilities and truthmakers with regard to parthood to be a guide to its metaphysical profile. I specify, further, a two-dimensional formula encoding the relation between the epistemic possibility and verification of essential properties obtaining and their metaphysical possibility or verification. I then generalize the approach to haecceitistic properties, and examine the Julius Caesar problem as a test case.

Philosophy of Mathematics by David Elohim

A Hyperintensional Two-Dimensionalist Solution to the Access Problem

I argue that the two-dimensional hyperintensions of epistemic topic-sensitive two-dimensional tru... more I argue that the two-dimensional hyperintensions of epistemic topic-sensitive two-dimensional truthmaker semantics provide a compelling solution to the access problem.

I countenance an abstraction principle for two-dimensional hyperintensions based on Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory. The truth of my first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable i.e. Turing computable and the Turing machine being physically implementable. I apply, further, modal rationalism in modal epistemology to solve the access problem. Epistemic possibility and hyperintensionality, i.e. conceivability, can be a guide to metaphysical possibility and hyperintensionality, when (i) epistemic worlds or epistemic hyperintensional states are interpreted as being centered metaphysical worlds or hyperintensional states, i.e. indexed to an agent, when (ii) the epistemic (hyper-)intensions and metaphysical (hyper-)intensions for a sentence coincide, i.e. the hyperintension has the same value irrespective of whether the worlds in the argument of the functions are considered as epistemic or metaphysical, and when (iii) sentences are said to consist in super-rigid expressions, i.e. rigid expressions in all epistemic worlds or states and in all metaphysical worlds or states. I argue that (i) and (ii) obtain in the case of the access problem.

Modality and Hyperintensionality in Mathematics

This paper aims to contribute to the analysis of the nature of mathematical modality and hyperint... more This paper aims to contribute to the analysis of the nature of mathematical modality and hyperintensionality, and to the applications of the latter to absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance a topic-sensitive epistemic two-dimensional truthmaker semantics, if hyperintensional approaches are to be preferred to possible worlds semantics. I examine the relation between two-dimensional hyperintensional states and epistemic set theory, providing two-dimensional hyperintensional formalizations of epistemic set theory, large cardinal axioms, the modal axioms governing $\Omega$-logic, and the Epistemic Church-Turing Thesis.

Hyperintensional Category Theory and Indefinite Extensibility

This essay endeavors to define the concept of indefinite extensibility in the setting of category... more This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths is identifiable with the Grothendieck Universe Axiom and Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility.

Hyperintensional Ω-Logic

Modal Ω-Logic, 2019

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theo... more This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical. Second, the foregoing provides a hyperintensional account of the interpretation of mathematical and metamathematical vocabulary.

Hyperintensional Foundations of Mathematical Platonism

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hal... more This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics.

Logic by David Elohim

Truth, Modality, and Paradox: Critical Review of Scharp, 'Replacing Truth'

This paper targets a series of potential issues for the discussion of, and modal resolution to, t... more This paper targets a series of potential issues for the discussion of, and modal resolution to, the alethic paradoxes advanced by Scharp (2013). I proffer four novel extensions of the theory, and detail six issues that the theory faces. I provide a counter-example to epistemic closure for reductio proofs.

Formal Semantics by David Elohim

Topic-Sensitive Two-Dimensional Truthmaker Semantics

This paper endeavors to establish foundations for the interaction between hyperintensional semant... more This paper endeavors to establish foundations for the interaction between hyperintensional semantics and two-dimensional indexing. I examine the significance of the semantics, by developing three, novel interpretations of the framework. The first interpretation provides a characterization of the distinction between fundamental and derivative truths. The interaction between the hyperintensional truthmaker semantics and modal ontology is further examined. The second interpretation demonstrates how the elements of decision theory are definable within the semantics, and provides a novel account of the interaction between probability measures and hyperintensional grounds. The third interpretation concerns the contents of the types of intentional action, and the semantics is shown to resolve a puzzle concerning the role of intention in action. Two-dimensional truthmaker semantics can be interpreted epistemically and metasemantically.

Philosophy of Mind by David Elohim

Grounding, Conceivability, and the Mind-Body Problem

Grounding, Conceivability, and the Mind-Body Problem, 2018

This paper challenges the soundness of the two-dimensional conceivability argument against the de... more This paper challenges the soundness of the two-dimensional conceivability argument against the derivation of phenomenal truths from physical truths in light of a hyperintensional, ground-theoretic regimentation of the ontology of consciousness. The regimentation demonstrates how ontological dependencies between truths about consciousness and about physics cannot be witnessed by epistemic constraints, when the latter are recorded by the conceivability—i.e., the epistemic possibility—thereof. Generalizations and other aspects of the philosophical significance of the hyperintensional regimentation are further examined.

Correction to: "Grounding, Conceivability, and the Mind-Body Problem"

Consciousness, Haecceitism, and Grounding

This paper aims to demonstrate that the ontology of consciousness is consistent with both the mod... more This paper aims to demonstrate that the ontology of consciousness is consistent with both the modal and the metaphysical versions of Haecceitism. I examine the varieties of Haecceitism, and I specify the intended versions that the arguments will vindicate. I define the property of 'being purely qualitative', and examine its relation to the properties of phenomenal consciousness. I draw, inter alia, on Bayesian perceptual psychology, in order to specify the identity-conditions of phenomenal properties in detail. I provide two, abductive arguments for the claim that the identity-conditions on some individuals are metaphysically haecceitistic, in virtue of the relations that hold between those individuals and the phenomenal properties that they instantiate. The first argument is corroborated by empirical results concerning the phenomenological effects of attention. The second argument is corroborated by empirical results from the study of color in vision science. The arguments vindicate a version of Metaphysical Haecceitism, because the individuals are shown to be typed by the phenomenal properties that they instantiate, although quantification over the individuals is an ineliminable condition on their identity and distinctness. I provide, then, a regimentation of the extant proposals in the ontology of consciousness, using the logic of hyperintensional ground, as augmented by the Bayesian probability calculus. The hyperintensional regimentation vindicates a version of Modal Haecceitism, because the probabilistic ontological dependence of the parts of worlds on other parts thereof provides an ineliminable condition on the identity and distinctness of worlds.

Intention: Hyperintensional Semantics and Decision Theory

This paper argues that the types of intention can be modeled both as modal operators and via a mu... more This paper argues that the types of intention can be modeled both as modal operators and via a multi-hyperintensional semantics. I delineate the semantic profiles of the types of intention, and provide a precise account of how the types of intention are unified in virtue of both their operations in a single, encompassing, epistemic space, and their role in practical reasoning. I endeavor to provide reasons adducing against the proposal that the types of intention are reducible to the mental states of belief and desire, where the former state is codified by subjective probability measures and the latter is codified by a utility function. I argue, instead, that each of the types of intention -- i.e., intention-in-action, intention-as-explanation, and intention-for-the-future -- has as its aim the value of an outcome of the agent's action, as derived by her partial beliefs and assignments of utility, and as codified by the value of expected utility in evidential decision theory.

Autobiographical Interview

I wasn't interviewed by "What is it like to be a Philosopher?", but I thought that it would be wo... more

Interview with Timothy Williamson

HK: The continuum hypothesis (CH) claims that any infinite set of real numbers is in 1-1 correspo... more HK: The continuum hypothesis (CH) claims that any infinite set of real numbers is in 1-1 correspondence with either the natural numbers or the real numbers. From the work of Kurt Gödel and Paul Cohen, we know that CH can't be proved or refuted by current methods of mathematical proof. Thus, given that CH is either true or false, either CH or its negation is a mathematical truth which can't be proved by current methods. However, in your paper "Absolute Provability and Safe Knowledge of Axioms", you argue that no mathematical truth is absolutely unprovable, by the following line of thought. Suppose that A is a true interpreted mathematical formula which eludes present human techniques of provability. First, you argue that since A is a mathematical truth, it is metaphysically necessary. You then enjoin one to consider the following scenario: a metaphysical possibility in which there is a species which can and does prove A. You write: "In current epistemological terms, their knowledge of A meets the condition of safety: they could not easily have been wrong in a relevantly similar case. Here the relevantly similar cases include cases in which the creatures are presented with sentences that are similar to, but still discriminably different from, A, and express different and false propositions; by hypothesis, the creatures refuse to accept such other sentences, although they may also refuse to accept their negations."

Forms of Luminosity: Epistemic Modality and Hyperintensionality in Mathematics

Forms of Luminosity: Epistemic Modality and Hyperintensionality in Mathematics, 2017

This book concerns the foundations of epistemic modality and hyperintensionality and their applic... more 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 and hyperintensional cognitivism and modal and hyperintensional expressivism. I also develop a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional 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 μ-calculus 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 μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus 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 ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting.

Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional 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 Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing 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 higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic 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 two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the 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 in 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.

$Research paper thumbnail of Fixed Points in the Epistemic Hyperintensional $\mu$-Calculus and the KK Principle$

Fixed Points in the Epistemic Hyperintensional $\mu$-Calculus and the KK Principle

This essay provides a novel account of iterated epistemic states. The essay argues that states of... more This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing iterated epistemic states on the model of fixed points in the modal $\mu$-calculus. Despite the epistemic indeterminacy witnessed by the invalidation of modal axiom 4 in the sorites paradox -- i.e. the KK principle: $\square$$\phi$ $\rightarrow$ $\square$$\square$$\phi$ -- a hyperintensional epistemic $\mu$-automaton permits fixed points to entrain a principled means by which to iterate epistemic states and account thereby for necessary conditions on self-knowledge. The hyperintensional epistemic $\mu$-calculus is applied to the iteration of the epistemic states of a single agent instead of the common knowledge of a group of agents, and is thus a novel contribution to the literature.

Cognitivism about Epistemic Modality and Hyperintensionality

This essay aims to vindicate the thesis that cognitive computational properties are abstract obje... more This essay aims to vindicate the thesis that cognitive computational properties are abstract objects implemented in physical systems. I avail of Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory, in order to specify an abstraction principle for two-dimensional (hyper-)intensions. The homotopic abstraction principle for two-dimensional (hyper-)intensions provides an epistemic conduit for our knowledge of (hyper-)intensions as abstract objects. Higher observational type theory might be one way to make first-order abstraction principles defined via inference rules, although not higher-order abstraction principles, computable. The truth of my first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable i.e. Turing computable and the Turing machine being physically implementable. Epistemic modality and hyperintensionality can thus be shown to be both a compelling and a materially adequate candidate for the fundamental structure of mental representational states, comprising a fragment of the language of thought.

A Modal Logic and Hyperintensional Semantics for Gödelian Intuition

This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the p... more This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the modal $\mu$-calculus. Via correspondence results between fixed point modal propositional logic and the bisimulation-invariant fragment of monadic second-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the notion of 'intuition-that'. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions and hyperintensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages. Hyperintensionality is countenanced via a topic-sensitive epistemic two-dimensional truthmaker semantics.

Modal and Hyperintensional Cognitivism and Modal and Hyperintensional Expressivism

This paper aims to provide a mathematically tractable background against which to model both moda... more This paper aims to provide a mathematically tractable background against which to model both modal and hyperintensional cognitivism and modal and hyperintensional expressivism. I argue that epistemic modal algebras, endowed with a hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics, comprise a materially adequate fragment of the language of thought. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are categorically dual. I examine five methods for modeling the dynamics of conceptual engineering for intensions and hyperintensions. I develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. I examine then the virtues unique to the modal and hyperintensional expressivist approaches here proffered in the setting of the foundations of mathematics, by contrast to competing approaches based upon both the inferentialist approach to concept-individuation and the codification of speech acts via intensional semantics.

Hyperintensional Conceivability, Grounding, and Consciousness

This paper provides a rebuttal to the argument in Elohim (2018) in `Synthese'. Elohim provides a ... more This paper provides a rebuttal to the argument in Elohim (2018) in `Synthese'. Elohim provides a novel hyperintensional, ground-theoretic regimentation of the proposals in the metaphysics of consciousness. He then argues that Chalmers' (2010) intensional two-dimensional conceivability argument against physicalism is unsound, in light of the hyperintensional metaphysics of consciousness. Thus, intensional conceivability cannot be a guide to hyperintensional metaphysics. This paper demonstrates that a multi-hyperintensional version of epistemic two-dimensional semantics can be countenanced, and is sufficient for conceivability to be a guide to metaphysics in the hyperintensional setting such that Chalmers' argument, hyperintensionally construed, is in fact sound.

Conceivability, Essence, and Haecceities

This essay aims to redress the contention that epistemic possibility cannot be a guide to the pri... more This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I introduce a novel epistemic two-dimensional truthmaker semantics. I argue that the interaction between the two-dimensional framework and the mereological parthood relation, which is super-rigid, enables epistemic possibilities and truthmakers with regard to parthood to be a guide to its metaphysical profile. I specify, further, a two-dimensional formula encoding the relation between the epistemic possibility and verification of essential properties obtaining and their metaphysical possibility or verification. I then generalize the approach to haecceitistic properties, and examine the Julius Caesar problem as a test case.

A Hyperintensional Two-Dimensionalist Solution to the Access Problem

I argue that the two-dimensional hyperintensions of epistemic topic-sensitive two-dimensional tru... more I argue that the two-dimensional hyperintensions of epistemic topic-sensitive two-dimensional truthmaker semantics provide a compelling solution to the access problem.

I countenance an abstraction principle for two-dimensional hyperintensions based on Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory. The truth of my first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable i.e. Turing computable and the Turing machine being physically implementable. I apply, further, modal rationalism in modal epistemology to solve the access problem. Epistemic possibility and hyperintensionality, i.e. conceivability, can be a guide to metaphysical possibility and hyperintensionality, when (i) epistemic worlds or epistemic hyperintensional states are interpreted as being centered metaphysical worlds or hyperintensional states, i.e. indexed to an agent, when (ii) the epistemic (hyper-)intensions and metaphysical (hyper-)intensions for a sentence coincide, i.e. the hyperintension has the same value irrespective of whether the worlds in the argument of the functions are considered as epistemic or metaphysical, and when (iii) sentences are said to consist in super-rigid expressions, i.e. rigid expressions in all epistemic worlds or states and in all metaphysical worlds or states. I argue that (i) and (ii) obtain in the case of the access problem.

Modality and Hyperintensionality in Mathematics

This paper aims to contribute to the analysis of the nature of mathematical modality and hyperint... more This paper aims to contribute to the analysis of the nature of mathematical modality and hyperintensionality, and to the applications of the latter to absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance a topic-sensitive epistemic two-dimensional truthmaker semantics, if hyperintensional approaches are to be preferred to possible worlds semantics. I examine the relation between two-dimensional hyperintensional states and epistemic set theory, providing two-dimensional hyperintensional formalizations of epistemic set theory, large cardinal axioms, the modal axioms governing $\Omega$-logic, and the Epistemic Church-Turing Thesis.

Hyperintensional Category Theory and Indefinite Extensibility

This essay endeavors to define the concept of indefinite extensibility in the setting of category... more This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths is identifiable with the Grothendieck Universe Axiom and Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility.

Hyperintensional Ω-Logic

Modal Ω-Logic, 2019

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theo... more This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical. Second, the foregoing provides a hyperintensional account of the interpretation of mathematical and metamathematical vocabulary.

Hyperintensional Foundations of Mathematical Platonism

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hal... more This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics.

Truth, Modality, and Paradox: Critical Review of Scharp, 'Replacing Truth'

This paper targets a series of potential issues for the discussion of, and modal resolution to, t... more This paper targets a series of potential issues for the discussion of, and modal resolution to, the alethic paradoxes advanced by Scharp (2013). I proffer four novel extensions of the theory, and detail six issues that the theory faces. I provide a counter-example to epistemic closure for reductio proofs.

Topic-Sensitive Two-Dimensional Truthmaker Semantics

This paper endeavors to establish foundations for the interaction between hyperintensional semant... more This paper endeavors to establish foundations for the interaction between hyperintensional semantics and two-dimensional indexing. I examine the significance of the semantics, by developing three, novel interpretations of the framework. The first interpretation provides a characterization of the distinction between fundamental and derivative truths. The interaction between the hyperintensional truthmaker semantics and modal ontology is further examined. The second interpretation demonstrates how the elements of decision theory are definable within the semantics, and provides a novel account of the interaction between probability measures and hyperintensional grounds. The third interpretation concerns the contents of the types of intentional action, and the semantics is shown to resolve a puzzle concerning the role of intention in action. Two-dimensional truthmaker semantics can be interpreted epistemically and metasemantically.

Grounding, Conceivability, and the Mind-Body Problem

Grounding, Conceivability, and the Mind-Body Problem, 2018

This paper challenges the soundness of the two-dimensional conceivability argument against the de... more This paper challenges the soundness of the two-dimensional conceivability argument against the derivation of phenomenal truths from physical truths in light of a hyperintensional, ground-theoretic regimentation of the ontology of consciousness. The regimentation demonstrates how ontological dependencies between truths about consciousness and about physics cannot be witnessed by epistemic constraints, when the latter are recorded by the conceivability—i.e., the epistemic possibility—thereof. Generalizations and other aspects of the philosophical significance of the hyperintensional regimentation are further examined.

Correction to: "Grounding, Conceivability, and the Mind-Body Problem"

Consciousness, Haecceitism, and Grounding

This paper aims to demonstrate that the ontology of consciousness is consistent with both the mod... more This paper aims to demonstrate that the ontology of consciousness is consistent with both the modal and the metaphysical versions of Haecceitism. I examine the varieties of Haecceitism, and I specify the intended versions that the arguments will vindicate. I define the property of 'being purely qualitative', and examine its relation to the properties of phenomenal consciousness. I draw, inter alia, on Bayesian perceptual psychology, in order to specify the identity-conditions of phenomenal properties in detail. I provide two, abductive arguments for the claim that the identity-conditions on some individuals are metaphysically haecceitistic, in virtue of the relations that hold between those individuals and the phenomenal properties that they instantiate. The first argument is corroborated by empirical results concerning the phenomenological effects of attention. The second argument is corroborated by empirical results from the study of color in vision science. The arguments vindicate a version of Metaphysical Haecceitism, because the individuals are shown to be typed by the phenomenal properties that they instantiate, although quantification over the individuals is an ineliminable condition on their identity and distinctness. I provide, then, a regimentation of the extant proposals in the ontology of consciousness, using the logic of hyperintensional ground, as augmented by the Bayesian probability calculus. The hyperintensional regimentation vindicates a version of Modal Haecceitism, because the probabilistic ontological dependence of the parts of worlds on other parts thereof provides an ineliminable condition on the identity and distinctness of worlds.

Intention: Hyperintensional Semantics and Decision Theory

This paper argues that the types of intention can be modeled both as modal operators and via a mu... more This paper argues that the types of intention can be modeled both as modal operators and via a multi-hyperintensional semantics. I delineate the semantic profiles of the types of intention, and provide a precise account of how the types of intention are unified in virtue of both their operations in a single, encompassing, epistemic space, and their role in practical reasoning. I endeavor to provide reasons adducing against the proposal that the types of intention are reducible to the mental states of belief and desire, where the former state is codified by subjective probability measures and the latter is codified by a utility function. I argue, instead, that each of the types of intention -- i.e., intention-in-action, intention-as-explanation, and intention-for-the-future -- has as its aim the value of an outcome of the agent's action, as derived by her partial beliefs and assignments of utility, and as codified by the value of expected utility in evidential decision theory.

Hyperintensionality in Epistemic Democracy and Welfare Economics

* All rights reserved. † I changed my name, from Hasen Joseph Khudairi and Timothy Alison Bowen, ... more

Epistemicism and Moral Vagueness

This essay defends an epistemicist response to the phenomenon of vagueness concerning moral terms... more This essay defends an epistemicist response to the phenomenon of vagueness concerning moral terms. I outline a traditional model of - and then two novel approaches to - epistemicism about moral predicates, and I demonstrate how the foregoing are able to provide robust explanations of the source of moral, as epistemic, indeterminacy. The first approach to moral epistemicism concerns the extensions of moral predicates, as witnessed by the non-transitivity of a value-theoretic sorites paradox. The second approach to moral epistemicism is induced by the status of moral dilemmas in the epistemic interpretation of two-dimensional semantics. I examine the philosophical significance of the foregoing, and compare the proposal to those of ethical expressivism, constructivism, and scalar act-consequentialism. Finally, I examine the status of moral relativism in light of the epistemicist models of moral vagueness developed in the paper.

Physical Necessitism

This paper aims to provide two abductive considerations adducing in favor of the thesis of Necess... more This paper aims to provide two abductive considerations adducing in favor of the thesis of Necessitism in modal ontology. I demonstrate how instances of the Barcan formula can be witnessed, when the modal operators are interpreted 'naturally' -- i.e., as including geometric possibilities -- and the quantifiers in the formula range over a domain of natural, or concrete, entities and their contingently non-concrete analogues. I argue that, because there are considerations within physics and metaphysical inquiry which corroborate modal relationalist claims concerning the possible geometric structures of spacetime, and dispositional properties are actual possible entities, the condition of being grounded in the concrete is consistent with the Barcan formula; and thus -- in the geometric setting -- merits adoption by the Necessitist.

Entanglement, Modality, and Indeterminacy

This paper aims to contribute to the metaphysical foundations of the Everett or 'many-worlds' int... more This paper aims to contribute to the metaphysical foundations of the Everett or 'many-worlds' interpretation of quantum mechanics (cf. Everett, 1957; Wallace, 2012). I focus on the nature of the indeterminacy countenanced by states of entanglement, and argue that an account which clarifies the nature of the possible worlds at issue might serve to elucidate both the notion of metaphysical indeterminacy as well as the status of probability in the interpretation. I endeavor to elucidate the claim that the compossible states exhibited by entangled superpositions are real. I advance, then, three interpretations of the reality of the worlds at issue, and examine their interaction with the actuality operator. Finally, I examine which combinations of the approaches are consistent, and I argue in favor of a property-based approach to possible worlds.