Mathematician and Philosopher; Feminist; CIA Recruit; Gay, Trans, Religious Jew
*Please note that I have changed my name, from Hasen Joseph Khudairi and Timothy Alison Bowen, to
David Elohim, effective April, 2024.
Correspondence
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)
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)
Interviews
(1) I wasn't interviewed by "What is it like to be a Philosopher?", but I thought that it would be worthwhile to answer the questions that they proffer for anyone interested in my autobiography. Their questions and my replies can be found here,
Books
Forms of Luminosity: Epistemic Modality and Hyperintensionality in Mathematics.
Ph.D. Dissertation, Arché Philosophical Research Centre for Logic, Language, Metaphysics, and Epistemology, University of St Andrews.
[PhilPapers][Published with Amazon. Amazon's site for the book can be found here]
This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines 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. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter 3 provides an abstraction principle for two-dimensional (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 Elohim's 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. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, 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. Elohim invents 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. Elohim provides a counter-example to epistemic closure for logical deduction. 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.
Articles and Book Chapters
Hyperintensional Ω-Logic. Originally published as "Modal Ω-Logic", in Don Berkich and Matteo Vincenzo d'Alfonso (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence -- Themes from IACAP 2016. Springer (2019).
This essay examines the philosophical significance of the consequence relation defined in the Ω-logic for set-theoretic languages. I argue that, as with second-order logic, the hyperintensional profile of validity in Ω-Logic enables the property to be epistemically tractable. Because of the duality between coalgebras and algebras, Boolean-valued models of set theory can be interpreted as coalgebras. In Section 2, I demonstrate how the hyperintensional profile of Ω-logical validity can be countenanced within a coalgebraic logic. Finally, in Section 3, the philosophical significance of the characterization of the hyperintensional profile of Ω-logical validity for the philosophy of mathematics is examined. I argue (i) that Ω-logical validity is genuinely logical, and (ii) that it provides a hyperintensional account of formal grasp of the concept of `set'. Section 4 provides concluding remarks.
Grounding, Conceivability, and the Mind-Body Problem. Early View: 2016. Synthese, 195 (2): 919–926 (2018), doi:10.1007/s11229-016-1254-2
[PhilPapers] [Correction]
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 epistemic possibility thereof. Generalizations and other aspects of the philosophical significance of the hyperintensional regimentation are further examined.
Submitted Papers
Epistemic Modality and Hyperintensionality:
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 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$ -- an epistemic hyperintensional $\mu$-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 epistemic hyperintensional $\mu$-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 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 compelling and materially adequate candidates for the fundamental structure of mental representational states, comprising fragments 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 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 modal 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 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.
Conceivability, Essence, and Haecceities
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:
A Hyperintensional Two-dimensionalist Solution to the Access Problem
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 to the applications of the latter to unrestricted quantification and 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 Foundations for Mathematical Platonism
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. 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.
Hyperintensional Category Theory and Indefinite Extensibility
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.
Logic:
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, the alethic paradoxes advanced by Scharp (2013). I proffer four novel extensions of the theory, and detail five issues that the theory faces.
Formal Semantics:
Topic-Sensitive Two-Dimensional Truthmaker Semantics
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 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 truthmakers. 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. Topic-sensitive two-dimensional truthmaker semantics can be interpreted epistemically and metasemantically as well.
Metaphysics:
Physical Necessitism
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' 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.
Philosophy of Mind and Cognitive Science:
Consciousness, Haecceitism, and Grounding
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.
Hyperintensional Conceivability, Grounding, and Consciousness
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
Ethics:
Epistemicism and Moral Vagueness
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 epistemic indeterminacy 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 intensional 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, and I argue that the rigidity of ethical value-theoretic concepts adduces in favor of an epistemic interpretation of the indeterminacy thereof.
Intention: Hyperintensional Semantics and Decision Theory
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.
Books in Progress
Hyperintensionality in Epistemic Democracy and Welfare Economics
Research Presentations
'Fritz and Goodman on Counting Incompossibles'. Arché Research Group in History and Philosophy of Logic and Mathematics, November 2016.
'Modal Cognitivism and Modal Expressivism'. History and Philosophy of Logic and Mathematics, Arché, October 2016.
'Grounding, Conceivability, and the Mind-Body Problem'. Grounding and Consciousness, University of Birmingham, June 2016. (Refereed)
'Modal Ω-Logic'. International Association for Computing and Philosophy -- Annual Meeting, University of Ferrara, June 2016. (Refereed)
'Goodness and Moral Obligation'. Kant, Metaethics, and Value, Trinity College Dublin, May 2016. (Refereed)
'Grounding, Conceivability, and the Mind-Body Problem'. The Science of Consciousness, University of Arizona, April 2016. (Refereed)
'Logical and Epistemic Modality'. Postgraduate Friday Seminar, Departments of Logic and Metaphysics and of Moral Philosophy, University of St Andrews, April 2016.
'Algebraic Metaphysical Semantics'. Uehiro Graduate Philosophy Conference, University of Hawai‘i at Mānoa, March 2016. (Refereed).
'Grounding and Fundamentality'. Metaphysics: Identity, Existence, and Structure, Arché, November 2015.
'Rules and Evolution' and 'Inference in Logic'. History and Philosophy of Logic and Mathematics, Arché, October 2015.
'Bisimulations'. Arché Logic Group, April 2015.
'Consciousness, Haecceitism, and Grounding'. Metaphysics: Identity, Existence, and Structure, Arché, November 2014.
'Haecceitism, Chance, and Counterfactuals'. Metaphysics: Identity, Existence, and Structure, Arché, November 2014.
'On Scharp's Resolution to the Alethic Paradoxes'. Models, Modality, and Meaning, Arché, October 2014.
Awards and Honors
St Leonard's College Ph.D. Research Scholarship. University of St Andrews, 2014 - 2017.
Departmental Honors in Philosophy. Johns Hopkins University, 2008.
University Honors. Johns Hopkins University, 2008.