Philosophy of Mathematics and Logic (modality and hyperintensionality in mathematics; modal logic, especially modal algebra and coalgebra, the modal mu-calculus, automata theory, and correspondence theory; set theory; abstractionism; mathematical practice; necessitism)

Epistemology (epistemic logic; conceivability; epistemic two-dimensional semantics; modal epistemology; epistemology of mathematics; the apriori)

Philosophy of Mind (epistemic modality; consciousness; computation and representation; attention)

Metaphysics (modal ontology; consciousness; grounding; hyperintensionality; metaphysical semantics)


Philosophy of Language and Linguistics;


Cognitive Science


(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,

(2) Interview with Timothy Williamson


Epistemic Modality, Mind, and Mathematics.

Ph.D. Dissertation, Arché Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology, University of St Andrews.


This book concerns the foundations of epistemic modality. 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 relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{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. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's "criterial" identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{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 epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapter \textbf{8} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the interaction between epistemic two-dimensional semantics and absolute decidability. Chapter \textbf{11} avails of modal coalgebraic automata to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. The hyperintensional, epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is applied in chapters \textbf{8}, \textbf{10}, and \textbf{11}. Chapter \textbf{12} provides a modal logic for rational intuition. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory.

Published Research Articles

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism. 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 Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ω-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω-logical validity correspond to those of second-order logical consequence, Ω-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets.

Grounding, Conceivability, and the Mind-Body Problem. Synthese 195 (2):919–926 (2018), doi:10.1007/s11229-016-1254-2


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 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.

Papers under review

Epistemic Modality:

Modal Cognitivism and Modal Expressivism


This paper aims to provide a mathematically tractable background against which to model both modal cognitivism and modal expressivism. I argue that epistemic modal algebras, endowed with a hyperintensional 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 dual. I examine, in particular, the virtues unique to the modal expressivist approach 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.

Cognitivism about Epistemic Modality


This paper aims to vindicate the thesis that cognitive computational properties are abstract objects implemented in physical systems. I avail of the equivalence relations countenanced in Homotopy Type Theory, in order to specify an abstraction principle for epistemic intensions. The homotopic abstraction principle for epistemic intensions provides an epistemic conduit into our knowledge of intensions as abstract objects. I examine, then, how intensional functions in Epistemic Modal Algebra are deployed as core models in the philosophy of mind, Bayesian perceptual psychology, and the program of natural language semantics in linguistics, and I argue that this provides abductive support for the truth of homotopic abstraction. Epistemic modality 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.

Non-Transitive Self-Knowledge: Luminosity via Modal μ-Automata


This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing self-knowledge on the model of fixed points in monadic second-order modal logic, i.e. the modal μ-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 interpretatation of the Kripke functors of a μ-automaton permits the iterations of the transition functions to entrain a principled means by which to account for necessary conditions on self-knowledge.

An Epistemicist Solution to Curry's Paradox


This paper targets a series of potential issues for the discussion of, and modal resolution to, the alethic paradoxes advanced by Scharp (2013). I aim, then, to provide a novel, epistemicist treatment to Curry's Paradox. The epistemicist solution that I advance enables the retention of both classical logic and the traditional rules for the alethic predicate: truth-elimination and truth-introduction.

Conceivability, Haecceities, and Essence


This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I argue that the interaction between the two-dimensional intensional framework and the mereological parthood relation enables epistemic possibilities to target the haecceitistic properties of individuals. I specify, then, a two-dimensional intensional formula encoding the relation between the epistemic possibility of haecceity comprehension and its metaphysical possibility. I examine the Julius Caesar problem as a test case. I then generalize the approach to essential properties. I conclude by addressing objections from the indeterminacy of ontological principles relative to the space of epistemic possibilities, and from the consistency of epistemic modal space.

Philosophy of Mathematics:

Abstracta and Possibilia: Modal Foundations for Mathematical Platonism


This paper aims to provide modal 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); examine cardinality issues which arise depending on whether Necessitism is accepted at first- and higher-order; and demonstrate how a two-dimensional intensional approach to the epistemology of mathematics, augmented with Necessitism, is consistent with Hale and Wright's notion of there being 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 an epistemic two-dimensional truthmaker semantics. 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 explaining our possible knowledge thereof.

Epistemic Modality and Absolute Decidability


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 an epistemic two-dimensional truthmaker semantics, if hyperintenisonal approaches are to be preferred to possible worlds semantics.

Grothendieck Universes 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 in the category-theoretic setting is identifiable with the Kripke functors of modal coalgebraic automata, where set-coalgebras model Grothendieck Universes and the functors are further inter-definable with the elementary embeddings of large cardinal axioms. The Kripke functors are argued to account for both reinterpretations of quantifier domains as well as the ontological expansion effected by the elementary embeddings in the category of sets. The interaction between the interpretational and metaphysical modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. By characterizing the modal profile of $\Omega$-logical validity, and thus the generic invariance of mathematical truth, modal coalgebraic automata are further capable of capturing the notion of definiteness, in order to yield a non-circular definition of indefinite extensibility.

A Modal Logic 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 axioms of a dynamic provability logic, which embeds GL within the modal $\mu$-calculus. Via correspondence results between modal logic and the bisimulation-invariant fragment of 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 of mathematical concepts that are formalizable in monadic first- and second-order formal languages.

Formal Semantics:

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 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 as well.


Algebraic Metaphysical Semantics


This paper argues that metaphysically fundamental truths ought to be defined within an algebraic language. In the first part of the paper, I provide examples of the algebraic structures used to define models of physical ontology (namely, quantum mechanics and field theory); the mathematical universe (set-theory); modal logic; and the metaphysics of consciousness. I outline, then, some explanatory desiderata concerning the relation between fundamental and derivative truths. I argue that a relation of apriori material implication, i.e. 'scrutability', cannot satisfy the relevant desiderata; and I propose in turn that -- given the model-theoretic uniformity between fundamental modal truths and the derivative truths concerning mental representational states -- a novel derivability relation can be specified. The relation is unique in having a purely model-theoretic characterization, and I examine the epistemic advantages accruing to the relation's model-theoretic profile.

Nomic 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 and nomological possibilities -- and the quantifiers in the formula range over a domain of natural, or concrete, entities and their contingently non-concrete analogues. The first abductive argument for Nomic Necessitism is that the modal profile of dispositional properties at higher-order requires the adoption of Necessitism at first-order. The second abductive argument for Nomic Necessitism is that the epistemology of necessary beings has a naturalistically adequate basis. 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 nomological 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:

Attention, Phenomenal Content, and Inexact Knowledge

This essay aims to provide a precise account of the manner in which the properties of phenomenal consciousness can be understood as possessing accuracy-conditions. The account is restricted to an examination of sensory phenomenal properties. I provide a critical discussion of an approach to phenomenal content, which avails of the epistemic interpretation of multi-dimensional intensional semantics. I examine several general issues for the foregoing approach, which I argue adduce against the sufficiency of the account as a theory of phenomenal representation. By contrast, I go on to provide a formally tractable account of the manner in which phenomenal properties can precisely compose via a category-theoretic semantics. The version of category-theoretic semantics that I proffer is inspired by Russell's (op. cit.) and Pettigrew's (ms) seminal work on the approach. I demonstrate how the proposal is empirically adequate, by constitutively incorporating some of the results concerning the kinds of attention and models of Bayesian perceptual psychology; and I argue that this concatenation of phenomenal properties into propositional form, assessable for accuracy or veridicality, explains the nature of the correctness-conditions for phenomenal consciousness. The second, central aim of the chapter is to explain how the phenomenological effects of attention interact with phenomenal representations, and to examine how the formal properties of phenomenal content relate to the logic of knowledge. I argue (i) that the phenomenological effects of attention do not affect the satisfaction of phenomenal knowledge, and thus do not affect the factivity thereof, and (ii) that the value-altering effects of attention bear on phenomenal content only by targeting its component parts, in a manner similar to how changes in the semantic values of sub-sentential expressions might affect the truth-condition for the sentence that the component expressions comprise.

Emotional Phenomenology, Desire, and Belief

This essay concerns the interaction between the emotions and sensory and cognitive phenomenal consciousness. In Section \textbf{2}, I argue that emotional phenomenology can be identified with the positively and negatively valenced information comprising a subject's memory stores, and that the valences modulate the subject's sensory and cognitive phenomenal awareness. I argue that a sufficient condition on the instantiation of emotional phenomenal properties is the subject's desires. In Section \textbf{3}, I defend the proposal from three arguments against non-reductive emotional phenomenology. In Section \textbf{4}, I argue that desire is able to modulate the cognitive intentional states at issue, by responding to Lewis' (1988, 1996) impossibility proofs concerning the identification of beliefs with desires in the setting of decision theory.


Moral Vagueness and Luminosity

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 model of epistemic indeterminacy concerns the extensions of moral predicates, as witnessed by the non-transitivity of a value-theoretic sorites paradox. The second model of moral epistemicism is induced by the status of moral dilemmas in the epistemic interpretation of multi-dimensional intensional semantics. The third model is argued to consist in the formal invalidation of modal axiom K -- and thus of epistemic closure -- in the derivation of Curry's paradox. 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, Modality, and Decision Theory

This paper argues that the types of intention can be modeled as modal operators. I delineate the intensional-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 modal 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

Recent Book Project

Epistemic Democracy: Epistemic Modality, Game Theory, and Welfare Economics.


Research Presentations

'Imagination and Knowledge of Necessary Existence'. Evidence and Imagination, University of Graz, November 2016. [declined, owing to illness]

'Modal Cognitivism and Modal Expressivism'. Arché Research Group in History and Philosophy of Logic and Mathematics, October 2016.

'Grothendieck Universes and Indefinite Extensibility'. Salzburg Conference for Young Analytic Philosophy, University of Salzburg, September 2016. [declined, owing to illness]

'Imagination and Knowledge of Necessary Existence'. The Logics of Image, International Symmetry Society Congress (Santorini, Greece), July 2016.

'Grounding, Conceivability, and the Mind-Body Problem'. Grounding and Consciousness, University of Birmingham, June 2016.

'Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism'. International Association for Computing and Philosophy -- Annual Meeting, University of Ferrara, June 2016.

'Goodness and Moral Obligation'. Kant, Metaethics, and Value, Trinity College Dublin, May 2016.

'Grounding, Conceivability, and the Mind-Body Problem'. The Science of Consciousness, University of Arizona, April 2016.

'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.

'Grounding and Fundamentality'. Identity, Existence, and Structure Research Group, Arché, November 2015.

'Rules and Evolution' and 'Inference in Logic'. History and Philosophy of Logic and Mathematics, Arché, October 2015.

'Modal Saturations via Ultrafilter Extensions'. Arché Logic Group, May 2015.

'Grounding beyond the Image of Causation'. Identity, Existence, and Structure, Arché, April 2015.

'Bisimulations'. Arché Logic Group, April 2015.

'An Epistemicist Solution to the Alethic Paradoxes'. Postgraduate 'Friday' Seminar, University of St Andrews, March 2015.

'Bayesian Apriority'. Models, Modality, and Meaning Research Group, Arché, December 2014.

'Consciousness, Haecceitism, and Grounding'. Identity, Existence, and Structure, Arché, November 2014.

'Haecceitism, Chance, and Counterfactuals'. Identity, Existence, and Structure, Arché, November 2014.

'On Oliver and Smiley's Mid-plural Logic'. Arché Logic Group, November 2014.

'On Gradational Accuracy Measures for 4-valued Logic'. Arché Graduate Conference, November 2014.

'On Scharp's Resolution to the Alethic Paradoxes'. Models, Modality, and Meaning, Arché, October 2014.

'Imagination and Knowledge of Necessary Existence'. Postgraduate 'Friday' Seminar, University of St Andrews, October 2014.

'On Second-order Logic and Mathematics'. Arché Logic Group, October 2014.

Awards and Honors

St Leonard's College PhD Research Scholarship. University of St Andrews, 2014 - 2017.

Departmental Honors in Philosophy. Johns Hopkins University, 2008.

University Honors. Johns Hopkins University, 2008.