Teruji Thomas

Reconstructing Arrhenius’s Impossibility Theorems
Note, 2016 (PDF)

Abstract. Gustaf Arrhenius has developed a series of ‘impossibility theorems’ in population ethics, aiming to show that there can be no population axiology satisfying various combinations of plausible conditions (Arrhenius (MS), Population Ethics: The Challenge of Future Generations; all of the theorems have been previously published in some version). These results are undoubtedly important, but the notation and the proofs themselves can be formidable. This note provides a much more concise reconstruction of the proofs, using a notation which, I hope, will be more generally accessible. This also allows me to correct what appears to be a problem with his sixth and favoured theorem.

Keywords. Population ethics, impossibility theorems, Gustaf Arrhenius

A Paradox for Tiny Probabilities and Enormous Values
Noûs (2024)

Abstract. We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. Reckless theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; timid theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; non-transitive theories deny the principle that, if A is better than B and B is better than C, then A must be better than C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty.

Keywords. Decision theory, axiology, recklessness, timidity, fanaticism, spectrum arguments, St Petersburg paradox, Pascal's wager, separability, normative uncertainty, risk aversion, Nicolausian discounting

The Asymmetry, Uncertainty, and the Long Term
Philosophy and Phenomenological Research (2023)
Extended working paper on GPI website (2019)

Note. The extended working paper contains some additional material about cyclic choice and also about ‘hard’ versions of the asymmetry, according to which harms to independently existing people cannot be justified by the creation of good lives. But for other material, please refer to and cite the published version in PPR.

Abstract. The asymmetry is the view in population ethics that, while we ought to avoid creating additional bad lives, there is no requirement to create additional good ones. The question is how to embed this intuitively compelling view in a more complete normative theory, and in particular one that treats uncertainty in a plausible way. While arguing against existing approaches, I present new and general principles for thinking about welfarist choice under uncertainty. Together, these reduce arbitrary choices to uncertainty-free ones, regardless of how the latter should be made. I illustrate these principles by developing two theories of the asymmetry, reflecting different views about the non-identity problem. In doing so, I clarify some other major choice-points, presenting new arguments that creating additional good lives can justify (but not require) doing harm. Finally, I consider what the developed theories have to say about the importance of extinction risk and the long-run future.

Keywords. Population ethics, procreation asymmetry, uncertainty, anti-natalism, extinction

Some Possibilities in Population Axiology
Mind 127 (2018): 807-832

Abstract. It is notoriously difficult to find an intuitively satisfactory rule for evaluating populations based on the welfare of the people in them. Standard examples, like total utilitarianism, either entail the Repugnant Conclusion or in some other way contradict common and deeply felt intuitions about the relative value of populations. Several philosophers have presented formal arguments that seem to show that this happens of necessity: our core intuitions stand in contradiction. This paper assesses the state of play, focusing on the most powerful of these ‘impossibility theorems’, as developed by Gustaf Arrhenius. I highlight two ways in which these theorems fall short of their goal: some of them appeal to a supposedly egalitarian condition which, however, does not properly reflect egalitarian intuitions; the others rely on a background assumption about the structure of welfare which cannot be taken for granted. Nonetheless, the theorems remain important: they give insight into the difficulty, if perhaps not the impossibility, of constructing a satisfactory population axiology. We should aim for reflective equilibrium between intuitions and more theoretical considerations. I conclude by highlighting one possible ingredient in this equilibrium, which, I argue, leaves open a still wider range of acceptable theories: the possibility of vague or otherwise indeterminate value relations.

Keywords. Population ethics, impossibility theorems, egalitarianism, vagueness

Are Spectrum Arguments Defused by Vagueness?
Australasian Journal of Philosophy (2021)

Abstract. I consider paradoxical spectrum arguments involving transitive relations like ‘better than’. I argue that, despite being formally different from sorites arguments, at least some spectrum arguments arise from vagueness, and that vagueness may often be the most natural diagnosis.

Keywords. Spectrum or continuum arguments, sorites arguments, moral vagueness, transitivity, parity

On Evaluative Imprecision
In Ethics and Existence: The Legacy of Derek Parfit (OUP 2022)

Abstract. This chapter presents several arguments related to Parfit’s notion of evaluative imprecision and his imprecisionist lexical view of population ethics. After sketching Parfit’s view, it argues that, contrary to Parfit, imprecision and lexicality are both compatible with thinking about goodness in terms of positions on a scale of value. Then, by examining the role that imprecision is meant to play in defusing spectrum argument, it suggests that imprecision should be identified with vagueness. Next, it argues that there is space for robust moral realists to think of evaluative vagueness as a semantic phenomenon, illustrating this view with a version of conceptual role semantics on which the precisifications of betterness are correctness conditions for the precisifications of preference. Finally, it gives a probability-based argument against the imprecisionist lexical view.

Keywords. Derek Parfit, population ethics, spectrum arguments, Imprecisionist Lexical View, moral vagueness, moral realism, conceptual role semantics, vague preference

Separability in Population Ethics
In The Oxford Handbook of Population Ethics (2022)

Abstract. Separability is roughly the principle that, in comparing the value of two outcomes, one can ignore any people whose existence and welfare are unaffected. Separability is both antecedently plausible, at least as a principle of beneficence, and surprisingly powerful; it is the key to some of the best positive arguments in population ethics. This chapter surveys the motivations for and consequences of separability. In particular, it presents an additivity theorem which explains how separability leads to total utilitarianism and closely related axiological views. It then examines systematically how this family of views can avoid the Repugnant Conclusion by incorporating lexicality, a critical level, or a neutral range.

Keywords. Separability, additivity, aggregation, utilitarianism, prioritarianism, measurement theory

Aggregation for Potentially Infinite Populations Without Continuity or Completeness
(with David McCarthy and Kalle Mikkola)
arXiv (2019)

Abstract. We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown to be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi's social aggregation theorem. The second generalizes a classic result about linear opinion pooling.

Keywords. Social aggregation; discontinuous preferences and comparative likelihood relations; incomplete preferences and comparative likelihood relations; infinite populations; Harsanyi’s social aggregation theorem; linear opinion pooling; partially ordered vector spaces

Expected utility theory on mixture spaces without the completeness axiom
(with David McCarthy and Kalle Mikkola)
Journal of Mathematical Economics (2021)

Abstract (Less Technical). This paper examines a foundational problem in decision theory. Standard expected utility theory assumes that an agent’s preferences are complete: for all relevant x and y, either x is weakly preferred to y, or vice versa. This, in conjunction with other axioms, entails that the agent can be represented as maximizing expected utility. Sometimes, at least, if the agent’s preferences are not complete, the agent can nonetheless be represented as maximizing expected utility with respect to a family of utility functions. (Similarly, in formal epistemology, an agent’s epistemic state can sometimes be represented by a family of probability functions, instead of a single one.) We examine the conditions under which this claim is true. We especially show that some common continuity conditions, while necessary, are not sufficient.

Abstract. A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.

Keywords. Expected utility, multi-representation, incompleteness, mixture continuity

Utilitarianism With and Without Expected Utility
(with David McCarthy and Kalle Mikkola)
Journal of Mathematical Economics (2020)

Abstract. We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single ‘individual preorder’. The theorems give axioms that uniquely determine a social preorder in terms of this individual preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems are also consistent with the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels. In that sense, expected utility is inessential to Harsanyi-style utilitarianism. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We then derive further results under the assumption of our basic axioms. First, the individual preorder satisfies the main expected utility axiom of strong independence if and only if the social preorder has a vector-valued expected total utility representation, covering Harsanyi’s utilitarian theorem as a special case. Second, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to strong independence. Third, if the individual preorder satisfies a ‘local expected utility’ condition popular in non-expected utility theory, then the social preorder has a ‘local expected total utility’ representation. Fourth, a wide range of non-expected utility theories nevertheless lead to social preorders of outcomes that have been seen as canonically egalitarian, such as rank-dependent social preorders. Although our aggregation theorems are stated under conditions of risk, they are valid in more general frameworks for representing uncertainty or ambiguity.

Keywords. Harsanyi, aggregation, utilitarianism, expected utility, nonexpected utility, incompleteness and discontinuity, uncertainty, egalitarianism, variable populations.

On the CPT Theorem (with Hilary Greaves)
Studies in History and Philosophy of Modern Physics 45 (2014) 46-66 / arXiv

Abstract. We provide a careful development and rigorous proof of the CPT theorem within the framework of mainstream (Lagrangian) quantum field theory. This is in contrast to the usual rigorous proofs in purely axiomatic frameworks, and non-rigorous proof-sketches in the mainstream approach. We construct the CPT transformation for a general field directly, without appealing to the enumerative classification of representations, and in a manner that is clearly related to the requirements of our proof. Our approach applies equally in Minkowski spacetimes of any dimension at least three, and is in principle neutral between classical and quantum field theories: the quantum CPT theorem has a natural classical analogue. The key mathematical tool is that of complexification; this tool is central to the existing axiomatic proofs, but plays no overt role in the usual mainstream approaches to CPT.

Keywords. Quantum field theory; CPT theorem; Discrete symmetries; Spacetime symmetries.

Weil Representation and Transfer Factor
Algebra and Number Theory 7 (2013), 1535-1570 / Preprint version

Abstract. This paper concerns the Weil representation of the semidirect product of the metaplectic and Heisenberg groups. First we present a canonical construction of the metaplectic group as a central extension of the symplectic group by a subquotient of the Witt group. This leads to simple formulas for the character, for the inverse Weyl transform, and for the transfer factor appearing in J. Adams’s work on character lifting. Along the way, we give formulas for outer automorphisms of the metaplectic group induced by symplectic similitudes. The approach works uniformly for finite and local fields.

Keywords. Metaplectic group, Weil representation, Weyl transform, transfer factor, Cayley transform, Maslov index.

Compatible Intertwiners for Representations of Finite Nilpotent Groups (with Masoud Kamgarpour)
Represent. Theory 15 (2011), 407-432 / arXiv version

Abstract. We sharpen the orbit method for finite groups of small nilpotence class by associating representations to functionals on the corresponding Lie rings. This amounts to describing compatible intertwiners between representations parameterized by an additional choice of polarization. Our construction is motivated by the theory of the linearized Weil representation of the symplectic group. In particular, we provide generalizations of the Maslov index and the determinant functor to the context of finite abelian groups.

Keywords. Nilpotent groups, orbit method, Weil representation, Maslov index, determinants.

The Character of the Weil Representation
J. LMS (2) 77 (2008), 221-239. doi: 10.1112/jlms/jdm098 / arXiv version

Abstract. Let V be a symplectic vector space over a finite or local field. We compute the character of the Weil representation of the metaplectic group Mp(V). The final formulas are overtly free of choices (for example, they do not involve the usual choice of a Lagrangian subspace of V). Along the way, in results similar to those of Maktouf, we relate the character to the Weil index of a certain quadratic form, which may be understood as a Maslov index. This relation also expresses the character as the pullback of a certain simple function from Mp(V\oplus V).

Keywords. Weil representation, Maslov index, character formula, metaplectic group.

Characteristic Cycle of the Theta Sheaf
Manuscript (2012)

Abstract. We extend from characteristic p to characteristic zero S. Lysenko’s theory of theta sheaves on the moduli stack of metaplectic bundles. The main tool is a Fourier transform for semi-homogeneous sheaves on vector bundles. We then calculate the characteristic cycles of the theta sheaves, showing that they lie in a small part of the global nilpotent cone.

Keywords. Metaplectic bundles, theta sheaf, characteristic cycles, Fourier transform.

The Maslov Index as a Quadratic Space
Math. Res. Lett. 13 no. 6 (2006), 985-999 / arXiv (expanded version)

Abstract. Kashiwara defined the Maslov index (associated to a collection of Lagrangian subspaces of a symplectic vector space over a field F) as a class in the Witt group W(F) of quadratic forms. We construct a canonical quadratic vector space in this class and show how to understand the basic properties of the Maslov index without passing to W(F) – that is, more or less, how to upgrade Kashiwara’s equalities in W(F) to canonical isomorphisms between quadratic spaces. We also show how our canonical quadratic form occurs naturally in the context of the Weil representation. The quadratic space is defined using elementary linear algebra. On the other hand, it has a nice interpretation in terms of sheaf cohomology, due to A. Beilinson.

Keywords. Maslov index, quadratic forms, sheaf cohomology, Witt group.