Papers by Wesley Holliday
Journal of Mathematical Economics, 2024
We propose a model of unawareness that remains close to the paradigm of Aumann's model for knowle... more We propose a model of unawareness that remains close to the paradigm of Aumann's model for knowledge [R. J. Aumann, International Journal of Game Theory 28 (1999) 263-300]: just as Aumann uses a correspondence on a state space to define an agent's knowledge operator on events, we use a correspondence on a state space to define an agent's awareness operator on events. This is made possible by three ideas. First, like the model of [A. Heifetz, M. Meier, and B. Schipper, Journal of Economic Theory 130
(2006) 78-94], ours is based on a space of partial specifications of the world, partially ordered by a relation of further specification or refinement, and the idea that agents may be aware of some coarser-grained specifications while unaware of some finer-grained specifications; however, our model is based on a different implementation of this idea, related to forcing in set theory. Second, we depart from a tradition in the literature, initiated by [S. Modica and A. Rustichini, Theory and Decision 37 (1994) 107-124] and adopted by Heifetz et al. and [J. Li, Journal of Economic Theory 144 (2009) 977-993], of taking awareness to be definable in terms of knowledge. Third, we show that the negative conclusion of a well-known impossibility theorem concerning unawareness in [Dekel, Lipman, and Rustichini, Econometrica 66 (1998) 159-173] can be escaped by a slight weakening of a key axiom. Together these points demonstrate that a correspondence on a partial-state space is sufficient to model unawareness of events. Indeed, we prove a representation theorem showing that any abstract Boolean algebra equipped with awareness, knowledge, and belief operators satisfying some plausible axioms is representable as the algebra of events arising from a partial-state space with awareness, knowledge, and belief correspondences.
Challenges to classical logic have emerged from several sources. According to recent work [24], t... more Challenges to classical logic have emerged from several sources. According to recent work [24], the behavior of epistemic modals in natural language motivates weakening classical logic to orthologic, a logic originally discovered by Birkhoff and von Neumann [3] in the study of quantum mechanics. In this paper, we consider a different tradition of thinking that the behavior of vague predicates in natural language motivates weakening classical logic to intuitionistic logic or even giving up some intuitionistic principles. We focus in particular on Fine's recent approach to vagueness [12]. Our main question is: what is a natural non-classical base logic to which to retreat in light of both the non-classicality emerging from epistemic modals and the non-classicality emerging from vagueness? We first consider whether orthologic itself might be the answer. We then discuss whether accommodating the nonclassicality emerging from epistemic modals and vagueness might point in the direction of a weaker system of fundamental logic [21].
Theory and Decision, 2024
There is an extensive literature in social choice theory studying the consequences of weakening t... more There is an extensive literature in social choice theory studying the consequences of weakening the assumptions of Arrow's Impossibility Theorem. Much of this literature suggests that there is no escape from Arrow-style impossibility theorems, while remaining in an ordinal preference setting, unless one drastically violates the Independence of Irrelevant Alternatives (IIA). In this paper, we present a more positive outlook. We propose a model of comparing candidates in elections, which we call the Advantage-Standard (AS) model. The requirement that a collective choice rule (CCR) be representable by the AS model captures a key insight of IIA but is weaker than IIA; yet it is stronger than what is known in the literature as weak IIA (two profiles alike on x,y cannot have opposite strict social preferences on x and y). In addition to motivating violations of IIA, the AS model makes intelligible violations of another Arrovian assumption: the negative transitivity of the strict social preference relation P. While previous literature shows that only weakening IIA to weak IIA or only weakening negative transitivity of P to acyclicity still leads to impossibility theorems, we show that jointly weakening IIA to AS representability and weakening negative transitivity of P leads to no such impossibility theorems. Indeed, we show that several appealing CCRs are AS representable, including even transitive CCRs.
Foundation models such as GPT-4 are fine-tuned to avoid unsafe or otherwise problematic behavior,... more Foundation models such as GPT-4 are fine-tuned to avoid unsafe or otherwise problematic behavior, such as helping to commit crimes or producing racist text. One approach to fine-tuning, called reinforcement learning from human feedback, learns from humans' expressed preferences over multiple outputs. Another approach is constitutional AI, in which the input from humans is a list of high-level principles. But how do we deal with potentially diverging input from humans? How can we aggregate the input into consistent data about "collective" preferences or otherwise use it to make collective choices about model behavior? In this paper, we argue that the field of social choice is well positioned to address these questions, and we discuss ways forward for this agenda, drawing on discussions in a recent workshop on Social Choice for AI Ethics and Safety held in Berkeley, CA, USA in December 2023.
Advances in Modal Logic, 2024
Non-classical generalizations of classical modal logic have been developed in the contexts of con... more Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation □, and a completely additive operation ◊. Such lattice expansions can be represented by means of a set X together with binary relations ◁, R, and Q, satisfying some first-order conditions, used to represent (L, ¬), □, and ◊, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative □, and an additive ◊ embeds into the lattice of propositions of a frame (X, ◁, R, Q). Building on our recent study of fundamental logic, we focus on the case where ¬ is dually self-adjoint (a ≤ ¬b implies b ≤ ¬a) and ◊¬a ≤ ¬□a. In this case, the representations can be constrained so that R = Q, i.e., we need only add a single relation to (X, ◁) to represent both □ and ◊. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X, ◁, R).
An obstacle to the implementation of Condorcet voting methods in political elections is the perce... more An obstacle to the implementation of Condorcet voting methods in political elections is the perceived complexity of these methods. In this note, we propose a simple Condorcet voting method for use in a Final Four election, i.e., after a preliminary process in which up to four candidates qualify for the election. In the Final Four election, voters submit rankings of the candidates. If one candidate beats each of the others in a head-to-head majority comparison using the voters' rankings, that candidate is elected; if not, then among the candidates with at most one head-to-head loss, the candidate with the smallest loss is elected. We analyze this voting method from the perspective of voting theory. It avoids some standard objections to the related Minimax voting method, and it has advantages over the Instant Runoff method that has already been implemented in a number of cities and states.
Economics Letters, 2024
In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive... more In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.
We give a proof-theoretic as well as a semantic characterization of a logic in the signature with... more We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations.
The Logica Yearbook 2023, ed. Igor Sedlar, College Publications, 2024
In recent work, we introduced a new semantics for conditionals, covering a large class of what we... more In recent work, we introduced a new semantics for conditionals, covering a large class of what we call preconditionals. In this paper, we undertake an axiomatic study of preconditionals and subclasses of preconditionals. We then prove that any bounded lattice equipped with a preconditional can be represented by a relational structure, suitably topologized, yielding a single relational semantics for conditional logics normally treated by different semantics, as well as generalizing beyond those semantics.
May’s Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two al... more May’s Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May’s axioms, we can uniquely determine how to vote on three alternatives (setting aside tiebreaking). In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, must choose from among the Minimax winners in all three- alternative elections. When applied to more than three alternatives, our axioms also distinguish Minimax from other known voting methods that coincide with or refine Minimax for three alternatives.
Journal of Philosophical Logic, 2024
Epistemic modals have peculiar logical features that are challenging to account for in a broadly ... more Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form p ∧ ♦¬p ('p, but it might be that not p') appears to be a contradiction, ♦¬p does not entail ¬p, which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under-or over-correct. Some theories predict that p∧♦¬p, a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not allow substitution of logical equivalents; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail, like distributivity and disjunctive syllogism, but also rules like non-contradiction, excluded middle, De Morgan's laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics, based on ortholattices instead of Boolean algebras, and then propose a more concrete possibility semantics, based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize. Then we show how to extend our semantics to explain parallel phenomena involving probabilities and conditionals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary.
Social Choice and Welfare
A number of rules for resolving majority cycles in elections have been proposed in the literature... more A number of rules for resolving majority cycles in elections have been proposed in the literature. Recently, Holliday and Pacuit (Journal of Theoretical Politics 33 (2021) 475-524) axiomatically characterized the class of rules refined by one such cycle-resolving rule, dubbed Split Cycle: in each majority cycle, discard the majority preferences with the smallest majority margin. They showed that any rule satisfying five standard axioms plus a weakening of Arrow’s Independence of Irrelevant Alternatives (IIA), called Coherent IIA, is refined by Split Cycle. In this paper, we go further and show that Split Cycle is the only rule satisfying the axioms of Holliday and Pacuit together with two additional axioms, which characterize the class of rules that refine Split Cycle: Coherent Defeat and Positive Involvement in Defeat. Coherent Defeat states that any majority preference not occurring in a cycle is retained, while Positive Involvement in Defeat is closely related to the well-known axiom of Positive Involvement (as in J. Pérez, Social Choice and Welfare 18 (2001) 601-616). We characterize Split Cycle not only as a collective choice rule but also as a social choice correspondence, over both profiles of linear ballots and profiles of ballots allowing ties.
The 2024 Conference on Empirical Methods in Natural Language Processing (EMNLP 2024), 2024
The reasoning abilities of large language models (LLMs) are the topic of a
growing body of resea... more The reasoning abilities of large language models (LLMs) are the topic of a
growing body of research in AI and cognitive science. In this paper, we probe
the extent to which twenty-nine LLMs are able to distinguish logically correct
inferences from logically fallacious ones. We focus on inference patterns
involving conditionals (e.g., 'If Ann has a queen, then Bob has a jack') and
epistemic modals (e.g., 'Ann might have an ace', 'Bob must have a king'). These
inferences have been of special interest to logicians, philosophers, and
linguists, since they play a central role in the fundamental human ability to
reason about distal possibilities. Assessing LLMs on these inferences is thus
highly relevant to the question of how much the reasoning abilities of LLMs
match those of humans. All the LLMs we tested make some basic mistakes with
conditionals or modals, though zero-shot chain-of-thought prompting helps them
make fewer mistakes. Even the best performing LLMs make basic errors in modal
reasoning, display logically inconsistent judgments across inference patterns
involving epistemic modals and conditionals, and give answers about complex
conditional inferences that do not match reported human judgments. These
results highlight gaps in basic logical reasoning in today's LLMs.
Appears at the 1st Workshop on Social Choice and Learning Algorithms (SCaLA 2024) held at the 23rd International Conference on Autonomous Agents and Multiagent Systems, organized by B. Armstrong, R. Fairstein, N. Mattei, and Z. Terzopoulou, May 6-7, 2024, Auckland, New Zealand.
By classic results in social choice theory, any reasonable preferential voting method sometimes g... more By classic results in social choice theory, any reasonable preferential voting method sometimes gives individuals an incentive to report an insincere preference. The extent to which different voting methods are more or less resistant to such strategic manipulation has become a key consideration for comparing voting methods. Here we measure resistance to manipulation by whether neural networks of varying sizes can learn to profitably manipulate a given voting method in expectation, given different types of limited information about how other voters will vote. We trained over 70,000 neural networks of 26 sizes to manipulate against 8 different voting methods, under 6 types of limited information, in committee-sized elections with 5--21 voters and 3--6 candidates. We find that some voting methods, such as Borda, are highly manipulable by networks with limited information, while others, such as Instant Runoff, are not, despite being quite profitably manipulated by an ideal manipulator with full information. For the two probability models for elections that we use, the overall least manipulable of the 8 methods we study are Condorcet methods, namely Minimax and Split Cycle.
Advances in Modal Logic, Vol. 14, 2022
In this paper, we study three representations of lattices by means of a set with a binary
relati... more In this paper, we study three representations of lattices by means of a set with a binary
relation of compatibility in the tradition of Ploscica. The standard representations of
complete ortholattices and complete perfect Heyting algebras drop out as special cases
of the first representation, while the second covers arbitrary complete lattices, as well
as complete lattices equipped with a negation we call a protocomplementation. The
third topological representation is a variant of that of Craig, Haviar, and Priestley. We
then extend each of the three representations to lattices with a multiplicative unary
modality; the representing structures, like so-called graph-based frames, add a second
relation of accessibility interacting with compatibility. The three representations
generalize possibility semantics for classical modal logics to non-classical modal logics,
motivated by a recent application of modal orthologic to natural language semantics.
Selected Topics from Contemporary Logics, ed. Melvin Fitting, Volume 2 of Landscapes in Logic, College Publications, London, ISBN 97-1-84890-350-0, 2021
In traditional semantics for classical logic and its extensions, such as modal logic, proposition... more In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.
Constitutional Political Economy, 2023
In this paper, we propose a new single-winner voting system using ranked ballots: Stable Voting. ... more In this paper, we propose a new single-winner voting system using ranked ballots: Stable Voting. The motivating principle of Stable Voting is that if a candidate A would win without another candidate B in the election, and A beats B in a head-to-head majority comparison, then A should still win in the election with B included (unless there is another candidate A' who has the same kind of claim to winning, in which case a tiebreaker may choose between A and A'). We call this principle Stability for Winners (with Tiebreaking). Stable Voting satisfies this principle while also having a remarkable ability to avoid tied outcomes in elections even with small numbers of voters.
Journal of Theoretical Politics, 2021
We propose six axioms concerning when one candidate should defeat another in a democratic electio... more We propose six axioms concerning when one candidate should defeat another in a democratic election involving two or more candidates. Five of the axioms are widely satisfied by known voting procedures. The sixth axiom is a weakening of Kenneth Arrow’s famous condition of the Independence of Irrelevant Alternatives (IIA). We call this weakening Coherent IIA. We prove that the five axioms plus Coherent IIA single out a method of determining defeats studied in our recent work: Split Cycle. In particular, Split Cycle provides the most resolute definition of defeat among any satisfying the six axioms for democratic defeat. In addition, we analyze how Split Cycle escapes Arrow’s Impossibility Theorem and related impossibility results.
Public Choice, 2023
We propose a Condorcet consistent voting method that we call Split Cycle. Split Cycle belongs to ... more We propose a Condorcet consistent voting method that we call Split Cycle. Split Cycle belongs to the small family of known voting methods satisfying the anti-vote-splitting criterion of independence of clones. In this family, only Split Cycle satisfies a new criterion we call immunity to spoilers, which concerns adding candidates to elections, as well as the known criteria of positive involvement and negative involvement, which concern adding voters to elections. Thus, in contrast to other clone-independent methods, Split Cycle mitigates both ``spoiler effects'' and ``strong no show paradoxes.''
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Papers by Wesley Holliday
(2006) 78-94], ours is based on a space of partial specifications of the world, partially ordered by a relation of further specification or refinement, and the idea that agents may be aware of some coarser-grained specifications while unaware of some finer-grained specifications; however, our model is based on a different implementation of this idea, related to forcing in set theory. Second, we depart from a tradition in the literature, initiated by [S. Modica and A. Rustichini, Theory and Decision 37 (1994) 107-124] and adopted by Heifetz et al. and [J. Li, Journal of Economic Theory 144 (2009) 977-993], of taking awareness to be definable in terms of knowledge. Third, we show that the negative conclusion of a well-known impossibility theorem concerning unawareness in [Dekel, Lipman, and Rustichini, Econometrica 66 (1998) 159-173] can be escaped by a slight weakening of a key axiom. Together these points demonstrate that a correspondence on a partial-state space is sufficient to model unawareness of events. Indeed, we prove a representation theorem showing that any abstract Boolean algebra equipped with awareness, knowledge, and belief operators satisfying some plausible axioms is representable as the algebra of events arising from a partial-state space with awareness, knowledge, and belief correspondences.
growing body of research in AI and cognitive science. In this paper, we probe
the extent to which twenty-nine LLMs are able to distinguish logically correct
inferences from logically fallacious ones. We focus on inference patterns
involving conditionals (e.g., 'If Ann has a queen, then Bob has a jack') and
epistemic modals (e.g., 'Ann might have an ace', 'Bob must have a king'). These
inferences have been of special interest to logicians, philosophers, and
linguists, since they play a central role in the fundamental human ability to
reason about distal possibilities. Assessing LLMs on these inferences is thus
highly relevant to the question of how much the reasoning abilities of LLMs
match those of humans. All the LLMs we tested make some basic mistakes with
conditionals or modals, though zero-shot chain-of-thought prompting helps them
make fewer mistakes. Even the best performing LLMs make basic errors in modal
reasoning, display logically inconsistent judgments across inference patterns
involving epistemic modals and conditionals, and give answers about complex
conditional inferences that do not match reported human judgments. These
results highlight gaps in basic logical reasoning in today's LLMs.
relation of compatibility in the tradition of Ploscica. The standard representations of
complete ortholattices and complete perfect Heyting algebras drop out as special cases
of the first representation, while the second covers arbitrary complete lattices, as well
as complete lattices equipped with a negation we call a protocomplementation. The
third topological representation is a variant of that of Craig, Haviar, and Priestley. We
then extend each of the three representations to lattices with a multiplicative unary
modality; the representing structures, like so-called graph-based frames, add a second
relation of accessibility interacting with compatibility. The three representations
generalize possibility semantics for classical modal logics to non-classical modal logics,
motivated by a recent application of modal orthologic to natural language semantics.
(2006) 78-94], ours is based on a space of partial specifications of the world, partially ordered by a relation of further specification or refinement, and the idea that agents may be aware of some coarser-grained specifications while unaware of some finer-grained specifications; however, our model is based on a different implementation of this idea, related to forcing in set theory. Second, we depart from a tradition in the literature, initiated by [S. Modica and A. Rustichini, Theory and Decision 37 (1994) 107-124] and adopted by Heifetz et al. and [J. Li, Journal of Economic Theory 144 (2009) 977-993], of taking awareness to be definable in terms of knowledge. Third, we show that the negative conclusion of a well-known impossibility theorem concerning unawareness in [Dekel, Lipman, and Rustichini, Econometrica 66 (1998) 159-173] can be escaped by a slight weakening of a key axiom. Together these points demonstrate that a correspondence on a partial-state space is sufficient to model unawareness of events. Indeed, we prove a representation theorem showing that any abstract Boolean algebra equipped with awareness, knowledge, and belief operators satisfying some plausible axioms is representable as the algebra of events arising from a partial-state space with awareness, knowledge, and belief correspondences.
growing body of research in AI and cognitive science. In this paper, we probe
the extent to which twenty-nine LLMs are able to distinguish logically correct
inferences from logically fallacious ones. We focus on inference patterns
involving conditionals (e.g., 'If Ann has a queen, then Bob has a jack') and
epistemic modals (e.g., 'Ann might have an ace', 'Bob must have a king'). These
inferences have been of special interest to logicians, philosophers, and
linguists, since they play a central role in the fundamental human ability to
reason about distal possibilities. Assessing LLMs on these inferences is thus
highly relevant to the question of how much the reasoning abilities of LLMs
match those of humans. All the LLMs we tested make some basic mistakes with
conditionals or modals, though zero-shot chain-of-thought prompting helps them
make fewer mistakes. Even the best performing LLMs make basic errors in modal
reasoning, display logically inconsistent judgments across inference patterns
involving epistemic modals and conditionals, and give answers about complex
conditional inferences that do not match reported human judgments. These
results highlight gaps in basic logical reasoning in today's LLMs.
relation of compatibility in the tradition of Ploscica. The standard representations of
complete ortholattices and complete perfect Heyting algebras drop out as special cases
of the first representation, while the second covers arbitrary complete lattices, as well
as complete lattices equipped with a negation we call a protocomplementation. The
third topological representation is a variant of that of Craig, Haviar, and Priestley. We
then extend each of the three representations to lattices with a multiplicative unary
modality; the representing structures, like so-called graph-based frames, add a second
relation of accessibility interacting with compatibility. The three representations
generalize possibility semantics for classical modal logics to non-classical modal logics,
motivated by a recent application of modal orthologic to natural language semantics.