Wesley Holliday
University of California, Berkeley, Philosophy, Faculty Member
- Stanford University, Philosophy, Alumnusadd
- Philosophy, Logic, Philosophical Logic, Modal Logic, Formal Epistemology, Epistemic Logic, and 84 moreLogic And Foundations Of Mathematics, Modality, Formal Philosophy, Analytic Philosophy, Non-Classical Logic, Epistemic Modals, Metalogic, Formal Logic, Qualitative Probability, Qualitative Probabilistic Reasoning, Reasoning about Uncertainty, Decision Making Under Uncertainty, Knowledge Representation and Reasoning, Uncertainty, Foundations of Probability, Symbolic Logic, Dynamic Epistemic Logic, Hybrid Logic, Temporal and Modal Logic, Quantified Epistemic Logic, Modals, Possible World Semantics, Social Choice Theory, Computational Social Choice, Logic and Social Choice, Voting Theory, Mathematics, Set Theory, Mathematical Logic, Boolean Algebra, Stone Duality, Topology, Constructive Mathematics, Duality Theory, Axiom of Choice, Pointfree Topology, Intuitionistic Logic, Inquisitive Semantics, Quantifiers, Algebraic Semantics, Probability, Probability Theory, Semantics, Conditionals, Formal Semantics, Conditionals (Philosophy), Epistemic modality, Indicative Conditionals, Philosophy Of Language, Counterfactuals, Algebraic Modal Logic, Algebraic Logic, Artificial Intelligence, Game Theory, Beliefs, Knowledge, Multi Agent System, Multi-Agent Systems, Voting, Decision Theory, Voting Systems, Elections and Voting Behavior, MultiAgent Systems (Computer Science), Elections, Decision And Game Theory, Economic Theory, Political Science, Judgment and decision making, Multiagent Systems, Decision Making, Political Philosophy, Welfare Economics, Rational Choice, Possible Worlds, Public Choice Theory, Rational Choice Theory, Democracy, Public Choice, Economics, Voting Behavior, Social Choice Mathematics, Social Choice, Computer Science, and Formal Methods (Formal Verification)edit
- Professor of Philosophy and Chair of the
Group in Logic and the Methodology of Science, University of California, Berkeley.edit
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... 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.
Research Interests:
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,... 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.
Research Interests: Computer Science, Artificial Intelligence, Philosophy, Reinforcement Learning, Machine Learning, and 15 moreSocial Choice Theory, Future of artificial intelligence, Decision Theory, Ethics of Artificial Intelligence, Computational Social Choice, Artificial Neural Networks, Philosophy of Artificial Intelligence, Revealed and Stated Preferences, Preferences, Artificial Intelligence And Techniques, Computer Science and Artificial Intelligence, Revealed Preference, Preference, Philosophy of Mind and Artificial Intelligence & AI, and Large language models
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... 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).
Research Interests: Lattice Theory (Order Theory), Modal Logic, Intuitionistic Logic, Philosophy, Logic, and 15 moreLattice Theory, Temporal and Modal Logic, Modality, Philosophical Logic, Metaphysics of Modality, Mathematical Logic, Non-Classical Logic, Quantum Logic, Modal and Temporal Logics, Epistemic modality, Algebraic Modal Logic, Symbolic Logic, Constructive Modal Logic, Algebraic Logic, and Intuitionistic modal logic
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... 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.
Research Interests: Political Philosophy, Political Theory, Democratic Theory, Political Science, Social Choice Theory, and 15 morePublic Choice, Electoral Systems, Democracy, Voting Theory, Elections, Voting, Elections and Voting Behavior, Electoral Studies, Electoral Reform, Democracy and Good Governance, Condorcet, Voting Systems, Democratic Governance, Local elections, and Local government elections
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... 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.
Research Interests: Applied Mathematics, Economics, Microeconomics, Welfare Economics, Philosophy, and 15 morePolitical Philosophy, Political Science, Economic Theory, Formal Theory, Decision And Game Theory, Social Choice Theory, Mechanism Design, Public Choice, Voting Theory, Decision Theory, Voting, Multi-Agent Systems, Computational Social Choice, Social Choice Mathematics, and Voting Systems
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... 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.
Research Interests: Mathematics, Modal Logic, Intuitionistic Logic, Philosophy, Logic, and 15 moreLattice Theory, Proof Theory, Philosophy Of Mathematics, Philosophical Logic, Philosophy of Logic, Proof-Theoretic Semantics, Logical Constants, Logical Consequence, Mathematical Logic, Non-Classical Logic, Quantum Logic, Formal Logic, Symbolic Logic, Logical Pluralism, and Algebraic Logic
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... 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.
Research Interests:
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... 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.
Research Interests: Mathematics, Applied Mathematics, Microeconomics, Political Philosophy, Political Theory, and 15 morePolitical Science, Economic Theory, Social Choice Theory, Formal Philosophy, Voting Theory, Voting, Elections and Voting Behavior, Computational Social Choice, Social Choice Mathematics, Social Choice, Voting Systems, Preferences, Mathematical Philosophy, Computational Social choice theory, and Logic and Social Choice
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... 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.
Research Interests:
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... 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.
Research Interests: Mathematics, Applied Mathematics, Economics, Political Science, Economic Theory, and 13 moreRational Choice, Social Choice Theory, Public Choice, Voting Theory, Rational Choice Theory, Voting, Elections and Voting Behavior, Computational Social Choice, Social Choice Mathematics, Social Choice, Voting Systems, Public Choice Theory, and Computational Social choice theory
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-five LLMs are able to distinguish logically correct... 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-five 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. Among the LLMs we tested, all but the GPT-4 model family often make basic mistakes with conditionals, though zero-shot chain-of-thought prompting helps them make fewer mistakes. Moreover, even the GPT-4 family displays logically inconsistent judgments across inference patterns involving epistemic modals, and almost all models give answers to certain complex conditional inferences widely discussed in the literature that do not match human judgments. These results highlight gaps in basic logical reasoning in today's LLMs.
Research Interests:
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... 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.
Research Interests: Game Theory, Voting Behavior, Machine Learning, Economic Theory, Social Choice Theory, and 10 moreApplications of Machine Learning, Mechanism Design, Neural Networks, Voting Theory, Neural Network, Voting, Elections and Voting Behavior, Computational Social Choice, Artificial Neural Networks, and Voting Systems
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... 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.
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.
Research Interests:
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... 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.
Research Interests: Mathematics, Modal Logic, Intuitionistic Logic, Set Theory, Philosophy, and 15 morePhilosophy Of Language, Logic, Lattice Theory, Temporal and Modal Logic, Philosophical Logic, Philosophy of Logic, Mathematical Logic, Possible World Semantics, Temporal Logics, Formal Semantics, Possible Worlds, Boolean Algebra, First-Order Logic, Algebraic Logic, and Provability logic
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... 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.
Research Interests: Mathematics, Economics, Philosophy, Political Philosophy, Social Philosophy, and 15 moreDemocratic Theory, Political Science, Social Choice Theory, Mechanism Design, Public Choice, Electoral Systems, Democracy, Voting Theory, Rational Choice Theory, Voting, Elections and Voting Behavior, Computational Social Choice, Voting Systems, Electoral System, and Public Choice Theory
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... 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.
Research Interests: Mathematics, Applied Mathematics, Philosophy, Political Philosophy, Decision Making, and 15 morePolitical Theory, Multiagent Systems, Judgment and decision making, Political Science, Economic Theory, Decision And Game Theory, Social Choice Theory, Democracy, Voting Theory, Elections, Decision Theory, MultiAgent Systems (Computer Science), Voting, Elections and Voting Behavior, and Voting Systems
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... 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.''
Research Interests: Mathematics, Applied Mathematics, Welfare Economics, Philosophy, Political Philosophy, and 15 moreDecision Making, Multiagent Systems, Judgment and decision making, Political Science, Economic Theory, Decision And Game Theory, Social Choice Theory, Voting Theory, Elections, Decision Theory, MultiAgent Systems (Computer Science), Voting, Elections and Voting Behavior, Computational Social Choice, and Voting Systems
A fundamental principle of individual rational choice is Sen's γ axiom, also known as expansion consistency, stating that any alternative chosen from each of two menus must be chosen from the union of the menus. Expansion consistency can... more
A fundamental principle of individual rational choice is Sen's γ axiom, also known as expansion consistency, stating that any alternative chosen from each of two menus must be chosen from the union of the menus. Expansion consistency can also be formulated in the setting of social choice. In voting theory, it states that any candidate chosen from two fields of candidates must be chosen from the combined field of candidates. An important special case of the axiom is binary expansion consistency, which states that any candidate chosen from an initial field of candidates and chosen in a head-to-head match with a new candidate must also be chosen when the new candidate is added to the field, thereby ruling out spoiler effects. In this paper, we study the tension between this weakening of expansion consistency and weakenings of resoluteness, an axiom demanding the choice of a single candidate in any election. As is well known, resoluteness is inconsistent with basic fairness conditions on social choice, namely anonymity and neutrality. Here we prove that even significant weakenings of resoluteness, which are consistent with anonymity and neutrality, are inconsistent with binary expansion consistency. The proofs make use of SAT solving, with the correctness of a SAT encoding formally verified in the Lean Theorem Prover, as well as a strategy for generalizing impossibility theorems obtained for special types of voting methods (namely majoritarian and pairwise voting methods) to impossibility theorems for arbitrary voting methods. This proof strategy may be of independent interest for its potential applicability to other impossibility theorems in social choice.
Research Interests:
Reasoning about what other people know is an important cognitive ability, known as epistemic reasoning, which has fascinated psychologists, economists, and logicians. In this paper, we propose a computational model of humans' epistemic... more
Reasoning about what other people know is an important cognitive ability, known as epistemic reasoning, which has fascinated psychologists, economists, and logicians. In this paper, we propose a computational model of humans' epistemic reasoning, including higher-order epistemic reasoning—reasoning about what one person knows about another person's knowledge—that we test in an experiment using a deductive card game called "Aces and Eights". Our starting point is the model of perfect higher-order epistemic reasoners given by the framework of dynamic epistemic logic. We modify this idealized model with bounds on the level of feasible epistemic reasoning and stochastic update of a player's space of possibilities in response to new information. These modifications are crucial for explaining the variation in human performance across different participants and different games in the experiment. Our results demonstrate how research on epistemic logic and cognitive models can inform each other.
Research Interests:
In the context of computational social choice, we study voting methods that assign a set of winners to each profile of voter preferences. A voting method satisfies the property of positive involvement (PI) if for any election in which a... more
In the context of computational social choice, we study voting methods that assign a set of winners to each profile of voter preferences. A voting method satisfies the property of positive involvement (PI) if for any election in which a candidate x would be among the winners, adding another voter to the election who ranks x first does not cause x to lose. Surprisingly, a number of standard voting methods violate this natural property. In this paper, we investigate different ways of measuring the extent to which a voting method violates PI, using computer simulations. We consider the probability (under different probability models for preferences) of PI violations in randomly drawn profiles vs. profile-coalition pairs (involving coalitions of different sizes). We argue that in order to choose between a voting method that satisfies PI and one that does not, we should consider the probability of PI violation conditional on the voting methods choosing different winners. We should also relativize the probability of PI violation to what we call voter potency, the probability that a voter causes a candidate to lose. Although absolute frequencies of PI violations may be low, after this conditioning and relativization, we see that under certain voting methods that violate PI, much of a voter's potency is turned against them—in particular, against their desire to see their favorite candidate elected.
Research Interests:
We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from... more
We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.
Research Interests:
There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on computer-aided methods such as SAT solving and interactive theorem proving. In this paper,... more
There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on computer-aided methods such as SAT solving and interactive theorem proving. In this paper, we report on the development of a framework for formalizing voting theory in the Lean theorem prover, which we have applied to verify properties of a recently studied voting method. While previous applications of interactive theorem proving to social choice (using Isabelle/HOL and Mizar) have focused on the verification of impossibility theorems, we aim to cover a variety of results ranging from impossibility theorems to the verification of properties of specific voting methods (e.g., Condorcet consistency, independence of clones, etc.). In order to formalize voting theoretic axioms concerning adding or removing candidates and voters, we work in a variable-election setting whose formalization makes use of dependent types in Lean.
Research Interests: Computer Science, Formal Methods (Formal Verification), Type Theory, Logic, Political Science, and 15 moreEconomic Theory, Social Choice Theory, Formal methods, Theorem Proving, Voting Theory, Voting, Computational Social Choice, Social Choice Mathematics, Voting Systems, Logical Methods in Computer Science, Automated Theorem Proving, Symbolic Logic, Interactive Theorem Proving, Logic and Social Choice, and Dependent type theory
Under the California gubernatorial recall system, it is possible for a Governor to be recalled and replaced by a candidate C even though every voter prefers the original Governor to C. The likelihood that at least a majority of voters... more
Under the California gubernatorial recall system, it is possible for a Governor to be recalled and replaced by a candidate C even though every voter prefers the original Governor to C. The likelihood that at least a majority of voters would prefer the original Governor to a recall winner is assessed in a toy probabilistic model.
Research Interests:
This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability and comparative probability. The goal is to identify complete logics for... more
This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability and comparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.
Research Interests: Mathematics, Probability Theory, Modal Logic, Artificial Intelligence, Philosophy, and 15 moreEpistemology, Logic, Reasoning about Uncertainty, Philosophy Of Probability, Formal Epistemology, Epistemic Logic, Decision Making Under Uncertainty, Mathematical Logic, Probability, Uncertainty, Epistemic modality, Imprecise Probability, Dynamic Epistemic Logic, Symbolic Logic, and Knowledge Representation and Reasoning
This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set- theoretic forcing, or doubly ordered structures, as in a recent... more
This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set- theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis.
Research Interests:
In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However,... more
In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the world part of possible world semantics—the atomicity of the algebra of propositions—but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic.
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Inquisitive logic is a research program seeking to expand the purview of logic beyond declarative sentences to include the logic of questions. To this end, inquisitive propositional logic extends classical propositional logic for... more
Inquisitive logic is a research program seeking to expand the purview of logic beyond declarative sentences to include the logic of questions. To this end, inquisitive propositional logic extends classical propositional logic for declarative sentences with principles governing a new binary connective of inquisitive disjunction, which allows the formation of questions. Recently inquisitive logicians have considered what happens if the logic of declarative sentences is assumed to be intuitionistic rather than classical. In short, what should inquisitive logic be on an intuitionistic base? In this paper, we provide an answer to this question from the perspective of nuclear semantics, an approach to classical and intuitionistic semantics pursued in our previous work. In particular, we show how Beth semantics for intuitionistic logic naturally extends to a semantics for inquisitive intuitionistic logic. In addition, we show how an explicit view of inquisitive intuitionistic logic comes via a translation into propositional lax logic, whose completeness we prove with respect to Beth semantics.
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In Arrovian social choice theory assuming the independence of irrelevant alternatives , Murakami (1968) proved two theorems about complete and transitive collective choice rules that satisfy strict non-imposition (citizens' sovereignty),... more
In Arrovian social choice theory assuming the independence of irrelevant alternatives , Murakami (1968) proved two theorems about complete and transitive collective choice rules that satisfy strict non-imposition (citizens' sovereignty), one being a dichotomy theorem about Paretian or anti-Paretian rules and the other a dictator-or-inverse-dictator impossibility theorem without the Pareto principle. It has been claimed in the later literature that a theorem of Malawski and Zhou (1994) is a generalization of Murakami's dichotomy theorem and that Wilson's (1972) impossibility theorem is stronger than Murakami's impossibility theorem, both by virtue of replacing Murakami's assumption of strict non-imposition with the assumptions of non-imposition and non-nullness. In this note, we first point out that these claims are incorrect: non-imposition and non-nullness are together equivalent to strict non-imposition for all transitive collective choice rules. We then generalize Murakami's dichotomy and impossibility theorems to the setting of incomplete social preference. We prove that if one drops completeness from Murakami's assumptions, his remaining assumptions imply (i) that a collective choice rule is either Paretian, anti-Paretian, or dis-Paretian (unanimous individual preference implies noncomparability) and (ii) that adding proposed constraints on noncomparability, such as the regularity axiom of Eliaz and Ok (2006), restores Murakami's dictator-or-inverse-dictator result.
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This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such... more
This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.
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Much of the theoretical work on strategic voting makes strong assumptions about what voters know about the voting situation. A strategizing voter is typically assumed to know how other voters will vote and to know the rules of the voting... more
Much of the theoretical work on strategic voting makes strong assumptions about what voters know about the voting situation. A strategizing voter is typically assumed to know how other voters will vote and to know the rules of the voting method. A growing body of literature explores strategic voting when there is uncertainty about how others will vote. In this paper, we study strategic voting when there is uncertainty about the voting method. We introduce three notions of manipulability for a set of voting methods: sure, safe, and expected manipulability. With the help of a computer program, we identify voting scenarios in which uncertainty about the voting method may reduce or even eliminate a voter's incentive to misrepresent her preferences. Thus, it may be in the interest of an election designer who wishes to reduce strategic voting to leave voters uncertain about which of several reasonable voting methods will be used to determine the winners of an election.
Research Interests:
Research Interests:
The early literature on epistemic logic in philosophy focused on reasoning about the knowledge or belief of a single agent, especially on controversies about "introspection axioms" such as the 4 and 5 axioms. By contrast, the later... more
The early literature on epistemic logic in philosophy focused on reasoning about the knowledge or belief of a single agent, especially on controversies about "introspection axioms" such as the 4 and 5 axioms. By contrast, the later literature on epistemic logic in computer science and game theory has focused on multi-agent epistemic reasoning, with the single-agent 4 and 5 axioms largely taken for granted. In the relevant multi-agent scenarios, it is often important to reason about what agent A believes about what agent B believes about what agent A believes; but it is rarely important to reason just about what agent A believes about what agent A believes. This raises the question of the extent to which single-agent introspection axioms actually matter for multi-agent epistemic reasoning. In this paper, we formalize and answer this question. To formalize the question, we first define a set of multi-agent formulas that we call agent-alternating formulas, including formulas like Box_a Box_b Box_a p but not formulas like Box_a Box_a p. We then prove, for the case of belief, that if one starts with multi-agent K or KD, then adding both the 4 and 5 axioms (or adding the B axiom) does not allow the derivation of any new agent-alternating formulas-in this sense, introspection axioms do not matter. By contrast, we show that such conservativity results fail for knowledge and multi-agent KT, though they hold with respect to a smaller class of agent-nonrepeating formulas.
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The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free... more
The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone's representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone's representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomor-phisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.
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We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints... more
We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev's logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces) [2]. In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic.
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In his classic monograph, Social Choice and Individual Values, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s... more
In his classic monograph, Social Choice and Individual Values, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s Impossibility Theorem has shown the importance for social choice theory of reasoning about coalitions of voters with different grades of decisiveness. The goal of this paper is a fine-grained analysis of reasoning about decisive coalitions, formalizing how the concept of a decisive coalition gives rise to a social choice theoretic language and logic all of its own. We show that given Arrow’s axioms of the Independence of Irrelevant Alternatives and Universal Domain, rationality postulates for social preference correspond to strong axioms about decisive coalitions. We demonstrate this correspondence with results of a kind familiar in economics—representation theorems—as well as results of a kind coming from mathematical logic—completeness theorems. We present a complete logic for reasoning about decisive coalitions, along with formal proofs of Arrow’s and Wilson’s theorems. In addition, we prove the correctness of an algorithm for calculating, given any social rationality postulate of a certain form in the language of binary preference, the corresponding axiom in the language of decisive coalitions. These results suggest for social choice theory new perspectives and tools from logic.
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Brouwer's views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent... more
Brouwer's views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the semantic hierarchy.
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In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same... more
In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same role as the compound modality ∀pA. Unlike the propositional quantifier by itself, the global quantificational modality can be straightforwardly interpreted in any Boolean Algebra Expansion (BAE). We present a logic GQM for this language and prove that it is complete with respect to the intended algebraic semantics. This logic enables a conceptual shift, as what have traditionally been called different "modal logics" now become [∀p]-universal theories over the base logic GQM: instead of defining a new logic with an axiom schema such as □ϕ → □□ϕ, one reasons in GQM about what follows from the globally quantified formula [∀p](□p → □□p).
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In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same... more
In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same role as the compound modality ∀pA. Unlike the propositional quantifier by itself, the global quantificational modality can be straightforwardly interpreted in any Boolean Algebra Expansion (BAE). We present a logic GQM for this language and prove that it is complete with respect to the intended algebraic semantics. This logic enables a conceptual shift, as what have traditionally been called different "modal logics" now become [∀p]-universal theories over the base logic GQM: instead of defining a new logic with an axiom schema such as □ϕ → □□ϕ, one reasons in GQM about what follows from the globally quantified formula [∀p](□p → □□p).
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The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the... more
The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation ≥ on events that does not preclude a probabilistic interpretation, in the sense that ≥ has extensions that are probabilistically representable, we characterize the extension ≥+ of ≥ that is exactly the intersection of all probabilistically representable extensions of ≥. This extension ≥+ gives us all the additional comparisons that we are entitled to infer from ≥, based on the assumption that there is some probability measure of which ≥ gives us partial qualitative information. We pay special attention to the problem of extending an order on states to an order on events. In addition to the probabilistic interpretation, this problem has a more general interpretation involving measurement of any additive quantity: e.g., given comparisons between the weights of individual objects, what comparisons between the weights of groups of objects can we infer?
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Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between... more
Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional via a full and faithful computable translation from their logic to the modal logic K45.
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While much of semantic theorizing is based on intuitions about logical phenomena associated with linguistic constructions—phenomena such as consistency and entailment—it is rare to see axiomatic treatments of linguistic fragments. Given a... more
While much of semantic theorizing is based on intuitions about logical phenomena associated with linguistic constructions—phenomena such as consistency and entailment—it is rare to see axiomatic treatments of linguistic fragments. Given a fragment interpreted in some class of formally specified models, it is often possible to ask for a characterization of the reasoning patterns validated by the class of models. Axiomatizations provide such a characterization, often in a perspicuous and efficient manner. In this paper, we highlight some of the benefits of providing axiomatizations for the purpose of semantic theorizing. We illustrate some of these benefits using three examples from the study of modality.
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In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have... more
In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. This result raises the question: for which normal modal logics L can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for L?
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Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about... more
Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about relative likelihood, with statements of the form φ ≥ ψ expressing that φ is at least as likely as ψ, a standard qualitative approach using preordered preferential structures yields a dramatically different logical system than a quantitative approach using probability measures. In fact, the standard preferential approach validates principles of reasoning that are incorrect from a probabilistic point of view. However, in this paper we show that a natural modification of the preferential approach yields exactly the same logical system as a probabilistic approach—not using single probability measures, but rather sets of probability measures. Thus, the same preferential structures used in the study of non-monotonic logics and belief revision may be used in the study of comparative probabilistic reasoning based on imprecise probabilities.
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In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem [1979], " Syntactic aspects of modal incompleteness theorems, " and a longstanding open question: whether every normal... more
In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem [1979], " Syntactic aspects of modal incompleteness theorems, " and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB [Japaridze, 1988, Boolos, 1993]. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the famed Blok Dichotomy [Blok, 1978] to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness.
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This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humberstone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic... more
This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humberstone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. The analogues of classical Kripke frames, i.e., full world frames, are full possibility frames, in which propositional variables may be interpreted as any regular open sets.
We develop the beginnings of duality theory, definability/correspondence theory, and completeness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] established a duality between the category of Kripke frames with p-morphisms and the category of complete (C), atomic (A), and completely additive (V) BAOs with complete BAO-homomorphisms (these categories being dually equivalent), we establish a duality between the category of full possibility frames with possibility morphisms and the category of CV-BAOs, i.e., allowing non-atomic BAOs, with complete BAO-homomorphisms (the latter category being dually equivalent to a reflective subcategory of the former). This parallels the connection between forcing posets and Boolean-valued models in set theory, but now with accessibility relations and modal operators involved. Generalizing further, we introduce a class of principal possibility frames that capture the generality of V-BAOs. If we do not require a full or principal frame, then every BAO has an equivalent possibility frame, whose possibilities are proper filters in the BAO. With this filter representation, which does not require the ultrafilter axiom, we are lead to a notion of filter-descriptive possibility frames. Whereas Goldblatt [1974] showed that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of descriptive world frames with p-morphisms, we show that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of filter-descriptive possibility frames with p-morphisms. Applying our duality theory to definability theory, we prove analogues for possibility semantics of theorems of Goldblatt [1974] and Goldblatt and Thomason [1975] characterizing modally definable classes of frames. In addition, we discuss analogues for possibility semantics of first-order correspondence results in the style of Lemmon and Scott [1977], Sahlqvist [1975], and van Benthem [1976a]. Finally, applying our duality theory to completeness theory, we show that there are continuum many normal modal logics that can be characterized by full possibility frames but not by Kripke frames, that all Sahlqvist logics can be characterized by full possibility frames that contain no worlds, and that all normal modal logics can be characterized by filter-descriptive possibility frames.
We develop the beginnings of duality theory, definability/correspondence theory, and completeness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] established a duality between the category of Kripke frames with p-morphisms and the category of complete (C), atomic (A), and completely additive (V) BAOs with complete BAO-homomorphisms (these categories being dually equivalent), we establish a duality between the category of full possibility frames with possibility morphisms and the category of CV-BAOs, i.e., allowing non-atomic BAOs, with complete BAO-homomorphisms (the latter category being dually equivalent to a reflective subcategory of the former). This parallels the connection between forcing posets and Boolean-valued models in set theory, but now with accessibility relations and modal operators involved. Generalizing further, we introduce a class of principal possibility frames that capture the generality of V-BAOs. If we do not require a full or principal frame, then every BAO has an equivalent possibility frame, whose possibilities are proper filters in the BAO. With this filter representation, which does not require the ultrafilter axiom, we are lead to a notion of filter-descriptive possibility frames. Whereas Goldblatt [1974] showed that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of descriptive world frames with p-morphisms, we show that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of filter-descriptive possibility frames with p-morphisms. Applying our duality theory to definability theory, we prove analogues for possibility semantics of theorems of Goldblatt [1974] and Goldblatt and Thomason [1975] characterizing modally definable classes of frames. In addition, we discuss analogues for possibility semantics of first-order correspondence results in the style of Lemmon and Scott [1977], Sahlqvist [1975], and van Benthem [1976a]. Finally, applying our duality theory to completeness theory, we show that there are continuum many normal modal logics that can be characterized by full possibility frames but not by Kripke frames, that all Sahlqvist logics can be characterized by full possibility frames that contain no worlds, and that all normal modal logics can be characterized by filter-descriptive possibility frames.
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Research Interests:
It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can... more
It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of "paths" to a poset, forming what we call a Dragalin frame. This allowed Dragalin to introduce a semantics for intuitionistic logic that generalizes Beth and Kripke semantics. He proved that every spatial locale (locale of open sets of a topological space) can be realized as fixpoints of the nucleus generated by a Dragalin frame. In this paper, we strengthen Dragalin's result and prove that every locale—not only spatial locales—can be realized as fixpoints of the nucleus generated by a Dragalin frame. In fact, we prove the stronger result that for every nucleus on the upsets of a poset, there is a Dragalin frame based on that poset that generates the given nucleus. We then compare Dragalin's approach to generating nuclei with the relational approach of Fairtlough and Mendler [11], based on what we call FM-frames. Surprisingly, every Dragalin frame can be turned into an equivalent FM-frame, albeit on a different poset. Thus, every locale can be realized as fixpoints of the nucleus generated by an FM-frame. Finally, we consider the relational approach of Goldblatt [13] and characterize the locales that can be realized using Goldblatt frames.
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In this paper we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modeling for classical propositional and modal logics [Humberstone, 1981, Holliday, 2015]. In... more
In this paper we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modeling for classical propositional and modal logics [Humberstone, 1981, Holliday, 2015]. In particular, we define a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and we show how to modulate this analysis by varying across logics and model classes that have independent topological motivations. This relates the two realms under comparison both semantically and syntactically at the level of derivations. Moreover, our analysis clarifies the interplay between the complexity of translations and axiomatizations of the corresponding logics: adding axioms to the target bimodal logic simplifies translations, or vice versa, complex translations can simplify frame conditions. We also investigate a transfer of first-order correspondence theory between possibility semantics and its bimodal counterpart. Finally, we discuss the conceptual trade-off between giving translations and giving new semantics for logical systems, and we identify a number of further research directions to which our analysis gives rise.
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Epistemic logic in the tradition of Hintikka provides, as one of its many applications, a toolkit for the precise analysis of certain epistemological problems. In recent years, dynamic epistemic logic has expanded this toolkit. Dynamic... more
Epistemic logic in the tradition of Hintikka provides, as one of its many applications, a toolkit for the precise analysis of certain epistemological problems. In recent years, dynamic epistemic logic has expanded this toolkit. Dynamic epistemic logic has been used in analyses of well-known epistemic "paradoxes", such as the Paradox of the Surprise Examination and Fitch's Paradox of Knowability, and related epistemic phenomena, such as what Hintikka called the "anti-performatory effect" of Moorean announcements. In this paper, we explore a variation on basic dynamic epistemic logic—what we call sequential epistemic logic—and argue that it allows more faithful and fine-grained analyses of those epistemological topics.
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In this talk at BLAST 2018 (https://www.cs.du.edu/~wesfussn/blast.html), I discuss my joint work with Nick Bezhanishvili in our paper "Choice-free Stone duality" (Journal of Symbolic Logic, 2020).
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In this talk at Topology, Algebra, and Categories in Logic (TACL) 2019 (https://math.unice.fr/tacl/2019/invited-speakers.html), I survey a recent research program of investigating “possibility semantics”, a generalization of possible... more
In this talk at Topology, Algebra, and Categories in Logic (TACL) 2019 (https://math.unice.fr/tacl/2019/invited-speakers.html), I survey a recent research program of investigating “possibility semantics”, a generalization of possible world semantics, for modal, superintuitionistic, and inquisitive logics.
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Poster for the paper at http://dx.doi.org/10.4204/EPTCS.297.17. Paper abstract: Much of the theoretical work on strategic voting makes strong assumptions about what voters know about the voting situation. A strategizing voter is typically... more
Poster for the paper at http://dx.doi.org/10.4204/EPTCS.297.17. Paper abstract: Much of the theoretical work on strategic voting makes strong assumptions about what voters know about the voting situation. A strategizing voter is typically assumed to know how other voters will vote and to know the rules of the voting method. A growing body of literature explores strategic voting when there is uncertainty about how others will vote. In this paper, we study strategic voting when there is uncertainty about the voting method. We introduce three notions of manipulability for a set of voting methods: sure, safe, and expected manipulability. With the help of a computer program, we identify voting scenarios in which uncertainty about the voting method may reduce or even eliminate a voter's incentive to misrepresent her preferences. Thus, it may be in the interest of an election designer who wishes to reduce strategic voting to leave voters uncertain about which of several reasonable voting methods will be used to determine the winners of an election.