•  8
    Moorean Phenomena in Epistemic Logic
    In Lev Beklemishev, Valentin Goranko & Valentin Shehtman (eds.), Advances in Modal Logic, Volume 8, Csli Publications. pp. 178-199. 2010.
    A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬ BOX p, read as “p is true but you do not know p.” Not only is the Moore sentence unsuccessful, it is self-refuting, for it …Read more
  •  1
    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 Moore…Read more
  •  1
    A Uniform Logic of Information Dynamics
    In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Csli Publications. pp. 348-367. 2012.
    Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitution core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class of all relational mod…Read more
  •  1
    Partiality and Adjointness in Modal Logic
    In Rajeev Goré, Barteld Kooi & Agi Kurucz (eds.), Advances in Modal Logic, Volume 10, Csli Publications. pp. 313-332. 2014.
    Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula…Read more
  •  12
    The Logic of Comparative Cardinality
    with Yifeng Ding and Matthew Harrison-Trainor
    Journal of Symbolic Logic. forthcoming.
    This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and omplementation 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.
  •  3
    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 genera…Read more
  •  8
    Arrow's Decisive Coalitions
    Social Choice and Welfare. forthcoming.
    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 …Read more
  •  11
    Strategic Voting Under Uncertainty About the Voting Method
    Electronic Proceedings in Theoretical Computer Science 297. 2019.
    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 se…Read more
  •  17
    When Do Introspection Axioms Matter for Multi-Agent Epistemic Reasoning?
    with Yifeng Ding and Cedegao Zhang
    Electronic Proceedings in Theoretical Computer Science 297. 2019.
    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 …Read more
  •  16
    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 tha…Read more
  •  25
    Choice-Free Stone Duality
    Journal of Symbolic Logic 1-48. forthcoming.
    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 o…Read more
  •  21
    A Note on Algebraic Semantics for S5 with Propositional Quantifiers
    Notre Dame Journal of Formal Logic 60 (2): 311-332. 2019.
    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. T…Read more
  •  20
    Complete additivity and modal incompleteness
    Review of Symbolic Logic 12 (3): 487-535. 2019.
    In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem, “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, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolv…Read more
  •  23
    A Semantic Hierarchy for Intuitionistic Logic
    with Guram Bezhanishvili
    Indagationes Mathematicae 30 (3): 403-469. 2019.
    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 incre…Read more
  •  27
    Axiomatization in the meaning sciences
    In Derek Ball & Brian Rabern (eds.), The Science of Meaning: Essays on the Metatheory of Natural Language Semantics, Oxford University Press. pp. 73-97. 2018.
    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…Read more
  •  7
    A Uniform Logic of Information Dynamics
    with Tomohiro Hoshi and Thomas F. Icard Iii
    In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence Moss (eds.), Advances in Modal Logic, Volume 9, Csli Publications. pp. 348-367. 2012.
    Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitu- tion core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class of all relational m…Read more
  • Logic, Rationality, and Interaction (edited book)
    with Wiebe van der Hoek and Wen-Fang Wang
    Springer-Verlag. 2016.
  •  26
    Indicative Conditionals and Dynamic Epistemic Logic
    with Thomas Icard
    Proceedings of the Sixteenth Conference on Theoretical Aspects of Rationality and Knowledge (TARK 2017), Liverpool, UK, 24-26 July 2017. 2017.
    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 consequen…Read more
  •  43
    Inferring Probability Comparisons
    with Matthew Harrison-Trainor and Thomas Icard
    Mathematical Social Sciences. forthcoming.
    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 …Read more
  •  84
    Moorean Phenomena in Epistemic Logic
    with Thomas F. Icard
    In Lev Beklemishev, Valentin Goranko & Valentin B. Shehtman (eds.), Advances in Modal Logic 8, College Publications. 2010.
    A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is the Moore sentence unsuccessful, it is self-refuting, for it ne…Read more
  •  15
    Locales, Nuclei, and Dragalin Frames
    with Guram Bezhanishvili
    In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11, College Publications. pp. 177-196. 2016.
    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 int…Read more
  •  50
    Response to Egré and Xu
    In Johan van Benthem Fenrong Liu (ed.), Logic Across the University: Foundations and Applications, College Publications. pp. 39-46. 2013.
    In this note, I respond to comments by Paul Egré and Xu Zhaoqing on my “Epistemic Closure and Epistemic Logic I: Relevant Alternatives and Subjunctivism” (Journal of Philosophical Logic).
  •  24
    A bimodal perspective on possibility semantics
    with Johan van Benthem and Nick Bezhanishvili
    Journal of Logic and Computation 27 (5). 2017.
    In this article, we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modelling for classical propositional and modal logics. 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 motivat…Read more
  •  135
    Fallibilism and Multiple Paths to Knowledge
    Oxford Studies in Epistemology 5 97-144. 2015.
    This chapter argues that epistemologists should replace a “standard alternatives” picture of knowledge, assumed by many fallibilist theories of knowledge, with a new “multipath” picture of knowledge. The chapter first identifies a problem for the standard picture: fallibilists working with this picture cannot maintain even the most uncontroversial epistemic closure principles without making extreme assumptions about the ability of humans to know empirical truths without empirical investigation. …Read more
  •  131
    Information dynamics and uniform substitution
    with Tomohiro Hoshi and Thomas F. Icard Iii
    Synthese 190 (1): 31-55. 2013.
    The picture of information acquisition as the elimination of possibilities has proven fruitful in many domains, serving as a foundation for formal models in philosophy, linguistics, computer science, and economics. While the picture appears simple, its formalization in dynamic epistemic logic reveals subtleties: given a valid principle of information dynamics in the language of dynamic epistemic logic, substituting complex epistemic sentences for its atomic sentences may result in an invalid pri…Read more
  •  159
    Measure Semantics and Qualitative Semantics for Epistemic Modals
    with Thomas F. Icard
    Proceedings of SALT 23 514-534. 2013.
    In this paper, we explore semantics for comparative epistemic modals that avoid the entailment problems shown to result from Kratzer’s (1991) semantics by Yalcin (2006, 2009, 2010). In contrast to the alternative semantics presented by Yalcin and Lassiter (2010, 2011), based on finitely additive probability measures, we introduce semantics based on qualitatively additive measures, as well as semantics based on purely qualitative orderings, including orderings on propositions derived from orderin…Read more