Split Cycle: A New Condorcet Consistent Voting Method Independent of Clones and Immune to Spoilers

We propose a Condorcet consistent voting method that we call Split Cycle. Split Cycle belongs to the small family of known voting methods that significantly narrow the choice of winners in the presence of majority cycles while also satisfying 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."Comment: 71 pages, 15 figures. Added a new explanation of Split Cycle in Section 1, updated the caption to Figure 2, the discussion in Section 3.3, and Remark 4.11, and strengthened Proposition 6.20 to Theorem 6.20 to cover single-voter resolvability in addition to asymptotic resolvability. Thanks to Nicolaus Tideman for helpful discussio

Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper 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, 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. 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 Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

A fundamental non-classical logic

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.Comment: added topological representation of bounded lattices with implications in Appendi

An impossibility theorem concerning positive involvement in voting

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

Indicative Conditionals and Dynamic Epistemic Logic

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.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Axioms for Defeat in Democratic Elections

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 voting procedure studied in our recent work: Split Cycle. In particular, Split Cycle is the most resolute voting procedure satisfying the six axioms for democratic defeat. In addition, we analyze how Split Cycle escapes Arrow's Impossibility Theorem and related impossibility results.Comment: 41 page

Impossibility theorems involving weakenings of expansion consistency and resoluteness in voting

A fundamental principle of individual rational choice is Sen's $\gamma$ 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.Comment: Forthcoming in Mathematical Analyses of Decisions, Voting, and Games, eds. M. A. Jones, D. McCune, and J. Wilson, Contemporary Mathematics, American Mathematical Society, 202

The Orthologic of Epistemic Modals

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 $p\wedge\Diamond\neg p$ ('$p$, but it might be that not $p$') appears to be a contradiction, $\Diamond\neg p$ does not entail $\neg p$, 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 predict that $p\wedge\Diamond\neg p$, a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not always allow us to replace $p\wedge\Diamond\neg p$ with a contradiction; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail 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.Comment: added Lemma 3.27, Fact 3.30, Definition 4.36, and Remark 4.3