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Choice-free Stone duality

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arXiv 2021年

作者： Bezhanishvili, Nick Holliday, Wesley H. Institute for Logic Language and Computation University of Amsterdam Amsterdam1090 GE Netherlands Department of Philosophy Group In Logic and The Methodology of Science University of California Berkeley BerkeleyCA94720 United States

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 homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality

关键词： Boolean algebra

Impossibility theorems involving weakenings of expansion consistency and resoluteness in voting

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arXiv 2022年

作者： Holliday, Wesley H. Norman, Chase Pacuit, Eric Zahedian, Saam Department of Philosophy and Group in Logic and the Methodology of Science University of California Berkeley United States Department of Electrical Engineering and Computer Science University of California Berkeley United States Department of Philosophy University of Maryland United States Stanford Institute for Economic Policy Research Stanford University United States

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 *** Codes 91B12, 91B14 Copyright © 2022, The Authors. All rights reserved.

关键词： Formal verification

Why and When to Expect Gaussian Error Distributions in Epoch of Reionization 21-cm Power Spectrum Measurements

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arXiv 2022年

作者： Wilensky, Michael J. Brown, Jordan Hazelton, Bryna J. Jodrell Bank Centre for Astrophysics University of Manchester ManchesterM13 9PL United Kingdom Group in Logic and the Methodology of Science University of California Berkeley BerkeleyCA94720 United States Department of Physics University of Washington SeattleWA98195 United States eScience Institute University of Washington SeattleWA98195 United States

We explore error distributions in Epoch of Reionization 21-cm power spectrum estimators using a combination of mathematical analysis and numerical simulations. We provide closed form solutions for the error distributions of individual bins in 3d-power spectra for two estimators currently in use in the field, which we designate as "straight-square" and "cross-multiply" estimators. We then demonstrate when the corresponding spherically binned power spectra should (and should not) have Gaussian error distributions, which requires appealing to nonstandard statements of the central limit theorem. This has important implications for how upper limits are reported, as well as how cosmological inferences are performed based on power spectrum measurements. Specifically, assuming a Gaussian error distribution can over or underestimate the upper limit depending on the type of estimator, and produces overly compact likelihood functions for the power spectrum. © 2022, CC BY.

关键词： Power spectrum

Choice-free duality for orthocomplemented lattices by means of spectral spaces

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arXiv 2020年

作者： McDonald, Joseph Yamamoto, Kentarô Institute for Logic Language Computation University of Amsterdam North Holland 1098 XG Netherlands Group in Logic and Methodology of Science University of California BerkeleyCA94720-3840 United States

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander's Subbase Theorem, which asserts that a topological space X is compact if every subbasic open cover of X admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem-whose proof depends upon Zorn's Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces;choice-free in the sense that our representation avoids use of Alexander's Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call upper Vietoris orthospaces in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday's choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó's choice-dependent orthospace approach to Stone duality for orthocomplemented *** Codes 06C15, 06E15, 03E25 Copyright © 2020, The Authors. All rights reserved.

关键词： Topology

Weakly Aggregative Modal logic: Characterization and Interpolation 1

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7th International Workshop on logic, Rationality, and Interaction, LORI 2019

作者： Liu, Jixin Wang, Yanjing Ding, Yifeng Department of Philosophy Sichuan University Chengdu China Department of Philosophy Peking University Beijing China Group in Logic and the Methodology of Science UC Berkeley Berkeley United States

ISBN: (数字)9783662602928

ISBN: (纸本)9783662602911

Weakly Aggregative Modal logic ((formula presented)) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has some interesting applications on epistemic logic and logic of games, so we study some basic model theoretical aspects of WAML in this paper. Specifically, we give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation and show that each basic WAML system (formula presented) lacks Craig Interpolation. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

关键词： Interpolation

A note on the consistency operator

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arXiv 2019年

作者： Walsh, James Group in Logic and Methodology of Science University of California Berkeley United States

It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory T, the next strongest natural theory is T ` ConT. We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories. Copyright © 2019, The Authors. All rights reserved.

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Correspondence, canonicity, and model theory for monotonic modal logics

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arXiv 2019年

作者： Yamamoto, Kentarô Group in Logic and the Methodology of Science University of California Berkeley BerkeleyCA United States

We investigate the role of coalgebraic predicate logic, a logic for neighborhood frames first proposed by Chang, in the study of monotonic modal logics. We prove analogues of the Goldblatt-Thomason Theorem and Fine’s Canonicity Theorem for classes of monotonic neighborhood frames closed under elementary equivalence in coalgebraic predicate logic. The elementary equivalence here can be relativized to the classes of monotonic, quasi-filter, augmented quasi-filter, filter, or augmented filter neighborhood frames, respectively. The original, Kripke-semantic versions of the theorems follow as a special case concerning the classes of augmented filter neighborhood frames. Copyright © 2019, The Authors. All rights reserved.

关键词： Formal logic

On the logics with propositional quantifiers extending S5Π 12

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On the logics with propositional quantifiers extending S5Π

12th Conference on "Advances in Modal logic", AiML 2018

作者： Ding, Yifeng Group in Logic and the Methodology of Science University of California BerkeleyCA United States

ISBN: (纸本)9781848902558

Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π. © 2018 College Publications. All rights reserved.

关键词： Algebra

Incompleteness and jump hierarchies

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arXiv 2019年

作者： Lutz, Patrick Walsh, James Department of Mathematics University of California Berkeley United States Group in Logic and the Methodology of Science University of California Berkeley United States

This paper is an investigation of the relationship between Gödel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation {(A, B) ∊ 2 : OA ≤H B} is well-founded. We provide an alternative proof of this fact that uses Gödel's second incompleteness theorem instead of the theory of admissible ordinals. We then use this result to derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any A ∊ , if the rank of A is α, then ω1A is the (1 + α)th admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of X is ω1X Copyright © 2019, The Authors. All rights reserved.

关键词： Semantics