Logic

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New submissions for Tue, 14 May 24

[1]  arXiv:2405.06852 [pdf, ps, other]

Title: Possibility Semantics

Authors: Wesley H. Holliday

Comments: Previous version in Selected Topics from Contemporary Logics, ed. Melvin Fitting, Volume 2 of Landscapes in Logic, College Publications, London, 2021, ISBN 97-1-84890-350-0, pp. 363-476. This version corrects Section 4.3

Subjects: Logic (math.LO); Logic in Computer Science (cs.LO)

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.

[2]  arXiv:2405.06961 [pdf, ps, other]

Title: Dimensionality and randomness

Authors: George Barmpalias, Xiaoyan Zhang

Subjects: Logic (math.LO); Discrete Mathematics (cs.DM); Information Theory (cs.IT)

Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and characterize this phenomenon for arrays and trees and its relationship to negligible classes.

[3]  arXiv:2405.07112 [pdf, other]

Title: Definable compactness in o-minimal structures

Authors: Pablo andújar Guerrero

Comments: This is a round-up of the topological content of arXiv:2111.03802

Subjects: Logic (math.LO); General Topology (math.GN)

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various definitions of definable compactness in the literature, including those in terms of definable curves, definable types and definable downward directed families of closed sets.

[4]  arXiv:2405.07114 [pdf, ps, other]

Title: Definable separability and second-countability in o-minimal structures

Authors: Pablo Andújar Guerrero

Subjects: Logic (math.LO); General Topology (math.GN)

We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and definable second-countability -- which make sense in a wider model-theoretic context. We prove that within o-minimality these notions have the desired properties, including their equivalence among definable metric spaces, and conjecture a definable version of Urysohn's Metrization Theorem.

Cross-lists for Tue, 14 May 24

[5]  arXiv:2405.07307 (cross-list from math.GR) [pdf, ps, other]

Title: Primitive permutation groups of finite Morley rank and affine type

Authors: Ayşe Berkman, Alexandre Borovik

Comments: 27 pages

Subjects: Group Theory (math.GR); Logic (math.LO)

We give a review of one of the lines in development of the theory of groups of finite Morley rank. These groups naturally appear in model theory as model-theoretic analogues of Galois groups, therefore their actions and their role as permutation groups is of primary interest. We restrict our story to the study of connected groups of finite Morley rank $G$ acting in a definably primitive way on a set $X$ and containing a definable abelian normal subgroup $V$ which acts on $X$ regularly -- the so-called \emph{primitive groups of affine type}. For reasons explained in the paper, this case plays a central role in the theory.

Replacements for Tue, 14 May 24

[6]  arXiv:2206.02228 (replaced) [pdf, ps, other]

Title: On automorphisms of $\mathcal P(λ)/[λ]^{<λ}$

Authors: Jakob Kellner, Anda Latif, Saharon Shelah

Subjects: Logic (math.LO)

[7]  arXiv:2305.11777 (replaced) [pdf, ps, other]

Title: State-based Modal Logics for Free Choice

Authors: Maria Aloni, Aleksi Anttila, Fan Yang

Comments: 46 pages

Subjects: Logic (math.LO); Logic in Computer Science (cs.LO)

[8]  arXiv:2308.10812 (replaced) [pdf, other]

Title: On countable isotypic structures

Authors: Pavel Gvozdevsky

Comments: 6 pages, Published in journal of Groups, Complexity, Cryptology

Subjects: Logic (math.LO)

[9]  arXiv:2310.19473 (replaced) [pdf, ps, other]

Title: Wetzel families and the continuum

Authors: Jonathan Schilhan, Thilo Weinert

Comments: 24 pages

Subjects: Logic (math.LO); Complex Variables (math.CV)

[10]  arXiv:2405.00618 (replaced) [pdf, ps, other]

Title: An axiomatisation of the temporal logic of two dimensional Minkowski spacetime

Authors: Robin Hirsch

Subjects: Logic (math.LO)

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