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The logic of comparative cardinality

Yifeng Ding, Matthew Harrison-Trainor & Wesley H. Holliday

Journal of Symbolic Logic 85 (3):972-1005 (2020)

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Probing the quantitative–qualitative divide in probabilistic reasoning.Duligur Ibeling, Thomas Icard, Krzysztof Mierzewski & Milan Mossé - 2024 - Annals of Pure and Applied Logic 175 (9):103339.details This paper explores the space of (propositional) probabilistic logical languages, ranging from a purely `qualitative' comparative language to a highly `quantitative' language involving arbitrary polynomials over probability terms. While talk of qualitative vs. quantitative may be suggestive, we identify a robust and meaningful boundary in the space by distinguishing systems that encode (at most) additive reasoning from those that encode additive and multiplicative reasoning. The latter includes not only languages with explicit multiplication but also languages expressing notions of dependence and (...) Download Export citation Bookmark 5 citations
Generalized Quantifiers Meet Modal Neighborhood Semantics.Dag Westerståhl & Johan van Benthem - 2021 - In Judit Madarász & Gergely Székely (eds.), Hajnal Andréka and István Németi on Unity of Science: From Computing to Relativity Theory Through Algebraic Logic. Springer. pp. 187-206.details In a mathematical perspective, neighborhood models for modal logic are generalized quantifiers, parametrized to points in the domain of objects/worlds. We explore this analogy further, connecting generalized quantifier theory and modal neighborhood logic. In particular, we find interesting analogies between conservativity for linguistic quantifiers and the locality of modal logic, and between the role of invariances in both fields. Moreover, we present some new completeness results for modal neighborhood logics of linguistically motivated classes of generalized quantifiers, and raise new types (...) Download Export citation Bookmark
Axiomatization of modal logic with counting.Xiaoxuan Fu & Zhiguang Zhao - forthcoming - Logic Journal of the IGPL.details Modal logic with counting is obtained from basic modal logic by adding cardinality comparison formulas of the form $ \#\varphi \succsim \#\psi $, stating that the cardinality of successors satisfying $ \varphi $ is larger than or equal to the cardinality of successors satisfying $ \psi $. It is different from graded modal logic where basic modal logic is extended with formulas of the form $ \Diamond _{k}\varphi $ stating that there are at least $ k$-many different successors satisfying $ (...) Download Export citation Bookmark 1 citation
The Logic of Cardinality Comparison Without the Axiom of Choice.Matthew Harrison-Trainor & Dhruv Kulshreshtha - forthcoming - Annals of Pure and Applied Logic.details Download Export citation Bookmark
Modal Logic with “Most”.Xiaoxuan Fu & Zhiguang Zhao - forthcoming - Studia Logica:1-41.details In this paper, we axiomatize modal logic extended with the modal operator $$M\varphi $$ saying that “there are strictly more $$\varphi $$ -successors than $$\lnot \varphi $$ -successors”, both in the class of image-finite Kripke frames and in the class of all Kripke frames. We follow the proof strategy of van der Hoek (Int J Uncertain Fuzziness Knowl Based Syst 4(1):45–60, 1996.), and prove a characterization result of finite majority structures which are capable of representing finite cardinality measures and a (...) Download Export citation Bookmark

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