A modal logic framework for reasoning about comparative distances and topology

Annals of Pure and Applied Logic 161 (4):534-559 (2010)

Abstract
We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances . Due to its greater expressive power, this logic turns out to behave quite differently from both and conditional logics. We provide finite axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line is not recursively enumerable. This result is proved by an encoding of Diophantine equations
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DOI 10.1016/j.apal.2009.04.001
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References found in this work BETA

The Algebra of Topology.J. C. C. Mckinsey & Alfred Tarski - 1944 - Journal of Symbolic Logic 9 (4):96-97.
« Everywhere » and « Here ».Valentin Shehtman - 1999 - Journal of Applied Non-Classical Logics 9 (2-3):369-379.
A Decision Method for Elementary Algebra and Geometry.Alfred Tarski - 1952 - Journal of Symbolic Logic 17 (3):207-207.
First-Order Logics for Comparative Similarity.Timothy Williamson - 1988 - Notre Dame Journal of Formal Logic 29 (4):457-481.

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Citations of this work BETA

Logic of Confidence.Pavel Naumov & Jia Tao - 2015 - Synthese 192 (6):1821-1838.

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