Interleaving Logic and Counting
Johan van Benthem, Thomas Icard

TL;DR
This paper explores the integration of logical and arithmetical reasoning in natural language, analyzing formal systems that combine quantifiers and counting, and examining their implications for linguistics and cognitive science.
Contribution
It introduces a monadic first-order logic with counting, normal forms, and explores the boundaries of definability and complexity, linking formal logic with linguistic and cognitive phenomena.
Findings
Normal forms enable axiomatization of counting logic.
Certain arithmetical notions are definable in finite models.
Extensions lead to undecidability and connections with Presburger Arithmetic and Diophantine equations.
Abstract
Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus "grassroots mathematics". We begin with a brief review of first-order logic with counting operators and cardinality comparisons. This system is known to be of high complexity, and drowns out finer aspects of the combination of logic and counting. We move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting. We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which…
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