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 FO(#), first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns … Show more
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 $ \varphi $. In this paper, we investigate the axiomatization of ML(#) with respect to different frame classes, such as image-finite frames and arbitrary frames. Drawing inspiration from existing works, we employ a similar proof strategy that uses the characterization of binary relations on finite Boolean algebras capable of representing generalized probability measures or finite (respectively arbitrary) cardinality measures. Our main result shows that any formula not provable in the Hilbert system can be refuted within a finite (respectively arbitrary) cardinality measure Kripke frame with a finite domain. We then transform this finite (respectively arbitrary) cardinality measure Kripke frame into a Kripke frame in the corresponding class, refuting the unprovable formula.
In this paper, we investigate the numerical expressive power of various logical languages, encompassing fragments of Presburger Arithmetic (PbA), monadic second-order logic with counting with respect to finite domains (MSO$^{\phi }(\#)$) and shallow second-order graded modal logic with counting with respect to image-finite frames (SOGML$^{\textsf{s},\phi }$(#)). We show that in their respective existential fragments, the $1$-free fragment of PbA, the =-free fragment of MSO$^{\phi }(\#)$ and the graded modality-free fragment of SOGML$^{\textsf{s},\phi }$(#) possess equivalent numerical expressive power, specifically defining strongly semilinear sets. When adding universal quantifiers or adding $1$, = and graded modality to these three languages, the resulting definable sets become semilinear sets.
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