Finite Spectra of Syllogistic Logic with Cardinality Comparisons

Abstract

We study the finite spectra of the syllogistic logic with cardinality comparisons $S^\dagger(card)$. Since the language does not contain conjunction at the sentence level, we consider the spectrum of a theory $\Gamma$, namely the set of positive integers $n$ for which $\Gamma$ is satisfiable in an $n$-element model. We give a complete classification of these spectra: besides the empty set, every $S^\dagger(card)$-spectrum is either an eventual tail of the positive integers or an eventual tail of the positive even integers. We then consider the extension of $S^\dagger(card)$ by Boolean connectives at the sentence level and show that the collection of its spectra forms the topology generated by the spectra of the original language.