Finite Spectra of Syllogistic Logic with Cardinality Comparisons

We study the finite spectra of the syllogistic logic with cardinality comparisons $S^{†}(card)$. Since the language does not contain conjunction at the sentence level, we consider the spectrum of a theory $\mathrm{\Gamma }$, namely the set of positive integers $n$ for which $\mathrm{\Gamma }$ is satisfiable in an $n$-element model. We give a complete classification of these spectra: besides the empty set, every $S^{†}(card)$-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 $S^{†}(card)$ 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.