Finite Spectra of Syllogistic Logic with Cardinality Comparisons

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 $Γ$, namely the set of positive integers $n$ for which $Γ$ is satisfiable in an $n$-element