A random permutation $Π_n$ of $\{1,\dots,n\}$ follows the $\DeclareMathOperator{\Mallows}{Mallows}\Mallows(n,q)$ distribution with parameter $q>0$ if $\mathbb{P} ( Π_n = π)$ is proportional to $\DeclareMathOperator{\inv}{inv} q^{\inv(π)}$ for all $π$. Here $\DeclareMathOperator{\inv}{inv} \inv(π) := |\{ i<j : π(i)> π(j) \}|$ denotes the number of inversions of $π$. We consider properties of permutations that can be expressed by the sentences of two different logical languages. Namely, the theory of one bijection ($\mathsf{TOOB}$), which describes permutations via a single binary relation, and the theory of two orders ($\mathsf{TOTO}$), where we describe permutations by two total orders. We say that the convergence law holds with respect to one of these languages if, for every sentence $φ$ in the language, the probability $\mathbb{P} (Π_n\text{ satisfies } φ)$ converges to a limit as $n\to\infty$. If moreover that limit is in the set $\{0,1\}$ for all sentences, then the zero-one law holds. We will show that with respect to $\mathsf{TOOB}$ the $\Mallows(n,q)$ distribution satisfies the zero-one law when $0<q<1$ is fixed, and for fixed $q>1$ the convergence law fails. (In the case when $q=1$ Compton has shown the convergence law holds but not the zero-one law.) We will prove that with respect to $\mathsf{TOTO}$ the $\Mallows(n,q)$ distribution satisfies the convergence law but not the zero-one law for any fixed $q\neq 1$, and that if $q=q(n)$ satisfies $1 - 1/\log^*n < q < 1 + 1/\log^*n$ then $\Mallows(n,q)$ fails the convergence law. Here $\log^*$ denotes the discrete inverse of the tower function.