We consider a sequence \(\mathcal {L}_n\) , \(n = 1, 2, 3, \ldots \) , of linear preorders, a finite relational signature \(\sigma \) including the signature \(\{\preceq \}\) of the linear preorders \(\mathcal {L}_n\) , and the set \(\textbf{W}_n\) of all expansions of \(\mathcal {L}_n\) to \(\sigma \) . A probability distribution \(\mathbb {P}_n\) is defined on each \(\textbf{W}_n\) (it can be e.g. the uniform distribution, but many other distributions are possible). We prove that if all equivalence classes of the preorder \(\mathcal {L}_n\) grow (as \(n\rightarrow \infty \) ) faster than every logarithm but slower than some polynomial, then every first-order formula is almost surely equivalent to a (in general) simpler formula that only expresses distances between variables and which atomic formulas they satisfy. As a corollary we get a convergence law for formulas, and zero-one law for sentences, of first-order logic. If we also assume that for some positive \(\lambda \in \mathbb {N}\) the number of equivalence classes of every \(\mathcal {L}_n\) is exactly \(\lambda \) and that all equivalence classes have roughly the same size, then we can also almost surely eliminate “proportion quantifiers” that express, for some \(0 < r < 1\) , that the proportion of elements satisfying a formula is larger than r; and we get a convergence law for first-order logic extended by such quantifiers.