We present a probabilistic model of the influence of language on the behavior of the U.S.~Supreme Court, specifically influence of amicus briefs on Court decisions and opinions. The approach assumes that amici are rational, utility-maximizing agents who try to win votes or affect the language of court opinions. Our model leads to improved predictions of justices' votes and perplexity of opinion language. It is amenable to inspection, allowing us to explore inferences about the persuasiveness of different amici and influenceability of different justices; these are consistent with earlier findings.