Contemporary semantics has uncovered a sophisticated typology of linguistic inferences, characterized by their conversational status and their behavior in complex sentences. This typology is usually thought to be specific to language and in part lexically encoded in the meanings of words. We argue that it is neither. Using a method involving "composite" utterances that include normal words alongside novel nonlinguistic iconic representations (gestures and animations), we observe successful "one-shot learning" of linguistic meanings, with four of the main inference types (implicatures, presuppositions, supplements, homogeneity) replicated with gestures and animations. The results suggest a deeper cognitive source for the inferential typology than usually thought: Domain-general cognitive algorithms productively divide both linguistic and nonlinguistic information along familiar parts of the linguistic typology.