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This paper investigates the semantics of the suffix -NE in Javanese (Austronesian; Indonesia), bringing to light new findings to bear on the composition of modal strength distinctions. In a transparent manner, this functional morpheme derives weak necessity modals from strong necessity modals, but cannot attach to possibility modals to derive weak possibility. Javanese thereby takes a different compositional route to weak necessity than most Indo-European languages, which might lexicalise modal strength distinctions or rely on counterfactual morphology for that purpose. We propose a new type of domain restriction analysis for weak necessity to capture both the defining properties of weak necessity as well as the restriction in Javanese to only necessity modals. Specifically, we propose that -NE requires quantification over a non-empty subset of a minimal witness set for the original quantification. The Javanese data thus show that weak necessity is not a unified phenomenon across languages, and our analysis contributes to a model of crosslinguistic variation concerning the relationship between gradability and modality, and the semantics of weak modal strength.