Quantity implicatures

Scalar implicatures with Horn setsReplace Horn scales with Horn sets, like {all, some, most,many}, {brilliant, clever}, etc.Let φ(α) be a sentence, where α is an expression in φ.φ(β) is an alternative to φ(α) iffα, β ∈ H, where H is a Horn set, andφ(β) is stronger than φ(α) (i.e. φ(β) entails φ(α) but notvice versa).Examples:Sentence: Betty ate many of the tartsAlternatives: Betty ate most of the tartsBetty ate all of the tartsSentence: Betty didn’t eat many of the tartsAlternatives: Betty didn’t eat some of the tartsQuantity implicatures