Magri (2009a,b) proposed a generalization according to which a sentence is infelicitous whenever exhaustification over the full set of formal alternatives of the sentence leads to contextual contradiction. While Magri proposes an account of obligatory implicatures which explains some cases where this generalization expects infelicity, he does not provide a general account of
this generalization. In this paper I argue for a perspective on the ‘pruning’ of alternatives which predicts this generalization, building on the counter-intuitive idea that contradictions are relevant in every context (Lewis 1988). I further argue, using disjunction in the scope of a universal quantifier as a test case, that an extension of this view to obligatory ignorance inferences provides a new perspective on the Logical Integrity Generalization put forward by Anvari (2018b), while avoiding some empirical problems for this generalization.