The paper further explores the role of different types of alternatives in pragmatic reasoning. Using experimental evidence from a novel priming paradigm, we look at the question whether the exclusion of contextual (focus) alternatives can prime different readings associated with simple and complex disjunction, as well as the converse. We find that while training people with exclusive readings of simple disjunction primes them to derive strong readings of other simple and complex disjunctions, it does not prime them to derive quantity-based implicatures with focus. However, the converse holds: priming them for strong readings of focus increases the rate of strengthened meanings for simple and complex disjunctions, as well as focused sentences. The main goal of the study is to inform models and theories of how pragmatic strengthening proceeds. Recently, Rees & Bott (2018) argued for a one-step model, where activation of the alternative is sufficient to activate the mechanism of strengthening. We suggest a revision of the model distinguishing between different types of alternatives based on our data. Rather than arguing for one specific theory, we will identify essential properties of a theory that is in line with this model.