At the heart of the theory of SI is the idea of alternative expressions, or alternatives for short. That alternatives do play a role in SI computation is supported by the findings of several behavioural studies, which have focused on the differences between children and adults in their ability to compute alternatives (Guasti et al. 2005, Barner et al. 2011, Tieu, …Romoli et al. 2014, 2016, 2017, Hochstein et al. 2016); and by the empirical and predictive power of the theoretical proposals enumerated above. Yet, many of these proposals rely on assumptions about the space of alternatives relative to a word or linguistic construction. They thus depend on a general theory of how alternatives are generated and selected for the computation of SIs. However, developing such a theory has proven to be entirely non-trivial (Breheny, Klinedinst, Romoli & Sudo 2017, Romoli 2012, Fox 2007, Fox & Katzir 2011, Katzir 2007, 2014, Kroch 1972). In particular, the central unsolved issue is the so-called symmetry problem. To illustrate the issue, consider the sentence (1) Some of the homework is difficult. This has a SI that not all of the homework is difficult, and under most theories, this SI is derived by negating the alternative sentence (2) All of the homework is difficult. The problem, however, is why this is an alternative for the generation of the SI while (3) Some but not all of the homework is difficult is not. After all, (3) states explicitly the meaning of (1) enriched by its SI. It is apparently as relevant as the attested Alternative, by any standard understanding of ‘relevance’. Notice that if (3) were an alternative to (1), a SI that all of the homework is difficult would be predicted, which is not actually observed. 

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