There are several recent approaches to this problem, the most developed of which are the structural theory of alternatives (Katzir, 2007, Fox & Katzir, 2011, Trinh & Haida 2016) and the Bayesian, Rational Speech Act (RSA) approach (see Frank & Goodman, 2016; Bergen et al. 2016). Our main theoretical focus will be on the predictions of these two approaches, though we will also consider recent proposals such as Buccola et al. (2018). These approaches have in common an idea that alternatives are selected on the basis of structural complexity or cost. For example, (3) is structurally more complex than (2), which is equally complex to (1). Both structural and RSA approaches use the differences in complexity/cost of potential alternatives in computing what alternatives are used. As we discuss in detail in Breheny, Klinedinst, Romoli & Sudo 2017, however, a theory based on complexity alone, while it can take care of the simple case above, does not handle more complex cases of the symmetry problem. A simple illustration of the problem involves (4) Not all of the homework is difficult. This implies that some of the homework is difficult, presumably because (5) None of the homework is difficult is the alternative. However, neither approach can rule out (1) as an alternative to (4) on the basis of complexity alone. A solution offered by RSA (one which is implementable also within a structural account) would be to select alternatives also on the basis of a measure of ‘informativity’. However, in Breheny et al. (2017) we also showed that an approach based on complexity and informativity has problems with other cases (see for instance (8), which we discuss below). While we do believe that structural complexity/cost, is a factor relevant for determining alternatives, we think that other factors are also involved in breaking symmetry between alternatives. Recent work recognises this point, and suggests that discourse saliency and frequency might also have roles to play (Katzir 2007, Fox & Katzir 2011, Katzir 2014, Swanson 2010, 2018, Trinh & Haida 2016; see also Russell 2012). However, it is theoretically left unclear to what extent these factors are relevant and whether and how they interact with each other. This is due in part to the fact that many of the discussed data points rest on subtle judgements and that, while SI in general has been investigated extensively from an experimental perspective, data points and predictions more specifically relevant for theories of alternatives have been paid much less attention.