David Kaplan famously formulated a logic for demonstratives (including terms like 'I', 'now', 'here' 'actually' etc.), LD, which is a version of first-order two-dimensional logic. Very roughly and very informally, LD's model theory defines two components: Firstly, an expression relation between formulas, contexts (tuples of contextual parameters such as worlds, individuals, times etc.) and propositions (sets of world-time pairs). Secondly a truth relation between indices (world-time pairs) and propositions. So, LD's model theory amounts to defining 'formula A is true with respect to context c and world w (relative to model M and assignment g)'.
Now LD also involves a one-place operator, called dthat. Syntactically, dthat takes terms to produce new terms. Semantically, dthat is a 'diagonal' operator from term intensions (functions from indices to individuals) to term intensions in that it shifts its argument intension along its 'diagonal': For s a term the denotation of dthat(s) w.r.t. c,(w, t) = the denotation of t w.r.t. c, (w', t'), where w', t' are the world and the time of c, respectively. Kaplan furthermore says that dthat(s) is directly referential: Its truth-conditional contribution amounts to the individual it (rigidly) denotes.
Kaplan thinks dthat is a quite useful operator. But I never understood its significance; can someone set me straight?