 Burgess (1982a) provided a complete axiomatic system of the Since-Until logic on the class of all linear orderings with reflexive precedence relation, which was further simplified by Xu (1988). Extensions of this axiomatization for strict linear orderings were obtained by Venema (1993) and Reynolds (1994; 1996). For details and related results see Burgess (1984); Zanardo (1991); Gabbay et al. (1994); Finger et al. (2002); and Hodkinson and Reynolds (2006). The Burgess-Xu axiomatic system and some of its extensions are presented  in the supplementary document: