  In a logic seminar he gave at the University of Warsaw between 1927 and 1929, Tarski proved several results that made reference to notions that later would be called “semantical”, in particular results about the notions of definability and of truth in a structure (see Vaught 1974 and 1986). Vaught informs us that Tarski found certain difficulties when trying to give a mathematically satisfactory form to the results presented in the seminar, and that this led him to look for a precise theory of the semantical notions (cf. Vaught 1974, pp. 160ff.  and 1986, pp. 870ff.). Such a theory did not exist at the time. In particular, there were no definitions of these notions in terms of concepts accepted in the foundational systems designed for the reconstruction of classical mathematics (for example, Russell and Whitehead’s theory of types or Zermelo’s set theory). Because of this, the existing results in which these notions appeared could not be reconstructed in the accepted foundational systems. And there was no rigorous axiomatic theory of the semantical notions in which these were taken as primitive, either.