 These truth or correctness definitions don’t fall out of a definition of satisfaction. In fact they go by a much less direct route, which Tarski describes as a ‘purely accidental’ possibility that relies on the ‘specific peculiarities’ of the particular object language. It may be helpful to give a few more of the technical details than Tarski does, in a more familiar notation than Tarski’s, in order to show what is involved. Tarski refers his readers to a paper of Thoralf Skolem in 1919 for the technicalities.