  Tarski presents his theory of logical consequence in Tarski (1936a) (German version, Tarski 1936b; English translation of the Polish, Tarski 2002; English translation of the German, Tarski 1983c). This classic paper begins with some general remarks on the possibility of a precise definition of the concept of consequence. The essence of these remarks is that since the common concept is vague, it seems certainly difficult, and perhaps impossible, to reconcile all features of its use in the definition of a corresponding precise concept. Nevertheless, Tarski says, logicians had thought until recently that they had managed to define a precise concept that coincided exactly in extension with the intuitive concept of consequence. Tarski mentions the extraordinary development of mathematical logic in recent decades, which had shown “how to present mathematical disciplines in the shape of formalized deductive theories” (Tarski 1983c, p. 409). In these theories, consequences are extracted from axioms and theorems by rules of inference, “such as the rules of substitution and detachment” (Tarski 1983c, p. 410), of a purely syntactical (or “structural”, in Tarski’s word) nature. “Whenever a sentence follows from others, it can be obtained from them—so it was thought—by means of the transformations prescribed by the rules” (Tarski 1983c, p. 410). According to Tarski, this belief of the logicians was justified by “the fact that they had actually succeeded in reproducing in the shape of formalized proofs all the exact reasonings which had ever been carried out in mathematics” (Tarski 1983c, p. 410).