 That the main concepts of deductive logic (“e.g., provability, derivability from given premises, logical independence, etc”.) are purely syntactical was argued primarily by showing how they could be defined in two exemplary languages without reference to the meanings of any terms. Language I is a form of primitive recursive arithmetic, and was intended to exemplify a constructivist kind of language, while Language II contains classical mathematics. (In modern terminology, both of these “languages” are really theories.) Of the book’s five parts, the first three are taken up with the development of these two languages. In part IV, Carnap goes beyond the treatment of specific languages to sketch a template for “general syntax”; he tries to give a framework for the syntactical development or description of any language whatever. He abstracts from the syntactical categories of the particular language under consideration, and does not even assume he is given its set of variables, the distinction between its logical and descriptive vocabulary, its negation symbol, or any other syntactic property. Instead, all these sets and concepts are to be defined solely from the transformation (inference) rules of the language (see §46–47); Carnap is developing here what we would now call a version of “proof-theoretic semantics” or “logical inferentialism” (see the supplement  Logical Syntax of Language  for further details). Throughout all parts of the Logical Syntax, Carnap distinguishes between the “d-terms” specifiable this way (derivable, proof, demonstrable, refutable, decidable) from the “c-terms” (consequence, valid, contravalid, determinate, incompatible, content, synonymous). Though both are considered to be “syntactic” terms by Carnap, the c-terms seem closer to what we would now call “semantic”. This raises the question, for the modern reader, what Carnap actually means by “syntactic” (or “formal”) here; and it turns out (as Carnap himself later recognized) that “syntax” includes much of what he would later call “semantics”. Indeed we find (Wagner 2009: 22) that languages I and II are not actually “formal” in our sense at all; they are interpreted languages. While the syntactic method, as Carnap here conceives of it, requires that the interpretations be disregarded, a particular fixed interpretation implicitly remains in place. Carnap often refers to the import of some syntactic result or attribute “in material interpretation [bei inhaltlicher Deutung]” (e.g., LSS: 30), and the material interpretation sometimes slips in to play a role in the argument. Accordingly, Tarski (1936 [2002]: §2) characterizes Carnap’s definition of consequence for Language II in the Logical Syntax as semantic, acknowledges that “The first attempt at the formulation of a precise definition for the proper concept of following comes from R. Carnap” (in the Logical Syntax), and regards his own model-theoretic definition of logical consequence as essentially equivalent to Carnap’s (though applicable to a broader range of formalized languages than just to Language-II-like languages; see the supplement  Logical Syntax of Language).  This character of the Logical Syntax as a transitional work between syntax and semantics raises many subtle interpretative and historical issues that require distillation of the actual argument from the (often newly-invented) terminology in which it is embedded. Fortunately, Pierre Wagner’s (2009) excellent handbook on the Logical Syntax provides the reader with a guide. 