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Extra info for Carnap's Logical Syntax of Language
The programme was of an epistemological nature: it aimed at proving properties about more and more powerful formalized mathematical theories (arithmetic, analysis, set theory . ) using metamathematical means that were so elementary that they did not themselves require any further proof of non-contradiction. Because Hilbert’s method consisted in metamathematical reasoning on formal expressions, it is sometimes called ‘syntactical’. In this context, the word ‘syntactical’ is associated with the use of a weak metamathematical language.
64 Many commentators deemed that Carnap was thus giving too much extension to the principle of tolerance and strongly objected to P-rules. 66 Formally, the concept of P-rules results from the syntactical deﬁnitions of ‘L-consequence’ and ‘P-consequence’ given in §51 (LSL, p. 181), which are the basis for further syntactical deﬁnitions (L-language, P-language, L-content, P-content . ), the most important of which is probably the deﬁnition of ‘analytic’. An analytic (or L-valid) sentence is deﬁned as an L-consequence of the null class (whereas valid sentences are consequences 36 Carnap’s Logical Syntax of Language of the null class, and P-valid sentences are valid sentences which are not L-valid).
91). Clearly, when Carnap wrote this, he neither considered zero and the successor function as primitive logical concepts nor any sentence of arithmetic as a logical axiom. In 1930, he still followed Frege and Russell in taking the concept of logical symbol in a narrow sense and he took the trouble to explain his reasons for sticking to this philosophical position (Carnap 1930b). In 1934, when the principle of tolerance was applied, the decision to include mathematical symbols among the primitive symbols (as Hilbert had done) became a question of purely technical expedience.
Carnap's Logical Syntax of Language by Pierre Wagner