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Additional info for Carnap's Logical Syntax of Language
In §39, Carnap consider the special case of the real numbers: The usual axioms of the arithmetic of real numbers need not be set up here in the form of new primitive sentences. These axioms – and hence the theorems derivable from them – are demonstrable in Language II. (LSL, p. 148) The inclusion of ‘0’ and ‘l ’ among the logical symbols, and of axioms of arithmetic among the logical rules of Language I and Language II, should also be contrasted with Carnap’s earlier view on logic and mathematics.
Sm in Language I’ is deﬁned on the basis of primitive sentences (Grundsätze) and of rules of inference (Schlußregeln) (LSL, p. 29). Primitive sentences are not given directly but through schemata, because no sentential variables are available in Language I. Schemata for the sentential calculus, the bounded quantiﬁers, identity, arithmetic, and the K-operator are laid down in §11 (LSL, p. 30) in such a way that mathematical primitive sentences are not distinguished from the ones we would call ‘logical’.
Other numerals (zz), predicates (pr), or functors (fu) can be introduced through deﬁnitions. 55 ‘Derivation [Ableitung] of a sentence Sn with premisses S1 . . Sm in Language I’ is deﬁned on the basis of primitive sentences (Grundsätze) and of rules of inference (Schlußregeln) (LSL, p. 29). Primitive sentences are not given directly but through schemata, because no sentential variables are available in Language I. Schemata for the sentential calculus, the bounded quantiﬁers, identity, arithmetic, and the K-operator are laid down in §11 (LSL, p.
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