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Additional resources for Carnap's Logical Syntax of Language
30). The informal proof that follows, conducted in the metalanguage, is no application of the syntactical method. It is only an informal argument showing the usefulness of Language I as well as Carnap’s motivation for choosing this language. 3 Logical symbols, logical rules, and mathematics A fundamental distinction is made in LSL between logical symbols and nonlogical or descriptive ones. We encounter here another difference between Carnap’s logic and ours. First, in Language I and Language II, there is nothing like what we usually call ‘individual constants’, which serve the purpose of naming individuals in a speciﬁc universe of discourse.
First, in Language I and Language II, there is nothing like what we usually call ‘individual constants’, which serve the purpose of naming individuals in a speciﬁc universe of discourse. When materially interpreted, the expressions ‘0’, ‘0 l ’, ‘0ll ’, etc. are not names for objects but for positions: if ‘0’ designates the ﬁrst position, ‘0 l ’, ‘0ll ’, etc. respectively designate the second position, the third position . . (LSL, p. 13). Second, both ‘0’ and ‘l ’ are listed among the logical symbols and ‘l ’ is not a function-symbol, although it has properties of a successor function in view of its material interpretation.
The fact that the intuitionistic concept of the continuum cannot be formalized in Language I suggests that Brouwer himself would not have found Language I very appealing. More generally, his philosophy of mathematics was clearly Introduction 29 incompatible with Carnap’s idea of the logic of science, so that he would not have accepted this idea in the ﬁrst place. Brouwer insisted on the open character of mathematics as a mental activity and unlike Carnap, he never held that ‘the problems dealt with by Intuitionism can be exactly formulated only by the means of the construction of a calculus’ (LSL, p.