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Additional info for Carnap’s Logical Syntax of Language
If he wants to formalize let us say arithmetic, he will probably equip language L with an individual constant to be interpreted by zero, a one-place function symbol to be interpreted by the successor function, a two-places function symbol for addition . . and he will also make clear that the intended universe of discourse is the set of natural numbers. Yet, L is a non-interpreted language and there are an inﬁnite number of L-structures which, except in special cases, are not isomorphic to the intended interpretation.
In each case, the presentation of the syntactical characteristics of the language precedes the deﬁnition of semantic concepts. Introduction 19 The vocabulary of languages for propositional logic contains i. parentheses and symbols for propositional connectives (¬: negation, ∧: conjunction, ∨: disjunction, →: conditional . ) as logical symbols and ii. propositional letters (p, q . ) as non-logical symbols. The vocabulary of languages for ﬁrst-order logic contains i. parentheses, comma, symbols for propositional connectives, individual variables (x, y .
E. e. function symbols) is restricted by the fact that only bounded operators are used in their deﬁnition, or in the deﬁnition of the expressions used in their deﬁnition. So Language I, in which all variables are numerical (z, Zahlvariablen) and all individual constants are numerals (‘0’, ‘0l ’, ‘0ll ’. . abbreviated as ‘0’, ‘1’, ‘2’ . ), has only bounded universal and existential quantiﬁers, and the only way to express unlimited universality is the use of free variables (LSL, p. 21). A ‘least number’ operator (K-operator) is introduced but also bounded: (K z1 )Z1 (S1 ) means: the smallest number z1 up to number Z1 such that sentence S1 is true, and if no such number exists, 0 (LSL, pp.