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The name relation and the logical antinomies

Published online by Cambridge University Press:  17 August 2023

K. Reach
K. Reach*
Affiliation:
Prague
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Every system of signs that is defined in logical syntax may be called a formal language. It need not always be a language in the ordinary sense of the word. The rules of chess, e.g., can be expressed in the terminology of the syntax, but it would not occur to anybody to call chess a language.

A language in the ordinary sense has a meaning. That is to say that certain words in it are names of things or states or properties or relationships. In general it can be said that in any language having meaning all words subject to the rule of types are names. (Even sentences might be called names, i.e. names of facts, but we will not go so far.)

A syntax of a language L is a language with meaning, for the words of a definite type (in this paper of the type of individuals) contained in it are names of expressions of L.

If to a language we add its syntax, we get a language containing the syntax of one of its parts. If we formulate the syntax of a language L, in so far as its means allow, in the language L itself, we get a language containing a part of its syntax. We call such languages autosyntactic. It is well known that languages containing the whole of their own syntax do not exist.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 3 , Issue 3 , September 1938 , pp. 97 - 111
Copyright
Copyright © Association for Symbolic Logic 1938

References

1 On the concept “abstraction class,” cf. Carnap, Rudolf, Abriss der Logistik 20 b.Google Scholar

2 Cf. Whitehead, and Russell, , Principia mathematica *33.Google Scholar

3 Carnap, Rudolf, Logische Syntax der Sprache Google Scholar. (Abbreviation: Synt. )

4 Tarski, Alfred, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia philosophica, vol. 1 (1935)Google Scholar. In this paper Tarski also gives a definition for the relation “x designates a” which is identical with Carnap's SZ.

5 The author formulated this idea in 1932. The same idea is to be found in a lecture by Helmer, in the Actes du Congrès International de Philosophie Scientifique, Paris 1936, part VIIGoogle Scholar. Unfortunately Helmer applies this idea in a way which the present writer cannot approve of.

6 “Real variable” in the terminology of Principia mathematica .

7 Cf. Principia mathematica *70–*72.

8 “Gliedzahlen,” cf. Synt.

9 Cf. Synt.

10 Principia mathematica *30.

11 Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931)Google Scholar.

12 Tarski, loc. cit.

13 In his paper “Satz V.”

14 “~Sb;K[z]” means “~(Sb;K[x]).”

15 Proof

(Sb[x](y).Sb[x](z)) ≡ (∃F, G)(Nm[F[ź]](x).Nm[G[ź]](x).Nm[F]](y).Nm[G[x]](z)).

“F[ź]” is cognate with “G[ź]” and thus we may, according to Dd 1, in the sentence after the equivalence sign put “F” for “G.” By axiom A 2 we get G 6.