References
The division into languages as given above is very rough. Strictly speaking, as Church puts it, “... a given language has a given and uniquely determined vocabulary. E.g., the introduction of one additional symbol into the vocabulary is sufficient to produce a new and different language. (Thus the English of 1849 is not the same language as the English of 1949, though it is convenient to call them by the same name, and to distinguish, by specifying the date, only in case where the distinction is essential.)” Cf.Alonzo Church,Introduction to Mathematical Logic, Vol. 1, Princeton 1956, p. 48. n. 111.
The symbol\(\dot \subseteq \) and\(\dot \subset \) stand for mereological inclusion, respectively general and proper.
Alfred Tarski,The Concept of Truth in Formalized Languages, inLogic, Semantics, Metamathematics, Oxford 1956, p. 209.
Abraham A. Fraenkel & Yehoshua Bar-Hillel,Foudations of Set Theory, Amsterdam 1958, p. 274.
Alonzo Church, op. cit., p. 50 n. 117. Cf. also p. 69.
Cf.C. F. Tweney & L. E. C. Hughes (Ed.),Chamber's Technical Dictionary, London 1958, pp. 415 and 1023. That usage of the hyphen is, however, not consistent throughout the dictionary.
D. Hilbert & W. Ackermann,Grundzüge der theoretischen Logik, Berlin 1928, pp. 44–45.
Cf.Hans Marchand,The Categories and Types of Present-day English Word-formation, Wiesbaden 1960.
Technically, (23) might be simplified by the adoption of the definition:\(Ex_j^i \mathop = \limits_{df} Ex \cap L_{j + 1}^{i + 1} \) which would reduce (23) to\(\left[ {(x \sim y) \wedge (x \varepsilon Ex_j^i ) \wedge (y \varepsilon Ex_j^i )} \right] \to (x \approx y).\) But the introduction of that additional symbol does not seem necessary.
Cf.R. J. Solomonoff,A New Method for Discovering the Grammars of Phrase Structure Languages, known to the present author in an offprint from the material prepared for the international conference on the numerical treatment of information, Paris, June 1959. See also the bibliography referred to in Solomonoff's paper.
Cf.T. N. Moloshnaya,Algoritm perevoda s angliyskogo yazyka na russkiy. I., inProblemy kibernetyki, No. 3, Moscow 1960, pp. 209–272.
The objection comes from DrW. Kotanski, of Warsaw University, as does also one of the examples given below (slightly modified.)
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Allatum est die 24 Maii 1961
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Wojtasiewicz, O. Towards a general theory of sign systems. I. Stud Logica 13, 81–99 (1962). https://doi.org/10.1007/BF02317261
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DOI: https://doi.org/10.1007/BF02317261