First of all it has to be noted, that if a term in the German language ends with "...zismus" it usually indicates an exaggeration with a negative connotation. According to my knowledge the term "Logizismus" has its origins in the theory of knowledge, especially in the 19th and early 20th century German discussion on psychologism. It was used by Wilhelm Wundt ("Psychologismus und Logizismus", in Kleine Schriften, vol. 1, 1910, 511sq.) writing against a reduction of the non-logical to the purely logical. He was especially against a logicism in psychology (ibid., 549--582). The term was also used by Theodor Ziehen (Lehrbuch der Logik, 1920, 172sq.) to characterize a logical movement opposed to psychologistic, sensualistic and inductive trends, referring especially to Bolzano, Husserl, Brentano and Meinong. According to Ziehen "logicism" indicates directions which grant the logical an independent existence. On p. 173, footnote, he gives a history of the term "Logizismus", missing, however, Wundt's use of 1910. According to Ziehen L. Busse used the term "Logismus" for Edmund Husserl's anti-psychologistic logic in a paper of 1903 (Zeitschrift fuer Psychologie und Physiologie der Sinnesorgane 33 (1903), 153). Busse changed the term to "Logizismus" in 1914 (Zum gegewaertigen Standpunkt der Erkenntnistheorie, Wiesbaden 1914, 33), which is clearly after Wundt. Ziehen additionally refers to a paper by K. Groos in Zeitschrift fuer Psychologie 62 (1912), 271.
In the modern sense "logicism" seems to be introduced by Rudolf Carnap only in 1931 (Die logizistische Grundlegung der Mathematik, Erkenntnis 2 (1931), 91--105) [Carnap31].
The following observations concerning the English use might be of interest. Joergen Joergensen speaks in his 1931 A Treatise of Formal Logic (vol. 1, p. 193) of the "'logicising' of mathematics" with reference to Russell's Introduction to Mathematical Philosophy of 1919 (p. 6sq.). The styles of the quotation and the reference seem to indicate that there were not many uses in between. It should be noted that the German term equivalent to "logicising" would be "logisieren" or "logificiren" [in modern orthography:'logifizieren'] as Benno Kerry called it in his criticism on Frege's Grundlagen der Arithmetik (B. Kerry, Ueber Anschauung und ihre psychische Verarbeitung. I, Vierteljahrsschrift für wissenschaftliche Philosophie 9 (1885), 433--493, on p. 463, note 1). Following the German inclination for using nouns for verbs this would lead to "Logismus" and not to "Logizismus". This doesn't exclude the possibility that Carnap might have been motivated by Russell's use to form the German noun, which then, as a noun, found its way back to the English speaking world.
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