From A Non-Aristotelian System and its Necessity . . (etc)., December 1931 by Alfred Korzybski

The Polish school of (a) 'intuitional' formalism . . .  Leśniewski produced Protothetic, a still more general 'logical' system, by introducing variable 'functors',.   (b) The restricted semantic school represented by Chwistek and his pupils, . . (etc).

Supplement III, Science and Sanity, p. 748.

From Bibliography, The Nature of Mathematics (May 1933) by Max Black

From Bibliography, Science and Sanity (August 1933) by Alfred Korzybski

( page 774 )

From Introductory Remarks . . . (etc) 1939 by Stanisław Leśniewski

Introduction to my article

The object of the paper is a succinct presentation of my system of the foundations of mathematics. This system consists of three deductive theories, whose union forms one of the possible bases of the whole structure of mathematics. The theories in question are the following : (1) What I call Protothetic, which is the result of a certain peculiar enlargement of the well-known theory which goes by the name of the 'propositional calculus', or 'theory of deduction'.   (2) What I call Ontology, which forms a type of modernized 'traditional logic' and which most closely resembles in its content and power Schröder's 'logic of classes', regarded as including the theory of 'individuals'.   (3) What I call Mereology, whose first outline was published by me in a work of 1916 entitled Die Grundlage der allgemeinen Mengenlehre. I.

§ 1 of the article

In 1912 Henry Maurice Sheffer showed that in the theory of deduction of Whitehead and Russell there could be defined two functions of two propositional variables, in terms of either of which as sole primitive the two primitive functions of Whitehead and Russell, namely alternation and negation, could be defined.   (Etc.)  In 1916 J. G. P. Nicod built up the theory of deduction from a single axiom, which apart from variables contained only the sign for the second of Sheffer's functions. For this sign Nicod used the vertical stroke '[vertical stroke]'.

In the definition of non-primitive functions in the theory of deduction, both Sheffer and Nicod make use of a special definitional sign of identity, which is not itself defined in terms of the primitive functions of the system. This fact makes it difficult to say that Nicod's theory of deduction is really based upon the sole primitive sign '[vertical stroke]'. In 1921 I remarked that if one wishes really to base a system of the theory of deduction which contains definitions upon a single primitive term, one must write definitions using this primitive term without resorting for a special definitional sign of identity.   (Etc.)

In 1922 Alfred Tarski established that, by employing functional variables and quantifiers, all the familiar functions of the theory of deduction could be defined using the equivalence function as the sole primitive function.   (Etc.)

§ 2 of the article

In 1922 I sketched my conception of 'semantic categories' and constructed for the fundamental mathematical theories, especially for 'Protothetic' and 'Ontology', directives for definition and inference adapted to this conception. In my axiomatic investigations concerning the directives of protothetic I concentrated upon the task of axiomatizing as simply as possible a system based upon the sign of equivalence as the only primitive term. Tarski's above-mentioned work had made such a system possible, but it had not yet been realized in fact.

(McCall 1967 pages 117-119).

 

* * *

Supplementary remark VII

"I was never able to conceive of a sense of the word 'class' in which I should be at all inclined to ascribe to classes the totality of properties postulated in these theses [i.e. the various 'set theories' and 'class theories' mentioned earlier]. Expressions of the type 'class of objects a' are, on the basis of my [theory], names denoting definite and quite ordinary objects. These expressions naturally have nothing in common either with any mythology of 'classes', considered as objects of some 'higher type', or 'higher order', or with a use of the word 'class' in which the latter is not the name of any object(s), but rather a surrogate facon de parler of some entirely different syntactical type, as for example in the system of Whitehead and Russell.¹"     (page 169)

¹   See Alfred North Whitehead and Bertrand Russell, Principia Mathematica, vol. i, 2nd edition, (Cambridge, 1925), pp. 71 and 72.

* INTRODUCTORY REMARKS TO THE CONTINUATION OF MY ARTICLE : 'GRUNDZÜGE EINES NEUEN SYSTEMS DER GRUNDLAGE DER MATHEMATIK'

This paper was to have appeared . . . in vol. 1 of the periodical Collectanea Logica (Warsaw), 1939), pp. 1-60.   (Etc.)   An offprint copy of Leśniewski's paper survives in the Harvard College Library, together with the continuation (§12) of the original article, which was also to have appeared in Collectanea Logica. Translated by W. Teichmann and S. McCall.

In Polish Logic 1920-1939, editor Storrs McCall,
Oxford 1967.

Bibliographic -- source : http://melvyl.cdlib.org

Author Le�niewski, Stanis�aw, 1886-1939. Title Podstawy og�lnej teoryi mnogo�ci I. [microform] : cz��. Ingredyens. Mnogo��. Klasa. Element. Podmnogo��. Niekt�re ciekawe rodzaje klas. / Stanis�aw Le�niewski. Publisher Moskwa : Druk. A.P. Pop�awskiego, 1916. Description v. Series Prace Polskiego Ko�a Naukowego w Moskwie. [? (WPT)] Sekcya Matematyczno-Przyrodnicza ;no. 2

Author Le�niewski, Stanis�aw, 1886-1939. Uniform Title [ Selections. English. 1987] Title S. Le�niewski's lecture notes in logic / translated and provided with supplementary notes by Zbigniew Stachniak ; edited by Zbigniew Stachniak and Jan Srzednicki. Publisher Dordrecht ; Boston : M. Nijhoff, 1987. Description xi, 183 p. ; 25 cm. Series Logic and applying logic Nijhoff international philosophy series ;v. 24 Note "List of seminars and courses delivered by Le�niewski at Warsaw University between 1919 and 1939": p. Note Bibliography: p. 181-183. ISBN 9024734169

Author Le�niewski, Stanis�aw, 1886-1939 Uniform Title [ Selections. English. 1991] Title Collected works / Stanis�aw Le�niewski ; edited by Stanis�aw J. Surma, Jan T. Srzednicki, and D.I. Barnett ; with an annotated bibliography by V. Frederick Rickey Publisher Dordrecht ; Boston : Kluwer Academic Publishers, c1992 Description 2 v. (xvi, 794 p.) : port. ; 24 cm Series Nijhoff international philosophy series ; v. 44 Note Translated from the Polish Note Includes bibliographical references and index ISBN 079231512X Language English Subject Logic, Symbolic and mathematical