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Leon Chwistek on the no-classes theory in Principia Mathematica

2003; Taylor & Francis; Volume: 25; Issue: 1 Linguagem: Inglês

10.1080/01445340310001614698

1464-5149

Bernard Linsky,

Logic, Reasoning, and Knowledge

Abstract Leon Chwistek's 1924 paper ‘The Theory of Constructive Types’ is cited in the list of recent ‘contributions to mathematical logic’ in the second edition of Principia Mathematica, yet its prefatory criticisms of the no-classes theory have been seldom noticed. This paper presents a transcription of the relevant section of Chwistek's paper, comments on the significance of his arguments, and traces the reception of the paper. It is suggested that while Russell was aware of Chwistek's points, they were not important in leading him to the adoption of extensionality that marks the second edition of PM. Rudolf Carnap seems to have independently rediscovered Chwistek's issue about the scope of class expressions in identity contexts in his Meaning and Necessity in 1947. Notes Whitehead and Russell Citation 1925 (p. xlv) (hereafter PM). Consider Russell's example of a higher order function: the property of ‘having all the predicates that make a great general’ (PM, p. 56), which is true of individuals. As it involves quantification over predicates (properties) of individuals, that property will itself be of a higher order than the properties in the range of those quantifiers, as well as the property of ‘being a general’. In a predicative theory of classes the class of those individuals having all the qualities that make a great general will be of a higher order than the class of individuals which are generals. The axiom of reducibility of PM is *12·1 ⊢ : (∃f) : φx. ≡ x .f!x where the ! indicates that f is predicative, of the lowest level or type compatible with its argument. Adopting this axiom, and defining classes in terms of predicative properties alone, the two classes just discussed will then be of the same type, that just above the individuals which they contain. It is repeated faithfully, preserving numerous small typographical errors and the typography, but not duplicating the German style quotation marks of the original. I follow the view expressed in Grattan-Guinness Citation 2000 (p. 441), that Russell was responsible for changes of the second edition. Grattan-Guinness reports that Russell was working on the new edition in the summers of 1923 and 1924, so that he would have had Chwistek's papers before him. These changes are presented in the Introduction and three appendices to a (newly typeset) reprinting of the first edition. The issue of assessing the modification of the theory of types proposed is discussed in Hazen and Davoren Citation 2000 . These are: Equation, and three definitions of definite descriptions, classes and ε for Greek class expressions; *20·072, *20·08 and *20·081 respectively. Propositional functions are expressed with the different use of the circumflexed variable, Equation. See Hazen Citation 1983 . That this argument is flawed is noted in Gödel Citation 1944 and discussed in detail in Myhill Citation 1974 . ‘Antynomje Logiki Formalnej’ from 1921, presenting the same argument, is available translated as ‘Antinomies of Formal Logic’ in McCall 1967. A revised version of the argument is presented in Copi Citation 1950 , discussed in Myhill Citation 1979 . Chwistek describes the argument as originating in Jadacki 1912 (p. 256). A referee for this journal has identified that as a reference to Zasada sprzeczności w świetle nowszych badań Bertranda Russella (The Principle of Contradiction in the Light of New Investigations of Bertrand Russell), Polska Akademia Umiejêtności, Krakó, 1912. In a letter of 21 October 1923 in Jadacki Citation 1986 (p. 255). Jadacki Citation 1986 (p. 256). Ramsey 1925 objects to Chwistek's argument, and instead rejects the axiom of reducibility on the grounds that it is contingent and hence not tautologous, and so not a principle of logic. The thought that a similar contradiction could be derived was much later resurrected by I. Copi Citation 1950 . Copi cites Chwistek's argument yet finds fault with it, proposing instead to revive the Grelling paradox of the predicate ‘Heterological’. This version was in turn criticised in Myhill Citation 1979 . PM (59–60 and xiv) in the second edition. Carnap Citation 1928 (§43), Ramsey Citation 1925 . In the letter of 29 October 1923, Jadacki Citation 1986 (p. 256). The survey article ‘The Development of Mathematical Logic in Poland Between the Wars’ by Z. Jordan, in McCall 1967 (p. 353), says that ‘Chwistek's simple theory of types is the final result of a thorough and detailed examination of Principia Mathematica in which he attempted to remedy some technical defects, remove some inconsistencies of symbolism, banish some metaphysical assumptions …’. While apparently alluding to the ‘inconsistency’ produced by the scope convention, Jordan does not give any details. The reference from Black Citation 1933 (p. 83), mentioned below is the only other explicit reference I can find from before 1950. The same criticisms are presented with slightly different wording in Polish in Chwistek Citation 1922b . (I am thankful to Dr Rafal Dymarz for providing me with a translation of 1922b.) Yet the English presentations of the arguments in 1924, cited in PM, were available. The analagous argument for descriptions, leading to a corresponding ‘paralogism’ that an arbitrary description is proper, could not be carried out if the scope indicators are explicitly included, as there are no expressions requiring the application of the convention. This problem seems to have been independently rediscovered by Böer ( Citation 1972 ) who asserts that no substitutions of classes can ever be made in belief contexts for either scope reading. His own solution, in ‘The Theory of Constructive Types’ is unclear. For one thing, he simply abandons defining identity in terms of indiscernibility arguing that ‘In a system of Logic and of Mathematics we have to deal as a matter of fact with statements concerning identity either of classes or of relations, and, as we shall see below there is a definition of identity to be given, which is quite sufficient for this purpose’ (Chwistek Citation 1924 , p. 18). The definition of identity for classes that is forthcoming is simply the principle of extensionality. How he can simply ‘define’ identity for classes with this sort of stipulation is one of the numerous obscure aspects of the remaining portion of the paper. That this is not the standard notion of scope was pointed out to me by Ali Kazmi (personal communication). Hochberg (1958) says that it should be individual occurrences of descriptions rather than simply descriptions that are given scope in PM. Provided that Equation is read as Equation. Again, reading the negation as having narrow scope, Equation. Once more, read as Equation The last reading is clearly that used in *14·28, as pointed out by Geach Citation 1950 (p. 87). That the two occurrences of a description might be analyzed separately is indicated in the proof of *14·18. Geach goes on to describe Chwistek as having shown that the scope conventions in PM are ‘inconsistent’ citing Black's Citation 1933 (p. 83) reference to Chwistek. Changing f and g in Chwistek where necessary to conform with PM. I present the details of the need for this definition in my 2002. Landini (1998) asserts that Whitehead and Russell are rejecting a suggested definition as inadequate when they notice the ‘peculiarity’. Indeed each of the points that I describe as insights on Chwistek's part are errors according to Landini, who makes a case that they are all signs that the no-classes theory is an incoherent remnant of an earlier theory based on the ‘substitutional’ theory that preceded the theory of types in PM. In particular, Landini asserts that there is no problem with the scope conventions in PM and that abstraction is not a term forming operator in PM. See his 1998 (chapter 10). pace Landini Citation 1998 (§10 and explicitly at p. 265). This inadequacy of the circumflex notation for functions is well known. See Hatcher Citation 1968 (p. 126), who uses a different example to show that the notation is unable to express relations of scope between abstracted variables. Indeed it is folklore among some that the origin of the symbol ‘λ’ in fact comes from struggles with a typesetter who first placed a caret to the left of the variable it was supposed to cover, and then down. Carnap Citation 1928 (§43). Carnap's later Logical Syntax of Language, from 1938 cites six works by Chwistek, but not ‘The Theory of Constructive Types’. Notes by Carnap (1932) on 1922a repeat Chwistek's description of 1924 as not published, suggesting that he was not aware of its subsequent publication. Black Citation 1933 (p. 83). It is clear from the introduction to Black's book that Chwistek sent him several papers, including presumably this one. See Geach Citation 1950 , cited in Hochberg Citation 1958 . Grattan Guinness (2000) in his exhaustive survey which includes a section on Chwistek (‘§8.8.4 Pole apart …’) discusses ‘The Theory of Constructive Types’, but does not mention the discussion of the no-classes theory. Chwistek, in Cracow, was not close to the ‘Warsaw–Lwow’ school of logicians, who apparently found his technical work obscure. Indeed, Wolenski (1989, p. 310 ) says that he can ‘in no way’ be included in that school. de Rouilhan Citation 1996 (p. 260), Landini Citation 1998 (p.170) and Linsky Citation 2002 seem independent. Quine and Carnap Citation 1990 (pp. 351–352). These numbers refer to a manuscript which has not been located. Though only after having been rejected by several journals including Fundamenta Mathematicae (Jadacki 1986 p. 254).