Comments on Stevens’ Review of The Cambridge Companion and Anellis on Truth Tables

Abstract

_Russell_ journal (home office): E:CPBRRUSSJOURTYPE2402\IVOR.242 : 2005-05-19 13:36 iscussion COMMENTS ON STEVENS’ REVIEW OF THE CAMBRIDGE COMPANION AND ANELLIS ON TRUTH TABLES I. G-G Mathematics / Middlesex U. at Enfield Middlesex  ,  @. n his review of the Griffin Companion on Russell, Stevens calls into quesItion several aspects of my reading of the mathematical aspects of Russell’s logic. Some clarifications are in order. (). One of the most innovative aspects of recent Russell studies is the upgrading by Landini of Russell’s substitutional logic, in which Russell had dispensed with propositional functions and relations and worked only with propositions , their truth values, and individuals; the latter were substitutable into a proposition, and propositions into compound propositions, by means of a notion called “matrix”. Russell developed the theory from about mid- to early in , in a large collection of manuscripts that we may be able to read one day complete in Papers . He came to see it as the means to achieve logicism; but then he found various difficulties with it, including a paradox. Thereafter it was only a residue in his definitive theories of types, where propositional functions and relations were back on centre stage. This is my view, recorded in my article in the book; but Stevens sees the  Graham Stevens, review of Nicholas Griffin, ed., The Cambridge Companion to Bertrand Russell, in Russell, n.s.  (): –.  A main source of this paradox is a letter by Russell to R. G. Hawtrey. Both Stevens (p. ) and Landini (p.  of the Companion) locate it in the Russell Archives; in fact, the original letter is among the Hawtrey Papers at Churchill College, Cambridge (Grattan-Guinness, The Search for Mathematical Roots, –. Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel [Princeton: Princeton U. P., ], p. ). russell: the Journal of Bertrand Russell Studies n.s.  (winter –): – The Bertrand Russell Research Centre, McMaster U.  - _Russell_ journal (home office): E:CPBRRUSSJOURTYPE2402\IVOR.242 : 2005-05-19 13:36  . - matrix in the type theory of Russell’s “Mathematical Logic as Based on the Theory of Types” () as “dramatically” (p. ) exhibiting a difference from the version in Principia Mathematica, where a matrix is redefined as a propositional function with no quantifiers (: –). In a companion on Russell my aim was capture Russell’s apparent position in the s and s: the treatment of substitution in  covers just one paragraph in art. , immediately followed by a declaration that it is “technically inconvenient”, while the redefinition of matrix in Principia does not deploy substitution at all. So maybe “residue” is a reasonable historical appraisal. Similarly, Stevens wonders what mathematics Russell might not have been achieved with the substitutional theory, given Landini’s revival of it (pp. –). The table in my article shows the range of mathematics that Principia was intended to cover: much (but not all) of Cantorian set theory and transfinite arithmetic, and a rather mixed bag from the foundations of real-variable analysis (including a second definition of integers and numbers), some intended for the Volume  of Principia on geometry that Whitehead was in the end to abandon. Russell himself did not get that far into this mathematical mountain in his substitutional manuscripts, and I am not aware of any more detailed explorations in Landini’s or anyone else’s writings. Given the considerable technical and philosophical difficulties that attend the exposition in Principia, such as the issues surrounding the multiplicative axiom, these claims for substitutional theories need further elaboration. In recent years I have advocated, as a fundament of historiography, the distinction between two ways of reading past knowledge, each one legitimate but independent of and quite different from each other: history, where the effort is to reconstruct what the historical figures did and did not do, with careful handling and often avoidance of later theories; and heritage, where those later theories can be readily deployed. In the passage cited, Stevens seems to read Principia in a heritage spirit inspired by Landini’s excellent proposals; but this leads to a “temptation to assimilate Russell’s views to more modern ones[, which] has been a cause of many misinterpretations of Russell’s logic...