Russell’s Hidden Substitutional Theory by Gregory Landini

eiews SUBSTITUTION AND THE THEORY OF TYPES G S Philosophy / U. of Manchester Oxford Road, Manchester, ..,   ..@.. Gregory Landini. Russell’s Hidden Substitutional Theory. Oxford and New York: Oxford U.P., . Pp. xi, . £.; .. hen, in , Russell communicated to Frege the famous paradox of the Wclass of all classes which are not members of themselves, Frege immediately located the source of the contradiction in his generous attitude towards the existence of classes. The discovery of Russell’s paradox exposed the error in Frege’s reasoning, but it was left to Whitehead and Russell to reformulate the logicist programme without the assumption of classes. Yet the “noclasses ” theory that eventually emerged in Principia Mathematica, though admired by many, has rarely been accepted without significant alterations. Almost without exception, these calls for alteration of the system have been induced by distaste for the theory of types in Principia. Much of the dissatisfaction that philosophers and logicians have felt about type-theory hinged on Whitehead and Russell’s decision to supplement the hierarchy of types (as applied to propositional functions) with a hierarchy of orders restricting the range of quantifiers so as to obtain the ramified theory of types. This ramified system was roundly criticized, predominantly because it necessitated the axiom of reducibility—famously denounced as having “no place in mathematics” by Frank Ramsey, who maintained that “anything which cannot be proved without it cannot be regarded as proved at all”. Russell himself, who had admitted that the axiom was not “self-evident” in the first edition of Principia (PM, : ), explored the possibility of removing the axiom in the  second edition. Ramsey’s extrication and expulsion of the order part of the ramified hierarchy, along with the offending axiom, placated  F. Ramsey, The Foundations of Mathematics and Other Logical Essays, ed. R. Braithwaite (London: Routledge, ), p. .  All page references to PM are to the nd edition, –. russell: the Journal of Bertrand Russell Studies n.s.  (winter –): – The Bertrand Russell Research Centre, McMaster U.  -  Reviews some of the opponents of type-stratified logic, but dissatisfaction remained. Even when confronted by only the simple theory of types, most remained unconvinced that Whitehead and Russell really had reduced mathematics to logic, rather than simply added, ad hoc, sufficient complexity to logic to provide it with the resources to express mathematical reasoning. Russell’s earlier Principles of Mathematics had presented the logicist thesis with extraordinary elegance. Logic was there presented as a universal science, applying indiscriminately to everything. This account of logic, reflected in the formal “doctrine of the unrestricted variable” (the requirement that the nonlogical constituents of every Russellian proposition be treated as belonging to one all-inclusive logical type), provided the philosophical appeal of the logicist project. By contrast, type-restricted variables appear devoid of any philosophical appeal other than their formal ability to block the paradoxes. Or so traditional interpretations of Russell’s philosophical and mathematical logic have maintained. Gregory Landini’s hugely important book is a determined , and largely successful, effort to expose this traditional picture as wholly inaccurate. According to Landini’s interpretation, previous attempts to make sense of the development of the theory of types have run into a locked door. The key that is required to unlock that door is Russell’s much neglected substitutional theory of classes and relations developed between  and . It is perhaps unsurprising that the substitutional theory has been neglected until very recently. Russell’s two most important papers on the theory, though both written in , were not made widely available until , and several important manuscripts on the subject went unstudied until very recently. These fascinating manuscripts, forthcoming in Volume  of The Collected Papers of Bertrand Russell, are subjected to a detailed analysis by Landini, which results in a comprehensive commentary on the theory. This achievement alone would be of great merit, but Landini’s study has more to offer than an explanation of the mechanics of the calculus of substitution. Russell’s Hidden Substitutional Theory divides into three sections, covering roughly the periods –, –, and –. The first section provides an exposition of the philosophy espoused in the Principles, extracting a Russellian calculus of propositions from the informal sketches made in that work. The...

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