VI. the demise of the substitutional theory Alas, the substitutional theory as sketched above is inconsistent. By April of 1906, Russell discovered a new paradox distinct from the propositional liar, the propositional paradox of Appendix B of 22 See the manuscript Miscellaneous notes on PM, #230.031230f1 in the Russell

Archives, McMaster University, Hamilton, Ontario. 271 272 g r e g o r y l a n d i n i Principles, and distinct from his paradoxes of classes and attributes.23 Its existence as a new paradox and its unique significance for the historical development of Russells ramified type-theory were largely unknown until I unearthed it from the archival manuscripts.24 I called it Russells po/ao paradox. Abbreviate by putting: po =df ( p, a, r )(ao = { p/a;b!q} .&. p/a;ao!r :: r )}. Then observe that: po/ao; { po/ao;b!q}!( p, a, r )({ po/ao;b!q} = { p/a;b!q} .& . p/a; { po/ao;b!q} !r :: r )} But given that, ( p, a)(r, c)(b, q)({ p/a;b!q} = {r/c;b!q} :: p = r .&. a = c), one arrives at the following contradiction: ( p, a, r )({ po/ao;b!q} = { p/a;b!q} .&. p/a; { po/ao;b!q} !r :: r ) ( p, a, r )({ po/ao;b!q} = { p/a;b!q} .&. p/a; { po/ao;b!q}!r :: r ).

The problematic axiom schema is (S17). Coupled with (S16) one arrives at the theorem schema (CPsub)1. Russells formulation is derived from the following instance: (t, w)(x)(t/w;x!( p, a, r )(x = { p/a;b!q} .&. p/a;x!r :: r )}). But Russell was aware that there are even simpler variants of the paradox. The following is also an instance of (CPsub)1: (t, w)(x)(t/w;x!( p, a, r )(x = { p a} .&. p/a;x!r :: r )}). By existential instantiation we arrive at: (x)(t/w;x!( p, a, r )(x = { p a} .&. p/a;x!r :: r )} 23 The paradox is the central theme of a number of Russells worknotes for 1906. See pp. 7, 57, 71 of On Substitution (April/May 1906) , Logic in Which Propositions are Not Entities, (p. 15f), and The Paradox of the Liar, (p. 72ff), all catalogued in the Russell Archives (McMaster University, Hamilton, Ontario, Canada).

24 Cocchiarella showed the way, proclaiming that Russell had blundered in thinking that substitution would be able to capture the Cantorian fact that there must be more classes of propositions than propositions. See Nino Cocchiarella, The Development of the Theory of Logical Types and the Notion of a Logical Subject in Russells Early Philosophy Synthese 45 (1980), pp. 71115. Russells Substitutional Theory Then by universal instantiation we have: t/w; {t w}!( p, a, r )({t w} = { p a} .&. p/a; {t w}!r :: r )} Since it is provable in substitution that: ( p, a)(r, c)({ p a} = {r c} :: p = r .&. a = c). We shall be able to deduce the following contradiction: ( p, a, r )({t w} = { p a} .&. p/a; {t w}!r :: r ) ( p, a, r )({t w} = { p a} .&. p/a; {t w}!r :: r ). The flaw in the substitutional system lies with schema (S17).