The Limits of Thought—and Beyond

The inclosure scheme and the solution to the paradoxes of self-reference

“…The answer to both questions is "yes". Priest (1994;and 2002, Chapters 9-11) has shown that all paradoxes considered by Russell in 1908 and by Ramsey in 1926 are instances of IS. In addition to these, he has considered other interesting cases.…”

“…According to Priest (1994Priest ( , 2002, Russell was on the right track when he stressed the relevance of structural similarities among paradoxes in order to understand them and to solve the problem they represent. What is more, in a paper published in 1905, he actually found the basic constituents of any paradox of self-reference and came close to find a general description of this phenomenon.…”

“…We have then δ( ) / ∈ (transcendence) and δ( ) ∈ (closure). If we avoid any mention to ψ in IS, we obtain Russell's original description, a scheme that captures the best known set theoretic paradoxes: Russell's, Burali-Forti's and Mirimanoff's (Priest 1994, 27-8;2002, 129-31). Priest introduces the property ψ in Russell's description in order to enlarge its scope and include, among others, semantic paradoxes like the liar and those due to König, Berry, Richard, etc.…”

“…Hence not only paradoxes fit into IS. The "paradox" of the barber who shaves all those who do not shave themselves is one of these "fake paradoxes" (see Priest 2002, 157, note 2). We can simply conclude that there is no such barber.…”

“…These paradoxes involve some totality of things satisfying the property ϕ (no new ϕ can be found out of ) and some operator δ that generates new ϕs from any previous collection of ϕs in its domain, including, paradoxically, itself. Priest (1994;2002, Chapters 9-10) distinguishes several subfamilies of inclosures according to the different possible interpretations of ψ in IS. Set theoretic paradoxes like Russell's, Mirimanoff's and Burali-Forti's form, together with other paradoxes, a relevant subfamily of inclosures (those where ψ is a universal property).…”