INCLOSURE SCHEMA

As we have seen . . . all the paradoxes of self‑reference (including Berkeley's Paradox and the Fifth Antinomy) are inclosure contradictions; that is, they all instantiate the Inclosure Schema, which, to remind the reader, concerns properties φ and ψ, and a function δ such that:

(1) Ω = {y; φ(y)} exists and ψ(Ω)                   Existence

(2) if x ⊂ Ω and ψ(x)     (a) δ(x) not-∈   x    Transcendence

                                       (b) δ(x) ∈ Ω           Closure

One might depict these conditions as in Figure 1. The large oval is Ω, the set of all φ things. x is any subset of Ω satisfying ψ, and δ applied to this takes us out of x but into Ω. Applying δ to Ω takes us both into and out of Ω. This is somewhat difficult to depict(!). I have done so by taking δ(Ω) to be a spot on the boundary of Ω. [1]

1 There is something rather appropriate about this. The boundary is, in its own way, paradoxical, both joining and separating the inside and the outside.


Source: Priest, Graham. Beyond the Limits of Thought, 2nd ed. (Oxford: Clarendon Press; New York: Oxford University Press, 2002), p. 156.

Note: I can't find a readable symbol for the following, so:

not-∈ = not an element of


Graham Priest, Paraconsistent Logic, and Philosophy, Or, Logic and Reality
by R. Dumain

Graham Priest vs Erwin Marquit on Contradiction by R. Dumain

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Philosophy of Paraconsistency & Associated Logics (Web Guide)


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