Graham Priest’s Inclosure Schema

Fantastic Depiction of the Solar System (woodcut) (later colouration)Woodblock print from the 17th Century (with Bridgeman Art Library Watermark)

Graham Priest maintains in Beyond the Limits of Thought that most of the paradoxes that have arisen like bad pennies in the history of philosophy;  the Liar, Russell’s paradox, Konig’s paradox, and many more; fall into a common pattern. Russell himself posed the formula for the set-theoretic version of these paradoxes (1905, “On Some Difficulties in the Theory of Transfinite Numbers and Order Types” with notation modified by Priest):

Given a property ϕ and a function δ, such that, if ϕ belongs to all members of u, δ(u) always exists, has the property ϕ, and is not a member of u; then the supposition that there is a class Ω of all terms having property ϕ and that δ(Ω) exists leads to the conclusion that δ(Ω) both has and has not the property ϕ.

Priest is a technical logician and likes to express this in functional notation:

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

(2)    For all x ↄ Ω such that ψ(x):

  1. δ(x) ~є  x (Transcendence)
  2. δ(x)  є  Ω (Closure)

My favorite example of this comes from a paper that Priest did with Jay Garfield, the translator of Nagarjuna, entitled “Nagarjuna and the limits of thought”, which is reprinted both in Garfield’s book Empty Words and Priest’s book under consideration here.  Nagarjuna says in the MulaMadhyamakakarika (Chapter XVIII; 8):

Everything is real and is not real,
Both real and not real,
Neither real nor not real.
This is the Lord Buddha’s teaching.

Nagarjuna says in the Vigrahavyavartani

By their nature, the things are not a determinate entity. Their nature is a non-nature; it is their no-nature that is their nature. For they have only one nature; no nature . . .

They call this Nagarjuna’s ontological contradiction. The Inclosure arises as follows:

ϕ(x) is ‘x is empty’

ψ(x) is ‘x is a set of things with some common nature’

δ(x) is ‘the nature of things in x’

If a thing belongs both to the set of things that have some common nature and to the set of things that are empty, it follows that the nature of things is to have no nature. All things both do and do not have a nature. Note that Nagarjuna implies that all things fall into this category. Priest doesn’t go that far. For it is only the special class of things that inhabit the margins of experience that fall into dialetheism, showing that Parmenides’s Law of Non-Contradiction is false for at least one category of situations. This, of course, is the position that Priest has helped to make famous and which has brought down upon him such a host of criticism. Yet, the paradoxes remain.

About Randal Samstag

Randal has an undergraduate degree in political philosophy, but has a graduate degree in engineering and has earned his bread for 30 years working on municipal and community water supply and wastewater collection and treatment systems in the US, Caribbean, Latin America, and Asia.
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2 Responses to Graham Priest’s Inclosure Schema

  1. Pingback: Do Contradictions Imply Everything? | Notes from my library

  2. Pingback: Free Market Fool | Notes from my library

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