Transcending the ultimate duality

Abstract

In many philosophical traditions, it is held that reality is non-dual. Of course, to be non-dual, as opposed to dual, is itself to partake of a certain duality. If reality really is non-dual, it must transcend this duality too. But what could this mean? Can one make coherent sense of it? To keep the discussion focussed, I will locate it in one specific tradition: the Mahāyāna Buddhist tradition. The idea that ultimate reality is non-dual goes back to the earliest Mahāyāna sūtras at the turn of the Common Era. Thereafter, the question of what it means to transcend duality plays a central role in Buddhist philosophy. The point that reality must transcend even the duality between duality and non-duality plays a significant role in the Chinese Sanlun philosopher Jizang (, 549–623). His discussion points the way to an answer to our problem which may be articulated with the techniques of contemporary paraconsistent logic, as we will see.

Appendix

In this Appendix, I spell out the full technical details of the model described informally in 3.2.

The language of second-order LP contains predicates and constants. The connectives are ∧, ∨, and $\lnot$; and there are first- and second-order variables and quantifiers. (We suppose, for simplicity, that the second-order predicates and variables are monadic.)²⁸

An interpretation is a triple, $\left \langle D_{1},D_{2},\delta \right \rangle$. For every term, t, δ(t) ∈ D₁. D₂ is a set of pairs, $\left \langle Y^{+},Y^{-}\right \rangle$, such that Y⁺ ∪ Y⁻ = D₁.²⁹ For every predicate, P, δ(P) ∈ D₂. Let us write δ(P) as $\left \langle \delta ^{+}(P),\delta ^{-}(P)\right \rangle$.

Write $\Vdash ^{+}A$ and $\Vdash ^{-}A$ to mean that A is true, resp. false, in an interpretation. Then the truth/falsity conditions are as follows:

• $\Vdash ^{+}Pt$ iff δ(t) ∈ δ⁺(P)

• $\Vdash ^{-}Pt$ iff δ(t) ∈ δ⁻(P)

• $\Vdash ^{+}\lnot A$ iff $\Vdash ^{-}A$

• $\Vdash ^{-}\lnot A$ iff $\Vdash ^{+}A$

• $\Vdash ^{+}A\wedge B$ iff $\Vdash ^{+}A$ and $\Vdash ^{+}B$

• $\Vdash ^{-}A\wedge B$ iff $\Vdash ^{-}A$ or $\Vdash ^{-}B$

• $\Vdash ^{+}A\vee B$ iff $\Vdash ^{+}A$ or $\Vdash ^{+}B$

• $\Vdash ^{-}A\vee B$ iff $\Vdash ^{-}A$ and $\Vdash ^{-}B$

For the quantifiers, we assume that the language has been augmented with constants, k_d for d ∈ D₁, such that δ(k_d) = d, and K_d for d ∈ D₂, such that δ(K_d) = d. A_x(d) is A with every fee occurrence of x replaced by d. (Similarly for second-order variables.) Then:

• $\Vdash ^{+}\forall xA$ iff for all d ∈ D₁ $\Vdash ^{+}A_{x}(k_{d})$

• $\Vdash ^{-}\forall xA$ iff for some d ∈ D₁ $\Vdash ^{-}A_{x}(k_{d})$

• $\Vdash ^{+}\forall XA$ iff for all d ∈ D₂ $\Vdash ^{+}A_{X}(K_{d})$

• $\Vdash ^{-}\forall XA$ iff for some d ∈ D₂ $\Vdash ^{-}A_{X}(K_{d})$

An inference is valid, Σ⊧A, iff for every interpretation, if $\Vdash ^{+}B$ for every B ∈Σ, $\Vdash ^{+}A$.

Now, let t (tathāta) be ultimate reality. To obtain the model, we require, $\mathfrak {M}$, take it to satisfy the conditions:

• t ∈ D₁

• for all $\left \langle Y^{+},Y^{-}\right \rangle \in D_{2}$, t ∈ Y⁻

• for every $Z\subseteq D_{1}$, there is a Y such that $\left \langle Z,Y\right \rangle \in D_{2}$.

It is immediate that in $\mathfrak {M}$, $\Vdash ^{+}\forall X\lnot Xk_{t}$, that is, $\Vdash ^{+}\lnot \exists X Xk_{t}$. t has no properties; it is ineffable. Let $Z=\{d\in D_{1}: \Vdash ^{+}\lnot \exists X Xk_{d}\}$. Then t ∈ Z. But for some Y, $d=\left \langle Z,Y\right \rangle \in D_{2}$, so in $\mathfrak {M}$, $\Vdash ^{+}K_{d}k_{t}$. Hence, $\Vdash ^{+}\exists X Xk_{t}$. t has some properties—notably, the property of being ineffable. So it is effable.

Note that the contradictions in $\mathfrak {M}$ need spread no further than t. If d is any other member of D₁, it may behave quite consistently.