Monthly Archives: December 2013

Singleton Sets and Divine Simplicity

Plantinga argues against the doctrine of divine simplicity (DDS). Basically, if DDS is true God = God’s omnipotence. But this seems absurd to Plantinga since God is a person and not a property. Of course, we might also point out that DDS seems to imply that some properties are persons. But, I think the better route is to reject Plantinga’s property-based metaphysics in favor of substance metaphysics. This won’t help my friends who are less inclined towards Aristotelian-Thomistic metaphysics. So, in lieu of razing contemporary metaphysics to the ground and providing a cogent defense of A-T metaphysics, here is my attempt to construct a Quinean response to Plantinga’s Argument against Divine Simplicity:

Let:
xÎy – x has a membership relation to y
O – the set of beings that have omnipotence
g – God

1. (∀x)(∀y)(xÎy) ≝ [(x = y) ∨ x∈y)] (df “Δ )
2. (∀x)~(x∈x) (pr)
3. (∀x)(∀y)(x = {y}) → (∀z)[(zÎx ↔ (z∈x ∨ z=y)] (pr)1
4. O = {g} (pr)
5. (∀y)(O = {y}) → (∀z)[(zÎO ↔ (z∈O ∨ z=y))] (3 UI)
6. (O = {g}) → (∀z)[(zÎO ↔ (z∈O ∨ z=g))] (5 UI)
7. (∀z)[(zÎO ↔ (z∈O ∨ z=g))] (4,6 MP)
8. OÎO ↔ (O∈O ∨ O=g) (7 UI)
→9. OÎO (CP)
↑10. O∈O ∨ O=g (8,9 MP)
↑11. ~(O∈O) (2 UI)
←12. O = g (10,11 DS)
13. OÎO → (O = g) (9-12 CP)
14. (∀x)(x = x) (IR)
15. O = O (14 UI)
16. (O = O) ∨ O∈O (15 Add)
17. OÎO (1,16 df “Δ)

From this we can conclude that God is identical to the set of beings that have omnipotence:

18. O = g (13,17 MP)

And we can generalize this result, as we can also conclude that God is identical to {God}:

19. {g} = g (4,18 SI)

So divine simplicity could be argued if the following holds:

20. (∀x)[xÎg → (x = {g})] (pr)
21. (∀x)[xÎg → (x = g)] (19,20 SI)

So, if every divine property has a corresponding set, and that set has a membership relationship with God such that God is the only member, then God is identical with the set and each set identical with God is identical with one another. This may seem implausible to those who insist that sets are causally inert abstract objects, but as Vallicella suggests that we should not insist on a “non-constituent” ontology. A constituent ontology views properties not as causally effete abstracta, but as constitutive of a being’s ontology:

Constituent ontology allows for a sort of ‘coalescence’ of the concrete and the abstract, the particular and the universal. Indeed, such a coalescence is what we find in the simple God who is in some sense both concrete and abstract in that he is a nature that is his own suppositum.2

This may not solve all of the difficulties associated with DDS, but I do think it is a plausible response to Plantinga. Interestingly, the drive in set theory to avoid “a = {a}” was considered the drive to clarify an Aristotelian ambiguity. Instead, it begs the question towards some sort of Platonic extreme realism, which still pervades contemporary metaphysics.

1This premise is based on H.A. Harris. 2011. God, Goodness and Philosophy. Burlington, VT: Ashgate Publishing (pg. 105). Harris argues against Quine, by saying that the membership-relation involves assimilation, which she takes to be asymmetrical. However, I think her complaint is misguided, since the issue isn’t whether the assimilation is symmetrical, but the asymmetrical assimilation entails something that is symmetrical itself. And in the case of the singleton, given that a set cannot belong to itself, symmetry is attained.

2W.F. Vallicella “Divine Simplicity”, The Stanford Encyclopedia of Philosophy (Fall 2010 Edition), Edward N. Zalta (ed.)

(For C’zar)

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