The Existence of an Omniscient Mind

In a previous post I presented an argument for a necessarily existing omniscient mind. However, two readers noted a similar problem in the argument, so that version doesn’t succeed. I used the predicate Kp to mean “p is known by someone”. This was to follow the presentation of the Knowability Paradox as I have seen it in the literature, including the encyclopedia entry to which I linked. However, when I reached (29) in the argument, □(∀p)(p ⊃ Kp) ⊃ □(∃x)(Mx & Ox), an ambiguity led me to move from “necessarily, every proposition is known by someone” to “necessarily, there exists an omniscient mind”. That is, “someone” is ambiguous and does not necessarily mean the same person knows every truth. I had initially hoped that switching to a two-place predicate would help me to avoid the ambiguity, but I don’t think it can be avoided, at least in the way I’ve formulated the argument. So this was a genuine weakness in my original version of the argument.

In the proof derived in this post I’ve employed a two-place predicate for knowability because I think it is more clear and helps me to avoid stumbling into that ambiguity. The additional premise is that there is a truth that may be called the whole truth. This is a truth that, were it known by a mind, that mind would qualify as omniscient. The whole truth could be understood as a maximal conjunction of all necessary truths, contingent truths, and possible truths. The possible truths are indexed according to the worlds in which they obtain such that part of this truth would be that I am bald in w1 and red-headed in w2, etc. This whole truth, or maximally big conjunctive fact, will be named “b.” Thus, my argument shows that the knowability paradox establishes the existence of at least one omniscient mind. This is to advance beyond the mere establishment of omniscience as, say, a collective feature of a world where truths happen to be knowable within a community of knowers.

I should also note that this version of the argument is modally weaker than the earlier post. While it does lead to the conclusion that there is an omniscient mind, it does not prove that this omniscient mind necessarily exists in every possible world. I could not make this move because, even if it is necessarily the case that all truths are known, I could not derive omniscience as a necessary feature of the mind, though I was able to derive that a mind that is actually omniscient necessarily exists. An odd result that falls out of assuming that the maximally big conjunctive fact is de re necessarily the maximally big conjunctive fact in every possible world. I want to revisit this in future posts, but I will not display the result here. Finally, since I am not trying to prove that an omniscient mind necessarily exists, my knowability premise is not modalized. It merely states that all truths are knowable, not that they are necessarily knowable.

Let:

Kxp – x knows p
Mx – x is a mind
Ox – x is omniscient
b – the big conjunctive fact

1. (∀x)[(Mx & Ox) ≝ Kxb] (definition)
2. (∀p)(p ⊃ ◊(∃x)Kxp) (premise)
3. (∃p)[p & (p = b)] (premise)
4. (∀p)(∀q)(∀x)[Kx(p & q) ⊢ ( Kxp & Kxq)] (theorem)
5. (∀p)(∀x)(Kxp ⊢ p) (theorem)
→6. (∀p)(p ⊃ ◊(∃x)Kxp) (assumption for CP)
↑→7. (∃p)(p &(∀y)~Kyp) (assumption for IP)
↑↑8. u & (∀y)~Kyu (7 EI)
↑↑9. (u & (∀y)~Kyu) ⊃ ◊(∃x)Kx(u & (∀y)~Kyu )(6 UI)
↑↑10. ◊(∃x)Kx(u & (∀y)~Kyu )(8,9 MP)
↑↑→11. (∃x)Kx(u & (∀y)~Kyu) (assumption for IP)
↑↑↑12. Kv(u & (∀y)~Kyu) (11 EI)
↑↑↑13. Kvu & Kv(∀y)~Kyu(4,12 theorem)
↑↑↑14. Kv(∀y)~Kyu (13 Simp)
↑↑↑15. (∀y)~Kyu (5,14 theorem)
↑↑↑16. Kvu (13 Simp)
↑↑↑17. ~Kvu (15 UI)
↑↑↑18. Kvu & ~Kvu (16,17 Conj)
↑↑←19. ~(∃x)Kx(u & (∀y)~Kyu) 11-18 IP)
↑↑20. □~(∃x)Kx(u & (∀y)~Kyu) (19 NI)
↑↑21. ~◊(∃x)Kx(u & (∀y)~Kyu) (20 MN)
↑↑22. ◊(∃x)[Kx(u & (∀y)~Kyu )] & ~◊(∃x)[Kx(u & (∀y)~Kyu)] (10,21 Conj)
↑←23. ~(∃p)(p & (∀y)~Kyp)) (7-22 IP)
↑24. (∀p)~(p & (∀y)~Kyp) (23 QN)
↑25. ~(u & (∀y)~Kyu) (24 UI)
↑26. ~u ∨ ~(∀y)~Kyu (25 DeM)
↑27. u ⊃ ~(∀y)~Kyu (26 Impl)
↑28. (∀p)(p ⊃ ~(∀y)~Kyp) (27 UG)
↑29. (∀p)(p ⊃ ~~(∃y)Kyp) (28 QN)
↑30. (∀p)(p ⊃ (∃y)Kyp) (29 DN)
←31. (∀p)(p ⊃ ◊(∃x)Kxp) ⊃ (∀p)(p ⊃(∃y)Kyp) (6-30 CP)
32. (∀p)(p ⊃(∃y)Kyp) (2,31 MP)
33. u & (u = b) (3, EI)
34. u (33 Simp)
35. b (34 ID)
36. b ⊃(∃y)Kyb (32, UI)
37. (∃y)Kyb (35,36 MP)
38. Kvb (37 EI)
39. (Mv & Ov) (1,38 Def)
40. (∃x)(Mx & Ox) (39 EG)

[Edit Janurary 9, 2013: Thank you to my friend Sam Priest who caught an error in an earlier draft of this argument regarding (1) and (3). Previously, the argument suggested that a mind was omniscient if it knew *that* there was a big conjunctive fact. I´ve since corrected the argument such that the mind must know the big conjunctive fact itself]

Posted on July 14, 2013, in Arguments for God and tagged , , . Bookmark the permalink. 3 Comments.

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