# That Necessarily an Omniscient Mind Exists

[Update: a revised version of the argument can be found here.]

[Update: Some readers have pointed out that “someone” may not mean the same person, and so the move in 29 from “necessrily, every proposition is known by someone (or other)” to “necessarily, there exists an omniscient mind seems illicit. I believe there may be a way to navigate around this objection.  I hope to have a better version of this argument up soon.  I thank my readers for pushing me to tighten my argument].

[Updated 7/13/2012: I found a problem in the original version of the proof.  A minor change simplified the proof and corrected the mistake]

I’ve been thinking about the knowability paradox. I think that it could be considered as a proof that, necessarily, there exists an omniscient mind. The first 26 steps of this proof are my attempt to work out a necessary implication between knowability and the proposition that all truths are known by someone, which is just a reworking of Fitch’s famous paradox to my own ends. From there, one need only to accept (27) the idea that truth is essentially knowable, i.e. that it is necessarily the case that all truths can be known by someone. Given the necessary implication between knowability and the proposition that all truths are known by someone along with the fact that truth is essentially knowable, it follows that (28) it is necessary that all truths are known by someone. I argue that if it is necessary that all truths are known by someone, then necessarily there exists an omniscient mind (29). Hence, necessarily there exists an omniscient mind. The proof is as follows:
Let:
Kp -p is known by someone
Mx – x is a mind
Ox – x is omniscient
1. (∀p)(∀q)[K(p & q) ⊢ ( Kp & Kq)] (theorem)
2. (∀p)(Kp ⊢ p) (theorem)
→3. (∀p)(p ⊃ ◊Kp) (assumption for CP)
↑→4. (∃p)(p & ~Kp) (assumption for IP)
↑↑5. u & ~Ku (4 EI)
↑↑6. (u & ~Ku) ⊃ ◊K(u & ~Ku) (3 UI)
↑↑7. ◊K(u & ~Ku) (5,6 MP)
↑↑→8. K(u & ~Ku) (assumption for IP)
↑↑↑9. Ku & K~Ku (1,8 theorem)
↑↑↑10. K~Ku (9 Simp)
↑↑↑11. ~Ku (2,10 theorem)
↑↑↑12. Ku (9 Simp)
↑↑↑13. Ku & ~Ku (11,12 Conj)
↑↑←14. ~K(u & ~Ku)(8-13 IP)
↑↑15. □~ K(u & ~Ku) (14 NR)
↑↑16. ~◊K(u & ~Ku) (15 MN)
↑↑17. ◊K(u & ~Ku) & ~◊K(u & ~Ku) (7,16 Conj)
↑←18. ~(∃p)(p & ~Kp) (4-17 IP)
↑19. (∀p)~(p & ~Kp) (18 QN)
↑20. ~(u & ~Ku) (19 UI)
↑21. ~u ∨ ~~Ku (20 DeM)
↑22. ~ u ∨ Ku (21 DN)
↑23. u ⊃ Ku (22 Impl)
↑24. (∀p)(p ⊃ Kp) (23 UG)
←25. (∀p)(p ⊃ ◊Kp) ⊃ (∀p)(p ⊃ Kp) (3-24 CP)
26. □[(∀p)(p ⊃ ◊Kp) ⊃ (∀p)(p ⊃ Kp)] (25 NR)
27. □(∀p)(p ⊃ ◊Kp) (premise)
28. □(∀p)(p ⊃ Kp) (26,27 MMP)
29. □(∀p)(p ⊃ Kp) ⊃ □(∃x)(Mx & Ox) (premise)
30. □(∃x)(Mx & Ox) (28,29 MP)

I think the controversial premise is going to be (27), that truth is essentially knowable. Some might object that certain paradoxes contain truths that cannot possibly be known. Or some may point to issues related to Gödel’s Incompleteness Theorems, which purport to show that some truths cannot be derived from within a given axiomatic system. I would respond by saying that the inscrutability of certain paradoxes and/or the non-deriviability of certain truths from within an axiomatic system merely demonstrate the limits of certain minds, or certain systems. But such truths may be knowable from the perspective of other minds, or other axiomatic systems. To say that all truths are essentially knowable is really to say something about truth rather than to say something of minds or systems. Since some truths are knowable, should we think that knowability is incidental to some truths and not others? I’m inclined to think that truth is necessarily knowable, and if you agree, I think you too should think that, necessarily, an omniscient mind exists, whom we call God.

Reference:

Brogaard, B. 2009. “Fitch’s Paradox of Knowability”. In The Stanford Encyclopedia of Philosophy. Winter 2012. Ed. E.N. Zalta.