In a previous post I argued that the knowability of truth entails an omniscient mind. But the whole argument is predicated on the knowability principle, KP, which states that truth can be known. But, who would defend such a position? “Only theists” one might suspect. As it turns out, KP follows upon some versions of anti-realism and idealism.1 Of course, in light of the paradox, many anti-realists have modified their form of anti-realism so that it does not fall prey to “paradox”.2 Whether these more subtle anti-realists escape the knowability paradox is still a matter hotly debated (Brogaard 2009).
The (naïve?) anti-realist is not the only one who commits herself to KP. Verificationists, like the logical positivists of yesteryear, also hold to KP, since meaning is predicated on analytic or empirical verifiability.3 If KP is expressed in the slogan “To be true is to be provable”, one can still faintly hear Ayer and his acolytes chanting their implicit consent to KP. Ironically, so long as the positivists admit that there is a maximally big conjunctive truth, they would have to concede that an omniscient mind exists. How embarrassing!
Perhaps less surprising, though also worthy of note, is that Aquinas, a metaphysical realist, would also be a proponent of KP:
There is nothing, however, that the divine intellect does not actually know, and nothing that the human intellect does not know potentially, for the agent intellect is said to be that “by which we make all things knowable,” and the possible intellect, as that “by which we become all things.” For this reason, one can place in the definition of a true thing its actually being seen by the divine intellect, but not its being seen by a human intellect, except potentially, as is clear from our earlier discussion (De Veritate Q. 1,A.2).4
I mention this because the literature on the knowability paradox seems to equate KP with antirealism. But, as Aquinas proves, there is no reason why KP should be restricted to anti-realism.5 It is clear that for Aquinas all things are known to some mind, i.e. God, and knowable to all other minds. Of course my non-theistic interlocutor should hardly be moved by any of this, unless she were to embrace the particularities of Thomistic psychology while eschewing his metaphysics. I suspect dodos would be less rare.
I would also like to consider some objections to KP. Some of my interlocutors have already countered my claims that all truths are knowable by referencing the Münchhausen trilemma or Gödel’s incompleteness theorems. Münchhausen trilemma offers us three options, each of which appear to be untoward for the tenability of epistemology. The trilemma runs as follows:
1. Justifications for some knowledge must be justified, ad infinitum, which abandons foundationalism.
2. Justifications are eventually circular, which means that knowledge is question begging.
3. Justifications are not sought for some truths, which abandons justification arbitrarily.
But before we despair, its readily apparent that the trilemma is answered by various epistemologies. Infinitists take the first horn. Infinitism does not abandon knowability, but argues that knowledge may be justified by an infinite chain. I wouldn’t personally take this horn, but it seems to me that many, especially non-theists, claim to have no problem with infinite regresses. For justification can emerge out of the infinite chain of reasoning much like an effect can arise out of an infinite sea of prior causes. So, if my non-theistic interlocutors reject knowability on the basis of this trilemma, I might ask them to reconsider the force of Thomas’s Five Ways.
The second horn is accepted by coherentists. They reject the charge that these, so called, loops of justification beg the question. Rather, they see knowledge in a holistic way. Beliefs fit within a reinforcing and consistent web. The literature is rife with defenses of this position, so it is at least a plausible alternative to utter skepticism.
Furthermore, it seems to me that the Reformed Epistemology argues that justifications are unnecessary in knowing some truths, i.e. those truths that are properly basic. But this is not to embrace the third horn, since there are specific conditions offered by which a belief can be considered properly basic. Hence, stopping points are not inherently arbitrary, or in violation of some PSR. Even the foundationalist typically will concede that the law of non-contradiction needs no support. But RE allows for stopping points that extend beyond mere logical principles and analytic truths, and so extends the foundation of knowledge out much further.
So the Münchhausen trilemma does not force us into some form of Pyrrhonism, nor does it explicitly assert that some truths are unknowable. It merely forces us to think more clearly about epistemology, which is a good thing. So in my assessment, the Münchhausen trilemma is not a good reason to reject KP.
As for Gödel’s incompleteness theorems, while they do seem to suggest that truth transcends proof, it should be noted that “proof” must be understood within the context of an axiomatic system, and does not mean justification or warrant in some broader epistemological sense. It is for this reason that we cannot simply concede to the highly controversial thesis that the results of the incompleteness theorems are applicable to human minds, or any other minds for that matter. In fact, Lucas and Penrose have argued that Gödel’s theorems are an indication that the mind is not a Turing-Machine, that human intelligence is not restricted to axiomatic proofs, and that truths that no machine could know, can be known to us. If so, AI will never be analogous to human intelligence. In sum, I do not think it is compelling to say that Gödel’s incompleteness theorems count against KP without the additional support of the premise that all minds are functionally equivalent to Turing machines.
So, I have suggested that there are philosophical defenders of the knowability thesis, both realists and anti-realists, theists and atheists. Also, I have defended KP against the two main objections raised by my interlocutors thus far. Still, are there any good reasons to endorse KP? I have no deductive argument or a priori argument to offer. I’m an optimist, and I’d like to think that all truths can be known in principle. Absent a defeater, I think there are good inductive reasons for accepting KP. They can be summarized as follows:
1. No specific instance of a truth that one might point to is unknowable.
2. Therefore, all truths are knowable.
(1) is equivalent to the fact that all specific instances of truth that one might point to are knowable. In other words, KP is continually confirmed, and counterexamples are not forthcoming, and never will be forthcoming. For to confirm the truth of a counterexample would require that we know that something unknowable is true, which is a contradiction. If induction may be used to support a principle, then it certainly offers abundant evidence in support of KP.
One last point, and this is for the naturalists:
Suppose the metaphysical naturalist were to reject the knowability principle in an attempt to escape my argument for an omniscient mind. The rejection of the knowability principle would entail that there are some truths that cannot be known, verified, etc. by our best, or even ideal, natural sciences. If a truth is defined as natural insofar as it comports with and can be subjected to analysis and empirical verification by a natural science, it follows that there would some non-natural truths. This raises an interesting dilemma for the metaphysical naturalist:
1. Either all truths are knowable, or not all truths are knowable. (LEM)
2. If all truths are knowable, then an omniscient mind exists. (see this post for proof)
3. If not all truths are knowable, then some truths cannot be verified by the natural sciences even in principle.
4. If some truths cannot be verified by the natural sciences in principle, then metaphysical naturalism is false.
5. Therefore, either an omniscient mind exists, or metaphysical naturalism is false.
I think both disjuncts are true, but it seems that we are forced to pick one! And its on odd metaphysical naturalist who admits that there is an omniscient mind.
1 B. Brogaard & J. Salerno, 2009. “Fitch’s Paradox of Knowability”, The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/entries/fitch-paradox/>.
2 S.A. Rasmussen. 2009. “The Paradox of Knowability and the Mapping Objection”. In New Essays on the Knowability Paradox. Ed. J. Salerno. New York: Oxford University Press. 53-54
3 J. Beall “Knowability and Possible Epistemic Oddities”. In New Essays on the Knowability Paradox. Ed. J. Salerno. New York: Oxford University Press. 113-4
4 Thomas Aquinas. 1952. Questiones Disputatae de Veritate. Trans. Robert W. Mulligan, S.J.Chicago: Henry Regnery Company. Ed. Joseph Kenny, O.P. Accessed July 20, 2013. URL = <http://dhspriory.org/thomas/QDdeVer1.htm>
5 Plantinga seems to identify Thomas as a theistic anti-realist, since Thomas thinks that truth would not exist without minds. However, Thomas is careful to note that “truth”, though found primarily in the intellect, is secondarily found in things. It is for this reason that I reject Plantinga’s assessment. See A. Plantinga. 1982. Proceedings and Addresses of the American Philosophical Association, Vol. 56, No. 1. 47-70
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.
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]
[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:
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.