Omniscience and Simplicity
November 9, 2010 — 19:19

Author: Kenny Pearce  Category: Books of Interest Christian Theology Concept of God  Tags: , , , ,   Comments: 5

The end of the semester is fast approaching, which means an even more hectic academic schedule, followed by a vacation. This post will be a brief remark on Sobel‘s treatment of omniscience, which completes his interlude on divine attributes. Following this, I will leave off until after the holidays, at which point I will deal with the remainder of the book, which treats arguments against the existence of God, and also ‘Pascalian’ practical arguments for belief in God.
The main puzzle Sobel finds with omniscience is one pushed by Patrick Grim. The thrust of the argument is this: (1) a Cantorian diagonalization argument shows that there can be no set of all truths. But, (2) for any being, there is a set containing all and only the propositions known by that being. Therefore, (3) no being knows all truths. (This is my simplified reconstruction; Sobel spells out some of the set-theoretic details related to (1).)
As Sobel rightly points out, there is no reason for the theist to accept (2) and, as a result, the argument fails. (Sobel also considers a similar argument from Grim to the effect that the sentence ‘there is a being who knows every proposition’ fails to express a proposition, because there are no propositions about all propositions. Sobel is, I think, correct in saying that Grim’s premises involve details of a theory of propositions, rather than just an intuitive definition of propositions and ‘aboutness’, and any theory of propositions that has this consequence is clearly unacceptable.) All I want to note here is that Sobel doesn’t point out what I take to be one of the more interesting reasons theists might reject the premise. Consider the following argument in support of (2):

(a) For every distinct proposition p known by a being S, S is in a distinct mental state which (partly) constitutes S’s knowledge that p.
(b) No being can be in a proper class of distinct mental states.
Therefore, (c) No being can know a proper class of propositions, i.e. (2) is true.

(a) is plausible insofar as knowledge either is itself a mental state (as Williamson says), or else is partly constituted by belief, which is a mental state. (b) seems plausible probably because we typically think of mental states as concrete entities, and we balk at the idea of a proper class of concrete entities. (Having countably or continually many concrete entities is mind-boggling enough.)
I think Sobel probably has an argument like this in the back of his mind, and this is why he offers the suggestion (pp. 384-388) that if we aren’t too wedded to pure actuality and atemporality as divine attributes, we might hold that only some set of propositions is before God’s mind at any given time, but these propositions are such that God can easily (instantaneously) deduce any of the other propositions from them whenever he likes. Sobel calls this ‘virtual’ knowledge.
But, as Sobel realizes, the theist is at liberty to reject (b), and so to continue rejecting (2). What Sobel doesn’t seem to realize, is that certain theists, those who accept the strong (Western) form of divine simplicity, are under independent pressure to reject (a). According to this view, God is identical to each of his attributes. Therefore, if God knows that p, and God knows that q, then God’s knowledge that p = God’s knowledge that q = God, and similarly for God’s belief in each of these propositions. If this idea makes any sense (and I suppose we shouldn’t just take for granted that it does), then God can know a proper class of propositions without being in a proper class of mental states.
[Cross-posted at blog.kennypearce.net]

Comments:
  • Omniscience and Simplicity

    The end of the semester is fast approaching, which means an even more hectic academic schedule, followed by a vacation. This post will be a brief remark on Sobel’s treatment of omniscience, which completes his interlude on divine attributes. Following …

    November 9, 2010 — 19:23
  • I’m not sure it’s necessary even to go so far as to appeal to simplicity to reject (a); it’s a pretty common thesis in scholastic philosophy, for instance, that God knows everything by the single act of knowledge by which He knows Himself (cf., for example, Aquinas, ST 1.14.5ad3 & 1.14.14, SCG 1.45-55; Bonaventure, In Sent I d. 39 a. 2 q. 1). Simplicity was often given as the reason for this, but there were secondary ways of arguing the point that didn’t depend on simplicity (e.g., based on what is the most suitable object of divine knowledge).
    The scholastics, I suspect, would take (a) (and (2)) to assume at the very least a mind that works by composition and division (which ours does, but which they argued God’s does not). Since unlike us God does not compose and divide propositions in knowing them (he knows them by thoroughly knowing what finite minds are capable of thinking) there’s nothing that requires that there be a distinct mental state of composition-and-division for each distinct proposition known as there plausibly is with us.

    November 9, 2010 — 20:28
  • I think it’s implausible to suppose that always when I know p and I know q there are distinct mental states underlying the two knowings. Cf. my doxins/orektins hypothesis. Here’s one way to get to this. It seems very plausible that our brains compress data in some way when they store it. If I were writing a computer-based reasoning system, I wouldn’t want the computer to have separate representational states corresponding to the beliefs that 0 is a number, that 1 is a number, that 2 is a number, that 3 is a number, …, that 65535 is a number. That would be a profligate use of storage space. Instead, I might compress it all into something like: “For all integers x between 0 and 65535: believe that x is a number.” So, I have 65536 propositions that the system “believes”, but they are stored in a compressed way using much fewer than 65536 bits.
    If one doesn’t like the computational model, just take this. Suppose you inform me that p and q, and I justifiably believe you (and nothing aberrant happens). Then I know that p and I know that q. But it is implausible to suppose that I automatically form three genuinely distinct mental states: one that p, another that q, and a third that p and q.
    In general, I suspect that the counting of mental states is a dubious enterprise. If Sam is a dog, and I am afraid of dogs, and hence afraid of Sam, do I have two fears, one of Sam and another of dogs?

    November 10, 2010 — 14:51
  • Kenny Pearce

    Hi Alex,
    Do you think your approach is similar to Sobel’s? That is, is it a version of the ‘virtual knowledge’ idea? Also, would you apply the same line of thought to God, or do you think that case is different?

    November 11, 2010 — 19:37
  • I think my approach is a generalization of Sobel’s. The generation of propositions isn’t just by entailment. Rather, the mind stores rules for generating propositional credences, without storing many, and maybe any, of the individual proposition-credence pairs. Entailment is one kind of rule. But there are others. For instance, one rule I probably have is something like: “Assign credence 0.74 to <x exists&rt; for any small to medium-sized positive integer x.” Then you ask me if the number 127 exists, and I say: “I have credence 0.74 for it.” This isn’t just an entailment relation, though. (For instance, I can believe <(x)(if x is a small to medium-sized positive integer, then x exists)> has credence 0.74, but it does not follow that <127 exists> has credence 0.74–it might have greater credence.)

    November 12, 2010 — 9:48