Consider two claims about God’s knowledge.
- For all p, if p, then God knows p.
- For all p, if p, and possibly God knows p, then God knows p.
It is an interesting fact that (2), combined with two uncontroversial premises, entails (1). I said this in an earlier post, but now I have a more elegant argument. Here are my uncontroversial premises:
- Necessarily, God’s knowledge is closed under conjunction and tautological implication (i.e., if God knows p and God knows q, then God knows (p and q), and if God knows p, and p tautologically implies q, then God knows q).
- There is at least one proposition p such that possibly God knows p and possibly God knows not-p.
Obviously, the proposition p in (4) is contingent, since knowledge entails truth.
Here is the argument that (2)-(4) entail (1). Fix any true p. By (4), let q be any proposition such that possibly God knows q and possibly God knows not-q. If q holds, then let r=q. If q does not hold, then let r=not-q. Note that r is true. Observe that possibly God knows not-r (if r=q, then this follows from the fact that God possibly knows not-p; if r=not-q, then this follows from the fact that God possibly knows q as well as (3), since q tautologically implies not-r). Let s be the proposition (p or not-r). Then, God possibly knows s. For God possibly knows not-r, and in any world where God knows not-r, God also knows (p or not-r) by (3). Now, s is true as p is true. Therefore, s is a proposition that is true and possibly known by God. Therefore, by (2), God knows s. Moreover, r is a true proposition, and God possibly knows r (since God possibly knows q and God possibly knows not-q). Therefore, God knows r, by (2). But s is (p or not-r). By (3), it follows that God knows p, since (s and r) tautologically implies p.
So if one attempts to limit omniscience by saying that omniscience only means that God knows things that God can know, or that God only knows things that possibly are known by someone (which also entails (2)), one hasn’t limited omniscience at all: God still ends up knowing all true propositions, assuming (3) and (4). Is there some other way of non-arbitrarily limiting omniscience? I am not sure. But, fortunately, there is no need to limit omniscience. God knows all truths.