A while ago I was obsessed with trying to show that God’s foreknowledge of our actions was incompatible with Human free will. I have had some time to reflect on the issue and I want to take another stab at it.
So, let ‘K’ be ‘knows that’ and ‘G’ stand for God, and ‘R’ for Richard Brown (me). Then (1) says that if God knows that I will do some action then it is necessary that I do that action.
(1) (x)(K(G,R,x) –> [](D(R,x))
(1) captures the intuition that God’s knowledge necessitates our actions. I think that this is true, so to prove it I tried to show that denying it leads to a contradiction and, since it can’t be false it must be true. Here is the proof.
1. ~(x)(K(G,R,x) –> []D(R,x)) assume
2. (Ex)~(K(G,R,x) –> []D(R,x)) 1, by definition
3. (Ex)~~(K(G,R,x) & ~[]D(R,x)) 2, by def
4. (Ex) (K(G,R,x) & ~[]D(R,x)) 3, by def
5. K(G,R,a) & ~[]D(R,a) 4, EI
6. K(G,R,a) 5, CE
7. []K(G,R,a) 6, necessitation
8. ~[]D(R,a) 5, CE
9. (x)[] (K(G,R,x) –> D(R,x)) assumption (2′)
10. [](K(G,R,a) –> D(R,a)) 9, UI.
11. []K(G,R,a) –> []D(R,a) 10, distribution
12. ~[]D(R,a) –> ~[]K(G,R,a) 11, contraposition
13. ~[]K(G,R,a) 8,11 MP
14. []K(G,R,a) & ~[]K(G,R,a) 7,13 CI
15. (x)(K(G,R,x) –> [](D,R,x)) 1-14 reductio
The main objection centered on step (lucky number) 7 and my use of the rule of necessitation. 7 says that it is necessay that God knows that I perform action a. That means that it would have to be true in every possible world that God (in that world) knows that you perform action a. This may seem unreasonable if one thinks that there is a possible world where you do not perform action a. But if actions are events that can be named then it is easy to show that they must necessarily exist, in which case I would have to perform that action in every world where I exist, and snce it is just as easy to show that I must necessarily exist it follows that God would indeed know that I perform action a in every possible world and so 7 comes out true. So if one accepts S5 then one should not have a problem with 7.
But suppose that one rejects, or modifys S5 to avoid the embaressment of necessary existence? Then 7 starts to look fishy again. But is it? Say that there is some world where I do in fact perform a and some other world where I do not. Call them ‘A’ and ‘~A’. The in A God knows that I perform a but in ~A He doesn’t know that I perform a because it is false that I perform a and God does not know falsehoods. But is it really true that in ~A God does not know that I perform a? He knows everything, so He knows what is possible and so He knows that there is a possible world where I do perform a. Yes, but that just means that He knows “possibly Richard performs a’ not ‘Richard performs a'”, or in symbols; he knows <>D(R,a) not D(R,a). This I admit, and so it seems that there is a conception of God’s foreknowledge that is compatible with Human free will. But there does seem to be a sense in which He still knows that I do a; He knows in which possible worlds I do it and in which I don’t. But maybe that isn’t enough to justify 7 and so enough to avoid the issue.
But notice that it is a conception of God as confined to particular possible worlds where he knows all and only the truths in that world that is the actual world. The possible worlds are not real worlds but formal descriptions or specifications of how the actual world could have been and God has maximal knowledge of that. If one were a modal realist and thought that the possible worlds were real worlds that exist then there would be a problem here. In each world God would know either that you perform action a in that world or that you perform it in world-x. In both cases He knows that you perform action a and so it will true in all worlds that He knows that you do a. So 7 would be true again.
So I conclude that there are some interpretations where 7 comes out true; in which case there are some metaphysical systems in which God’s omniscience is incompatible with Human free will. Or He’s a dialetheist…
Richard Brown 
Richard, here’s why (7) just can’t be right: there are possible worlds where you do not perform a, and hence where K(G,R,a) is false. So []K(G,R,a) is false.
(More generally, you can see the general strategy is misguided because the sort of necessity you’re worried about is surely not of the “truth in all possible worlds” variety. It’s plainly not true that you do a in all possible worlds, and God’s knowledge that you will actually do a is not any kind of reason to think otherwise. So there’s just no way you’re going to get anywhere with these argument from modal logic, given the PW interpretation. It’s not latching on to the real basis of the intuitive argument.)
Towards the end of your post you argue that God could have necessary knowledge that you do a in world-x, but that’s something completely different! That’s an argument for []K(G,R,a-in-x), not for []K(G,R,a).
I’m also unsure how you get from the necessary existence of an action/event to the claim that you necessarily perform it! (Again, that just can’t be right given that there clearly are actions we perform contingently!)
Hi Richard, thanks for the comment!
I agree that IF there are worlds where I do not perform a then 7 is trick-ier, but if one accepts standard modal logic then it is easy to prove that your actions are not contingent. Take some action of mine, call it ‘lisa’. It is just as easy to prove that lisa’s existence is necessary as it is to prove that mine is. If lisa’s existence is necessary then in all possible worlds there is an action that is done by me. How could my action exist without my performing it? If it were to be performed by someone else it wouldn’t be lisa since that names my action, not the action type that my action is a token of. But I grant that this problem can be solved by modifying modal logic, but to some people that’s a big give…
Yeah, I know, but the idea is that if God knows that I do a in x then he knows that I do a. Compare: If I know that Harry brushes his teath in the bathroom, then I know that he brushes his teath. So, if possible worlds are real places, then the inference should hold. It might even hold for fictional worlds; If I know that Harry Potter defeats Valdermort in the story then I know that Harry Potter beats Valdermort, don’t I?
“the idea is that if God knows that I do a in x then he knows that I do a.”
But the former might be true whilst the latter is false. That is, you might do a in x even though you do not do a simpliciter, i.e. in the context world. Note that – at least on the usual understanding – a proposition P is true just in case it is true at the world of context (not just in case it is true at some world, as your argument presupposes).
Compare: Suppose I know there is beer in your fridge, but I’m looking at mine and it’s empty. Do I know that there’s no beer? Sure I do, since there’s an implicit quantifier restriction, such that “there’s no beer” is true just in case there’s no beer around here. Modal realists tell a similar story about truths that are local to worlds.
Why not just bring in the necessity of the past? Simply admit that 7 isn’t true (with the necessity understood strictly). After all, things could have gone differently, with God knowing you’ll do something else, and you (accordingly) doing that something else. But then insist that 7 is true when the necessity is understood differently: namely, the past is now fixed and therefore beyond your control. This ‘necessity per accidens’ gets the results you want, and unlike strict necessity, it doesn’t make 7 end up obviously false.
I agree, but in this case we are considering the special case of what an omniscient being would know. So, I don’t think that p would be true in world(1) just because it were true at world(n)…But if God were in world(n) He would know that p.
Angain, agreed; but you also know that there is beer. The restrictions on the quantifier allows you to believe ‘there’s no beer’ and ‘there is beer’ without contradiction. So, there is noting contradictory about the kind of knowledge that God is supposed to have, but I don’t see how you can get out of the claim that God knows that it happens whether or not it (actually) happens…
Hi Dave2, thanks for the comment!!!
Yeah, I agree that if one interprets ‘[]’ as ‘it is true at all times’ instead of ‘true at every possible world’ 7 is intuitively in better shape and the argument is fine. But I don’t agree that otherwise 7 is obviously false! 🙂 You say,
it is precisely because this is debatable that 7 is not obviously false, though I admit that there is a way to get it to come out false…