Third Time’s the Charm (or: This Time I Really Got It!!!)

OK, so I am basically obsessed with this stuff about God’s omniscience and Free Will. I have been having some very interesting, and helpful, discussion about whether Plantinga’s defense, which I take it is the standard defense, of their compatibility is any good or not. I have a sneaking suspicion that the two are incompatible and I have been trying to construct a poof to that effect, with mixed results…but I think I got it this time…if it turns out that I don’t then I promise that I will give up!

It seems to me that the problem is that “If God knows what I will do before I do it then it is necessary that I do it” does not really capture what the person who says that God’s foreknowledge is incompatible with our free will is trying to say. This is because, as we have seen, it must be the case that all my actions are necessary, but this doesn’t sound right at all (however, I do think some people are committed to it).

So, to make it clearer what I am actually trying to say, let me introduce a new modal operator ‘@’  with the following truth condition, where ‘v(x,w)’ is the valuation of x at world w,

v(@a, w)=T iff v(a, actual world)=T

v(~@a,w)=T iff v(a, actual world)=F

this says that @a is true if a is in fact true at the actual world and ~@a is true if a is in fact false at the actual world (~@a<–>@~a) so there is no need to introduce a fourth operator). ‘[]’ and ‘<>’ are given their usual interpretations.

Then I can say that God actually knows before I do a certain action that I will in fact actually do it. To avoid getting involved in tense logic let us introduce a predicate ‘B’ for before (though I think we could define ‘B’ in terms of the standard tense operators F, P, H, and G). Let ‘k’ be ‘God knows that’ and ‘a’ be some action of mine, then I can symbolize ‘God actually knows before I do action a that I will in fact actually do action a’ as @B(k,a), then the proof goes as follows

1. @a & @B(k,a)   (this says that God actually knows what I did before I did it)

2. []@B(k,a) –> []@a  (necessary truth)

3. @a –> []@B(k,a)      (necessary truth)

4. @a       (from 1)

5. []@B(k,a)     (4,3 MP)

6. []@a     (5,2 MP)

7. (@a & @B(k,a)) –> []@a   (1-6 conditional proof)

Since 7 says that if it is the case that I actually do a and God knows beforehand that I actually do a then it is necessary that I actually do a, and God’s actually knowing that I do a entails that I actually do a (7) reduces to

7′ @B(k,a) –> []@a

which says that if God actually knows what I will do beforehand then it is necessary that I actually do it.

Now one may wonder what the difference between ‘a’ and ‘@a’ is. Ordinarily there will be no difference, but there will be a huge difference when we examine the modal properties of the two. []a will be true iff a is true in all possible worlds, whereas []@a will be true if @a is true in all possible worlds, or in other words if it is the case that at every possible world it is true that, in the actual world, I do a. This is why (3) above is a necessary truth but (3′) is not,

(3′) a –> []B(k,a)

(3′) says that if I do a then in every possible world God knows beforehand that I will do a. This can be false because there are possible worlds where the antecedant turns out false because in that world I do not do a and so God does not know it. But (3) can’t be false. For if it were then it would be the case both that I actually do a and that God did not actually know beforehand that I did a, which is just to deny that God is omnicient (so enigman will be happy).

Whew! So, if this is right then God’s foreknowledge is indeed incompatible with my having free will. If not then I will finally have to admit that there is at least one metaphysical interpretation on which it can both be true that God knows what I will do before I do it and that I am free…and I will then actually be very depressed!

17 thoughts on “Third Time’s the Charm (or: This Time I Really Got It!!!)

  1. I’m still having difficulty with the rationale for (3). Certainly @a -> @B(k,a) is an uncontroversial necessary truth (someone might well say, “Necessarily if a actually happens God actually knows about it beforehand”)but your (3) seems to make the very controversial claim that if anything is actually done, then there is no possible world in which God does not actually know that it will actually be done. That is, it seems to say not that if a actually happens it is (conditional on that) necessary that God actually know it beforehand (which would be the uncontroversial necessary truth previously mentioned) but instead that if a actually happens it is necessary (no condition, simpliciter) that God actually know it beforehand. Which seems very strong, and more than a little controversial. Am I misreading something here?

  2. Hi Brandon,

    Thanks for the comment!

    Yes, 3 is very strong, but I don’t think there is anything wrong with it…what it says is that if a actually happens then it is true in all possible worlds that in the actual world God knows beforehand that a will happen, which is just to say that from []@a we get @a…I don’t think that this is controversial.

    Consider what would happen if we denied 3….it says @a –> []@B(k,a) so to deny it would be to say ‘@a & ~[]B(k,a)’ which is equivelent to ‘@a & ◊@~B(k,a)’ but this says that a happens in the actual worlds and that it is possible that in the actual world God does not know that it happens before it does. The second clause means that there is a possible world where it is true that in the actual world God doesn’t know something but if God is omniscient then this shouldn’t be possible (given, of course @a…the second clause would be no problem if the thing that it were possible that He didn’t actually know didn’t actually happen). So it seems to me that if yoiu deny 3 then you deny that God is omniscient

    But actually (:)) I think that there is a simplere proof we can give…

    a. @B(k,a) (assume that God actually knows beforehand what I will do)
    b. @a (from a…if He actually knows beforehand that a, then a actual)
    c. @a –> []@a (if a actually happens, then it is true in all possible worlds that a happens in the actual world)
    d. []@a (b,c, MP)
    3. @B(k,a) –> []@a (if God knows what I will do before I do it then it is necessary that I do it in the actual world)

    You may be womdering about c…but again consider what happens if you deny it…it would then be the case that a happens in the actual world and that there is some possible world where it is true in the actual world that a does not happen…does this help?

  3. So, by the way, the last bit of the comment appeals to the following theorem for ‘@’

    ◊@a –> @a

    If it is true in some possible world that a is true at the actual world, then a is true at the actual world (this is similar to the familiar ◊[]a –> []a, the axiom called B)

    and the shortend proof appeals to the theorem

    @a –> []@a

    If it is true in the actual world that a then it is true in all possible worlds that a is true in the actual world (this is similar tot he more familiar ◊a –> []◊a)

  4. Consider what would happen if we denied 3….it says @a –> []@B(k,a) so to deny it would be to say ‘@a & ~[]B(k,a)’ which is equivelent to ‘@a & ◊@~B(k,a)’ but this says that a happens in the actual worlds and that it is possible that in the actual world God does not know that it happens before it does. The second clause means that there is a possible world where it is true that in the actual world God doesn’t know something but if God is omniscient then this shouldn’t be possible (given, of course @a…the second clause would be no problem if the thing that it were possible that He didn’t actually know didn’t actually happen). So it seems to me that if yoiu deny 3 then you deny that God is omniscient.

    Is this really so? What we are really saying, I would have thought, when we say (@a & M@~B(k,a)) [I’m using M instead of lozenge because I can’t remember offhand how to do the latter] is just that as a matter of fact (i.e., in the actual world) a obtains, and it is possible for matters of fact (the actual world) to be such that B(k,a) does not obtain. But this will be true (and compatible with omniscience) if @a is not necessary, since if @is not necessary, then it is possible for @~a to obtain (even though in fact it does not), and thus possible for an omniscient being not to believe beforehand that @a. In other words, both 3 and c look like they make the assumption that for any a, @a is a necessary truth. Then it certainly is true that God’s foreknowledge of @a is incompatible with free will, but that’s not particularly surprising.

    The proposition that I think you may be thinking of that would involve denial of omniscience is not (@a & M@~B(k,a)) but M@(a & ~B(k,a)).

  5. Whoops; I seem to be having a run of bad luck in your comments section. I had forgotten that M plus @ is a mail-to abbreviation. Well, at least it’s still readable, even with the odd links.

  6. In other words, both 3 and c look like they make the assumption that for any a, @a is a necessary truth.

    Sorry to clutter up your comments box, but I accidentally dropped something out of this; it should read, “In other words, both 3 and c look like they make the assumption that for any a such that @a, @a is a necessary truth.”

  7. Hi Brandon,

    It is not possible that @a and @~a both be true…so if one is true the other isn’t. So if @a is true then it is not possible that @~a is true…what is true is that it is possible that ~a is true, which is a different matter…

  8. Quite right; it is not possible for both @a and @~a to be true. But this is not relevant here. The conjunction is (with [loz] for the lozenge):

    (3) @a & [loz]@(~B(k,a))

    Under conditions of omniscience this is equivalent to:

    (3′) @a & [loz]@~a

    This is not a contradiction, because of the possibility operator; all it says is that in the actual world a obtains and it is possible for not-a to obtain in the actual world. For (3) to be a problem for omniscience, we have to assume that @a is necessary.

  9. That’s not what it says.

    3′ says “a is actually true and it is possible that ~a is actually true” That means that there is a possible world where ~a is true in the actual world. But given that a is the one that is actually true it cannot be the case that there is a possible world where it is false that a is true in the actual world. So if @a is true, then it is necessarily true…how could there be a possible world where it was false that at the actual world a is true? That would mean that a is false at the actual world, but we have already agreed that a is true…unless you are thinking dialetheism?

    The way that you say it is ‘@◊~a’ which says that it is true in the actual world that ~ a is possible…that, I agree, is fine.

  10. But given that a is the one that is actually true it cannot be the case that there is a possible world where it is false that a is true in the actual world.

    Precisely: this is to assume that for any a such that @a, @a is a necessary truth.

    Think of it this way: possible worlds, or descriptions of them, are entirely constituted by ways the actual world could possibly be, or accounts of the ways the actual world could possibly be. That’s what makes them possible worlds rather than impossible ones. And each possible world encodes (or its description encodes, if you prefer) nothing more about the actual world than one maximally compossible way it could be. Possible worlds, as such, don’t tell us anything about the way the actual world is, only about the way it could be, except for cases where the actual world could not possibly be any other way. Understood this way, what you are effectively claiming in the above comment is that it is a truth in all possible worlds that this particular possible world is the actual world (since for any a such that @a, there could be no possible world in which it was false that @a). But this can only be the case if there is one and only one possible world, the actual one; for every possible world posits itself, so to speak, as the actual world, being one way the actual world could be — the only difference between the actual world and non-actual worlds is that the actual world is right in so positing.

    Richard Chappell had some lovely conversations about this general sort of issue a while back on his blog. Unfortunately, they are scattered over a number of posts, but these three posts and comments threads are the major ones:

    Actual Ambiguities

    Impossibly Conceivable Counteractual

    The Actual World Is Not a Possible World

  11. Well, I don’t think that I am merely assuming that @a is neccessary. I mean I do appeal to @a –> []@a as an axiom for ‘@’ and I think that there is a good argument for this…

    You say, “what you are effectively claiming in the above comment is that it is a truth in all possible worlds that this particular possible world is the actual world”

    Well, I don’t really like the way that you describe possible worlds, but basically, yes, that is the idea behind the ‘@’ operator. It is supposed to be a modal operator just like [] and ◊. Whereas their ‘job’ is to take you from the actual to the possible, the ‘@’ is supposed to just take you to the actual. As I say in the other post, the basic idea behind hybrid logic is that one can name a world and then specify that certain sentences’s truth depend on the way THAT world is…so ‘@’, the way I defined it, is supposed to name the real world, the one that we live in…

    So when you go on to say “But this can only be the case if there is one and only one possible world, the actual one;” I half agree. The real world is priveleged in a way that other possible worlds are not. They are not real things that we can see in a telescope. They are merely a formal device for giving a semantics to a certain fragment of logic (as per Kripke).

    So when you say, in support of what you said above, “for every possible world posits itself, so to speak, as the actual world, being one way the actual world could be — the only difference between the actual world and non-actual worlds is that the actual world is right in so positing,” I just couldn’t disagree more! This can’t seriously be the ONLY difference between the actual world and possible ones! The way you describe it makes it sound like you are a hradcore modal realist…is that the case?

    Thanks for the links. I have read ‘The Actual World is Not a Possible World’ but not the others, (though now I plan to!) and I think that Richard’s view is pretty close to the view I am defending here (which, is just Kripke’s view)…

  12. Richard,

    I am aware of the point of the @ operator. But it can be used in two very different ways, and you seem to be trying to use it in both, unless I am missing something here.

    In the first way, the @ is simply to provide a frame of reference within the manifold of possible worlds: it indexes one possible world, the one that happens to be actual, and serves no further function but to carry you to that one possible world that happens to be actual. Then it is true that any p involved in its complete description is true across possible worlds, because that is the way all possible worlds work; but the @ doesn’t add anything to the world, since it just marks one possible world over many that could be actual. In this sense we get the basic conjunction:

    @a & [loz]@~a

    And you’re right that this is a contradiction, since the @ takes you to one possible world, and only one possible world. But the contradiction is due simply to the fact that in taking that one possible world you are assuming every truth within it. Therefore on this use of the @ operator it is of course impossible with respect to one and only possible world for a to obtain and an omniscient being in that possible world not to know it. But this is not relevant to free will, because free will is not the claim that you can do something other than a given that your total history is described by the possible world in which you do a, but rather that you can do something other than a rather than a. That is, a description of free will requires comparison of the actual world with non-actual worlds; but no non-actual world is compared. (3) remains difficult to motivate on this use of the operator, since it would be more natural to say [](@a->@B(k,a)) (and this is all that omniscience requires). Your original (3) becomes a very convoluted statement; it says that a obtains in that possible world that happens to be actual and, given this, it is a truth in all possible worlds that a obtains in that possible world that happens to be actual. And this is not a threat to free will. So on this use of the operator, your (3) is true, and we are not assuming that @a is necessary, but we have no incompatibility between omniscience and free will.

    In the second way, the @ is not used to index any particular possible world, but to take you to the actual world, whatever possible world it may happen to be. But if this is the way the operator is being used, your original (3) is certainly false unless we are already assuming that @a is a necessary truth. We’d get the incompatibility; but the reason would be that the view of what @a implies has already treated it as a necessary truth (and thus it is impossible for an omniscient being to believe anything other than it). That is, we’d be assuming the impossibility of a’s being otherwise.

    I’m not a hardcore modal realist; I think possible worlds are purely hypothetical constructs, and not particularly useful ones for most modal reasoning, for that matter. But the only sense in which we can pick out the actual world from among possible worlds requires that we are treating it in terms of our hypothetical construct, and not privileging it at all (except perhaps extrinsically by taking it as a reference point).

  13. Hey Brandon, I just saw that this comment had escaped my notice! Sorry about that…

    Yeah, I see the ambiguity that you and Richard keep talking about…I think it roughly corresponds to what Chalmers calls the primary and secondary intension of a word. So the first way you talk about the @ operator is used to indicate that wer are evaluating a sentence using the meaning of the words in that world (for we are considering it to be the actual world)…the @ operator I was using was meant to take you back to the real world (the second way you and Richard talk about)…so when we go to any other possible world the @ operator tells us to check the sentences truth in the real world. So what I have been trying to argue is that God knwoing beforehand means that it is already actually true (before I do it) that I will do it…so in any possible world it will also be true that back in the real world it is true that I do action a…to be free, I have been arguing, means that there must be a time when both a and not a are accesible and it is my free choice that decides which happens…but at the time of my action it is already true that I do a since God actually knew what I would do beforehand…so normally it would not be a problem it is just that God’s knowledge ‘collapses the wave packet’ so to speak…

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