Weak actualization
January 7, 2010 — 17:42

Author: Alexander Pruss  Category: Molinism Problem of Evil  Comments: 28

Central to Plantinga’s formulation of the FWD is the notion of “weak actualization”. In the Profiles volume, Plantinga defines this as follows:

1. God weakly actualizes S iff there is an S* such that God strongly actualizes S* and S* â S, where â is “counterfactual implication”.

I think this is a problematic definition. Here is the basic problem. Say that a conditional C is “centered” iff pCp holds whenever both p and q hold. Then, trivially:

Theorem 1. If (1), and â is centered, then if God strongly actualizes any actual state of affairs, God weakly actualizes every actual state of affairs.

(Proof: Let S* be any actual state of affairs that God strongly actualizes. Let S be any actual state of affairs. Then, by centering S*âS, and so by (1), God weakly actualizes S.)

Theorem 1 is clearly problematic, as we can see by substituting “Al” for “God”. Since Al strongly actualizes some state of affairs (say, the writing of The Nature of Necessity), it follows that he weakly actualizes the Battle of Waterloo.

In light of Theorem 1, we could simplify the concept of “weakly actualizes”: God weakly actualizes S iff S is actual and there is an S* such that God strongly actualizes. But if that is what “weakly actualizes” comes down to, it is not a very interesting concept. It is a pretty trivial concept, and I think it does not seem to support the proof that Plantinga gives of Lewis’s Lemma.

The crucial assumption in the above is that â is centered. Here are some reasons to think so. First, the Lewis/Stalnaker conditional is centered. But I’ve heard Plantinga be friendly to the idea that â fits into the Lewis/Stalnaker framework for an appropriate closeness relation. Second, one’s reason for believing in conditionals of free will might be that one believes conditional excluded middle (pâq or pâ~q for all p and q), but every conditional satisfying conditional excluded middle and modus ponens is centered (proof: suppose p and q; then pâq or pâ~q by conditional excluded middle; if pâ~q, then by modus ponens ~q, since p, but that contradicts q; so, pâq).

Moreover, even if centering does not hold in general, we get implausible conclusions. Consider, for instance, back-tracking conditionals, like: “Were Al not to have written The Nature of Necessity, the Battle of Waterloo would (still) have taken place.” If that back-tracking conditional holds, surely so does: “Were Al to have written The Nature of Necessity, the Battle of Waterloo would have taken place”, albeit it is awkward. (This awkwardness may simply be due to the awkwardness of “pâq”, for the counterfactual conditional â, when p holds. However, Plantinga cannot complain about this awkwardness, because (1) defines weak actualization precisely in terms of a conditional whose antecedent is true.) But then if we extend weak actualization to Al, we get the conclusion that because Al strongly actualized The Nature of Necessity, he weakly actualized the Battle of Waterloo.

• For what it’s worth: Al has denied that the counterfactual conditional is centered. I believe he does this in “Respondeo” (in _Warrant in Contemporary Epistemology_).

January 7, 2010 — 18:25
• Mike Almeida

[Reposted comment (from PR post)]
Alex, you write,
Notice that the definition of “it is the case at W that God weakly actualizes S” makes no reference to conditionals holding at worlds other the world where God weakly actualizes S. But LL3 makes reference to such condiitonals.
No, it doesn’t, not at all. Where are you getting that from? LL3 make reference to just the CCF’s that obtain at the world W at which God weakly actualizes S. LL3 says this,
LL3. It is true at @ that, for every world W in which God exists, God could have weakly actualized W only if G(T(W))-> W.
This says that, given the CFF’s that actually obtain (i.e., that obtain at @), it is true that God could have weakly actualized the world W only if, were God to strongly actualize T of world W (i.e., some possible non-actual world, W) he would have actualized W. This is how we determine whether it is true at our world that God could have weakly actualized some world other than ours. If he could not have done so, given the actual CCF’s, then it is false that God could have actualized just any world he wished.

January 8, 2010 — 7:10
• Andrew:
Thanks for that!
I wonder if he’s changed his mind about that, given his greater friendliness towards Lewisian counterfactuals.
Mike:
“it is true that God could have weakly actualized the world W only if, were God to strongly actualize T of world W (i.e., some possible non-actual world, W) he would have actualized W.”
You are evaluating the conditional “were God to strongly actualize T of W, he would have actualized W” at the actual world, not at W. That’s what I meant by “But LL3 makes reference to such conditionals”.

January 8, 2010 — 9:18
• Mike Almeida

You are evaluating the conditional “were God to strongly actualize T of W, he would have actualized W” at the actual world, not at W. That’s what I meant by “But LL3 makes reference to such conditionals”.
No, it doesn’t. Look again. LL3 makes no reference at all to CCF’s holding in worlds other than those that actually obtain in evaluating the claim that God could have actualized another world. LL3 says this,
LL3. It is true at @ that, for every world W in which God exists, God could have weakly actualized W only if G(T(W))-> W.
Once again, LL3 says that, given the CFF’s that actually obtain (i.e., that obtain at @), it is true that God could have weakly actualized the world W only if, were God to strongly actualize T of world W (i.e., some possible non-actual world, W) he would have actualized W. This is how we determine whether it is true at our world that God could have weakly actualized some world other than ours.
It’s obvious that what could have been weakly actualized at our world has nothing to do with CCF’s that do not obtain at our world, and LL3 does not suggest anything to the contrary.

January 8, 2010 — 14:13
• Mike Almeida

Ok, I think I might see where you’re misreading LL3. It says this,
LL3. It is true at @ that, for every world W in which God exists, God could have weakly actualized W only if G(T(W))-> W.
You are reading it this way,
It is true at @ that, for every world W in which God exists, God could have in that world W weakly actualized W only if G(T(W))->W.
But that’s a very strange reading since God weakly actualizes every world (some degenerately, with no free creatures) at which he exists. But that is not what LL3 says, and I can’t think of any reason one might have to make such an assertion. Since we both agree that God necessarily exists, we could simply rewrite LL3 in a way that leaves no doubt,
LL3.1. It is true at @ that God could have weakly actualized W only if G(T(W))-> W.

January 8, 2010 — 14:54
• Mike:
I agree that LL3.1 is the right reading of what Plantinga is intending, and I was not thinking of the question of what God can do at W (except when I was formulating LL1).
But the problem is this: How do we derive LL3.1, in a logically valid way, from the definition of “weakly actualizes”? What are the steps in the derivation?

January 8, 2010 — 23:50
• Mike Almeida

It’s true that we can trivially and uninterestingly weakly actualize all sorts of states of affairs. But that doesn’t trivialize the notion of weak actualization any more than it trivializes the notion of counterfactual dependence. Counterfactual dependence is an interesting and important notion, too, despite the fact that there are trivial instances (given centering). Similarly for weak actualization. Here’s a modified version of the argument for LL.
1. God could have weakly actualized a state of affairs S iff. there is a state of affairs S* such that (1) it was within God’s power to strongly actualize S* and (2) if he had strongly actualized S*, then S would have been actual. (Definition)
2. LL. For every world W in which God exists, God could have weakly actualized W only if G(T(W))-> W. [there is a largest state of affairs T such that had God actualized T, W would have been actual].
3. Let W be any arbitrary world that God could have weakly actualized. Assumption for conditional proof.
4. G(A) -> W (from 3 and 1) [there is some state of affairs A such that had God strongly actualized A it W would have been actual.].
5. G(A) -> G(T(W)) (from 4) [for every world W in which God exists, W entails G(T(W))].
6. G(T(W)) -> G(A) (from 4 and def. T(W)) (God strongly actualize A in W, and T includes every state of affairs God strongly actualizes in W, so T entails G(A)].
7. G(T(W)) -> W (from 4,5,6, counterfactual logic).
8. /:. W is a world that God could have weakly actualized only if G(T(W)) -> W (3,7, discharge conditional proof).

January 9, 2010 — 9:44
• A. I agree that LL follows from your (1). But where do you get (1) from?
You mark it as “definition”. Do you mean it is a definition or it follows from a definition? If it follows, presumably from the definition of “weakly actualizes”, then I would like to see the argument for how it follows. I just don’t see how to make it follow.
If it is a definition, then what is it a definition of? Of “can weakly actualize”, I suppose. But then what if I say: “I am not interested in whether God can weakly actualize W in this apparently stipulated sense of ‘can weakly actualize’, but in whether God really can weakly actualize W, in the standard agency sense of ‘can’.” Of course if you can offer an independent argument as to why (1) should be accepted, that’s fine.
B. The problem is that it seems that there are no non-trivial cases of weak actualization. For in every case where God strongly actualizes S* and S*→ S, both S* and S obtain, and so S*→S is trivial by centering. Now maybe one can introduce the notion of S*→S holding non-trivially, provided that they hold “for some reason other than mere centering.” (Parallel case: Whenever p is impossible, p→q holds. But sometimes p→q holds merely trivially, as when we note that were torturing the innocent for fun right, then two plus two would be five, and sometimes it holds both trivially and non-trivially as when we note that were it the case that 2+1=4, then it would be the case that 2+2=5.) But I do not know if an account can be given of the “some reason”. That would require a deeper analysis of conditionals of free will than, to my knowledge, has yet been done.
Everybody:
Even without centering, consider this plausible principle. No major event of WW2 would be any different no matter what I now do. The following seems a special case of this: (x)(If I can now x, then (I now x)→(Poland is invaded in 1939)), where the quantification is over actions. If I now x, then I can now x. But I am now strongly actualizing this comment. Therefore, (I now strongly actualize this comment)→(Poland is invaded in 1939). Hence, I am weakly actualizing the invasion of Poland in 1939. Which is absurd.
Now it may be thoughts like this that made Plantinga restrict the definition to the case of God. So maybe someone can make an argument that although the definition is a bad one in the case of humans, it’s a fine one in the case of God, assuming centering doesn’t hold. I’d like to see that argument.

January 9, 2010 — 10:17
• Mike Almeida

Of course if you can offer an independent argument as to why (1) should be accepted, that’s fine.
Alex, it is the definition that AP provides on page 49, Profiles.
The problem is that it seems that there are no non-trivial cases of weak actualization. For in every case where God strongly actualizes S* and S*â S, both S* and S obtain, and so S*âS is trivial by centering.
What do you mean they’re all trivial? It is non-trivially true at our world @ that had God strongly actualized T’ he would have weakly actualized (some other world) W’. So G(T’)[]-> W’ is true and certainly not trivial.
Now maybe one can introduce the notion of S*âS holding non-trivially, provided that they hold “for some reason other than mere centering.
Again, it holds non-trivially here, in the actual world, that had God actualized a state of affairs T that included no rational beings, he would have actualized some other world W. So G(T)[]-> W is non-trivially true in our world (and of course in many others).
Therefore, (I now strongly actualize this comment)â(Poland is invaded in 1939). Hence, I am weakly actualizing the invasion of Poland in 1939. Which is absurd.
I don’t find it absurd, since I also know that had you failed to actualize that comment, it would still be true that Poland is invaded in 1939. So there are some trivial instances of weak actualization. So what? There are also trivial instances of counterfactual dependence. We recognize these for what they are: degenerate instances of an otherwise important relation. By the way, in the case of the argument I offered, I do not appeal to any degenerate instances of weak actualization.

January 9, 2010 — 11:12
• 1. I read page 49 differently. Plantinga says: “To simplify matters, let’s adopt the following definitions. [A] Let us say that God strongly actualizes a state of affairs S if and only if he causes S to be actual and causes to be actual every contingent state of affairs S* such that S includes S*; and [B] let’s say that God weakly actualizes a state of affairs S if and only if he strongly actualizes a state of affairs S* that counterfactually implies S. (Strong actualization is thus a special case of weak actualization.) [C] Then God could have weakly actualized a state of affairs S if and only if there is a state of affairs S* such that (1) it was within his power to strongly actualize S*, and (2) if he had strongly actualized S*, then S would have been actual.”
I am keeping Plantinga’s italics, but I added the letters in square brackets. My reading is that “the following definitions” applies to [A] and [B], but not to [C]. Here is my not completely conclusive evidence:
(a) In both [A] and [B], the crucial word in the definition is italicized. In [C], nothing is italicized. If [C] were a definition of “could have weakly actualized”, one would expect “could have” to be italicized.
(b) Point [C] begins with “Then”, which suggests a consequence relation of some sort.
(c) Definitions [A] and [B] are in the same sentence, and it is natural to take the “the following definitions” to apply to just what is in that sentence, and hence not to [C].
(d) If [C] were a definition, it is not clear that the FWD would be of much relevance. The point of the FWD is to prove that possibly God could not weakly actualize a world containing significantly free agents none of whom sin. If the “could … weakly actualize” is merely stipulative, this accomplishment is not very interesting. (So even if [C] were a definition, an argument would be needed that it’s the right definition.)
So, it seems to me that Plantinga means [C] to follow from [A] and [B] in some way. But I don’t see how it does.
2. As for triviality, I don’t know that “(I now strongly actualize this comment)â(Poland is invaded in 1939)” holds trivially. It seems to be a non-trivial consequence of the non-trivial principle that nothing that I can now do can counterfactually affect major human events in 1939.
3. I guess we can say this: Some of the cases of weak actualization hold trivially, i.e., simply due to centering, and some hold both trivially and non-trivially, i.e., they hold both due to centering and for an additional reason beyond centering. Is that a fair compromise? 🙂 If so, then the interesting concept is not “weak actualization” but “weak actualization holding non-trivially”.

January 10, 2010 — 18:32
• Back to that pesky Respondeo where Plantinga denies centering. There, he says that p and q do not suffice for p→q. But what does suffice for p→q is this: p holds in the actual world and q holds in all sufficiently close worlds in which p holds. In other words, there must be some robustness.
But my temporally backwards examples satisfy this boosted condition. So if we extend the definition of weak actualization to non-divine beings, Plantinga is committed to saying that I weakly actualized Napoleon’s defeat, because in all worlds sufficiently close to this one where I write this comment, Napoleon is defeated.
Actually, that “all sufficiently close worlds” business is problematic for other reasons. If q is something really major, like the formation of the Local Group of galaxies, then any world where q does not hold will automatically be quite far from this world, and so q will of course hold in all worlds that are sufficiently close, and in particular all sufficiently close p-worlds. Maybe the measure of “sufficient closeness”, though, has to be proportionate to how far the nearest ~q-world is? But then it would be really hard to have p→q where p is actual and the nearest ~q-world is very distant, and that’s not right.
So, in summary, denying centering does not get Plantinga out of trouble as his preferred alternative to centering allows some of my counterexamples, and anyway his preferred alternative to centering doesn’t work.

January 10, 2010 — 18:48
• Mike Almeida

If the “could … weakly actualize” is merely stipulative, this accomplishment is not very interesting. (So even if [C] were a definition, an argument would be needed that it’s the right definition.)
Plantinga introduced the notion of God weakly actualizing a world, and C is exactly what he meant by it. That’s evident not merely from the current argument, but also from the NN version, which clearly appeals to C, and frankly every other version of the proof he offers (perhaps setting aside the earliest version in _God and Other Minds_). There is no Platonic version of weak actualization that Plantinga is striving and failing to capture. It’s a curious position to hold that he’s fumbling to get at the concept that he himself introduced. He is introducing the concept of actualizing a world w without causing a certain set S of states of affairs that obtain in w, where S includes all and only states of affairs that are strongly actualized by libertarian free agents. That is exactly what C does; nothing more or less than that. But if we are now going to press further (though it is getting late in the day and I think Plantinga’s argument is pretty much vindicated) to demand that C is proved by yet earlier definitions, I’ll bet it can be done and I’ll work out that proof. But there is nothng trivial or uninteresting in this proof, in my estimation.

January 10, 2010 — 19:51
• “Plantinga introduced the notion of God weakly actualizing a world, and C is exactly what he meant by it.”
But he’s already defined “weakly actualizing” in B. C is not a definition of “weakly actualizing” but a necessary and sufficient condition for being able to weakly actualize.

January 11, 2010 — 7:43
• Maybe the temporally backward examples are irrelevant to the FWD, because for the FWD, Plantinga does not need to give a necessary and sufficient condition for “weakly actualizes”–all he needs to give is a necessary condition. Still, the question of working out a concept of “weakly actualizes” has independent interest.

January 11, 2010 — 8:39
• Mike Almeida

Alex,
There are three definitions offered in Profiles. Strictly, I wouldn’t hold Plantinga to the position that these are definitions, since he simply invites us to adopt these as definitions for the sake of simplifying discussion (49, 9 down). The definitions are not related by way of inference, since the third is modalized and the initial two are not. In the sequence in which they are introduced, they are these,
SA. God strongly actualizes a state of affairs S iff. he causes S to be actual and causes to be actual every contingent states of affairs S* such that S includes S*.
WA. God weakly actualizes a state of affairs S iff. he strongly actualizes a state of affairs S* that counterfactually implies S.
CA. God could have weakly actualized a state of affairs S iff. there is a state of affairs S* such that (1) it was within God’s power to strongly actualize S* and (2) if he had strongly actualized S*, then S would have been actual.
It is CA that plays the important role in FWD, WA and SA are just working up the CA. How do we know that? In Profiles this is evident from the first full paragraph on 49 where he begins to describe the atheologian’s appeal to counterfactuals of freedom. The atheologian claims that “. . .for every possible world W…there is something God could have done to bring about it’s actuality”. This is what he is trying to clarify as he moves through the simplifying definitions. Obviously, it is CA that he aims to present, since it is CA that is necessary to capturing the atheologian’s claim and it is CA that matters to the debate between the atheologian and the theist on the compossibility of God & evil.
I agree, as I’ve said above, that WA has some unusual implications on the assumption of centering. Still, things are no where near as bad as you suggest. Quick example: I say to you “don’t weakly actualize the world in which my watch is broken”. Knowing the relevant CCF’s, you tell little Tim that he might like to break my watch, and Tim does just that. We are now in W, the world you weakly actualized. I complain to you that you should not have weakly actualized W. You say, “what’s your problem, it’s just trivially true that in W I weakly actualized W. After all, it is true in W that I told Tim you might like that, and it is true in W that Tim broke it. So there’s no interesting sense in which I weakly actualized W.”
Wrong! You’re confusing C1 with C2. (C1) is false in W even though (C2) is true in W.
C1. It is trivially true that I weakly actualized W.
C2. It is trivially true that I weakly actualized W in W.
There is of course a very interesting sense in which you weakly actualized W, a sense in which you are now responsible for my broken watch and for corrupting the youth.

January 11, 2010 — 9:26
• Thanks, this is helpful. You are right about C1 and C2. I think the reason C1 is not trivial is that it is not trivial that W is actual.
Here’s what I’m thinking, though. Normally, once we define an action A, we would not want to give a separate definition of “can A”. Rather, we’d like to proceed as follows: define A, then take a general story about “can”, and derive the conditions for “can A” from the definition of A and the story about “can”. It would be unfortunate to have to first define “sleeps”, “eats” and “weakly actualizes”, and then to give separate definitions of “can sleep”, “can eat” and “can weakly actualize”.
But on thinking this through, maybe this is too ambitious a request. After all, maybe we can give an account of “sleeps”, “eats” and “can sleep” without being able to give an account of “can eat”. Maybe in fact there is no general story to be told about “can” that, when combined with definitions of “sleep”, “eat” and “weakly actualize”, lets one understand “can sleep”, “can eat” and “can weakly actualize”, just as maybe there is no general story to be told about “good” that, when combined with definitions of “basketball player” and “person”, lets one understand “good basketball player” and “good person”.
Still, at least an argument is needed that C captures the intuitive idea of what God can do. As it stands, if we substitute me for God, it follows from C that I can weakly actualize past states of affairs by snapping my fingers. This does suggest that something about the concept of what God can do has not been captured. Perhaps one can argue, though, that C provides a necessary condition for what God can do, and for the FWD that’s all that’s needed.

January 11, 2010 — 10:19
• Mike Almeida

Normally, once we define an action A, we would not want to give a separate definition of “can A”. Rather, we’d like to proceed as follows: define A, then take a general story about “can”, and derive the conditions for “can A” from the definition of A and the story about “can”.
This seems right to me. Things get a little messier here, though, since AP consistently uses ‘could have’ + pp instaed of ‘can’, so there are tense worries, too. But as a general way to proceed, this seems right.
Still, at least an argument is needed that C captures the intuitive idea of what God can do.
That’s certainly an issue. Looking back at Mackie’s last response to FWD in TMOT, (1982), having the benefit of reading NN, he still doesn’t take (C) or it’s ilk as providing an interesting basis for limits on what God can do. His argument is odd, though. He says that, in every possible world, God can instantiate any logically possible individual essence (IE)–Plantinga’s surrogate individuals– including of course those that do not suffer from TWD. The implicit argument seems to be (I couldn’t swear to it) that properties are necessarily existing objects, IE’s are nothing more than collections of properties, so the relevant collections of properties exist in every world and God can instantiate them. There are lots of mistakes here, primarily that an individual essence won’t have the property of not being TWD in every world, even if it is true that “having the property of not being TWD” is a property that exists in every world and “not being TWD” is a property that exists in every world. But that’s Mackie’s bid against C mattering in the debate over what God can do.

January 11, 2010 — 14:20
• So, let’s play with the strategy I suggest. We want a story about “can” for an agent. If the agent is in time and proceeds like we do in a changing way, we need to talk about the agent at a time. If the agent is not in time or is unchanging, we instead need to talk about the agent at “an explanatory moment”. I will use “moment” for both.
We want to know the circumstances under which x at moment t “can” A.
Suggestion: x at t can A iff x’s Aing at t is compatible with everything prior (temporally if t is a time, explanatorily if t is an explanatory moment) to t.
This captures a possibility sense of “can” but not an agency sense of “can”. The agency sense of “can” requires a more stringent condition. Satisfying definition 1 will be a necessary condition for an agency sense of “can” but not a sufficient one.
Now, can we at least derive the necessity of condition C for “can weakly actualize” from A and B, and my suggestion? If →-facts are not explanatorily prior to t, then the answer seems to me to be negative. If →-facts are explanatorily prior to t, then I think the answer is positive, though I think we’ll need some auxiliary hypotheses, such as that there are “enough” →-facts.
I really am lost here, and I think there is some interesting and non-trivial work to be done.

January 11, 2010 — 16:17
• Mike Almeida

By condition C, you mean the following?
[C] Then God could have weakly actualized a state of affairs S if and only if there is a state of affairs S* such that (1) it was within his power to strongly actualize S*, and (2) if he had strongly actualized S*, then S would have been actual.”
How is this as a (rough) start on an argument from right to left. Let W be a possible, non-actual world.
1. Let W be the closest world to @ at which God strongly actualizes S* (as SA is defined in [A]). Assume
2. In W, it is true that if God strongly actualized S*, then S would have been actual. Assume condition (2).
3. In W, God weakly actualizes S. from B.
4. /:. It is true that God could have weakly actualized some state of affairs S (from S5).
In premise (1) I take God’s power to A as entailing there is a world in which God A’s. That’s controversial, since there are unmanifested powers. But I’m not sure there are unmanifested powers in the sense that there is no possible world at all in which they are manifested. In premise (2) I specify that the countefactual in condition (2) of C is true in some world. In premise (3) I conclude that God weakly actualizes that world as specified in [B].

January 11, 2010 — 17:12
• Mike Almeida

Better, something like this, right to left.
1. Let W be the closest world to @ at which God strongly actualizes S* (as SA is defined in [A]). Assume condition (1).
2. It is true at @ that if God strongly actualized S*, then S would have been actual. Assume condition (2).
3. In W, God weakly actualizes S. from 1,2 & B.
4. /:. It is true at @ that God could have weakly actualized some state of affairs S (1,2,3, S5).

January 12, 2010 — 11:18
• Mike:
If not, I am having trouble at step 3 of your revised argument. It is true at @ that if God strongly actualized S*, then S would have been actual. But what we need for 3 is this: It is true at W that if God strongly actualized S*, then S would have been actual. If we assume centering, then of course we do get this (since both S* and S are actual at W).
Step 3 will work if we assume this axiom: If p→q, then p→(p→q). (This axiom follows from centering, but Plantinga doesn’t like centering.)
As for Step 4, I am not sure how that works. Are you reading “It is true at @ that God could have weakly actualized S” as “It is true at @ that: Possibly(God weakly actualizes S)”? If so, then step 4 does work, but I don’t think this is what it means to say “God could have weakly actualized S”.

January 13, 2010 — 9:30
• By the way, Al told me by email yesterday that the definition of “weakly actualizes” should be thought of as a mere abbreviation. He hasn’t told me the status of C, though.

January 13, 2010 — 9:43
• Here is another technical problem for Plantinga’s replacement for centering. The replacement is that if p, then p→q iff q at all sufficiently close p-worlds.
Now, suppose that w1, w2 and w3 are three worlds such that both w2 and w3 are p-worlds, and w1 is a ~p-world, and w2 is the p-world closest to w1. Suppose that p is something within God’s power of strong actualization. Suppose also that w3 is sufficiently close to w2. Let q be the proposition that w2 is actual. Then, it is true at w1 that p→q, because q holds at the p-world closest to w1 (namely at w2). But it is not true at w2 that q holds at all sufficiently close p-worlds, because w3 is sufficiently close to w2, and q does not hold at w2.
So:
– At w1: p→q and God can strongly actualize the truth of p.
– At w2: ~(p→q) and hence God does not weakly actualize the truth of q.
In fact, there is no possible world at which God weakly actualizes the truth of q, because there is only one q-world, namely w2, and God does not actualize the truth of q there.

January 13, 2010 — 9:58
• Mike Almeida

Alex,
The argument does assume centering since, as I’ve been reading you, centering is a problem for AP. I’m trying to get right to left in the proof of C without begging any questions against your objection.

January 13, 2010 — 10:50
• Mike Almeida

– At w1: pâq and God can strongly actualize the truth of p. – At w2: ~(pâq) and hence God does not weakly actualize the truth of q.
I’m not sure this model describes a situation that is possible. The similarity relation induced makes salient–really of top importance–having the property of being identitical with w2 (i.e. all of the most similar worlds in which p is true have the property of being identical with w2). But obviously w3 does not have that important property, so it cannot be among the most similar worlds to w2; that is, it is not sufficiently close to w2. Only w2 has that property, hence p->q is true at w2.

January 13, 2010 — 11:15
• If centering is true, I think it’s easier to go right to left. But left to right is more problematic. And it’s left to right that matters for the FWD, I think.

January 13, 2010 — 12:07
• Surely the property of having q hold is not overridingly salient for closeness, because if it is, then the condition that p→q hold in all the really clsoe worlds is trivialized and we get back to centering, as the worlds where q holds will automatically count as the close ones. So, if the property of q holding matters at all for closeness, it doesn’t decisively so matter. Well, then, we can suppose that w3 is very close to w2 in all other respects.

January 14, 2010 — 8:42
• Mike Almeida

Surely the property of having q hold is not overridingly salient for closeness, because if it is, then the condition that pâq hold in all the really clsoe worlds is trivialized and we get back to centering,
Hard to see how it would not be, Alex. Consider the counterfactual,
If it were the case that p then it would be the case that w2 is actualized.
That is true only if the closest worlds in which p is true are worlds that have the property of being identical with w2. In fact, no world w3 that fails to have the property of bieng identical with w2–no matter how otherwise similar to w2 it might be, no matter if indiscernible from w2–would verify that counterfactual. If p holds in the indiscernible world w3, then the counterfactual that gets verified is,
If it were the case that p then it would be the case that w3 is actualized.
So the property of being identical to w2 is overridingly salient, as far as I can see, to world similarity in this case.

January 14, 2010 — 10:50