God's infallible knowledge and free will part II

Originally posted by lucifershammer
(Thanks for the clarification re: premise 4.)

But what, then, is the modal status of ~P in wp'? If I say 'necessity', then we end up with the reductio above. But the very definition of wp' as a world where ~P holds implies it is something stronger than just possibility.

Alternatively, one can say that 'modal status of ~P in wp' ' is incoh ak of 'necessity' in a way that does not imply 'cuts across all possible worlds'.)

what, then, is the modal status of ~P in wp'?

What exactly what does "in wp'" impart here? It adds absolutely nothing. The modal status of ~P is that it is possible (supposing there even is some wp'😉. The status of ~P would also be that it is contingent since it is true in some possible world (again supposing there is some wp'😉 but false in another (it's false in the actual world, for example). The status of ~P in wp' is that it is true.

I don't understand the problem here.

Originally posted by lucifershammer
Consider a function M:U^2x[b]P -> O such that

U = set of all possible worlds
P = set of all possible (meaningful) propositions
O = {necessary, possible, impossible} (the modal operators)
and
M(W C U,p E P) = {'necessary' iff Necessarily p in W (i.e. p holds in all worlds of W); 'possible' iff Possibly p in W (at least one worl oming back to my example)
Clearly M(U,~P) = 'possible'. But M(WP',~P) = 'necessary'.[/b]

So there may be some subset of U (the wp' worlds, if any exist) where ~P is true in every member of the subset. Regardless, you asked about the modal status of ~P in wp'. Again, this is nothing more than asking for the modal status of ~P (full stop). And, that is given by your M(U, ~P) = possible (again, supposing there even is some wp'😉. We know that it cannot be that ~P is necessary, since ~P is false in our world.

I'm failing to understand what you are trying to show with all this. I think you are trying to bring doubt on bbarr's criticism of a point within Conrau K's position earlier (where bbarr mentioned necessity cutting across all worlds). But bbarr's criticism was the right one there I think, and nothing you have presented here makes me doubt that. Or maybe you are just trying to clarify exactly what kinds of necessity must cut across the worlds, or something like that. Is that more what you have in mind?

Originally posted by lucifershammer
Since the discussion has already yielded viable challenges to premise 4, the omniscience-free will debate does not depend on our side-discussion on libertarian free will. Nevertheless, I am still interested in a discussion on exactly what L entails (my instinct is that your premise (4) can, in fact, be accommodated in a 'mere libertarianism' as Conrau ...[text shortened]... e that should move to another thread (even another forum since it's not really Spirituality).

OK. I think your instinct there is wrong. But, as you mention, that might be better left for a separate discussion.

Originally posted by LemonJello
So there may be some subset of U (the wp' worlds, if any exist) where ~P is true in every member of the subset. Regardless, you asked about the modal status of ~P in wp'. Again, this is nothing more than asking for the modal status of ~P (full stop). And, that is given by your M(U, ~P) = possible (again, supposing there even is some wp'😉. We know that ...[text shortened]... must cut across the worlds, or something like that. Is that more what you have in mind?

I have not been following bbarr's discussion with Conrau (particularly not the S5 stuff).

No, I am interested in why necessity is said to cut across all worlds and what that means. Is cutting across all worlds a necessary and sufficient condition for necessity? Is it meaningless to speak of necessity, possibility etc. when we are considering just a subset of possible worlds (as any discussion involving counterfactuals does) without reference to the universal set?

Mere intellectual doodling, if you will.

Originally posted by lucifershammer
Consider a function M:U^2x[b]P -> O such that

U = set of all possible worlds
P = set of all possible (meaningful) propositions
O = {necessary, possible, impossible} (the modal operators)
and
M(W C U,p E P) = {'necessary' iff Necessarily p in W (i.e. p holds in all worlds of W); 'possible' iff Possibly p in W (at least one worl ...[text shortened]... oming back to my example)
Clearly M(U,~P) = 'possible'. But M(WP',~P) = 'necessary'.[/b]

Originally posted by lucifershammer
Consider a function M:U^2x[b]P -> O such that[/b]
I'm not sure what this means, could you clarify?

I think we are talking about logical necessity. We can say necessarily p if to deny p involves a logical contradiction. One of the reasons that I think S5 is the right system for necessity is precisely because it cuts across all possible worlds. I'm really not sure what necessity involving a subset of possible worlds would mean really. If you could give an example of 'necessity lite', that might help.

Originally posted by lucifershammer
I have not been following bbarr's discussion with Conrau (particularly not the S5 stuff).

No, I am interested in why necessity is said to cut across all worlds and what that means. Is cutting across all worlds a necessary and sufficient condition for necessity? Is it meaningless to speak of necessity, possibility etc. when we are considering ...[text shortened]... tuals does) without reference to the universal set?

Mere intellectual doodling, if you will.

When we talk of necessity, we mean logical necessity (or perhaps, more broadly, analytic and metaphysical necessity). So, take some logical theorem like the law of non-contradiction:

(LNC) ~(P&~P)

To say that LNC is necessary is just to say that it must be the case; that it's denial yields a contradiction (logical, analytical, metaphysical...). Any world that is possible is such that this rule holds there. In fact, the complementary notions of necessity and possibility are inter-definable. It is necessary that P iff it is not possible that ~P. It is not necessary that P iff it is possible that ~P. It is possible that P iff it is not necessary that ~P. Is is not possible that P iff it is necessary that ~P. Think of the relationship between universal and existential quantifiers in first-order logic. The translation relations are isomorphic. What this means is that if a proposition is necessary, then there is no possible world where that proposition is false. So, necessity cuts across possible worlds. Perhaps a better way of saying this is that the set of necessary propositions determines which worlds are possible.

Originally posted by Lord Shark

This might be the old A-theorist versus B-theorist thing I suppose. My intuition is that nothing can be gained by considering temporal logic though. Just as from 'I am eating cornflakes' it follows necessarily that 'I am eating cornflakes', yet nothing about the necessity of my eating cornflakes flows from this, similarly 'whatever will be, will be' is ...[text shortened]... ich I started this thread, however if you can come up with a counter I'll be quite happy.

I think your intuition may be based on the assumption that we have to characterize the libertarian commitments in terms of possibility (i.e., If L, then it is possible for S to have done otherwise than he did). I agree that this won't work. But what if we characterize the libertarian commitments differently? We could use counter-factuals and the notion of causal power, for instance:

(L) S acts freely by A-ing only if it was within S's power to refrain from A-ing.

We limit the set of possible worlds under consideration to just those identical to the actual world until the time at which S A's. If God has knowledge of future contingents, and if it is true at T1 that S will A at T2, then God knows at T1 that S will A at T2. But then it would only be in S's power to refrain from A-ing only if it is in S's power to, at T2, make God's belief at T1 false. That's backward causation...

Meh, I don't know.

Originally posted by lucifershammer
I have not been following bbarr's discussion with Conrau (particularly not the S5 stuff).

No, I am interested in why necessity is said to cut across all worlds and what that means. Is cutting across all worlds a necessary and sufficient condition for necessity? Is it meaningless to speak of necessity, possibility etc. when we are considering ...[text shortened]... tuals does) without reference to the universal set?

Mere intellectual doodling, if you will.

No, I am interested in why necessity is said to cut across all worlds and what that means. Is cutting across all worlds a necessary and sufficient condition for necessity? Is it meaningless to speak of necessity, possibility etc. when we are considering just a subset of possible worlds (as any discussion involving counterfactuals does) without reference to the universal set?

Well, the reason I brought up S5, and S4, is to illustrate that notions of necessity really depend on the axioms we allow beforehand. If we reject axiom 5, we get a situation in which for one world A is necessary but for another world not-A is true. In natural language semantics, there may be good grounds for this weaker form of necessity (and I have given an example of this -- which bbar criticised as nomological, though that doesn't really matter.)

Originally posted by bbarr
Again, that's not what the libertarian wants. The libertarian wants it to be the case that the actual world could have been otherwise than it in fact is; that one could have done otherwise than one in fact did. We can reformulate the argument accordingly:

P: God knows that you will A in W1.
Q: You do not A in W1.
L: Libertarianism is true.

You end u im that something can be necessary in one world and not necessary in another. What will it be?

P: God knows that you will A in W1.
Q: You do not A in W1.
L: Libertarianism is true.

I don't see why libertarianism should require that you do not in w1. For not-A to be possible, it only has to be granted that you do not in some world (not necessarily the same one.) I would prefer Q, then, to be, You do not in W2. Whether that contradicts P only depends on how you define necessity. Again, I don't see why a libertarian would accept such a strong definition of necessity that it 'cuts across all worlds'.

What the libertarian would want is some weaker form of necessity that 'in all worlds you A, God knows you A' which obviously excludes the set of worlds in which you not-A. If you A, then necessarily God knows you A -- but it does not follow that God knows you A in the world in which you not-A. That is clearly ridiculous. God's knowledge of how you will act in one world should not exclude possibilities in other worlds.

Originally posted by Conrau K
No, I am interested in why necessity is said to cut across all worlds and what that means. Is cutting across all worlds a necessary and sufficient condition for necessity? Is it meaningless to speak of necessity, possibility etc. when we are considering just a subset of possible worlds (as any discussion involving counterfactuals does) without ref an example of this -- which bbar criticised as nomological, though that doesn't really matter.)

Again, how are you deriving this result from the rejection of S5's definitive axiom? Here is the axiom:

(5) If possibly P, then necessarily possibly P.

So, show me how the negation of this axiom yields the result you claim. If you can't, then drop the stuff about S5.

Look, what you are really objecting to is the combination of modal logic K and possible world semantics. It is K that construes the modal operators as behaving just like the quantifiers of first-order logic, such that necessity is definable as "it is not possible that ~ P". Now, combine that with the possible world semantics and you get "There is no possible world where ~P", which is what you want to deny as an entailment (definition) or necessity. You're barking up the wrong tree with your rejection of S5.

Originally posted by Conrau K
[b]
P: God knows that you will A in W1.
Q: You do not A in W1.
L: Libertarianism is true.

I don't see why libertarianism should require that you do not in w1. For not-A to be possible, it only has to be granted that you do not in some world (not necessarily the same one.) I would prefer Q, then, to be, You do not in W2. Whether that contra ...[text shortened]... wledge of how you will act in one world should not exclude possibilities in other worlds.[/b]

I have no idea what you're arguing anymore.

Originally posted by bbarr
Again, how are you deriving this result from the rejection of S5's definitive axiom? Here is the axiom:

(5) If possibly P, then necessarily possibly P.

So, show me how the negation of this axiom yields the result you claim. If you can't, then drop the stuff about S5.

Look, what you are really objecting to is the combination of modal logic K and poss t (definition) or necessity. You're barking up the wrong tree with your rejection of S5.

(5) If possibly P, then necessarily possibly P.

If you reject this axiom then, using world-semantics, it is possible to have a situation in which P is possible in w1 but not possible in w2. You might have a sample of worlds:

w1: v(P) = T
w2: v(P) = F
w3: v(P) = F

and the following relations (which are both licensed by T and K),

w1 R w2
w2 R w3
(but not admitting the converse, w2 R w1, which 5 would require.)

In that case, in w1, P is possible but in w2 P is impossible. In other words, in one world, not-P is necessary even though in another world P is possible. Rejecting 5 allows a situation in which necessity does not cut across all worlds.

Originally posted by bbarr
I have no idea what you're arguing anymore.

I am arguing that simply because something is necessary in one world does not mean that it is necessary across all worlds and, similarly, just because something is possible in one world it does not have be true of that world. I take issue with your claims that P entails that you will A in all worlds and that Q means that you will not-A in the same world.

Originally posted by Conrau K
[b](5) If possibly P, then necessarily possibly P.

If you reject this axiom then, using world-semantics, it is possible to have a situation in P is possible in w1 but not possible in w2. You might have a sample of worlds:

w1: v(P) = T
w2: v(P) = F
w3: v(P) = F

and the following relations,

w1 R w2
w2 R w3
(but not admitting the conv ...[text shortened]... P is possible. Rejecting 5 allows a situation in which necessity does not cut across all worlds.[/b]

Sorry, I just don't see how this follows from the rejection of axiom (5).

Originally posted by Conrau K
I am arguing that simply because something is necessary in one world does not mean that it is necessary across all worlds and, similarly, just because something is possible in one world it does not have be true of that world. I take issue with your claims that P entails that you will A in all worlds and that Q means that you will not-A in the same world.

Yes, I know that just because P is true in W1 it does not follow that P must be true of W1. I am familiar with the notion of contingent truths. What I don't see is how the rejection of axiom (5) yields the result that a proposition can be logically necessary of one world and not logically necessary of another. I'll need an argument or derivation of that claim.